Ordnung Gruppe Echte Untergruppen [1] Eigenschaften Zykel-Graph 1 Z 1 ≅ S 1 ≅ A 2 {\displaystyle \mathbb {Z} _{1}\cong S_{1}\cong A_{2}} (triviale Gruppe ) - abelsch, zyklisch 2 Z 2 ≅ S 2 ≅ D 1 {\displaystyle \mathbb {Z} _{2}\cong S_{2}\cong D_{1}} (Gruppe Z2 ) - abelsch, einfach , zyklisch, kleinste nichttriviale Gruppe 3 Z 3 ≅ A 3 {\displaystyle \mathbb {Z} _{3}\cong A_{3}} - abelsch, einfach, zyklisch 4 Z 4 ≅ D i c 1 {\displaystyle \mathbb {Z} _{4}\cong \mathrm {Dic} _{1}} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch V 4 ≅ Z 2 2 ≅ D 2 {\displaystyle V_{4}\cong \mathbb {Z} _{2}^{2}\cong D_{2}} (Kleinsche Vierergruppe ) 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} abelsch, die kleinste nichtzyklische Gruppe 5 Z 5 {\displaystyle \mathbb {Z} _{5}} - abelsch, einfach, zyklisch 6 Z 6 ≅ Z 2 × Z 3 {\displaystyle \mathbb {Z} _{6}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{3}} Z 3 {\displaystyle \mathbb {Z} _{3}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch S 3 ≅ D 3 {\displaystyle S_{3}\cong D_{3}} (Symmetrische Gruppe) Z 3 {\displaystyle \mathbb {Z} _{3}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} kleinste nichtabelsche Gruppe 7 Z 7 {\displaystyle \mathbb {Z} _{7}} - abelsch, einfach, zyklisch 8 Z 8 {\displaystyle \mathbb {Z} _{8}} Z 4 {\displaystyle \mathbb {Z} _{4}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 2 × Z 4 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}} 2 ⋅ Z 4 {\displaystyle 2\cdot \mathbb {Z} _{4}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} , D 2 {\displaystyle D_{2}} abelsch Z 2 3 ≅ D 2 × Z 2 {\displaystyle \mathbb {Z} _{2}^{3}\cong D_{2}\times \mathbb {Z} _{2}} 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} , 7 ⋅ D 2 {\displaystyle 7\cdot D_{2}} abelsch D 4 {\displaystyle D_{4}} Z 4 {\displaystyle \mathbb {Z} _{4}} , 2 ⋅ D 2 {\displaystyle 2\cdot D_{2}} , 5 ⋅ Z 2 {\displaystyle 5\cdot \mathbb {Z} _{2}} nichtabelsch Q 8 ≅ D i c 2 {\displaystyle Q_{8}\cong \mathrm {Dic} _{2}} (Quaternionengruppe ) 3 ⋅ Z 4 {\displaystyle 3\cdot \mathbb {Z} _{4}} , Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch; die kleinste hamiltonsche Gruppe 9 Z 9 {\displaystyle \mathbb {Z} _{9}} Z 3 {\displaystyle \mathbb {Z} _{3}} abelsch, zyklisch Z 3 2 {\displaystyle \mathbb {Z} _{3}^{2}} 4 ⋅ Z 3 {\displaystyle 4\cdot \mathbb {Z} _{3}} abelsch 10 Z 10 ≅ Z 2 × Z 5 {\displaystyle \mathbb {Z} _{10}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{5}} Z 5 {\displaystyle \mathbb {Z} _{5}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch D 5 {\displaystyle D_{5}} Z 5 {\displaystyle \mathbb {Z} _{5}} , 5 ⋅ Z 2 {\displaystyle 5\cdot \mathbb {Z} _{2}} nichtabelsch 11 Z 11 {\displaystyle \mathbb {Z} _{11}} - abelsch, einfach, zyklisch 12 Z 12 ≅ Z 4 × Z 3 {\displaystyle \mathbb {Z} _{12}\cong \mathbb {Z} _{4}\times \mathbb {Z} _{3}} Z 6 {\displaystyle \mathbb {Z} _{6}} , Z 4 {\displaystyle \mathbb {Z} _{4}} , Z 3 {\displaystyle \mathbb {Z} _{3}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 2 × Z 6 ≅ Z 2 2 × Z 3 ≅ D 2 × Z 3 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{6}\cong \mathbb {Z} _{2}^{2}\times \mathbb {Z} _{3}\cong D_{2}\times \mathbb {Z} _{3}} 3 ⋅ Z 6 {\displaystyle 3\cdot \mathbb {Z} _{6}} , Z 3 {\displaystyle \mathbb {Z} _{3}} , D 2 {\displaystyle D_{2}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} abelsch D 6 ≅ D 3 × Z 2 {\displaystyle D_{6}\cong D_{3}\times \mathbb {Z} _{2}} Z 6 {\displaystyle \mathbb {Z} _{6}} , 2 ⋅ D 3 {\displaystyle 2\cdot D_{3}} , 3 ⋅ D 2 {\displaystyle 3\cdot D_{2}} , Z 3 {\displaystyle \mathbb {Z} _{3}} , 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch A 4 {\displaystyle A_{4}} (Gruppe A4 ) D 2 {\displaystyle D_{2}} , 4 ⋅ Z 3 {\displaystyle 4\cdot \mathbb {Z} _{3}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch; kleinste Gruppe, die zeigt, dass die Umkehrung des Satzes von Lagrange nicht stimmt: keine Untergruppe der Ordnung 6 D i c 3 {\displaystyle \mathrm {Dic} _{3}} (hier Verknüpfungstafel ) Z 6 {\displaystyle \mathbb {Z} _{6}} , 3 ⋅ Z 4 {\displaystyle 3\cdot \mathbb {Z} _{4}} , Z 3 {\displaystyle \mathbb {Z} _{3}} , Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch 13 Z 13 {\displaystyle \mathbb {Z} _{13}} - abelsch, einfach, zyklisch 14 Z 14 ≅ Z 2 × Z 7 {\displaystyle \mathbb {Z} _{14}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{7}} Z 7 {\displaystyle \mathbb {Z} _{7}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch D 7 {\displaystyle D_{7}} Z 7 {\displaystyle \mathbb {Z} _{7}} , 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch 15 Z 15 ≅ Z 3 × Z 5 {\displaystyle \mathbb {Z} _{15}\cong \mathbb {Z} _{3}\times \mathbb {Z} _{5}} Z 5 {\displaystyle \mathbb {Z} _{5}} , Z 3 {\displaystyle \mathbb {Z} _{3}} abelsch, zyklisch (siehe „Jede Gruppe der Ordnung 15 ist zyklisch.“ ) 16 Z 16 {\displaystyle \mathbb {Z} _{16}} Z 8 {\displaystyle \mathbb {Z} _{8}} , Z 4 {\displaystyle \mathbb {Z} _{4}} , Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 2 4 {\displaystyle \mathbb {Z} _{2}^{4}} 15 ⋅ Z 2 {\displaystyle 15\cdot \mathbb {Z} _{2}} , 35 ⋅ D 2 {\displaystyle 35\cdot D_{2}} , 15 ⋅ Z 2 3 {\displaystyle 15\cdot \mathbb {Z} _{2}^{3}} abelsch Z 4 × Z 2 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}^{2}} 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} , 4 ⋅ Z 4 {\displaystyle 4\cdot \mathbb {Z} _{4}} , 7 ⋅ D 2 {\displaystyle 7\cdot D_{2}} , Z 2 3 {\displaystyle \mathbb {Z} _{2}^{3}} , 6 ⋅ Z 4 × Z 2 {\displaystyle 6\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}} abelsch Z 8 × Z 2 {\displaystyle \mathbb {Z} _{8}\times \mathbb {Z} _{2}} 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} , 2 ⋅ Z 4 {\displaystyle 2\cdot \mathbb {Z} _{4}} , D 2 {\displaystyle D_{2}} , 2 ⋅ Z 8 {\displaystyle 2\cdot \mathbb {Z} _{8}} , Z 4 × Z 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}} abelsch Z 4 2 {\displaystyle \mathbb {Z} _{4}^{2}} 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} , 6 ⋅ Z 4 {\displaystyle 6\cdot \mathbb {Z} _{4}} , D 2 {\displaystyle D_{2}} , 3 ⋅ Z 4 × Z 2 {\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}} abelsch D 8 {\displaystyle D_{8}} Z 8 {\displaystyle \mathbb {Z} _{8}} , 2 ⋅ D 4 {\displaystyle 2\cdot D_{4}} , 4 ⋅ D 2 {\displaystyle 4\cdot D_{2}} , Z 4 {\displaystyle \mathbb {Z} _{4}} , 9 ⋅ Z 2 {\displaystyle 9\cdot \mathbb {Z} _{2}} nichtabelsch D 4 × Z 2 {\displaystyle D_{4}\times \mathbb {Z} _{2}} 4 ⋅ D 4 {\displaystyle 4\cdot D_{4}} , Z 4 × Z 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}} , 2 ⋅ Z 2 3 {\displaystyle 2\cdot \mathbb {Z} _{2}^{3}} , 13 ⋅ Z 2 2 {\displaystyle 13\cdot \mathbb {Z} _{2}^{2}} , 2 ⋅ Z 4 {\displaystyle 2\cdot \mathbb {Z} _{4}} , 11 ⋅ Z 2 {\displaystyle 11\cdot \mathbb {Z} _{2}} nichtabelsch Q 16 ≅ D i c 4 {\displaystyle Q_{16}\cong \mathrm {Dic_{4}} } Z 8 {\displaystyle \mathbb {Z} _{8}} , 2 ⋅ Q 8 {\displaystyle 2\cdot Q_{8}} , 5 ⋅ Z 4 {\displaystyle 5\cdot \mathbb {Z} _{4}} , Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch Q 8 × Z 2 {\displaystyle Q_{8}\times \mathbb {Z} _{2}} 3 ⋅ Z 2 × Z 4 {\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}} , 4 ⋅ Q 8 {\displaystyle 4\cdot Q_{8}} , 6 ⋅ Z 4 {\displaystyle 6\cdot \mathbb {Z} _{4}} , Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch, hamiltonsche Gruppe Quasi-Diedergruppe Z 8 {\displaystyle \mathbb {Z} _{8}} , Q 8 {\displaystyle Q_{8}} , D 4 {\displaystyle D_{4}} , 3 ⋅ Z 4 {\displaystyle 3\cdot \mathbb {Z} _{4}} , 2 ⋅ Z 2 × Z 2 {\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 5 ⋅ Z 2 {\displaystyle 5\cdot \mathbb {Z} _{2}} nichtabelsch Nichtabelsche nicht-hamiltonsche modulare Gruppe 2 ⋅ Z 8 {\displaystyle 2\cdot \mathbb {Z} _{8}} , Z 4 × Z 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}} , 2 ⋅ Z 4 {\displaystyle 2\cdot \mathbb {Z} _{4}} , Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch Semidirektes Produkt Z 4 ⋊ Z 4 {\displaystyle \mathbb {Z} _{4}\rtimes \mathbb {Z} _{4}} (siehe hier ) 3 ⋅ Z 2 × Z 4 {\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}} , 6 ⋅ Z 4 {\displaystyle 6\cdot \mathbb {Z} _{4}} , Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch Die durch Pauli-Matrizen erzeugte Gruppe. 3 ⋅ Z 2 × Z 4 {\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}} , 3 ⋅ D 4 {\displaystyle 3\cdot D_{4}} , Q 8 {\displaystyle Q_{8}} , 4 ⋅ Z 4 {\displaystyle 4\cdot \mathbb {Z} _{4}} , 3 ⋅ Z 2 × Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch G 4 , 4 = V 4 ⋊ Z 4 {\displaystyle G_{4,4}=V_{4}\rtimes \mathbb {Z} _{4}} 2 ⋅ Z 2 × Z 4 {\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}} , Z 2 × Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 4 ⋅ Z 4 {\displaystyle 4\cdot \mathbb {Z} _{4}} , 7 ⋅ Z 2 × Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch 17 Z 17 {\displaystyle \mathbb {Z} _{17}} - abelsch, einfach, zyklisch 18 Z 18 ≅ Z 9 × Z 2 {\displaystyle \mathbb {Z} _{18}\cong \mathbb {Z} _{9}\times \mathbb {Z} _{2}} Z 9 , {\displaystyle \mathbb {Z} _{9},} Z 6 , {\displaystyle \mathbb {Z} _{6},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 6 × Z 3 {\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{3}} Z 3 2 , {\displaystyle \mathbb {Z} _{3}^{2},} 4 ⋅ Z 6 , {\displaystyle 4\cdot \mathbb {Z} _{6},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch D 9 {\displaystyle D_{9}} Z 9 , {\displaystyle \mathbb {Z} _{9},} 3 ⋅ D 3 , {\displaystyle 3\cdot D_{3},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 9 ⋅ Z 2 {\displaystyle 9\cdot \mathbb {Z} _{2}} nichtabelsch S 3 × Z 3 {\displaystyle S_{3}\times \mathbb {Z} _{3}} Z 3 2 , {\displaystyle \mathbb {Z} _{3}^{2},} D 3 , {\displaystyle D_{3},} 3 ⋅ Z 6 , {\displaystyle 3\cdot \mathbb {Z} _{6},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch ( Z 3 × Z 3 ) ⋊ α Z 2 {\displaystyle (\mathbb {Z} _{3}\times \mathbb {Z} _{3})\rtimes _{\alpha }\mathbb {Z} _{2}} mit α ( 1 ) = ( 2 0 0 2 ) {\displaystyle \alpha (1)={\begin{pmatrix}2&0\\0&2\end{pmatrix}}} Z 3 2 , {\displaystyle \mathbb {Z} _{3}^{2},} 12 ⋅ D 3 , {\displaystyle 12\cdot D_{3},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} 9 ⋅ Z 2 {\displaystyle 9\cdot \mathbb {Z} _{2}} nichtabelsch 19 Z 19 {\displaystyle \mathbb {Z} _{19}} - abelsch, einfach, zyklisch 20 Z 20 ≅ Z 5 × Z 4 {\displaystyle \mathbb {Z} _{20}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{4}} Z 10 , {\displaystyle \mathbb {Z} _{10},} Z 5 , {\displaystyle \mathbb {Z} _{5},} Z 4 , {\displaystyle \mathbb {Z} _{4},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 10 × Z 2 ≅ Z 5 × Z 2 × Z 2 {\displaystyle \mathbb {Z} _{10}\times \mathbb {Z} _{2}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}} 3 ⋅ Z 10 , {\displaystyle 3\cdot \mathbb {Z} _{10},} Z 5 , {\displaystyle \mathbb {Z} _{5},} D 2 , {\displaystyle D_{2},} 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} abelsch Q 20 ≅ D i c 5 {\displaystyle Q_{20}\cong \mathrm {Dic} _{5}} Z 10 , {\displaystyle \mathbb {Z} _{10},} Z 5 , {\displaystyle \mathbb {Z} _{5},} 5 ⋅ Z 4 , {\displaystyle 5\cdot \mathbb {Z} _{4},} Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch Z 5 ⋊ Z 4 ≅ {\displaystyle \mathbb {Z} _{5}\rtimes \mathbb {Z} _{4}\cong } AGL1 (5) D 5 , {\displaystyle D_{5},} Z 5 , {\displaystyle \mathbb {Z} _{5},} 5 ⋅ Z 4 , {\displaystyle 5\cdot \mathbb {Z} _{4},} 5 ⋅ Z 2 {\displaystyle 5\cdot \mathbb {Z} _{2}} nichtabelsch D 10 ≅ D 5 × Z 2 {\displaystyle D_{10}\cong D_{5}\times \mathbb {Z} _{2}} Z 10 , {\displaystyle \mathbb {Z} _{10},} D 5 , {\displaystyle D_{5},} Z 5 , {\displaystyle \mathbb {Z} _{5},} 5 ⋅ V 4 , {\displaystyle 5\cdot V_{4},} 11 ⋅ Z 2 {\displaystyle 11\cdot \mathbb {Z} _{2}} nichtabelsch 21 Z 21 ≅ Z 7 × Z 3 {\displaystyle \mathbb {Z} _{21}\cong \mathbb {Z} _{7}\times \mathbb {Z} _{3}} Z 7 , {\displaystyle \mathbb {Z} _{7},} Z 3 {\displaystyle \mathbb {Z} _{3}} abelsch, zyklisch Z 7 ⋊ Z 3 {\displaystyle \mathbb {Z} _{7}\rtimes \mathbb {Z} _{3}} Z 7 , {\displaystyle \mathbb {Z} _{7},} 7 ⋅ Z 3 {\displaystyle 7\cdot \mathbb {Z} _{3}} nichtabelsch 22 Z 22 ≅ Z 11 × Z 2 {\displaystyle \mathbb {Z} _{22}\cong \mathbb {Z} _{11}\times \mathbb {Z} _{2}} Z 11 , {\displaystyle \mathbb {Z} _{11},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch D 11 {\displaystyle D_{11}} Z 11 , {\displaystyle \mathbb {Z} _{11},} 11 ⋅ Z 2 {\displaystyle 11\cdot \mathbb {Z} _{2}} nichtabelsch 23 Z 2 3 {\displaystyle \mathbb {Z} _{2}3} - abelsch, einfach, zyklisch 24 Z 24 ≅ Z 8 × Z 3 {\displaystyle \mathbb {Z} _{24}\cong \mathbb {Z} _{8}\times \mathbb {Z} _{3}} Z 12 , {\displaystyle \mathbb {Z} _{12},} Z 8 , {\displaystyle \mathbb {Z} _{8},} Z 6 , {\displaystyle \mathbb {Z} _{6},} Z 4 , {\displaystyle \mathbb {Z} _{4},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch, zyklisch Z 12 × Z 2 ≅ Z 6 × Z 4 ≅ {\displaystyle \mathbb {Z} _{12}\times \mathbb {Z} _{2}\cong \mathbb {Z} _{6}\times \mathbb {Z} _{4}\cong } Z 4 × Z 3 × Z 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{3}\times \mathbb {Z} _{2}} Z 12 , {\displaystyle \mathbb {Z} _{12},} Z 6 , {\displaystyle \mathbb {Z} _{6},} Z 4 , {\displaystyle \mathbb {Z} _{4},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch Z 6 × D 2 ≅ Z 3 × Z 2 3 {\displaystyle \mathbb {Z} _{6}\times D_{2}\cong \mathbb {Z} _{3}\times \mathbb {Z} _{2}^{3}} Z 6 , {\displaystyle \mathbb {Z} _{6},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} abelsch Z 3 ⋊ Z 8 {\displaystyle \mathbb {Z} _{3}\rtimes \mathbb {Z} _{8}} Z 12 , {\displaystyle \mathbb {Z} _{12},} 3 ⋅ Z 8 , {\displaystyle 3\cdot \mathbb {Z} _{8},} Z 6 , {\displaystyle \mathbb {Z} _{6},} Z 4 , {\displaystyle \mathbb {Z} _{4},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch SL (2,3) ≅ Q 8 ⋊ Z 3 {\displaystyle \cong Q_{8}\rtimes \mathbb {Z} _{3}} Q 8 , {\displaystyle Q_{8},} 4 ⋅ Z 6 , {\displaystyle 4\cdot \mathbb {Z} _{6},} 3 ⋅ Z 4 , {\displaystyle 3\cdot \mathbb {Z} _{4},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch Q 24 ≅ Z 3 × Q 8 {\displaystyle Q_{24}\cong \mathbb {Z} _{3}\times Q_{8}} Z 12 , {\displaystyle \mathbb {Z} _{12},} 2 ⋅ Q 12 , {\displaystyle 2\cdot Q_{12},} 3 ⋅ Q 8 , {\displaystyle 3\cdot Q_{8},} Z 6 , {\displaystyle \mathbb {Z} _{6},} 7 ⋅ Z 4 , {\displaystyle 7\cdot \mathbb {Z} _{4},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch D 3 × Z 4 ≅ S 3 × Z 4 {\displaystyle D_{3}\times \mathbb {Z} _{4}\cong S_{3}\times \mathbb {Z} _{4}} Z 12 , {\displaystyle \mathbb {Z} _{12},} Q 12 , {\displaystyle Q_{12},} D 6 , {\displaystyle D_{6},} 3 ⋅ Z 4 × Z 2 , {\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2},} Z 6 , {\displaystyle \mathbb {Z} _{6},} 2 ⋅ D 3 , {\displaystyle 2\cdot D_{3},} 4 ⋅ Z 4 , {\displaystyle 4\cdot \mathbb {Z} _{4},} 3 ⋅ D 2 , {\displaystyle 3\cdot D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch D 12 {\displaystyle D_{12}} Z 12 , {\displaystyle \mathbb {Z} _{12},} 2 ⋅ D 6 , {\displaystyle 2\cdot D_{6},} 3 ⋅ D 4 , {\displaystyle 3\cdot D_{4},} Z 6 , {\displaystyle \mathbb {Z} _{6},} 4 ⋅ D 3 , {\displaystyle 4\cdot D_{3},} Z 4 , {\displaystyle \mathbb {Z} _{4},} 6 ⋅ D 2 , {\displaystyle 6\cdot D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 13 ⋅ Z 2 {\displaystyle 13\cdot \mathbb {Z} _{2}} nichtabelsch Q 12 × Z 2 ≅ ( Z 3 ⋊ Z 4 ) × Z 2 {\displaystyle Q_{12}\times \mathbb {Z} _{2}\cong (\mathbb {Z} _{3}\rtimes \mathbb {Z} _{4})\times \mathbb {Z} _{2}} Z 6 × Z 2 , {\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},} 2 ⋅ Q 12 , {\displaystyle 2\cdot Q_{12},} 3 ⋅ Z 4 × Z 2 , {\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2},} 3 ⋅ Z 6 , {\displaystyle 3\cdot \mathbb {Z} _{6},} 6 ⋅ Z 4 , {\displaystyle 6\cdot \mathbb {Z} _{4},} D 2 , {\displaystyle D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 3 ⋅ Z 2 {\displaystyle 3\cdot \mathbb {Z} _{2}} nichtabelsch ( Z 6 × Z 2 ) ⋊ Z 2 ≅ Z 3 ⋊ D 4 {\displaystyle (\mathbb {Z} _{6}\times \mathbb {Z} _{2})\rtimes \mathbb {Z} _{2}\cong \mathbb {Z} _{3}\rtimes D_{4}} Z 6 × Z 2 , {\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},} Q 12 , {\displaystyle Q_{12},} D 3 , {\displaystyle D_{3},} 3 ⋅ D 4 , {\displaystyle 3\cdot D_{4},} 3 ⋅ Z 6 , {\displaystyle 3\cdot \mathbb {Z} _{6},} 2 ⋅ D 3 , {\displaystyle 2\cdot D_{3},} 3 ⋅ Z 4 , {\displaystyle 3\cdot \mathbb {Z} _{4},} 4 ⋅ D 2 , {\displaystyle 4\cdot D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 9 ⋅ Z 2 {\displaystyle 9\cdot \mathbb {Z} _{2}} nichtabelsch D 4 × Z 3 {\displaystyle D_{4}\times \mathbb {Z} _{3}} Z 12 , {\displaystyle \mathbb {Z} _{12},} 2 ⋅ Z 6 × Z 2 , {\displaystyle 2\cdot \mathbb {Z} _{6}\times \mathbb {Z} _{2},} D 4 , {\displaystyle D_{4},} 5 ⋅ Z 6 , {\displaystyle 5\cdot \mathbb {Z} _{6},} Z 4 , {\displaystyle \mathbb {Z} _{4},} 2 ⋅ D 2 , {\displaystyle 2\cdot D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 5 ⋅ Z 2 {\displaystyle 5\cdot \mathbb {Z} _{2}} nichtabelsch Q 8 × Z 3 {\displaystyle Q_{8}\times \mathbb {Z} _{3}} 3 ⋅ Z 12 , {\displaystyle 3\cdot \mathbb {Z} _{12},} Q 8 , {\displaystyle Q_{8},} Z 6 , {\displaystyle \mathbb {Z} _{6},} 3 ⋅ Z 4 , {\displaystyle 3\cdot \mathbb {Z} _{4},} Z 3 , {\displaystyle \mathbb {Z} _{3},} Z 2 {\displaystyle \mathbb {Z} _{2}} nichtabelsch S 4 {\displaystyle S_{4}} A 4 , {\displaystyle A_{4},} 3 ⋅ D 4 , {\displaystyle 3\cdot D_{4},} 4 ⋅ D 3 , {\displaystyle 4\cdot D_{3},} 3 ⋅ Z 4 , {\displaystyle 3\cdot \mathbb {Z} _{4},} 4 ⋅ D 2 , {\displaystyle 4\cdot D_{2},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} 9 ⋅ Z 2 {\displaystyle 9\cdot \mathbb {Z} _{2}} nichtabelsch A 4 × Z 2 {\displaystyle A_{4}\times \mathbb {Z} _{2}} A 4 , {\displaystyle A_{4},} Z 2 3 , {\displaystyle \mathbb {Z} _{2}^{3},} 4 ⋅ Z 6 , {\displaystyle 4\cdot \mathbb {Z} _{6},} 7 ⋅ D 2 , {\displaystyle 7\cdot D_{2},} 4 ⋅ Z 3 , {\displaystyle 4\cdot \mathbb {Z} _{3},} 7 ⋅ Z 2 {\displaystyle 7\cdot \mathbb {Z} _{2}} nichtabelsch D 6 × Z 2 {\displaystyle D_{6}\times \mathbb {Z} _{2}} Z 6 × Z 2 , {\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2},} 6 ⋅ D 6 , {\displaystyle 6\cdot D_{6},} 3 ⋅ Z 2 3 , {\displaystyle 3\cdot \mathbb {Z} _{2}^{3},} 3 ⋅ Z 6 , {\displaystyle 3\cdot \mathbb {Z} _{6},} 4 ⋅ D 3 , {\displaystyle 4\cdot D_{3},} 19 ⋅ D 2 , {\displaystyle 19\cdot D_{2},} Z 3 , {\displaystyle \mathbb {Z} _{3},} 15 ⋅ Z 2 {\displaystyle 15\cdot \mathbb {Z} _{2}} nichtabelsch