List of Small Non-abelian Groups
| Order | Group | Subgroups | Properties | Cycle Graph |
|---|---|---|---|---|
| 6 | S3 = Dih3 | Z3, Z2 (3) | the smallest non-abelian group | |
| 8 | Dih4 | Z4, Z22 (2), Z2 (5) | ||
| quaternion group, Q8 = Dic2 | Z4 (3), Z2 | the smallest Hamiltonian group; smallest group demonstrating that all subgroups may be normal without the group being abelian; the smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G | ||
| 10 | Dih5 | Z5, Z2 (5) | ||
| 12 | Dih6 = Dih3 × Z2 | Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) | ||
| A4 | Z22, Z3 (4), Z2 (3) | smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See Lagrange's theorem and the Sylow theorems.) | ||
| Dic3 = Z3 Z4 | Z2, Z3, Z4 (3), Z6 | |||
| 14 | Dih7 | Z7, Z2 (7) | ||
| 16 | Dih8 | Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) | ||
| Dih4 × Z2 | Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) | |||
| generalized quaternion group, Q16 = Dic4 | ||||
| Q8 × Z2 | Hamiltonian | |||
| The order 16 quasidihedral group | ||||
| The order 16 modular group | ||||
| Z4 Z4 | ||||
| The group generated by the Pauli matrices | ||||
| G4,4 = Z22 Z4 |
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