List of Small Groups - List of Small Non-abelian Groups

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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