Finite Group DatabasebetaFinite Group DatabasebetaOrder 54.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 2 | C2×C32 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C32 | C2×C3 | 1 | 4 | yes | complemented |
| 4 | C2×C3 | — | 3 | 4 | no | — |
| 5 | C2×C3 | C32 | 1 | 1 | yes | characteristic |
| 6 | C3 | — | 3 | 4 | no | — |
| 7 | C3 | C2×C32 | 1 | 1 | yes | characteristic |
| 8 | C2 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3) | C2 | direct (N × H) |
| 2. C2×C32 | C3 | semidirect |
| 3. C32 | C2×C3 | semidirect |
| 8. C2 | H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3) | 2 |
| C2×C32 | 3 |