Finite Group DatabasebetaFinite Group DatabasebetaOrder 54.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C3⋊D3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×D3 | — | 3 | 1 | no | maximal |
| 4 | C32 | — | 2 | 1 | no | — |
| 5 | C32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C32 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×C3 | — | 9 | 1 | no | — |
| 8 | D3 | — | 3 | 1 | no | — |
| 9 | D3 | — | 9 | 1 | no | — |
| 10 | C3 | — | 3 | 1 | no | — |
| 11 | C3 | — | 3 | 1 | no | — |
| 12 | C3 | — | 6 | 1 | no | — |
| 13 | C3 | C3×D3 | 1 | 1 | yes | characteristic |
| 14 | C2 | — | 9 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32 | C2×C3 |
| Frattini, socle | C3 | C3×D3 |
| Fitting | H(3,𝔽3) | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3) | C2 | semidirect |
| 2. C3⋊D3 | C3 | semidirect |
| 5. C32 | D3 | semidirect |
| 6. C32 | C2×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3) | 2 |
| C3⋊D3 | 3 |
| C3×D3 | 3 |