Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
48 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#732 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | (C3⋊Dic3)⋊C3 | — | 3 | 3 | no | maximal |
| 3 | C3×Dic9 | — | 3 | 1 | no | maximal |
| 4 | #732 | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 5 | C2×H(3,𝔽3) | D3 | 1 | 3 | yes | — |
| 6 | C2×C3×C9 | D3 | 1 | 1 | yes | characteristic |
| 7 | C3⋊Dic3 | — | 3 | 3 | no | — |
| 8 | C3×Dic3 | — | 9 | 1 | no | — |
| 9 | Dic9 | — | 3 | 1 | no | — |
| 10 | H(3,𝔽3) | Dic3 | 1 | 3 | yes | complemented |
| 11 | C3×C9 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C32 | — | 3 | 3 | no | — |
| 13 | C2×C32 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 14 | C2×C9 | — | 3 | 1 | no | — |
| 15 | C3×C4 | — | 27 | 1 | no | — |
| 16 | Dic3 | — | 9 | 1 | no | — |
| 17 | Dic3 | — | 27 | 3 | no | — |
| 18 | C32 | — | 3 | 3 | no | — |
| 19 | C32 | C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 20 | C9 | — | 3 | 1 | no | — |
| 21 | C2×C3 | — | 3 | 1 | no | — |
| 22 | C2×C3 | — | 9 | 3 | no | — |
| 23 | C2×C3 | #272 | 1 | 1 | yes | characteristic |
| 24 | C4 | — | 27 | 1 | no | Sylow |
| 25 | C3 | — | 3 | 1 | no | — |
| 26 | C3 | — | 9 | 3 | no | — |
| 27 | C3 | #1084 | 1 | 1 | yes | characteristic |
| 28 | C2 | #4185 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4185 |
| commutator, hypocenter | #732 | C4 |
| Frattini | C2×C32 | C3⋊D3 |
| Fitting, prime radical, cosocle | C2×#732 | C2 |
| socle | C2×C3 | #272 |
| LCS | G ▷ #732 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #732 | C4 | semidirect |
| 10. H(3,𝔽3) | Dic3 | semidirect |
| 11. C3×C9 | Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#732 | 2 |
| (C3⋊Dic3)⋊C3 | 3 |
| C3×Dic9 | 3 |