Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
178 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×C3⋊Dic3 | C2 | 1 | 3 | yes | maximal |
| 2 | Dic32 | — | 3 | 3 | no | maximal |
| 3 | C22×C33 | C22 | 1 | 1 | yes | characteristic |
| 4 | C3×C3⋊Dic3 | C4 | 1 | 6 | yes | complemented |
| 5 | C2×C3⋊Dic3 | D3 | 1 | 3 | yes | — |
| 6 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 7 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 8 | C2×C33 | C2×C4 | 1 | 3 | yes | — |
| 9 | C4×Dic3 | — | 9 | 3 | no | — |
| 10 | C22×C32 | — | 2 | 3 | no | — |
| 11 | C22×C32 | — | 4 | 1 | no | — |
| 12 | C22×C32 | C2×D3 | 1 | 3 | yes | — |
| 13 | C3⋊Dic3 | Dic3 | 1 | 6 | yes | complemented |
| 14 | C3×Dic3 | — | 3 | 12 | no | — |
| 15 | C3×Dic3 | — | 6 | 6 | no | — |
| 16 | C33 | C42 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C2×C3×C4 | — | 9 | 3 | no | — |
| 18 | C2×Dic3 | — | 3 | 6 | no | — |
| 19 | C2×Dic3 | — | 6 | 3 | no | — |
| 20 | C2×C32 | — | 2 | 6 | no | — |
| 21 | C2×C32 | — | 2 | 3 | no | — |
| 22 | C2×C32 | — | 4 | 3 | no | — |
| 23 | C2×C32 | C4×D3 | 1 | 6 | yes | — |
| 24 | C2×C32 | C2×Dic3 | 1 | 3 | yes | — |
| 25 | C42 | — | 27 | 1 | no | Sylow |
| 26 | C22×C3 | — | 2 | 3 | no | — |
| 27 | C22×C3 | — | 4 | 1 | no | — |
| 28 | C22×C3 | D32 | 1 | 3 | yes | — |
| 29 | C3×C4 | — | 9 | 6 | no | — |
| 30 | Dic3 | — | 3 | 12 | no | — |
| 31 | Dic3 | — | 6 | 6 | no | — |
| 32 | C32 | — | 2 | 3 | no | — |
| 33 | C32 | — | 4 | 1 | no | — |
| 34 | C32 | C4×Dic3 | 1 | 3 | yes | complemented |
| 35 | C2×C4 | — | 9 | 3 | no | — |
| 36 | C2×C3 | — | 2 | 3 | no | — |
| 37 | C2×C3 | — | 2 | 6 | no | — |
| 38 | C2×C3 | — | 4 | 3 | no | — |
| 39 | C2×C3 | D3×Dic3 | 1 | 6 | yes | — |
| 40 | C2×C3 | #624 | 1 | 3 | yes | — |
| 41 | C22 | #1113 | 1 | 1 | yes | characteristic |
| 42 | C4 | — | 9 | 6 | no | — |
| 43 | C3 | — | 2 | 3 | no | — |
| 44 | C3 | — | 4 | 1 | no | — |
| 45 | C3 | Dic32 | 1 | 3 | yes | complemented |
| 46 | C2 | #6294 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | #1113 |
| commutator, hypocenter | C33 | C42 |
| Fitting, socle, prime radical, cosocle | C22×C33 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×C3⋊Dic3 | C4 | semidirect |
| 13. C3⋊Dic3 | Dic3 | semidirect |
| 16. C33 | C42 | semidirect |
| 34. C32 | C4×Dic3 | semidirect |
| 45. C3 | Dic32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C3⋊Dic3 | 2 |
| Dic32 | 3 |