Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
51 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(#41⋊C3) | — | 3 | 1 | no | maximal |
| 3 | #3701 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×Dic9 | — | 4 | 1 | no | maximal |
| 5 | C3×SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 6 | Q8×C9 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | D36 | — | 3 | 1 | no | — |
| 8 | C4×D9 | — | 3 | 1 | no | — |
| 9 | Q8⋊C9 | — | 2 | 1 | no | — |
| 10 | C2×C3×C9 | — | 4 | 1 | no | — |
| 11 | #239 | — | 3 | 1 | no | — |
| 12 | #41⋊C3 | — | 9 | 1 | no | — |
| 13 | C3×Dic3 | — | 12 | 1 | no | — |
| 14 | C2×D9 | — | 3 | 1 | no | — |
| 15 | C4×C9 | — | 3 | 1 | no | — |
| 16 | Dic9 | A4 | 1 | 1 | yes | characteristic |
| 17 | C3×C9 | — | 4 | 1 | no | Sylow |
| 18 | C3×Q8 | C3×D3 | 1 | 1 | yes | characteristic |
| 19 | D12 | — | 9 | 1 | no | — |
| 20 | C4×D3 | — | 9 | 1 | no | — |
| 21 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 22 | SL(2,𝔽3) | D9 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×C32 | — | 4 | 1 | no | — |
| 24 | C2×C9 | — | 8 | 1 | no | — |
| 25 | C2×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 26 | D9 | — | 6 | 1 | no | — |
| 27 | #41 | — | 9 | 1 | no | Sylow |
| 28 | C2×D3 | — | 9 | 1 | no | — |
| 29 | C3×C4 | — | 3 | 1 | no | — |
| 30 | C3×C4 | — | 36 | 1 | no | — |
| 31 | Dic3 | — | 3 | 1 | no | — |
| 32 | C32 | — | 4 | 1 | no | — |
| 33 | C9 | — | 8 | 1 | no | — |
| 34 | C9 | #41⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 35 | Q8 | C3×D9 | 1 | 1 | yes | characteristic |
| 36 | D4 | — | 27 | 1 | no | — |
| 37 | C2×C4 | — | 27 | 1 | no | — |
| 38 | C2×C3 | — | 4 | 1 | no | — |
| 39 | C2×C3 | — | 8 | 1 | no | — |
| 40 | C2×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 41 | D3 | — | 18 | 1 | no | — |
| 42 | C22 | — | 27 | 1 | no | — |
| 43 | C4 | — | 3 | 1 | no | — |
| 44 | C4 | — | 9 | 1 | no | — |
| 45 | C3 | — | 4 | 1 | no | — |
| 46 | C3 | — | 8 | 1 | no | — |
| 47 | C3 | C3⋊(#41⋊C3) | 1 | 1 | yes | characteristic |
| 48 | C2 | — | 54 | 1 | no | — |
| 49 | C2 | A4×D9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×D9 |
| commutator, Fitting, cosocle, hypocenter | Q8×C9 | C2×C3 |
| Frattini, socle | C2×C3 | D3×A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×SL(2,𝔽3) | C2 | semidirect |
| 3. #3701 | C3 | semidirect |
| 22. SL(2,𝔽3) | D9 | semidirect |
| 34. C9 | #41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×SL(2,𝔽3) | 2 |
| C3⋊(#41⋊C3) | 3 |
| #3701 | 3 |
| C3×Dic9 | 4 |