Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
65 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6198 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #6198 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #6188 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×(C3⋊D3)⋊C3 | — | 2 | 1 | no | — |
| 5 | C2×(C3⋊D3)⋊C3 | — | 2 | 1 | no | — |
| 6 | #1084 | C22 | 1 | 1 | yes | characteristic |
| 7 | #626 | — | 6 | 1 | no | — |
| 8 | #626 | — | 6 | 1 | no | — |
| 9 | C2×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 10 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 11 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 12 | D24 | — | 9 | 1 | no | maximal |
| 13 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 14 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 15 | C2×C3×D3 | — | 6 | 1 | no | — |
| 16 | C2×C3×D3 | — | 6 | 1 | no | — |
| 17 | C3×Dic3 | — | 6 | 1 | no | — |
| 18 | C3×Dic3 | — | 6 | 1 | no | — |
| 19 | H(3,𝔽3) | D8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | #67 | — | 18 | 1 | no | — |
| 21 | #67 | — | 18 | 1 | no | — |
| 22 | D12 | — | 9 | 1 | no | — |
| 23 | D12 | — | 9 | 1 | no | — |
| 24 | C3×C8 | — | 9 | 1 | no | — |
| 25 | C2×C32 | — | 2 | 1 | no | — |
| 26 | C2×C32 | — | 2 | 1 | no | — |
| 27 | C3⋊D3 | — | 4 | 1 | no | — |
| 28 | C3⋊D3 | — | 4 | 1 | no | — |
| 29 | C3×D3 | — | 12 | 1 | no | — |
| 30 | C3×D3 | — | 12 | 1 | no | — |
| 31 | D8 | — | 27 | 1 | no | Sylow |
| 32 | C22×C3 | — | 18 | 1 | no | — |
| 33 | C22×C3 | — | 18 | 1 | no | — |
| 34 | C2×D3 | — | 6 | 1 | no | — |
| 35 | C2×D3 | — | 6 | 1 | no | — |
| 36 | C2×D3 | — | 36 | 1 | no | — |
| 37 | C2×D3 | — | 36 | 1 | no | — |
| 38 | C3×C4 | — | 9 | 1 | no | — |
| 39 | Dic3 | — | 6 | 1 | no | — |
| 40 | Dic3 | — | 6 | 1 | no | — |
| 41 | C32 | — | 2 | 1 | no | — |
| 42 | C32 | — | 2 | 1 | no | — |
| 43 | D4 | — | 27 | 1 | no | — |
| 44 | D4 | — | 27 | 1 | no | — |
| 45 | C8 | — | 9 | 1 | no | — |
| 46 | C2×C3 | — | 6 | 1 | no | — |
| 47 | C2×C3 | — | 12 | 1 | no | — |
| 48 | C2×C3 | — | 36 | 1 | no | — |
| 49 | C2×C3 | — | 36 | 1 | no | — |
| 50 | C2×C3 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 51 | D3 | — | 12 | 1 | no | — |
| 52 | D3 | — | 12 | 1 | no | — |
| 53 | D3 | — | 36 | 1 | no | — |
| 54 | D3 | — | 36 | 1 | no | — |
| 55 | C22 | — | 18 | 1 | no | — |
| 56 | C22 | — | 18 | 1 | no | — |
| 57 | C4 | — | 9 | 1 | no | — |
| 58 | C3 | — | 6 | 1 | no | — |
| 59 | C3 | — | 12 | 1 | no | — |
| 60 | C3 | #3774 | 1 | 1 | yes | characteristic |
| 61 | C2 | — | 36 | 1 | no | — |
| 62 | C2 | — | 36 | 1 | no | — |
| 63 | C2 | #1088⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1088⋊C2 |
| commutator, cosocle | #1084 | C22 |
| Frattini, socle | C2×C3 | GO+(4,𝔽2) |
| Fitting | C2×H(3,𝔽3) | D4 |
| hypocenter | H(3,𝔽3) | D8 |
| LCS | G ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6198 | C2 | semidirect |
| 2. #6198 | C2 | semidirect |
| 3. #6188 | C2 | semidirect |
| 19. H(3,𝔽3) | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6198 | 2 |
| #6188 | 2 |
| D24 | 9 |