Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
164 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | A4×C3⋊D3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×C3⋊S4 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D3×S4 | — | 3 | 2 | no | maximal |
| 4 | #3811 | — | 3 | 1 | no | maximal |
| 5 | C32×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | #1113 | — | 4 | 1 | no | maximal |
| 7 | C22×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×D32 | — | 3 | 1 | no | — |
| 9 | D3×A4 | — | 3 | 2 | no | — |
| 10 | D3×A4 | — | 6 | 1 | no | — |
| 11 | C3⋊S4 | D3 | 1 | 2 | yes | complemented |
| 12 | C3×S4 | — | 3 | 2 | no | — |
| 13 | C3×S4 | — | 6 | 2 | no | — |
| 14 | C3×#67 | — | 3 | 2 | no | — |
| 15 | #626 | — | 3 | 2 | no | — |
| 16 | #624 | — | 3 | 1 | no | — |
| 17 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 18 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 19 | C2×S4 | — | 9 | 1 | no | — |
| 20 | D3×D4 | — | 9 | 2 | no | — |
| 21 | C22×C32 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 23 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 24 | C2×C3×D3 | — | 3 | 2 | no | — |
| 25 | C3×A4 | — | 2 | 2 | no | — |
| 26 | C3×A4 | — | 2 | 1 | no | — |
| 27 | C3×A4 | — | 4 | 1 | no | — |
| 28 | C3×A4 | C2×D3 | 1 | 2 | yes | complemented |
| 29 | D32 | — | 6 | 1 | no | — |
| 30 | D32 | — | 6 | 1 | no | — |
| 31 | D32 | — | 12 | 2 | no | — |
| 32 | C3×Dic3 | — | 3 | 2 | no | — |
| 33 | C33 | — | 4 | 1 | no | Sylow |
| 34 | C22×D3 | — | 3 | 2 | no | — |
| 35 | C22×D3 | — | 6 | 1 | no | — |
| 36 | C22×D3 | — | 9 | 2 | no | — |
| 37 | C2×A4 | — | 9 | 1 | no | — |
| 38 | S4 | — | 3 | 2 | no | — |
| 39 | S4 | — | 6 | 2 | no | — |
| 40 | C3×D4 | — | 9 | 2 | no | — |
| 41 | #67 | — | 3 | 2 | no | — |
| 42 | #67 | — | 9 | 2 | no | — |
| 43 | D12 | — | 9 | 2 | no | — |
| 44 | C4×D3 | — | 9 | 2 | no | — |
| 45 | C2×C32 | — | 3 | 1 | no | — |
| 46 | C3⋊D3 | — | 3 | 1 | no | — |
| 47 | C3⋊D3 | — | 4 | 2 | no | — |
| 48 | C3⋊D3 | S4 | 1 | 1 | yes | characteristic, complemented |
| 49 | C3×D3 | — | 6 | 2 | no | — |
| 50 | C3×D3 | — | 12 | 2 | no | — |
| 51 | C3×D3 | — | 12 | 2 | no | — |
| 52 | C3×D3 | — | 24 | 1 | no | — |
| 53 | C3×D3 | — | 24 | 2 | no | — |
| 54 | C2×D4 | — | 27 | 1 | no | Sylow |
| 55 | C22×C3 | — | 2 | 1 | no | — |
| 56 | C22×C3 | — | 9 | 2 | no | — |
| 57 | C22×C3 | D32 | 1 | 2 | yes | complemented |
| 58 | C2×D3 | — | 3 | 2 | no | — |
| 59 | C2×D3 | — | 9 | 2 | no | — |
| 60 | C2×D3 | — | 9 | 2 | no | — |
| 61 | C2×D3 | — | 18 | 1 | no | — |
| 62 | C2×D3 | — | 18 | 1 | no | — |
| 63 | C2×D3 | — | 18 | 2 | no | — |
| 64 | C2×D3 | — | 18 | 2 | no | — |
| 65 | C2×D3 | — | 36 | 1 | no | — |
| 66 | A4 | — | 2 | 2 | no | — |
| 67 | A4 | — | 4 | 1 | no | — |
| 68 | A4 | D32 | 1 | 1 | yes | characteristic, complemented |
| 69 | C3×C4 | — | 9 | 2 | no | — |
| 70 | Dic3 | — | 3 | 2 | no | — |
| 71 | C32 | — | 4 | 2 | no | — |
| 72 | C32 | — | 8 | 1 | no | — |
| 73 | C32 | — | 8 | 2 | no | — |
| 74 | C32 | — | 16 | 1 | no | — |
| 75 | C32 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 76 | C23 | — | 9 | 1 | no | — |
| 77 | C23 | — | 27 | 1 | no | — |
| 78 | D4 | — | 9 | 2 | no | — |
| 79 | D4 | — | 27 | 2 | no | — |
| 80 | C2×C4 | — | 27 | 1 | no | — |
| 81 | C2×C3 | — | 3 | 2 | no | — |
| 82 | C2×C3 | — | 6 | 1 | no | — |
| 83 | C2×C3 | — | 18 | 2 | no | — |
| 84 | C2×C3 | — | 36 | 1 | no | — |
| 85 | D3 | — | 3 | 2 | no | — |
| 86 | D3 | — | 6 | 1 | no | — |
| 87 | D3 | — | 6 | 2 | no | — |
| 88 | D3 | — | 9 | 2 | no | — |
| 89 | D3 | — | 12 | 2 | no | — |
| 90 | D3 | — | 18 | 1 | no | — |
| 91 | D3 | — | 24 | 2 | no | — |
| 92 | C22 | — | 9 | 2 | no | — |
| 93 | C22 | — | 27 | 1 | no | — |
| 94 | C22 | — | 27 | 1 | no | — |
| 95 | C22 | — | 54 | 1 | no | — |
| 96 | C22 | — | 54 | 1 | no | — |
| 97 | C22 | #1113 | 1 | 1 | yes | characteristic, complemented |
| 98 | C4 | — | 9 | 2 | no | — |
| 99 | C3 | — | 2 | 1 | no | — |
| 100 | C3 | — | 4 | 1 | no | — |
| 101 | C3 | — | 8 | 2 | no | — |
| 102 | C3 | — | 16 | 1 | no | — |
| 103 | C3 | D3×S4 | 1 | 2 | yes | complemented |
| 104 | C2 | — | 3 | 1 | no | — |
| 105 | C2 | — | 9 | 1 | no | — |
| 106 | C2 | — | 18 | 2 | no | — |
| 107 | C2 | — | 27 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A4 | C22 |
| Fitting, socle | C22×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×C3⋊D3 | C2 | semidirect |
| 2. C3×C3⋊S4 | C2 | semidirect |
| 5. C32×A4 | C22 | semidirect |
| 7. C22×C3⋊D3 | D3 | semidirect |
| 11. C3⋊S4 | D3 | semidirect |
| 21. C22×C32 | C2×D3 | semidirect |
| 28. C3×A4 | C2×D3 | semidirect |
| 48. C3⋊D3 | S4 | semidirect |
| 57. C22×C3 | D32 | semidirect |
| 68. A4 | D32 | semidirect |
| 75. C32 | C2×S4 | semidirect |
| 97. C22 | #1113 | semidirect |
| 103. C3 | D3×S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×C3⋊D3 | 2 |
| C3×C3⋊S4 | 2 |
| D3×S4 | 3 |
| #3811 | 3 |
| #1113 | 4 |