Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
107 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 107 conjugacy classes of subgroups — too many to draw. Only its 17 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×D9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C9⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×S4 | — | 3 | 1 | no | maximal |
| 5 | D4×D9 | — | 3 | 1 | no | maximal |
| 6 | C9×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | D3×D9 | — | 4 | 1 | no | maximal |
| 8 | D3×A4 | — | 3 | 1 | no | — |
| 9 | C3⋊S4 | — | 3 | 1 | no | — |
| 10 | C3×S4 | D3 | 1 | 1 | yes | characteristic |
| 11 | C22×D9 | — | 3 | 1 | no | — |
| 12 | C22×D9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | (C22⋊C9)⋊C2 | — | 6 | 1 | no | — |
| 14 | D4×C9 | — | 3 | 1 | no | — |
| 15 | #611 | — | 3 | 1 | no | — |
| 16 | #611 | — | 3 | 1 | no | — |
| 17 | D36 | — | 3 | 1 | no | — |
| 18 | C4×D9 | — | 3 | 1 | no | — |
| 19 | C3⋊D9 | — | 4 | 1 | no | — |
| 20 | D3×C9 | — | 4 | 1 | no | — |
| 21 | C3×D9 | — | 4 | 1 | no | — |
| 22 | C2×S4 | — | 9 | 1 | no | — |
| 23 | D3×D4 | — | 9 | 1 | no | — |
| 24 | C3×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 25 | D32 | — | 12 | 1 | no | — |
| 26 | C22×C9 | — | 3 | 1 | no | — |
| 27 | C22×C9 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×D9 | — | 3 | 1 | no | — |
| 29 | C2×D9 | — | 3 | 1 | no | — |
| 30 | C2×D9 | — | 3 | 1 | no | — |
| 31 | C2×D9 | — | 6 | 1 | no | — |
| 32 | C2×D9 | — | 6 | 1 | no | — |
| 33 | C22⋊C9 | — | 2 | 1 | no | — |
| 34 | C4×C9 | — | 3 | 1 | no | — |
| 35 | Dic9 | — | 3 | 1 | no | — |
| 36 | C3×C9 | — | 4 | 1 | no | Sylow |
| 37 | C22×D3 | — | 3 | 1 | no | — |
| 38 | C22×D3 | — | 9 | 1 | no | — |
| 39 | C2×A4 | — | 9 | 1 | no | — |
| 40 | S4 | — | 9 | 1 | no | — |
| 41 | S4 | — | 18 | 1 | no | — |
| 42 | S4 | D9 | 1 | 1 | yes | characteristic, direct product factor |
| 43 | C3×D4 | — | 3 | 1 | no | — |
| 44 | #67 | — | 9 | 1 | no | — |
| 45 | #67 | — | 9 | 1 | no | — |
| 46 | D12 | — | 9 | 1 | no | — |
| 47 | C4×D3 | — | 9 | 1 | no | — |
| 48 | C3⋊D3 | — | 12 | 1 | no | — |
| 49 | C3×D3 | — | 4 | 1 | no | — |
| 50 | C3×D3 | — | 12 | 1 | no | — |
| 51 | C2×C9 | — | 3 | 1 | no | — |
| 52 | C2×C9 | — | 6 | 1 | no | — |
| 53 | D9 | — | 3 | 1 | no | — |
| 54 | D9 | — | 6 | 1 | no | — |
| 55 | D9 | — | 24 | 1 | no | — |
| 56 | D9 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 57 | C2×D4 | — | 27 | 1 | no | Sylow |
| 58 | C22×C3 | — | 3 | 1 | no | — |
| 59 | C22×C3 | D32 | 1 | 1 | yes | characteristic |
| 60 | C2×D3 | — | 9 | 1 | no | — |
| 61 | C2×D3 | — | 9 | 1 | no | — |
| 62 | C2×D3 | — | 9 | 1 | no | — |
| 63 | C2×D3 | — | 18 | 1 | no | — |
| 64 | C2×D3 | — | 18 | 1 | no | — |
| 65 | C2×D3 | — | 36 | 1 | no | — |
| 66 | A4 | — | 2 | 1 | no | — |
| 67 | A4 | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 68 | C3×C4 | — | 3 | 1 | no | — |
| 69 | Dic3 | — | 9 | 1 | no | — |
| 70 | C32 | — | 4 | 1 | no | — |
| 71 | C9 | — | 8 | 1 | no | — |
| 72 | C9 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 73 | C23 | — | 9 | 1 | no | — |
| 74 | C23 | — | 27 | 1 | no | — |
| 75 | D4 | — | 3 | 1 | no | — |
| 76 | D4 | — | 27 | 1 | no | — |
| 77 | D4 | — | 27 | 1 | no | — |
| 78 | D4 | — | 27 | 1 | no | — |
| 79 | C2×C4 | — | 27 | 1 | no | — |
| 80 | C2×C3 | — | 3 | 1 | no | — |
| 81 | C2×C3 | — | 6 | 1 | no | — |
| 82 | C2×C3 | — | 36 | 1 | no | — |
| 83 | D3 | — | 3 | 1 | no | — |
| 84 | D3 | — | 4 | 1 | no | — |
| 85 | D3 | — | 9 | 1 | no | — |
| 86 | D3 | — | 18 | 1 | no | — |
| 87 | D3 | — | 36 | 1 | no | — |
| 88 | D3 | — | 72 | 1 | no | — |
| 89 | C22 | — | 3 | 1 | no | — |
| 90 | C22 | — | 27 | 1 | no | — |
| 91 | C22 | — | 27 | 1 | no | — |
| 92 | C22 | — | 27 | 1 | no | — |
| 93 | C22 | — | 54 | 1 | no | — |
| 94 | C22 | — | 54 | 1 | no | — |
| 95 | C22 | D3×D9 | 1 | 1 | yes | characteristic, complemented |
| 96 | C4 | — | 3 | 1 | no | — |
| 97 | C4 | — | 27 | 1 | no | — |
| 98 | C3 | — | 4 | 1 | no | — |
| 99 | C3 | — | 8 | 1 | no | — |
| 100 | C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 101 | C2 | — | 3 | 1 | no | — |
| 102 | C2 | — | 6 | 1 | no | — |
| 103 | C2 | — | 9 | 1 | no | — |
| 104 | C2 | — | 27 | 1 | no | — |
| 105 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C9×A4 | C22 |
| Frattini | C3 | D3×S4 |
| Fitting | C22×C9 | C2×D3 |
| socle | C22×C3 | D32 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×D9 | C2 | semidirect |
| 2. C9⋊S4 | C2 | semidirect |
| 3. C9×S4 | C2 | semidirect |
| 6. C9×A4 | C22 | semidirect |
| 12. C22×D9 | D3 | semidirect |
| 27. C22×C9 | C2×D3 | semidirect |
| 42. S4 | D9 | direct (N × H) |
| 56. D9 | S4 | direct (N × H) |
| 67. A4 | C2×D9 | semidirect |
| 72. C9 | C2×S4 | semidirect |
| 95. C22 | D3×D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×D9 | 2 |
| C9⋊S4 | 2 |
| C9×S4 | 2 |
| D3×S4 | 3 |
| D4×D9 | 3 |
| D3×D9 | 4 |