Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
270 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C2×C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C23 |
| hypercenter | C3×C4 | D32 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D32 | C2 | semidirect |
| 2. C3×C4⋊(C3⋊D3) | C2 | semidirect |
| 3. C32×D12 | C2 | semidirect |
| 4. C32×C4×D3 | C2 | semidirect |
| 5. C3×#626 | C2 | semidirect |
| 8. D3×D12 | C3 | direct (N × H) |
| 9. C2×C3×C3⋊D3 | C22 | semidirect |
| 11. C2×C32×D3 | C22 | semidirect |
| 13. C33×C4 | C22 | semidirect |
| 14. C32×Dic3 | C22 | semidirect |
| 17. C2×D32 | C2×C3 | semidirect |
| 20. C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 27. C3×D12 | D3 | direct (N × H) |
| 28. C3×D12 | C2×C3 | semidirect |
| 30. C3×C4×D3 | D3 | semidirect |
| 31. C3×C4×D3 | C2×C3 | semidirect |
| 32. #626 | C2×C3 | semidirect |
| 36. C32×D3 | D4 | semidirect |
| 42. C2×C3⋊D3 | C22×C3 | semidirect |
| 51. C2×C3×D3 | C2×D3 | semidirect |
| 53. C2×C3×D3 | C22×C3 | semidirect |
| 60. C32×C4 | C2×D3 | semidirect |
| 61. C32×C4 | C2×D3 | semidirect |
| 62. C32×C4 | C22×C3 | semidirect |
| 64. C3×Dic3 | C2×D3 | semidirect |
| 65. C3×Dic3 | C22×C3 | semidirect |
| 66. C33 | C2×D4 | semidirect |
| 83. D12 | C3×D3 | direct (N × H) |
| 84. C4×D3 | C3×D3 | semidirect |
| 102. C3×D3 | D12 | direct (N × H) |
| 103. C3×D3 | C3×D4 | semidirect |
| 119. C2×D3 | C2×C3×D3 | semidirect |
| 127. C3×C4 | D32 | semidirect |
| 128. C3×C4 | C2×C3×D3 | semidirect |
| 129. C3×C4 | C2×C3×D3 | semidirect |
| 130. Dic3 | C2×C3×D3 | semidirect |
| 135. C32 | C2×D12 | semidirect |
| 136. C32 | D3×D4 | semidirect |
| 137. C32 | C2×C3×D4 | semidirect |
| 161. D3 | C3×D12 | direct (N × H) |
| 167. C4 | C3×D32 | semidirect |
| 172. C3 | D3×D12 | direct (N × H) |
| 173. C3 | C2×C3×D12 | semidirect |
| 174. C3 | C3×D3×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D32 | 2 |
| C3×C4⋊(C3⋊D3) | 2 |
| C32×D12 | 2 |
| C32×C4×D3 | 2 |
| C3×#626 | 2 |
| C3×D3×D4 | 3 |
| C2×C3×D12 | 3 |
| D3×D12 | 3 |