Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C3×#41 |
| 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. C3×#634 | C2 | semidirect |
| 2. C32×D12 | C2 | semidirect |
| 3. C32×C4×D3 | C2 | semidirect |
| 4. C3×#625 | C2 | semidirect |
| 5. C3×D3×Dic3 | C2 | semidirect |
| 8. #3795 | C3 | direct (N × H) |
| 16. #634 | C2×C3 | semidirect |
| 21. C3×D12 | D3 | semidirect |
| 22. C3×D12 | C2×C3 | semidirect |
| 25. C3×C4×D3 | D3 | semidirect |
| 26. C3×C4×D3 | C2×C3 | semidirect |
| 30. #625 | C2×C3 | semidirect |
| 31. D3×Dic3 | C2×C3 | semidirect |
| 61. C33 | #41 | semidirect |
| 74. D12 | C3×D3 | semidirect |
| 76. C4×D3 | C3×D3 | semidirect |
| 123. C32 | #235 | semidirect |
| 124. C32 | #237 | semidirect |
| 125. C32 | C3×#41 | semidirect |
| 155. C3 | #3795 | direct (N × H) |
| 156. C3 | C3×#235 | semidirect |
| 157. C3 | C3×#237 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#634 | 2 |
| C32×D12 | 2 |
| C32×C4×D3 | 2 |
| C3×#625 | 2 |
| C3×D3×Dic3 | 2 |
| C3×#237 | 3 |
| C3×#235 | 3 |
| #3795 | 3 |