Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
218 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 218 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 | C22×C33 | C22 |
| socle | C2×C33 | C23 |
| hypercenter | C22×C3 | D32 |
| hypocenter | C32 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#638 | C2 | semidirect |
| 2. C32×#67 | C2 | semidirect |
| 3. C2×C32×Dic3 | C2 | semidirect |
| 4. C3×#627 | C2 | semidirect |
| 5. C3×#626 | C2 | semidirect |
| 6. C3×#624 | C2 | semidirect |
| 7. C3×D3×Dic3 | C2 | semidirect |
| 10. #3804 | C3 | direct (N × H) |
| 20. #638 | C2×C3 | semidirect |
| 26. C3×#67 | D3 | semidirect |
| 27. C3×#67 | C2×C3 | semidirect |
| 30. C2×C3×Dic3 | D3 | semidirect |
| 31. C2×C3×Dic3 | C2×C3 | semidirect |
| 38. #627 | C2×C3 | semidirect |
| 39. #626 | C2×C3 | semidirect |
| 40. #624 | C2×C3 | semidirect |
| 41. D3×Dic3 | C2×C3 | semidirect |
| 80. C33 | #41 | semidirect |
| 96. #67 | C3×D3 | semidirect |
| 98. C2×Dic3 | C3×D3 | semidirect |
| 163. C32 | #235 | semidirect |
| 164. C32 | #237 | semidirect |
| 165. C32 | C3×#41 | semidirect |
| 206. C3 | #3804 | direct (N × H) |
| 207. C3 | C3×#235 | semidirect |
| 208. C3 | C3×#237 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#638 | 2 |
| C32×#67 | 2 |
| C2×C32×Dic3 | 2 |
| C3×#627 | 2 |
| C3×#626 | 2 |
| C3×#624 | 2 |
| C3×D3×Dic3 | 2 |
| C3×#237 | 3 |
| C3×#235 | 3 |
| #3804 | 3 |