Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
7314 conjugacy classes of subgroups; only the 1047 normal subgroups are shown.
This group has 7314 conjugacy classes of subgroups — too many to draw. Only its 1047 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×#1113 |
| Fitting, socle | C24×C33 | C22 |
| hypocenter | C33 | C23×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C3⋊D3 | C2 | semidirect |
| 2. C23×C3×C3⋊Dic3 | C2 | semidirect |
| 3. C22×#6295 | C2 | direct (N × H) |
| 8. C23×C3×C3⋊D3 | C22 | semidirect |
| 9. C22×C3×C3⋊Dic3 | C22 | semidirect |
| 10. C2×#6295 | C22 | direct (N × H) |
| 11. C24×C3⋊D3 | D3 | semidirect |
| 15. C23×C3⋊Dic3 | D3 | semidirect |
| 20. C23×C33 | D4 | semidirect |
| 23. C22×C3×C3⋊D3 | C23 | semidirect |
| 24. C2×C3×C3⋊Dic3 | C23 | semidirect |
| 25. #6295 | C23 | direct (N × H) |
| 34. C23×C3⋊D3 | C2×D3 | semidirect |
| 41. C22×C3⋊Dic3 | C2×D3 | semidirect |
| 46. C22×C33 | C2×D4 | semidirect |
| 49. C2×C3×C3⋊D3 | C24 | semidirect |
| 50. C3×C3⋊Dic3 | C24 | semidirect |
| 66. C23×C32 | D12 | semidirect |
| 67. C23×C32 | #67 | semidirect |
| 71. C22×C3⋊D3 | C22×D3 | semidirect |
| 78. C2×C3⋊Dic3 | C22×D3 | semidirect |
| 84. C2×C33 | C22×D4 | semidirect |
| 111. C22×C32 | C2×D12 | semidirect |
| 112. C22×C32 | C2×#67 | semidirect |
| 116. C2×C3⋊D3 | C23×D3 | semidirect |
| 123. C3⋊Dic3 | C23×D3 | semidirect |
| 129. C33 | C23×D4 | semidirect |
| 138. C23×C3 | #625 | semidirect |
| 139. C23×C3 | #626 | semidirect |
| 159. C2×C32 | C22×D12 | semidirect |
| 160. C2×C32 | C22×#67 | semidirect |
| 180. C22×C3 | C2×#625 | semidirect |
| 181. C22×C3 | C2×#626 | semidirect |
| 196. C32 | C23×D12 | semidirect |
| 197. C32 | C23×#67 | semidirect |
| 200. C23 | #6295 | direct (N × H) |
| 211. C2×C3 | C22×#625 | semidirect |
| 212. C2×C3 | C22×#626 | semidirect |
| 220. C22 | C2×#6295 | direct (N × H) |
| 226. C3 | C23×#625 | semidirect |
| 227. C3 | C23×#626 | semidirect |
| 229. C2 | C22×#6295 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C3⋊D3 | 2 |
| C23×C3×C3⋊Dic3 | 2 |
| C22×#6295 | 2 |
| C23×#626 | 3 |
| C23×#625 | 3 |