Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4834 conjugacy classes of subgroups; only the 913 normal subgroups are shown.
This group has 4834 conjugacy classes of subgroups — too many to draw. Only its 913 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 | D3×D9 |
| commutator, cosocle | C2×C3×C9 | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C24×C32 | C2×D3 |
| hypocenter | C3×C9 | C23×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3⋊D9 | C2 | semidirect |
| 2. C24×C3×D9 | C2 | semidirect |
| 3. C23×C9×Dic3 | C2 | semidirect |
| 4. C22×#6192 | C2 | direct (N × H) |
| 9. C23×C3⋊D9 | C22 | semidirect |
| 11. C23×C3×D9 | C22 | semidirect |
| 12. C22×C9×Dic3 | C22 | semidirect |
| 13. C2×#6192 | C22 | direct (N × H) |
| 20. C24×D9 | D3 | semidirect |
| 23. C23×C3×C9 | D4 | semidirect |
| 26. C22×C3⋊D9 | C23 | semidirect |
| 28. C22×C3×D9 | C23 | semidirect |
| 29. C2×C9×Dic3 | C23 | semidirect |
| 30. #6192 | C23 | direct (N × H) |
| 47. C23×D9 | C2×D3 | semidirect |
| 50. C22×C3×C9 | C2×D4 | semidirect |
| 53. C2×C3⋊D9 | C24 | semidirect |
| 55. C2×C3×D9 | C24 | semidirect |
| 56. C9×Dic3 | C24 | semidirect |
| 64. C23×Dic3 | D9 | semidirect |
| 76. C23×C9 | #67 | semidirect |
| 83. C22×D9 | C22×D3 | semidirect |
| 87. C2×C3×C9 | C22×D4 | semidirect |
| 106. C22×Dic3 | C2×D9 | semidirect |
| 117. C22×C9 | C2×#67 | semidirect |
| 124. C2×D9 | C23×D3 | semidirect |
| 130. C3×C9 | C23×D4 | semidirect |
| 135. C23×C3 | D36 | semidirect |
| 149. C2×Dic3 | C22×D9 | semidirect |
| 157. C2×C9 | C22×#67 | semidirect |
| 173. C22×C3 | C2×D36 | semidirect |
| 186. Dic3 | C23×D9 | semidirect |
| 189. C9 | C23×#67 | semidirect |
| 194. C23 | #6192 | direct (N × H) |
| 201. C2×C3 | C22×D36 | semidirect |
| 213. C22 | C2×#6192 | direct (N × H) |
| 217. C3 | C23×D36 | semidirect |
| 221. C2 | C22×#6192 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3⋊D9 | 2 |
| C24×C3×D9 | 2 |
| C23×C9×Dic3 | 2 |
| C22×#6192 | 2 |
| C23×#626 | 3 |
| C23×D36 | 3 |