Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3559 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 3559 conjugacy classes of subgroups — too many to draw. Only its 297 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×D9 |
| commutator, cosocle, hypocenter | C22×C9 | C24×C3 |
| Frattini | C3 | C23×D3×A4 |
| Fitting | C25×C9 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| LCS | G ▷ C22×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C9×A4 | C2 | semidirect |
| 2. C22×A4×D9 | C2 | direct (N × H) |
| 4. C25×D9 | C3 | semidirect |
| 6. C22×C9×A4 | C22 | semidirect |
| 7. C2×A4×D9 | C22 | direct (N × H) |
| 10. C25×C9 | C2×C3 | semidirect |
| 14. C24×D9 | C2×C3 | semidirect |
| 18. C2×C9×A4 | C23 | semidirect |
| 19. A4×D9 | C23 | direct (N × H) |
| 27. C24×C9 | C22×C3 | semidirect |
| 35. C23×D9 | A4 | direct (N × H) |
| 36. C23×D9 | C22×C3 | semidirect |
| 40. C9×A4 | C24 | semidirect |
| 48. C23×A4 | D9 | direct (N × H) |
| 58. C23×C9 | C2×A4 | semidirect |
| 59. C23×C9 | C23×C3 | semidirect |
| 67. C22×D9 | C2×A4 | direct (N × H) |
| 68. C22×D9 | C23×C3 | semidirect |
| 88. C22×A4 | C2×D9 | direct (N × H) |
| 97. C22×C9 | C22×A4 | semidirect |
| 98. C22×C9 | C24×C3 | semidirect |
| 104. C2×D9 | C22×A4 | direct (N × H) |
| 110. C25 | C3×D9 | semidirect |
| 132. C2×A4 | C22×D9 | direct (N × H) |
| 138. C2×C9 | C23×A4 | semidirect |
| 140. D9 | C23×A4 | direct (N × H) |
| 152. C24 | C2×C3×D9 | semidirect |
| 169. A4 | C23×D9 | direct (N × H) |
| 172. C9 | C24×A4 | semidirect |
| 186. C23 | A4×D9 | direct (N × H) |
| 187. C23 | C22×C3×D9 | semidirect |
| 206. C22 | C2×A4×D9 | direct (N × H) |
| 207. C22 | C23×C3×D9 | semidirect |
| 215. C2 | C22×A4×D9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C9×A4 | 2 |
| C22×A4×D9 | 2 |
| C23×D3×A4 | 3 |
| C25×D9 | 3 |
| C23×C3×D9 | 4 |