Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4322 conjugacy classes of subgroups; only the 1425 normal subgroups are shown.
This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 1425 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, hypocenter | C3×C9 | C24×C4 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×C3×C9 | C22 |
| socle | C24×C32 | C2×D3 |
| prime radical | C24×C3×D9 | C2 |
| cosocle | C2×C3×C9 | C25 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C3⋊Dic9 | C2 | semidirect |
| 3. C23×C9×Dic3 | C2 | semidirect |
| 4. C22×Dic3×D9 | C2 | direct (N × H) |
| 8. C23×C3×D9 | C4 | semidirect |
| 10. C22×C3⋊Dic9 | C22 | semidirect |
| 11. C22×C9×Dic3 | C22 | semidirect |
| 12. C2×Dic3×D9 | C22 | direct (N × H) |
| 22. C23×C3×C9 | C2×C4 | semidirect |
| 24. C22×C3×D9 | C2×C4 | semidirect |
| 26. C2×C3⋊Dic9 | C23 | semidirect |
| 27. C2×C9×Dic3 | C23 | semidirect |
| 28. Dic3×D9 | C23 | direct (N × H) |
| 39. C23×D9 | Dic3 | direct (N × H) |
| 45. C22×C3×C9 | C22×C4 | semidirect |
| 47. C2×C3×D9 | C22×C4 | semidirect |
| 49. C3⋊Dic9 | C24 | semidirect |
| 50. C9×Dic3 | C24 | semidirect |
| 58. C23×Dic3 | D9 | direct (N × H) |
| 69. C23×C9 | C2×Dic3 | semidirect |
| 71. C22×D9 | C2×Dic3 | direct (N × H) |
| 78. C2×C3×C9 | C23×C4 | semidirect |
| 80. C3×D9 | C23×C4 | semidirect |
| 93. C22×Dic3 | C2×D9 | direct (N × H) |
| 103. C22×C9 | C22×Dic3 | semidirect |
| 105. C2×D9 | C22×Dic3 | direct (N × H) |
| 114. C3×C9 | C24×C4 | semidirect |
| 119. C23×C3 | C4×D9 | semidirect |
| 130. C2×Dic3 | C22×D9 | direct (N × H) |
| 137. C2×C9 | C23×Dic3 | semidirect |
| 139. D9 | C23×Dic3 | direct (N × H) |
| 150. C22×C3 | C2×C4×D9 | semidirect |
| 160. Dic3 | C23×D9 | direct (N × H) |
| 163. C9 | C24×Dic3 | semidirect |
| 166. C23 | Dic3×D9 | direct (N × H) |
| 174. C2×C3 | C22×C4×D9 | semidirect |
| 181. C22 | C2×Dic3×D9 | direct (N × H) |
| 186. C3 | C23×C4×D9 | semidirect |
| 189. C2 | C22×Dic3×D9 | 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 |
| C23×C3⋊Dic9 | 2 |
| C23×C9×Dic3 | 2 |
| C22×Dic3×D9 | 2 |
| C23×D3×Dic3 | 3 |
| C23×C4×D9 | 3 |