Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1096 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×#936 | C2 | semidirect |
| 3. C32×#909 | C2 | semidirect |
| 4. C33×#100 | C2 | semidirect |
| 5. C32×#825 | C2 | semidirect |
| 7. C3×#4746 | C3 | direct (N × H) |
| 13. C2×C32×Dic6 | C22 | semidirect |
| 14. C33×M16 | C22 | semidirect |
| 16. C32×C4×Dic3 | C22 | semidirect |
| 23. C3×#936 | C2×C3 | semidirect |
| 24. C3×#909 | C2×C3 | semidirect |
| 27. C32×#100 | D3 | semidirect |
| 28. C32×#100 | C2×C3 | semidirect |
| 29. C3×#825 | C2×C3 | semidirect |
| 45. #4746 | C32 | direct (N × H) |
| 58. C2×C3×Dic6 | C22×C3 | semidirect |
| 62. C32×M16 | C2×D3 | semidirect |
| 63. C32×M16 | C22×C3 | semidirect |
| 67. C3×C4×Dic3 | C22×C3 | semidirect |
| 83. #936 | C2×C32 | semidirect |
| 84. #909 | C2×C32 | semidirect |
| 87. C3×#100 | C3×D3 | semidirect |
| 88. C3×#100 | C2×C32 | semidirect |
| 89. #825 | C2×C32 | semidirect |
| 139. C2×Dic6 | C22×C32 | semidirect |
| 143. C3×M16 | C2×C3×D3 | semidirect |
| 144. C3×M16 | C22×C32 | semidirect |
| 148. C4×Dic3 | C22×C32 | semidirect |
| 177. #100 | C32×D3 | semidirect |
| 178. C33 | #455 | semidirect |
| 230. M16 | C2×C32×D3 | semidirect |
| 256. C32 | #4746 | direct (N × H) |
| 257. C32 | C3×#455 | semidirect |
| 293. C3 | C3×#4746 | direct (N × H) |
| 294. C3 | C32×#455 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#1004 | 2 |
| C32×#936 | 2 |
| C32×#909 | 2 |
| C33×#100 | 2 |
| C32×#825 | 2 |
| C32×#455 | 3 |
| C3×#4746 | 3 |