Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
596 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 186 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 |
|---|---|---|
| 1. C32×D3×Q16 | C2 | semidirect |
| 3. C33×SD32 | C2 | semidirect |
| 4. C32×#829 | C2 | semidirect |
| 5. C32×#827 | C2 | semidirect |
| 6. C32×Dic24 | C2 | semidirect |
| 7. C32×#798 | C2 | semidirect |
| 9. C3×#4915 | C3 | direct (N × H) |
| 23. C3×D3×Q16 | C2×C3 | semidirect |
| 29. C32×SD32 | D3 | semidirect |
| 30. C32×SD32 | C2×C3 | semidirect |
| 32. C3×#829 | C2×C3 | semidirect |
| 33. C3×#827 | C2×C3 | semidirect |
| 34. C3×Dic24 | C2×C3 | semidirect |
| 35. C3×#798 | C2×C3 | semidirect |
| 48. #4915 | C32 | direct (N × H) |
| 83. D3×Q16 | C2×C32 | semidirect |
| 89. C3×SD32 | C3×D3 | semidirect |
| 90. C3×SD32 | C2×C32 | semidirect |
| 92. #829 | C2×C32 | semidirect |
| 93. #827 | C2×C32 | semidirect |
| 94. Dic24 | C2×C32 | semidirect |
| 95. #798 | C2×C32 | semidirect |
| 172. SD32 | C32×D3 | semidirect |
| 174. C33 | #510 | semidirect |
| 237. C32 | #4915 | direct (N × H) |
| 238. C32 | C3×#510 | semidirect |
| 266. C3 | C3×#4915 | direct (N × H) |
| 267. C3 | C32×#510 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×D3×Q16 | 2 |
| C32×#912 | 2 |
| C33×SD32 | 2 |
| C32×#829 | 2 |
| C32×#827 | 2 |
| C32×Dic24 | 2 |
| C32×#798 | 2 |
| C32×#510 | 3 |
| C3×#4915 | 3 |