Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1404 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1404 conjugacy classes of subgroups — too many to draw. Only its 306 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. C33×#127 | C2 | semidirect |
| 2. C2×C32×#213 | C2 | semidirect |
| 3. C32×#874 | C2 | semidirect |
| 4. C32×C4×C3⋊C8 | C2 | semidirect |
| 6. C3×#5073 | C3 | direct (N × H) |
| 17. C32×#127 | D3 | semidirect |
| 18. C32×#127 | C2×C3 | semidirect |
| 19. C2×C3×#213 | C2×C3 | semidirect |
| 20. C3×#874 | C2×C3 | semidirect |
| 22. C3×C4×C3⋊C8 | C2×C3 | semidirect |
| 31. C32×C3⋊C8 | D4 | semidirect |
| 33. #5073 | C32 | direct (N × H) |
| 56. C33×C4 | D8 | semidirect |
| 61. C3×#127 | C3×D3 | semidirect |
| 62. C3×#127 | C2×C32 | semidirect |
| 63. C2×#213 | C2×C32 | semidirect |
| 64. #874 | C2×C32 | semidirect |
| 66. C4×C3⋊C8 | C2×C32 | semidirect |
| 89. C3×C3⋊C8 | C3×D4 | semidirect |
| 127. C32×C4 | #213 | semidirect |
| 128. C32×C4 | C3×D8 | semidirect |
| 133. #127 | C32×D3 | semidirect |
| 135. C33 | #493 | semidirect |
| 158. C3⋊C8 | C32×D4 | semidirect |
| 192. C3×C4 | C3×#213 | semidirect |
| 193. C3×C4 | C32×D8 | semidirect |
| 197. C32 | #5073 | direct (N × H) |
| 198. C32 | C3×#493 | semidirect |
| 224. C4 | C32×#213 | semidirect |
| 227. C3 | C3×#5073 | direct (N × H) |
| 228. C3 | C32×#493 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#127 | 2 |
| C2×C32×#213 | 2 |
| C32×#874 | 2 |
| C32×C4×C3⋊C8 | 2 |
| C32×#493 | 3 |
| C3×#5073 | 3 |