Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1296 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 258 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×#214 | C2 | semidirect |
| 3. C2×C32×#213 | C2 | semidirect |
| 4. C32×#875 | C2 | semidirect |
| 5. C32×#803 | C2 | semidirect |
| 7. C3×#5074 | C3 | direct (N × H) |
| 20. C32×#127 | D3 | semidirect |
| 21. C32×#127 | C2×C3 | semidirect |
| 23. C2×C3×#214 | C2×C3 | semidirect |
| 24. C2×C3×#213 | C2×C3 | semidirect |
| 25. C3×#875 | C2×C3 | semidirect |
| 27. C3×#803 | C2×C3 | semidirect |
| 38. C32×C3⋊C8 | D4 | semidirect |
| 40. #5074 | C32 | direct (N × H) |
| 73. C3×#127 | C3×D3 | semidirect |
| 74. C3×#127 | C2×C32 | semidirect |
| 76. C2×#214 | C2×C32 | semidirect |
| 77. C2×#213 | C2×C32 | semidirect |
| 78. #875 | C2×C32 | semidirect |
| 80. #803 | C2×C32 | semidirect |
| 108. C3×C3⋊C8 | C3×D4 | semidirect |
| 158. #127 | C32×D3 | semidirect |
| 161. C33 | #496 | semidirect |
| 189. C3⋊C8 | C32×D4 | semidirect |
| 232. C32 | #5074 | direct (N × H) |
| 233. C32 | C3×#496 | semidirect |
| 265. C3 | C3×#5074 | direct (N × H) |
| 266. C3 | C32×#496 | 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×#214 | 2 |
| C2×C32×#213 | 2 |
| C32×#875 | 2 |
| C32×#803 | 2 |
| C32×#496 | 3 |
| C3×#5074 | 3 |