Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1168 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 198 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×#124 | C2 | semidirect |
| 2. C2×C3×#3755 | C2 | semidirect |
| 3. C2×C3×#3753 | C2 | semidirect |
| 4. C3×#64083 | C2 | semidirect |
| 5. C3×#63630 | C2 | semidirect |
| 6. #10592757 | C3 | direct (N × H) |
| 18. C32×#124 | D3 | semidirect |
| 19. C32×#124 | C2×C3 | semidirect |
| 20. C2×#3755 | C2×C3 | semidirect |
| 21. C2×#3753 | C2×C3 | semidirect |
| 22. #64083 | C2×C3 | semidirect |
| 26. #63630 | C2×C3 | semidirect |
| 38. C3×C3⋊(C3⋊C8) | D4 | semidirect |
| 75. C3×#124 | C3×D3 | semidirect |
| 76. C3×#124 | C3⋊D3 | semidirect |
| 108. C3⋊(C3⋊C8) | C3×D4 | semidirect |
| 162. #124 | C3×C3⋊D3 | semidirect |
| 164. C33 | #496 | semidirect |
| 236. C32 | #5058 | semidirect |
| 237. C32 | C3×#496 | semidirect |
| 269. C3 | C3×#5058 | semidirect |
| 270. C3 | #10592757 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#124 | 2 |
| C2×C3×#3755 | 2 |
| C2×C3×#3753 | 2 |
| C3×#64083 | 2 |
| C3×#63630 | 2 |
| #10592757 | 3 |
| C3×#5058 | 3 |