Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
526 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 116 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. #11724368 | C2 | semidirect |
| 2. C2×#88946 | C2 | semidirect |
| 3. C3×C22⋊PSU(3,𝔽2) | C2 | semidirect |
| 6. #11724154 | C2 | semidirect |
| 7. #11724152 | C2 | semidirect |
| 8. #11724150 | C2 | semidirect |
| 9. #11724149 | C2 | semidirect |
| 10. Dic3×PSU(3,𝔽2) | C2 | semidirect |
| 11. D3×#3777 | C2 | semidirect |
| 41. C22⋊PSU(3,𝔽2) | D3 | semidirect |
| 63. #6312 | Q8 | semidirect |
| 64. C32×#67 | Q8 | semidirect |
| 65. #6292 | Q8 | semidirect |
| 66. #6290 | Q8 | semidirect |
| 228. C33 | #554 | semidirect |
| 246. #67 | PSU(3,𝔽2) | semidirect |
| 333. C32 | #5635 | semidirect |
| 393. C3 | #10596545 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724368 | 2 |
| C2×#88946 | 2 |
| C3×C22⋊PSU(3,𝔽2) | 2 |
| #67⋊#157 | 2 |
| #67×#157 | 2 |
| #11724154 | 2 |
| #11724152 | 2 |
| #11724150 | 2 |
| #11724149 | 2 |
| Dic3×PSU(3,𝔽2) | 2 |
| D3×#3777 | 2 |
| #10596545 | 3 |
| #5635 | 9 |