Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
810 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 140 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. C22×C3⋊PSU(3,𝔽2) | C2 | semidirect |
| 3. C2×#88945 | C2 | semidirect |
| 4. C2×#88945 | C2 | semidirect |
| 5. C2×C3×#3777 | C2 | semidirect |
| 8. #11724151 | C2 | direct (N × H) |
| 18. C2×C3⋊PSU(3,𝔽2) | C22 | semidirect |
| 21. #88945 | C22 | semidirect |
| 22. #88945 | C22 | semidirect |
| 23. C3×#3777 | C22 | semidirect |
| 33. C2×#3777 | D3 | semidirect |
| 34. C22×C3⋊(C3⋊D3) | Q8 | semidirect |
| 36. C22×C32×D3 | Q8 | semidirect |
| 39. C2×D3×C3⋊D3 | Q8 | semidirect |
| 72. #3777 | C2×D3 | semidirect |
| 75. C2×C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 81. C2×C32×D3 | C2×Q8 | semidirect |
| 129. C2×C33 | C22⋊Q8 | semidirect |
| 132. C3×C3⋊D3 | C22⋊Q8 | semidirect |
| 194. C33 | C2×C22⋊Q8 | semidirect |
| 209. C22×D3 | PSU(3,𝔽2) | semidirect |
| 229. C2×C32 | #896 | semidirect |
| 234. C3⋊D3 | #896 | semidirect |
| 283. C2×D3 | C2×PSU(3,𝔽2) | semidirect |
| 292. C32 | C2×#896 | semidirect |
| 322. C2×C3 | C22⋊PSU(3,𝔽2) | semidirect |
| 351. C3 | C2×C22⋊PSU(3,𝔽2) | semidirect |
| 357. C2 | #11724151 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊PSU(3,𝔽2) | 2 |
| C22×D3×#157 | 2 |
| C2×#88945 | 2 |
| C2×C3×#3777 | 2 |
| C2×(C2×D3)⋊#157 | 2 |
| #11724151 | 2 |
| C2×C22⋊PSU(3,𝔽2) | 3 |
| C2×#896 | 9 |