Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
666 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 188 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 |
|---|---|---|
| 6. #11724148 | C2 | direct (N × H) |
| 12. C2×C3⋊PSU(3,𝔽2) | C4 | semidirect |
| 34. C3⋊PSU(3,𝔽2) | C2×C4 | semidirect |
| 35. C2×C3⋊(C3⋊Dic3) | Q8 | semidirect |
| 36. C2×C32×Dic3 | Q8 | semidirect |
| 37. Dic3×C3⋊D3 | Q8 | semidirect |
| 61. C3⋊(C3⋊Dic3) | C2×Q8 | semidirect |
| 62. C32×Dic3 | C2×Q8 | semidirect |
| 90. C2×C33 | C4×Q8 | semidirect |
| 92. C3×C3⋊D3 | C4×Q8 | semidirect |
| 133. C33 | C2×C4×Q8 | semidirect |
| 150. C2×Dic3 | PSU(3,𝔽2) | semidirect |
| 158. C2×C32 | #887 | semidirect |
| 160. C3⋊D3 | #887 | semidirect |
| 196. Dic3 | C2×PSU(3,𝔽2) | semidirect |
| 199. C32 | C2×#887 | semidirect |
| 221. C2×C3 | C4×PSU(3,𝔽2) | semidirect |
| 239. C3 | C2×C4×PSU(3,𝔽2) | semidirect |
| 243. C2 | #11724148 | 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 |
| C2×#88945 | 2 |
| C2×C3×#3777 | 2 |
| C2×Dic3×#157 | 2 |
| #11724148 | 2 |
| C2×C4×PSU(3,𝔽2) | 3 |
| C2×#887 | 9 |