Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
394 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 89 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 |
|---|---|---|
| 4. #11724105 | C2 | direct (N × H) |
| 10. C2×C3⋊PSU(3,𝔽2) | C4 | semidirect |
| 30. C3⋊PSU(3,𝔽2) | C2×C4 | semidirect |
| 83. C2×C33 | #103 | semidirect |
| 85. C3×C3⋊D3 | #103 | semidirect |
| 123. C33 | C2×#103 | semidirect |
| 148. C2×C32 | #816 | semidirect |
| 150. C3⋊D3 | #816 | semidirect |
| 188. C32 | C2×#816 | semidirect |
| 210. C2×C3 | PSU(3,𝔽2)⋊C4 | semidirect |
| 227. C3 | C2×PSU(3,𝔽2)⋊C4 | semidirect |
| 231. C2 | #11724105 | 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×C3×AGL(1,𝔽9) | 2 |
| C2×#88940 | 2 |
| #11724105 | 2 |
| C2×PSU(3,𝔽2)⋊C4 | 3 |
| C2×#816 | 9 |