Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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. C2×D3×PSU(3,𝔽2) | C2 | semidirect |
| 2. #11724365 | C2 | semidirect |
| 3. C3×C4×PSU(3,𝔽2) | C2 | semidirect |
| 5. #11724150 | C2 | semidirect |
| 21. C4×PSU(3,𝔽2) | D3 | semidirect |
| 27. C3×PSU(3,𝔽2) | D4 | semidirect |
| 29. C4⋊(C3⋊(C3⋊D3)) | Q8 | semidirect |
| 31. C32×D12 | Q8 | semidirect |
| 32. #6291 | Q8 | semidirect |
| 45. C2×C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 47. C2×C32×D3 | C2×Q8 | semidirect |
| 53. C33×C4 | C2×Q8 | semidirect |
| 54. C3×C3⋊Dic3 | C2×Q8 | semidirect |
| 67. PSU(3,𝔽2) | D12 | direct (N × H) |
| 110. C32×C4 | D3×Q8 | semidirect |
| 111. C3⋊Dic3 | D3×Q8 | semidirect |
| 118. C33 | D4×Q8 | semidirect |
| 132. D12 | PSU(3,𝔽2) | direct (N × H) |
| 168. C2×D3 | C2×PSU(3,𝔽2) | semidirect |
| 174. C3×C4 | C2×PSU(3,𝔽2) | semidirect |
| 179. C32 | Q8×D12 | semidirect |
| 212. C4 | D3×PSU(3,𝔽2) | semidirect |
| 215. C3 | D4×PSU(3,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×PSU(3,𝔽2) | 2 |
| #11724365 | 2 |
| C3×C4×PSU(3,𝔽2) | 2 |
| D12×#157 | 2 |
| #11724150 | 2 |
| D4×PSU(3,𝔽2) | 3 |
| Q8×D12 | 9 |