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. C22×C3×PSU(3,𝔽2) | C2 | semidirect |
| 2. C2×D3×PSU(3,𝔽2) | C2 | semidirect |
| 3. #11724369 | C2 | semidirect |
| 5. #11724153 | C2 | semidirect |
| 6. #11724149 | C2 | semidirect |
| 7. Dic3×PSU(3,𝔽2) | C2 | semidirect |
| 24. C22×PSU(3,𝔽2) | D3 | semidirect |
| 39. C3×PSU(3,𝔽2) | D4 | semidirect |
| 41. #6312 | Q8 | semidirect |
| 42. C32×#67 | Q8 | semidirect |
| 43. #6292 | Q8 | semidirect |
| 44. #6290 | Q8 | semidirect |
| 62. C22×C33 | C2×Q8 | semidirect |
| 63. C2×C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 66. C2×C3×C3⋊D3 | C2×Q8 | semidirect |
| 68. C2×C32×D3 | C2×Q8 | semidirect |
| 75. C3⋊(C3⋊Dic3) | C2×Q8 | semidirect |
| 76. C32×Dic3 | C2×Q8 | semidirect |
| 96. PSU(3,𝔽2) | #67 | direct (N × H) |
| 134. C22×C32 | D3×Q8 | semidirect |
| 142. C2×C3⋊D3 | D3×Q8 | semidirect |
| 167. C33 | D4×Q8 | semidirect |
| 184. #67 | PSU(3,𝔽2) | direct (N × H) |
| 230. C22×C3 | C2×PSU(3,𝔽2) | semidirect |
| 242. C2×D3 | C2×PSU(3,𝔽2) | semidirect |
| 251. Dic3 | C2×PSU(3,𝔽2) | semidirect |
| 254. C32 | Q8×#67 | semidirect |
| 296. C22 | D3×PSU(3,𝔽2) | semidirect |
| 305. 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 |
|---|---|
| C22×C3×PSU(3,𝔽2) | 2 |
| C2×D3×PSU(3,𝔽2) | 2 |
| #11724369 | 2 |
| #67×#157 | 2 |
| #11724153 | 2 |
| #11724149 | 2 |
| Dic3×PSU(3,𝔽2) | 2 |
| D4×PSU(3,𝔽2) | 3 |
| Q8×#67 | 9 |