Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
630 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 124 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. #11724368 | C2 | semidirect |
| 4. C3×C22⋊PSU(3,𝔽2) | C2 | semidirect |
| 7. #11724154 | C2 | semidirect |
| 8. #11724153 | C2 | semidirect |
| 9. #11724151 | C2 | semidirect |
| 10. #11724150 | C2 | semidirect |
| 11. #11724148 | C2 | semidirect |
| 45. C22⋊PSU(3,𝔽2) | D3 | semidirect |
| 64. C3⋊PSU(3,𝔽2) | D4 | semidirect |
| 68. #6312 | Q8 | semidirect |
| 69. C32×#67 | Q8 | semidirect |
| 70. #6292 | Q8 | semidirect |
| 71. #6290 | Q8 | semidirect |
| 100. C22×C33 | C2×Q8 | semidirect |
| 101. C2×C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 104. C2×C3×C3⋊D3 | C2×Q8 | semidirect |
| 106. C2×C32×D3 | C2×Q8 | semidirect |
| 117. C3⋊(C3⋊Dic3) | C2×Q8 | semidirect |
| 118. C32×Dic3 | C2×Q8 | semidirect |
| 199. C22×C32 | D3×Q8 | semidirect |
| 207. C2×C3⋊D3 | D3×Q8 | semidirect |
| 240. C33 | D4×Q8 | semidirect |
| 258. #67 | PSU(3,𝔽2) | semidirect |
| 322. C22×C3 | C2×PSU(3,𝔽2) | semidirect |
| 334. C2×D3 | C2×PSU(3,𝔽2) | semidirect |
| 347. Dic3 | C2×PSU(3,𝔽2) | semidirect |
| 350. C32 | #5632 | semidirect |
| 400. C22 | D3×PSU(3,𝔽2) | semidirect |
| 413. 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 |
| #11724368 | 2 |
| C3×C22⋊PSU(3,𝔽2) | 2 |
| #67⋊#157 | 2 |
| #67×#157 | 2 |
| #11724154 | 2 |
| #11724153 | 2 |
| #11724151 | 2 |
| #11724150 | 2 |
| #11724148 | 2 |
| D4×PSU(3,𝔽2) | 3 |
| #5632 | 9 |