Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
650 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 156 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. #11724153 | C2 | direct (N × H) |
| 28. C2×C3⋊#157 | Q8 | semidirect |
| 37. C2×C3⋊(C3⋊Dic3) | Q8 | semidirect |
| 38. C2×C32×Dic3 | Q8 | semidirect |
| 39. Dic3×C3⋊D3 | Q8 | semidirect |
| 61. C3⋊#157 | C2×Q8 | semidirect |
| 64. C3⋊(C3⋊Dic3) | C2×Q8 | semidirect |
| 65. C32×Dic3 | C2×Q8 | semidirect |
| 94. C2×C33 | C4⋊Q8 | semidirect |
| 96. C3×C3⋊D3 | C4⋊Q8 | semidirect |
| 138. C33 | C2×C4⋊Q8 | semidirect |
| 156. C2×Dic3 | PSU(3,𝔽2) | semidirect |
| 164. C2×C32 | #944 | semidirect |
| 166. C3⋊D3 | #944 | semidirect |
| 203. Dic3 | C2×PSU(3,𝔽2) | semidirect |
| 206. C32 | C2×#944 | semidirect |
| 229. C2×C3 | C4⋊PSU(3,𝔽2) | semidirect |
| 247. C3 | C2×C4⋊PSU(3,𝔽2) | semidirect |
| 251. C2 | #11724153 | 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×PSU(3,𝔽2) | 2 |
| C2×#88945 | 2 |
| C2×Dic3⋊#157 | 2 |
| C2×Dic3×#157 | 2 |
| #11724153 | 2 |
| C2×C4⋊PSU(3,𝔽2) | 3 |
| C2×#944 | 9 |