Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
242 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 66 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. #10593561 | C3 | direct (N × H) |
| 26. C33×Q8 | Q8 | semidirect |
| 30. C3×#634 | Q8 | semidirect |
| 64. C32×Q8 | C3×Q8 | semidirect |
| 68. #634 | C3×Q8 | semidirect |
| 109. C33 | #475 | semidirect |
| 115. C3×Q8 | PSU(3,𝔽2) | semidirect |
| 161. C32 | C3×#475 | semidirect |
| 165. Q8 | C3×PSU(3,𝔽2) | semidirect |
| 191. C3 | #10593561 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q8×#157 | 2 |
| C3×C4⋊PSU(3,𝔽2) | 2 |
| C3×#63784 | 2 |
| C3×#63745 | 2 |
| C3×#63744 | 2 |
| #10593561 | 3 |
| C3×#475 | 9 |