Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
924 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊#1113 |
| commutator, socle, cosocle | C2×C34 | C23 |
| Fitting | C22×C34 | C22 |
| hypercenter | C22 | C3⋊#1113 |
| hypocenter | C34 | #41 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6312 | C2 | semidirect |
| 2. C2×C32×C3⋊Dic3 | C2 | semidirect |
| 3. C3⋊#6296 | C2 | semidirect |
| 4. #10620156 | C2 | semidirect |
| 5. #10620154 | C2 | semidirect |
| 13. #6312 | D3 | semidirect |
| 17. C2×C3×C3⋊Dic3 | D3 | semidirect |
| 52. C34 | #41 | semidirect |
| 57. C2×C3⋊Dic3 | C3⋊D3 | semidirect |
| 121. C33 | #235 | semidirect |
| 122. C33 | #237 | semidirect |
| 187. C32 | #3804 | semidirect |
| 188. C32 | #3805 | semidirect |
| 189. C32 | #3828 | semidirect |
| 217. C3 | #89031 | semidirect |
| 218. C3 | #89046 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6312 | 2 |
| C2×C32×C3⋊Dic3 | 2 |
| C3⋊#6296 | 2 |
| #10620156 | 2 |
| #10620154 | 2 |
| #89046 | 3 |
| #89031 | 3 |