Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
488 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #66431 |
| commutator, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | C2×#1113 |
| Fitting | C22×C32×C9 | C22 |
| socle | C22×C33 | C2×D3 |
| hypocenter | C32×C9 | C2×D4 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×C3⋊D3 | C2 | semidirect |
| 2. C22×C3×C3⋊D9 | C2 | semidirect |
| 3. C2×C3×C3⋊Dic9 | C2 | semidirect |
| 4. #10619899 | C2 | direct (N × H) |
| 10. C2×C9×C3⋊D3 | C22 | semidirect |
| 12. C2×C3×C3⋊D9 | C22 | semidirect |
| 13. C3×C3⋊Dic9 | C22 | semidirect |
| 18. C22×C3⋊D9 | D3 | semidirect |
| 24. C2×C3⋊Dic9 | D3 | semidirect |
| 29. C2×C32×C9 | D4 | semidirect |
| 50. C2×C3⋊D9 | C2×D3 | semidirect |
| 61. C3⋊Dic9 | C2×D3 | semidirect |
| 64. C32×C9 | C2×D4 | semidirect |
| 66. C22×C3⋊D3 | D9 | semidirect |
| 98. C2×C3×C9 | D12 | semidirect |
| 99. C2×C3×C9 | #67 | semidirect |
| 121. C2×C3⋊D3 | C2×D9 | semidirect |
| 155. C3×C9 | C2×D12 | semidirect |
| 156. C3×C9 | C2×#67 | semidirect |
| 180. C2×C32 | #611 | semidirect |
| 201. C2×C9 | #626 | semidirect |
| 237. C32 | C2×#611 | semidirect |
| 243. C9 | C2×#626 | semidirect |
| 258. C2×C3 | #6194 | semidirect |
| 259. C2×C3 | #6195 | semidirect |
| 280. C3 | C2×#6194 | semidirect |
| 281. C3 | C2×#6195 | semidirect |
| 285. C2 | #10619899 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×C3⋊D3 | 2 |
| C22×C3×C3⋊D9 | 2 |
| C2×C3×C3⋊Dic9 | 2 |
| #10619899 | 2 |
| C2×#6295 | 3 |
| C2×#6195 | 3 |
| C2×#6194 | 3 |