Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
742 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 125 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 | D9×C3⋊D3 |
| commutator, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | C2×D3×C3⋊D3 |
| 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×C3⋊(C3⋊D9) | C2 | semidirect |
| 2. C22×C9×C3⋊D3 | C2 | semidirect |
| 3. C2×C32×Dic9 | C2 | semidirect |
| 4. #10619878 | C2 | direct (N × H) |
| 9. C2×C3⋊(C3⋊D9) | C22 | semidirect |
| 11. C2×C9×C3⋊D3 | C22 | semidirect |
| 12. C32×Dic9 | C22 | semidirect |
| 20. C2×C3×Dic9 | D3 | semidirect |
| 22. C2×C32×C9 | D4 | semidirect |
| 43. C3×Dic9 | C2×D3 | semidirect |
| 44. C32×C9 | C2×D4 | semidirect |
| 48. C22×C3⋊D3 | D9 | semidirect |
| 58. C2×Dic9 | C3⋊D3 | semidirect |
| 65. C2×C3×C9 | D12 | semidirect |
| 82. C2×C3⋊D3 | C2×D9 | semidirect |
| 95. Dic9 | C2×C3⋊D3 | semidirect |
| 98. C3×C9 | C2×D12 | semidirect |
| 110. C2×C32 | #611 | semidirect |
| 122. C2×C9 | C4⋊(C3⋊D3) | semidirect |
| 143. C32 | C2×#611 | semidirect |
| 146. C9 | C2×C4⋊(C3⋊D3) | semidirect |
| 154. C2×C3 | #6195 | semidirect |
| 169. C3 | C2×#6195 | semidirect |
| 173. C2 | #10619878 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊(C3⋊D9) | 2 |
| C22×C9×C3⋊D3 | 2 |
| C2×C32×Dic9 | 2 |
| #10619878 | 2 |
| C2×#6292 | 3 |
| C2×#6195 | 3 |