Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
340 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10619880 |
| commutator, Fitting, cosocle | C32×C4×C9 | C22 |
| Frattini | C3×C4 | D3×C3⋊D3 |
| socle | C2×C33 | #67 |
| hypercenter | C4 | D3×C3⋊D9 |
| hypocenter | C32×C9 | D8 |
| LCS | G ▷ C32×C4×C9 ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊(C3⋊(C3⋊D9)) | C2 | semidirect |
| 2. C3×C9×D12 | C2 | semidirect |
| 3. C3×C3⋊(C9⋊C8) | C2 | semidirect |
| 18. C9×D12 | D3 | semidirect |
| 19. C3⋊(C9⋊C8) | D3 | semidirect |
| 41. C32×C9 | D8 | semidirect |
| 48. C3×D12 | D9 | semidirect |
| 94. C3×C9 | D24 | semidirect |
| 95. C3×C9 | #213 | semidirect |
| 103. D12 | C3⋊D9 | semidirect |
| 141. C32 | #3673 | semidirect |
| 145. C9 | #3714 | semidirect |
| 167. C3 | #88422 | semidirect |
| 168. C3 | #88544 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊(C3⋊(C3⋊D9)) | 2 |
| C3×C9×D12 | 2 |
| C3×C3⋊(C9⋊C8) | 2 |
| #88790 | 3 |
| #88544 | 3 |
| #88422 | 3 |