Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
378 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#4184 |
| commutator, cosocle | C2×#731 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C2×C4×#731 | C2 |
| socle | C22×C3 | C2×#272 |
| hypercenter | C2×C4 | #4184 |
| hypocenter | #731 | C2×D4 |
| LCS | G ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#4184 | C2 | semidirect |
| 2. C2×C4×#731 | C2 | semidirect |
| 3. C4⋊#4184 | C2 | direct (N × H) |
| 9. C2×#4184 | C22 | semidirect |
| 10. C4×#731 | C22 | semidirect |
| 14. C2×C4×3−1+2 | D3 | semidirect |
| 16. C2×C3×C4×C9 | D3 | semidirect |
| 20. C2×#731 | D4 | semidirect |
| 36. C4×3−1+2 | C2×D3 | semidirect |
| 38. C3×C4×C9 | C2×D3 | semidirect |
| 39. #731 | C2×D4 | semidirect |
| 52. C2×3−1+2 | D12 | semidirect |
| 56. C2×C3×C9 | D12 | semidirect |
| 80. 3−1+2 | C2×D12 | semidirect |
| 82. C3×C9 | C2×D12 | semidirect |
| 123. C2×C4 | #4184 | semidirect |
| 136. C4 | C2×#4184 | semidirect |
| 141. C2 | C4⋊#4184 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#4184 | 2 |
| C2×C4×#731 | 2 |
| C4⋊#4184 | 2 |
| C2×C4⋊AGL(1,ℤ/9) | 3 |
| C2×C4⋊((C3⋊D3)⋊C3) | 3 |
| C2×C3×D36 | 3 |