Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
432 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 432 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×#4185 |
| commutator, cosocle | C2×#732 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C23×#732 | C2 |
| socle | C22×C3 | C2×#272 |
| hypercenter | C23 | #4185 |
| hypocenter | #732 | C2×D4 |
| LCS | G ▷ C2×#732 ▷ #732 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#732 | C2 | semidirect |
| 2. C22×#4185 | C2 | semidirect |
| 3. #10619683 | C2 | direct (N × H) |
| 4. C2×#66332 | C2 | semidirect |
| 9. C22×#732 | C22 | semidirect |
| 11. C2×#4185 | C22 | semidirect |
| 12. #66332 | C22 | semidirect |
| 13. C23×H(3,𝔽3) | D3 | semidirect |
| 14. C23×C3×C9 | D3 | semidirect |
| 22. C2×#732 | D4 | semidirect |
| 30. C22×H(3,𝔽3) | C2×D3 | semidirect |
| 33. C22×C3×C9 | C2×D3 | semidirect |
| 40. #732 | C2×D4 | semidirect |
| 54. C2×H(3,𝔽3) | #67 | semidirect |
| 57. C2×C3×C9 | #67 | semidirect |
| 82. H(3,𝔽3) | C2×#67 | semidirect |
| 83. C3×C9 | C2×#67 | semidirect |
| 131. C23 | #4185 | semidirect |
| 150. C22 | C2×#4185 | semidirect |
| 157. C2 | #10619683 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#732 | 2 |
| C22×#4185 | 2 |
| #10619683 | 2 |
| C2×#66332 | 2 |
| C2×#638⋊C3 | 3 |
| C2×C3×#611 | 3 |