Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
351 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 351 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×#733⋊C2 |
| commutator, cosocle | C2×#733 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C23×#733 | C2 |
| socle | C22×C3 | C2×#272 |
| hypercenter | C23 | #733⋊C2 |
| hypocenter | #733 | C2×D4 |
| LCS | G ▷ C2×#733 ▷ #733 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#733 | C2 | semidirect |
| 2. C22×#733⋊C2 | C2 | semidirect |
| 3. #10619685 | C2 | direct (N × H) |
| 4. C2×#733⋊C4 | C2 | semidirect |
| 9. C22×#733 | C22 | semidirect |
| 11. C2×#733⋊C2 | C22 | semidirect |
| 12. #733⋊C4 | C22 | semidirect |
| 22. C2×#733 | D4 | semidirect |
| 40. #733 | C2×D4 | semidirect |
| 120. C23 | #733⋊C2 | semidirect |
| 135. C22 | C2×#733⋊C2 | semidirect |
| 141. C2 | #10619685 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#733 | 2 |
| C22×#733⋊C2 | 2 |
| #10619685 | 2 |
| C2×#733⋊C4 | 2 |
| C2×#611⋊C3 | 3 |
| C2×C3×#611 | 3 |