Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
146 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10620045 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C2×C3 |
| Frattini, socle | C2×C3 | C3×C3⋊S4 |
| Fitting | C3×Q8×C9 | D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10619912 | C2 | semidirect |
| 2. C9⋊GL(2,𝔽3) | C3 | semidirect |
| 7. C3×Q8×C9 | D3 | semidirect |
| 10. #6204 | D3 | semidirect |
| 14. C9×SL(2,𝔽3) | C2×C3 | semidirect |
| 66. C3×C9 | GL(2,𝔽3) | semidirect |
| 100. Q8 | #4207 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619912 | 2 |
| C9⋊GL(2,𝔽3) | 3 |
| (C3⋊GL(2,𝔽3))⋊C3 | 3 |
| #3689⋊C3 | 3 |
| C3×#3675 | 3 |
| C2×#4207 | 4 |