Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
121 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 121 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 | #66440⋊C2 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C2×C3 |
| Frattini, socle | C2×C3 | C3×C3⋊S4 |
| Fitting | Q8×3−1+2 | D3 |
| prime radical | Q8⋊H(3,𝔽3) | 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 |
|---|---|---|
| 2. C9⋊#226 | C3 | semidirect |
| 69. 3−1+2 | #226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619916 | 2 |
| C9⋊#226 | 3 |
| (C3⋊#226)⋊C3 | 3 |
| C3×#3688 | 3 |
| #3688⋊C3 | 3 |
| #3674⋊C3 | 3 |
| #66409 | 4 |