Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
127 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C9 |
| Frattini, socle | C2×C3 | C2×C3×S4 |
| Fitting | C9×#41 | D3 |
| prime radical | C3×#985 | C3 |
| cosocle | C3×SL(2,𝔽3) | C22×C3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#41⋊C3 | C2 | semidirect |
| 2. C9×GL(2,𝔽3) | C2 | semidirect |
| 3. C9×#226 | C2 | semidirect |
| 12. C9×#41 | D3 | semidirect |
| 22. #985 | C9 | direct (N × H) |
| 41. #41⋊C3 | C2×C9 | semidirect |
| 42. GL(2,𝔽3) | C2×C9 | semidirect |
| 43. #226 | C2×C9 | semidirect |
| 79. #41 | D3×C9 | semidirect |
| 96. C9 | #985 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#41⋊C3 | 2 |
| C9×GL(2,𝔽3) | 2 |
| C9×#226 | 2 |
| C3×#985 | 3 |
| C9×#135 | 3 |
| C4×D3×C9 | 4 |