Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
292 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22×C32 |
| Frattini | C2 | C2×C32×S4 |
| Fitting | C32×#41 | D3 |
| hypercenter | C32×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#41⋊C3 | C2 | semidirect |
| 2. C32×GL(2,𝔽3) | C2 | semidirect |
| 3. C3×#986 | C3 | direct (N × H) |
| 6. C32×SL(2,𝔽3) | C22 | semidirect |
| 7. C32×#41 | D3 | semidirect |
| 10. C3×#41⋊C3 | C2×C3 | semidirect |
| 11. C3×GL(2,𝔽3) | C2×C3 | semidirect |
| 17. #986 | C32 | direct (N × H) |
| 27. C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 31. C3×#41 | C3×D3 | semidirect |
| 34. #41⋊C3 | C2×C32 | semidirect |
| 35. GL(2,𝔽3) | C2×C32 | semidirect |
| 57. SL(2,𝔽3) | C22×C32 | semidirect |
| 65. #41 | C32×D3 | semidirect |
| 80. C32 | #986 | direct (N × H) |
| 101. C3 | C3×#986 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41⋊C3 | 2 |
| C32×GL(2,𝔽3) | 2 |
| C3×#986 | 3 |
| C32×AGL(1,ℤ/8) | 3 |
| C32×D12 | 4 |