Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2309 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 2309 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C22×C4×Q8 | D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×GL(2,𝔽3) | C2 | semidirect |
| 2. C22×C4×SL(2,𝔽3) | C2 | semidirect |
| 3. C2×C4⋊GL(2,𝔽3) | C2 | direct (N × H) |
| 8. C22×GL(2,𝔽3) | C22 | semidirect |
| 9. C2×C4×SL(2,𝔽3) | C22 | semidirect |
| 10. C4⋊GL(2,𝔽3) | C22 | direct (N × H) |
| 13. C22×C4×Q8 | D3 | semidirect |
| 22. C22×SL(2,𝔽3) | D4 | semidirect |
| 25. C2×GL(2,𝔽3) | C23 | semidirect |
| 26. C4×SL(2,𝔽3) | C23 | semidirect |
| 39. C2×C4×Q8 | C2×D3 | semidirect |
| 55. C2×SL(2,𝔽3) | C2×D4 | semidirect |
| 61. C22×Q8 | D12 | semidirect |
| 85. C4×Q8 | C22×D3 | semidirect |
| 104. SL(2,𝔽3) | C22×D4 | semidirect |
| 115. C2×Q8 | C2×D12 | semidirect |
| 132. C22×C4 | GL(2,𝔽3) | semidirect |
| 166. Q8 | C22×D12 | semidirect |
| 179. C2×C4 | C2×GL(2,𝔽3) | semidirect |
| 191. C22 | C4⋊GL(2,𝔽3) | direct (N × H) |
| 200. C4 | C22×GL(2,𝔽3) | semidirect |
| 203. C2 | C2×C4⋊GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×GL(2,𝔽3) | 2 |
| C22×C4×SL(2,𝔽3) | 2 |
| C2×C4⋊GL(2,𝔽3) | 2 |
| C22×#460 | 3 |
| C23×D12 | 4 |