Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
6685 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 6685 conjugacy classes of subgroups — too many to draw. Only its 559 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, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C23×#41 | D3 |
| hypercenter | C23×C4 | S4 |
| LCS | G ▷ 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×#41⋊C3 | C2 | semidirect |
| 2. C22×#986 | C2 | direct (N × H) |
| 3. C23×GL(2,𝔽3) | C2 | semidirect |
| 6. C22×#41⋊C3 | C22 | semidirect |
| 7. C23×SL(2,𝔽3) | C22 | semidirect |
| 8. C2×#986 | C22 | direct (N × H) |
| 9. C22×GL(2,𝔽3) | C22 | semidirect |
| 10. C23×#41 | D3 | semidirect |
| 20. C2×#41⋊C3 | C23 | semidirect |
| 21. C22×SL(2,𝔽3) | C23 | semidirect |
| 22. #986 | C23 | direct (N × H) |
| 23. C2×GL(2,𝔽3) | C23 | semidirect |
| 26. C22×#41 | C2×D3 | semidirect |
| 46. #41⋊C3 | C24 | semidirect |
| 47. C2×SL(2,𝔽3) | C24 | semidirect |
| 48. GL(2,𝔽3) | C24 | semidirect |
| 55. C2×#41 | C22×D3 | semidirect |
| 82. SL(2,𝔽3) | C25 | semidirect |
| 92. #41 | C23×D3 | semidirect |
| 124. C23 | #986 | direct (N × H) |
| 148. C22 | C2×#986 | direct (N × H) |
| 156. C2 | C22×#986 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41⋊C3 | 2 |
| C22×#986 | 2 |
| C23×GL(2,𝔽3) | 2 |
| C23×AGL(1,ℤ/8) | 3 |
| C23×D12 | 4 |