Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
12670 conjugacy classes of subgroups; only the 1429 normal subgroups are shown.
This group has 12670 conjugacy classes of subgroups — too many to draw. Only its 1429 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | A4 |
| commutator, cosocle, hypocenter | Q8 | C25×C3 |
| Frattini | C2 | C25×A4 |
| Fitting | C24×#41 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41⋊C3 | C2 | direct (N × H) |
| 2. C24×SL(2,𝔽3) | C2 | semidirect |
| 3. C24×#41 | C3 | semidirect |
| 5. C22×#41⋊C3 | C22 | direct (N × H) |
| 6. C23×SL(2,𝔽3) | C22 | semidirect |
| 8. C23×#41 | C2×C3 | semidirect |
| 14. C2×#41⋊C3 | C23 | direct (N × H) |
| 15. C22×SL(2,𝔽3) | C23 | semidirect |
| 19. C22×#41 | C22×C3 | semidirect |
| 31. #41⋊C3 | C24 | direct (N × H) |
| 32. C2×SL(2,𝔽3) | C24 | semidirect |
| 38. C2×#41 | C23×C3 | semidirect |
| 52. SL(2,𝔽3) | C25 | semidirect |
| 55. C24 | #41⋊C3 | direct (N × H) |
| 59. #41 | C24×C3 | semidirect |
| 75. C23 | C2×#41⋊C3 | direct (N × H) |
| 91. C22 | C22×#41⋊C3 | direct (N × H) |
| 98. C2 | C23×#41⋊C3 | 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 |
| C24×SL(2,𝔽3) | 2 |
| C24×#41 | 3 |
| C24×C3×C4 | 4 |