Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
13310 conjugacy classes of subgroups; only the 6571 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 6571 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×C3×C8 | C2 | semidirect |
| 3. C24×C3⋊C8 | C2 | semidirect |
| 4. C23×ΓL(1,𝔽25) | C2 | direct (N × H) |
| 11. C23×C3×C8 | C22 | semidirect |
| 12. C23×C3⋊C8 | C22 | semidirect |
| 13. C22×ΓL(1,𝔽25) | C22 | direct (N × H) |
| 17. C24×C8 | D3 | semidirect |
| 26. C22×C3×C8 | C23 | semidirect |
| 27. C22×C3⋊C8 | C23 | semidirect |
| 28. C2×ΓL(1,𝔽25) | C23 | direct (N × H) |
| 36. C23×C8 | C2×D3 | semidirect |
| 37. C24×C3 | M16 | semidirect |
| 46. C2×C3×C8 | C24 | semidirect |
| 47. C2×C3⋊C8 | C24 | semidirect |
| 48. ΓL(1,𝔽25) | C24 | direct (N × H) |
| 59. C22×C8 | C22×D3 | semidirect |
| 60. C23×C3 | C2×M16 | semidirect |
| 69. C3×C8 | C25 | semidirect |
| 70. C3⋊C8 | C25 | semidirect |
| 73. C24 | ΓL(1,𝔽25) | direct (N × H) |
| 82. C2×C8 | C23×D3 | semidirect |
| 83. C22×C3 | C22×M16 | semidirect |
| 92. C23 | C2×ΓL(1,𝔽25) | direct (N × H) |
| 100. C8 | C24×D3 | semidirect |
| 101. C2×C3 | C23×M16 | semidirect |
| 106. C22 | C22×ΓL(1,𝔽25) | direct (N × H) |
| 111. C3 | C24×M16 | semidirect |
| 113. C2 | C23×ΓL(1,𝔽25) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4×D3 | 2 |
| C24×C3×C8 | 2 |
| C24×C3⋊C8 | 2 |
| C23×ΓL(1,𝔽25) | 2 |
| C24×M16 | 3 |