Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1734 conjugacy classes of subgroups; only the 915 normal subgroups are shown.
This group has 1734 conjugacy classes of subgroups — too many to draw. Only its 915 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 |
|---|---|---|
| 3. C3×C42×C8 | C2 | semidirect |
| 4. C4×#815 | C2 | semidirect |
| 5. C42×C3⋊C8 | C2 | semidirect |
| 16. C4×ΓL(1,𝔽25) | C4 | direct (N × H) |
| 21. C42×C8 | D3 | semidirect |
| 34. C3×C4×C8 | C2×C4 | semidirect |
| 35. #815 | C2×C4 | semidirect |
| 36. C4×C3⋊C8 | C2×C4 | semidirect |
| 56. C3×C42 | M16 | semidirect |
| 62. ΓL(1,𝔽25) | C42 | direct (N × H) |
| 77. C4×C8 | C4×D3 | semidirect |
| 89. C3×C8 | C2×C42 | semidirect |
| 90. C3⋊C8 | C2×C42 | semidirect |
| 105. C42 | ΓL(1,𝔽25) | direct (N × H) |
| 111. C3×C4 | C4×M16 | semidirect |
| 127. C8 | C42×D3 | semidirect |
| 137. C4 | C4×ΓL(1,𝔽25) | direct (N × H) |
| 139. C3 | C42×M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×ΓL(1,𝔽25) | 2 |
| C43×D3 | 2 |
| C3×C42×C8 | 2 |
| C4×#815 | 2 |
| C42×C3⋊C8 | 2 |
| C42×M16 | 3 |