Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
3086 conjugacy classes of subgroups.
The subgroup lattice has 262 nodes — too large to display as a diagram.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#4925 | C2 | direct (N × H) |
| 4. C22×C4⋊(C3⋊C8) | C2 | semidirect |
| 5. C22×C4×C3⋊C8 | C2 | semidirect |
| 18. #4925 | C22 | direct (N × H) |
| 19. C2×C4⋊(C3⋊C8) | C22 | semidirect |
| 20. C2×C4×C3⋊C8 | C22 | semidirect |
| 29. C23×C3×C4 | C8 | semidirect |
| 51. C4⋊(C3⋊C8) | C23 | semidirect |
| 52. C4×C3⋊C8 | C23 | semidirect |
| 79. C22×C3×C4 | C2×C8 | semidirect |
| 104. C23×C4 | C3⋊C8 | semidirect |
| 145. C2×C3×C4 | C22×C8 | semidirect |
| 168. C22×C4 | C2×C3⋊C8 | semidirect |
| 193. C22×C3 | #431 | semidirect |
| 222. C2×C4 | C22×C3⋊C8 | semidirect |
| 235. C2×C3 | C2×#431 | semidirect |
| 242. C22 | #4925 | direct (N × H) |
| 255. C3 | C22×#431 | semidirect |
| 257. C2 | C2×#4925 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C42 | 2 |
| C22×#830 | 2 |
| C2×#4925 | 2 |
| C22×C4⋊(C3⋊C8) | 2 |
| C22×C4×C3⋊C8 | 2 |
| C22×#431 | 3 |