Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2494 conjugacy classes of subgroups.
The subgroup lattice has 368 nodes — too large to display as a diagram.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#4554 | C2 | direct (N × H) |
| 10. C22×C3×M16 | C4 | semidirect |
| 21. C22×#208 | C4 | semidirect |
| 24. #4554 | C22 | direct (N × H) |
| 47. C2×C3×M16 | C2×C4 | semidirect |
| 63. C2×#208 | C2×C4 | semidirect |
| 75. C22×M16 | Dic3 | semidirect |
| 124. C3×M16 | C22×C4 | semidirect |
| 133. #208 | C22×C4 | semidirect |
| 159. C2×M16 | C2×Dic3 | semidirect |
| 257. M16 | C22×Dic3 | semidirect |
| 273. C22×C3 | #343 | semidirect |
| 333. C2×C3 | C2×#343 | semidirect |
| 344. C22 | #4554 | direct (N × H) |
| 359. C3 | C22×#343 | semidirect |
| 363. C2 | C2×#4554 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×M16 | 2 |
| C23×C4⋊Dic3 | 2 |
| C23×#208 | 2 |
| C2×#4554 | 2 |
| C22×#343 | 3 |