Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
4334 conjugacy classes of subgroups.
The subgroup lattice has 498 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×#4527 | C2 | direct (N × H) |
| 9. C24×Dic3 | C4 | semidirect |
| 12. C22×C3×C22⋊C4 | C4 | semidirect |
| 20. C22×#217 | C4 | semidirect |
| 21. C22×#217 | C4 | semidirect |
| 24. #4527 | C22 | direct (N × H) |
| 50. C23×Dic3 | C2×C4 | semidirect |
| 54. C2×C3×C22⋊C4 | C2×C4 | semidirect |
| 66. C2×#217 | C2×C4 | semidirect |
| 67. C2×#217 | C2×C4 | semidirect |
| 86. C22×C22⋊C4 | Dic3 | semidirect |
| 141. C22×Dic3 | C22×C4 | semidirect |
| 144. C3×C22⋊C4 | C22×C4 | semidirect |
| 151. #217 | C22×C4 | semidirect |
| 191. C2×C22⋊C4 | C2×Dic3 | semidirect |
| 321. C22⋊C4 | C22×Dic3 | semidirect |
| 348. C22×C3 | #342 | semidirect |
| 439. C2×C3 | C2×#342 | semidirect |
| 468. C22 | #4527 | direct (N × H) |
| 484. C3 | C22×#342 | semidirect |
| 494. C2 | C2×#4527 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C22⋊C4 | 2 |
| C23×#217 | 2 |
| C2×#4527 | 2 |
| C22×#342 | 3 |