Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5206 conjugacy classes of subgroups.
The subgroup lattice has 388 nodes — too large to display as a diagram.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#948 | C2 | semidirect |
| 3. #84075 | C2 | direct (N × H) |
| 4. C2×#5234 | C2 | semidirect |
| 5. C2×#5207 | C2 | semidirect |
| 6. C22×#830 | C2 | semidirect |
| 19. C2×#948 | C22 | semidirect |
| 20. C22×#208 | C22 | semidirect |
| 21. C23×C3⋊C8 | C22 | semidirect |
| 22. #5234 | C22 | semidirect |
| 23. #5207 | C22 | semidirect |
| 26. C2×#830 | C22 | semidirect |
| 27. C2×#830 | C22 | semidirect |
| 28. C2×#830 | C22 | semidirect |
| 68. C2×#208 | C23 | semidirect |
| 72. C22×C3⋊C8 | C23 | semidirect |
| 75. #830 | C23 | semidirect |
| 76. #830 | C23 | semidirect |
| 204. C23×C3 | #131 | semidirect |
| 300. C22×C3 | C2×#131 | semidirect |
| 319. C23 | #948 | semidirect |
| 353. C2×C3 | #2880 | semidirect |
| 369. C22 | C2×#948 | semidirect |
| 379. C3 | C2×#2880 | semidirect |
| 383. C2 | #84075 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×#41 | 2 |
| C22×#948 | 2 |
| #84075 | 2 |
| C2×#5234 | 2 |
| C2×#5207 | 2 |
| C22×#830 | 2 |
| C2×#2880 | 3 |