Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
11422 conjugacy classes of subgroups.
The subgroup lattice has 138 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 | direct (N × H) |
| 3. C23×#208 | C2 | semidirect |
| 4. C24×C3⋊C8 | C2 | semidirect |
| 11. C2×#948 | C22 | direct (N × H) |
| 12. C22×#208 | C22 | semidirect |
| 15. C23×C3⋊C8 | C22 | semidirect |
| 29. #948 | C23 | direct (N × H) |
| 30. C2×#208 | C23 | semidirect |
| 33. C22×C3⋊C8 | C23 | semidirect |
| 56. #208 | C24 | semidirect |
| 59. C2×C3⋊C8 | C24 | semidirect |
| 75. C23×C3 | #131 | semidirect |
| 104. C22×C3 | C2×#131 | semidirect |
| 111. C23 | #948 | direct (N × H) |
| 124. C2×C3 | C22×#131 | semidirect |
| 127. C22 | C2×#948 | direct (N × H) |
| 133. C3 | C23×#131 | semidirect |
| 135. C2 | C22×#948 | 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 |
| C23×#208 | 2 |
| C24×C3⋊C8 | 2 |
| C23×#131 | 3 |