Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2398 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C24×C3 | C2×D4 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ C24×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#1735 | C2 | direct (N × H) |
| 4. C2×C3×#340 | C2 | semidirect |
| 5. C2×#1735 | C3 | direct (N × H) |
| 8. C23×C3×C8 | C4 | semidirect |
| 13. C2×C3×#117 | C4 | semidirect |
| 18. C3×#340 | C22 | semidirect |
| 21. #1735 | C2×C3 | direct (N × H) |
| 22. C2×#340 | C2×C3 | semidirect |
| 37. C22×C3×C8 | C2×C4 | semidirect |
| 41. C3×#117 | C2×C4 | semidirect |
| 52. C23×C8 | C3×C4 | semidirect |
| 57. C2×#117 | C3×C4 | semidirect |
| 62. #340 | C22×C3 | semidirect |
| 112. C22×C8 | C2×C3×C4 | semidirect |
| 116. #117 | C2×C3×C4 | semidirect |
| 237. C2×C3 | #1735 | direct (N × H) |
| 257. C3 | C2×#1735 | direct (N × H) |
| 259. C2 | C3×#1735 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C8 | 2 |
| C22×C3×#117 | 2 |
| C3×#1735 | 2 |
| C2×C3×#340 | 2 |
| C2×#1735 | 3 |