Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2830 conjugacy classes of subgroups.
The subgroup lattice has 262 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#432 | C2 | direct (N × H) |
| 3. C22×C3×C4⋊C8 | C2 | semidirect |
| 5. C22×C3×#97 | C2 | semidirect |
| 6. C22×#432 | C3 | direct (N × H) |
| 17. C3×#432 | C22 | direct (N × H) |
| 18. C2×C3×C4⋊C8 | C22 | semidirect |
| 20. C2×C3×#97 | C22 | semidirect |
| 22. C2×#432 | C2×C3 | direct (N × H) |
| 23. C22×C4⋊C8 | C2×C3 | semidirect |
| 25. C22×#97 | C2×C3 | semidirect |
| 50. C3×C4⋊C8 | C23 | semidirect |
| 52. C3×#97 | C23 | semidirect |
| 63. #432 | C22×C3 | direct (N × H) |
| 64. C2×C4⋊C8 | C22×C3 | semidirect |
| 66. C2×#97 | C22×C3 | semidirect |
| 125. C4⋊C8 | C23×C3 | semidirect |
| 127. #97 | C23×C3 | semidirect |
| 193. C22×C3 | #432 | direct (N × H) |
| 235. C2×C3 | C2×#432 | direct (N × H) |
| 242. C22 | C3×#432 | direct (N × H) |
| 255. C3 | C22×#432 | direct (N × H) |
| 257. C2 | C2×C3×#432 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C42 | 2 |
| C2×C3×#432 | 2 |
| C22×C3×C4⋊C8 | 2 |
| C22×C3×C22⋊C8 | 2 |
| C22×C3×#97 | 2 |
| C22×#432 | 3 |