Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2190 conjugacy classes of subgroups.
The subgroup lattice has 374 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C2×C22⋊C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#1742 | C2 | direct (N × H) |
| 4. C2×C3×#343 | C2 | semidirect |
| 5. C2×#1742 | C3 | direct (N × H) |
| 10. C22×C3×M16 | C4 | semidirect |
| 22. C3×#343 | C22 | semidirect |
| 25. #1742 | C2×C3 | direct (N × H) |
| 26. C2×#343 | C2×C3 | semidirect |
| 46. C2×C3×M16 | C2×C4 | semidirect |
| 70. C22×M16 | C3×C4 | semidirect |
| 82. #343 | C22×C3 | semidirect |
| 153. C2×M16 | C2×C3×C4 | semidirect |
| 340. C2×C3 | #1742 | direct (N × H) |
| 365. C3 | C2×#1742 | direct (N × H) |
| 370. C2 | C3×#1742 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×M16 | 2 |
| C22×C3×#117 | 2 |
| C3×#1742 | 2 |
| C2×C3×#343 | 2 |
| C2×#1742 | 3 |