Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2190 conjugacy classes of subgroups.
The subgroup lattice has 500 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 | #95 |
| 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 |
|---|---|---|
| 4. C2×C3×#344 | C2 | direct (N × H) |
| 5. C22×#344 | C3 | direct (N × H) |
| 14. C22×C3×M16 | C4 | semidirect |
| 15. C22×C3×M16 | C4 | semidirect |
| 16. C22×C3×M16 | C4 | semidirect |
| 20. C23×C3×C8 | C4 | semidirect |
| 28. C3×#344 | C22 | direct (N × H) |
| 32. C2×#344 | C2×C3 | direct (N × H) |
| 62. C2×C3×M16 | C2×C4 | semidirect |
| 63. C2×C3×M16 | C2×C4 | semidirect |
| 64. C2×C3×M16 | C2×C4 | semidirect |
| 72. C22×C3×C8 | C2×C4 | semidirect |
| 92. C22×M16 | C3×C4 | semidirect |
| 93. C22×M16 | C3×C4 | semidirect |
| 94. C22×M16 | C3×C4 | semidirect |
| 98. C23×C8 | C3×C4 | semidirect |
| 106. #344 | C22×C3 | direct (N × H) |
| 167. C3×M16 | C22×C4 | semidirect |
| 173. C2×C3×C8 | C22×C4 | semidirect |
| 209. C2×M16 | C2×C3×C4 | semidirect |
| 210. C2×M16 | C2×C3×C4 | semidirect |
| 211. C2×M16 | C2×C3×C4 | semidirect |
| 219. C22×C8 | C2×C3×C4 | semidirect |
| 352. M16 | C22×C3×C4 | semidirect |
| 358. C2×C8 | C22×C3×C4 | semidirect |
| 376. C22×C3 | #344 | direct (N × H) |
| 460. C2×C3 | C2×#344 | direct (N × H) |
| 473. C22 | C3×#344 | direct (N × H) |
| 490. C3 | C22×#344 | direct (N × H) |
| 497. C2 | C2×C3×#344 | 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 |
| C2×C3×#344 | 2 |
| C22×#344 | 3 |