Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
11422 conjugacy classes of subgroups.
The subgroup lattice has 138 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C8 | C22 |
| commutator | C2 | C25×C3×C4 |
| Frattini, cosocle | C4 | C26×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C23×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×#131 | C2 | direct (N × H) |
| 3. C23×C3×M16 | C2 | semidirect |
| 4. C24×C3×C8 | C2 | semidirect |
| 5. C23×#131 | C3 | direct (N × H) |
| 11. C2×C3×#131 | C22 | direct (N × H) |
| 12. C22×C3×M16 | C22 | semidirect |
| 15. C23×C3×C8 | C22 | semidirect |
| 17. C22×#131 | C2×C3 | direct (N × H) |
| 18. C23×M16 | C2×C3 | semidirect |
| 19. C24×C8 | C2×C3 | semidirect |
| 29. C3×#131 | C23 | direct (N × H) |
| 30. C2×C3×M16 | C23 | semidirect |
| 33. C22×C3×C8 | C23 | semidirect |
| 39. C2×#131 | C22×C3 | direct (N × H) |
| 40. C22×M16 | C22×C3 | semidirect |
| 43. C23×C8 | C22×C3 | semidirect |
| 56. C3×M16 | C24 | semidirect |
| 59. C2×C3×C8 | C24 | semidirect |
| 69. #131 | C23×C3 | direct (N × H) |
| 70. C2×M16 | C23×C3 | semidirect |
| 73. C22×C8 | C23×C3 | semidirect |
| 75. C23×C3 | #131 | direct (N × H) |
| 99. M16 | C24×C3 | semidirect |
| 102. C2×C8 | C24×C3 | semidirect |
| 104. C22×C3 | C2×#131 | direct (N × H) |
| 111. C23 | C3×#131 | direct (N × H) |
| 124. C2×C3 | C22×#131 | direct (N × H) |
| 127. C22 | C2×C3×#131 | direct (N × H) |
| 133. C3 | C23×#131 | direct (N × H) |
| 135. C2 | C22×C3×#131 | 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×C3×#131 | 2 |
| C23×C3×M16 | 2 |
| C24×C3×C8 | 2 |
| C23×#131 | 3 |