Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
13310 conjugacy classes of subgroups; only the 11166 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 11166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C3×C4 | C22 |
| commutator | C2 | C25×C3×C4 |
| Frattini, cosocle | C4 | C26×C3 |
| socle | C25×C3 | C2×C4 |
| prime radical | C26×C3 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C3×M16 | C2 | direct (N × H) |
| 3. C24×C3×C8 | C2 | semidirect |
| 4. C24×M16 | C3 | direct (N × H) |
| 10. C22×C3×M16 | C22 | direct (N × H) |
| 11. C23×C3×C8 | C22 | semidirect |
| 13. C23×M16 | C2×C3 | direct (N × H) |
| 14. C24×C8 | C2×C3 | semidirect |
| 23. C2×C3×M16 | C23 | direct (N × H) |
| 24. C22×C3×C8 | C23 | semidirect |
| 30. C22×M16 | C22×C3 | direct (N × H) |
| 31. C23×C8 | C22×C3 | semidirect |
| 33. C24×C3 | M16 | direct (N × H) |
| 41. C3×M16 | C24 | direct (N × H) |
| 42. C2×C3×C8 | C24 | semidirect |
| 51. C2×M16 | C23×C3 | direct (N × H) |
| 52. C22×C8 | C23×C3 | semidirect |
| 54. C23×C3 | C2×M16 | direct (N × H) |
| 62. C3×C8 | C25 | semidirect |
| 64. C24 | C3×M16 | direct (N × H) |
| 72. M16 | C24×C3 | direct (N × H) |
| 73. C2×C8 | C24×C3 | semidirect |
| 75. C22×C3 | C22×M16 | direct (N × H) |
| 82. C23 | C2×C3×M16 | direct (N × H) |
| 90. C8 | C25×C3 | semidirect |
| 92. C2×C3 | C23×M16 | direct (N × H) |
| 95. C22 | C22×C3×M16 | direct (N × H) |
| 101. C3 | C24×M16 | direct (N × H) |
| 103. C2 | C23×C3×M16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C3×C4 | 2 |
| C23×C3×M16 | 2 |
| C24×C3×C8 | 2 |
| C24×M16 | 3 |