Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
375068 conjugacy classes of subgroups.
The subgroup lattice has 172 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C64 | C27×C3 |
| socle, prime radical | C27×C3 | C64 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C25×C3×C128 | C2 | direct (N × H) |
| 3. C26×C128 | C3 | direct (N × H) |
| 7. C24×C3×C128 | C22 | direct (N × H) |
| 9. C25×C128 | C2×C3 | direct (N × H) |
| 15. C23×C3×C128 | C23 | direct (N × H) |
| 19. C24×C128 | C22×C3 | direct (N × H) |
| 27. C22×C3×C128 | C24 | direct (N × H) |
| 33. C23×C128 | C23×C3 | direct (N × H) |
| 43. C2×C3×C128 | C25 | direct (N × H) |
| 51. C22×C128 | C24×C3 | direct (N × H) |
| 63. C3×C128 | C26 | direct (N × H) |
| 73. C2×C128 | C25×C3 | direct (N × H) |
| 74. C26×C3 | C128 | direct (N × H) |
| 97. C128 | C26×C3 | direct (N × H) |
| 98. C25×C3 | C2×C128 | direct (N × H) |
| 108. C26 | C3×C128 | direct (N × H) |
| 120. C24×C3 | C22×C128 | direct (N × H) |
| 128. C25 | C2×C3×C128 | direct (N × H) |
| 138. C23×C3 | C23×C128 | direct (N × H) |
| 144. C24 | C22×C3×C128 | direct (N × H) |
| 152. C22×C3 | C24×C128 | direct (N × H) |
| 156. C23 | C23×C3×C128 | direct (N × H) |
| 162. C2×C3 | C25×C128 | direct (N × H) |
| 164. C22 | C24×C3×C128 | direct (N × H) |
| 168. C3 | C26×C128 | direct (N × H) |
| 169. C2 | C25×C3×C128 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C3×C64 | 2 |
| C25×C3×C128 | 2 |
| C26×C128 | 3 |