Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
23068 conjugacy classes of subgroups.
The subgroup lattice has 660 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4×C64 | C25×C3 |
| socle, prime radical | C25×C3 | C4×C64 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C3×C8×C128 | C2 | direct (N × H) |
| 4. C23×C8×C128 | C3 | direct (N × H) |
| 13. C2×C3×C8×C128 | C22 | direct (N × H) |
| 16. C22×C8×C128 | C2×C3 | direct (N × H) |
| 33. C23×C3×C128 | C8 | direct (N × H) |
| 35. C3×C8×C128 | C23 | direct (N × H) |
| 44. C2×C8×C128 | C22×C3 | direct (N × H) |
| 72. C22×C3×C128 | C2×C8 | direct (N × H) |
| 92. C23×C128 | C3×C8 | direct (N × H) |
| 96. C8×C128 | C23×C3 | direct (N × H) |
| 133. C2×C3×C128 | C22×C8 | direct (N × H) |
| 162. C22×C128 | C2×C3×C8 | direct (N × H) |
| 229. C3×C128 | C23×C8 | direct (N × H) |
| 266. C2×C128 | C22×C3×C8 | direct (N × H) |
| 278. C23×C3×C8 | C128 | direct (N × H) |
| 383. C128 | C23×C3×C8 | direct (N × H) |
| 394. C22×C3×C8 | C2×C128 | direct (N × H) |
| 432. C23×C8 | C3×C128 | direct (N × H) |
| 498. C2×C3×C8 | C22×C128 | direct (N × H) |
| 527. C22×C8 | C2×C3×C128 | direct (N × H) |
| 563. C23×C3 | C8×C128 | direct (N × H) |
| 576. C3×C8 | C23×C128 | direct (N × H) |
| 594. C2×C8 | C22×C3×C128 | direct (N × H) |
| 615. C22×C3 | C2×C8×C128 | direct (N × H) |
| 624. C23 | C3×C8×C128 | direct (N × H) |
| 637. C8 | C23×C3×C128 | direct (N × H) |
| 643. C2×C3 | C22×C8×C128 | direct (N × H) |
| 646. C22 | C2×C3×C8×C128 | direct (N × H) |
| 655. C3 | C23×C8×C128 | direct (N × H) |
| 656. C2 | C22×C3×C8×C128 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C8×C64 | 2 |
| C23×C3×C4×C128 | 2 |
| C22×C3×C8×C128 | 2 |
| C23×C8×C128 | 3 |