Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
2460 conjugacy classes of subgroups.
The subgroup lattice has 496 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C4×C128 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C4×C128 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C4×C8×C256 | C3 | direct (N × H) |
| 12. C3×C8×C256 | C4 | direct (N × H) |
| 29. C3×C4×C256 | C8 | direct (N × H) |
| 38. C8×C256 | C3×C4 | direct (N × H) |
| 77. C4×C256 | C3×C8 | direct (N × H) |
| 111. C3×C256 | C4×C8 | direct (N × H) |
| 208. C256 | C3×C4×C8 | direct (N × H) |
| 305. C3×C4×C8 | C256 | direct (N × H) |
| 402. C4×C8 | C3×C256 | direct (N × H) |
| 426. C3×C8 | C4×C256 | direct (N × H) |
| 460. C3×C4 | C8×C256 | direct (N × H) |
| 474. C8 | C3×C4×C256 | direct (N × H) |
| 486. C4 | C3×C8×C256 | direct (N × H) |
| 491. C3 | C4×C8×C256 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C8×C128 | 2 |
| C2×C3×C8×C256 | 2 |
| C3×C42×C256 | 2 |
| C4×C8×C256 | 3 |