Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
5916 conjugacy classes of subgroups.
The subgroup lattice has 468 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C128 | C24×C3 |
| socle, prime radical | C24×C3 | C22×C128 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C42×C256 | C2 | direct (N × H) |
| 4. C2×C42×C256 | C3 | direct (N × H) |
| 11. C2×C3×C4×C256 | C4 | direct (N × H) |
| 15. C42×C256 | C2×C3 | direct (N × H) |
| 30. C3×C4×C256 | C2×C4 | direct (N × H) |
| 36. C2×C4×C256 | C3×C4 | direct (N × H) |
| 61. C2×C3×C256 | C42 | direct (N × H) |
| 77. C4×C256 | C2×C3×C4 | direct (N × H) |
| 108. C3×C256 | C2×C42 | direct (N × H) |
| 128. C2×C256 | C3×C42 | direct (N × H) |
| 198. C256 | C2×C3×C42 | direct (N × H) |
| 278. C2×C3×C42 | C256 | direct (N × H) |
| 352. C3×C42 | C2×C256 | direct (N × H) |
| 368. C2×C42 | C3×C256 | direct (N × H) |
| 393. C2×C3×C4 | C4×C256 | direct (N × H) |
| 421. C42 | C2×C3×C256 | direct (N × H) |
| 433. C3×C4 | C2×C4×C256 | direct (N × H) |
| 440. C2×C4 | C3×C4×C256 | direct (N × H) |
| 452. C2×C3 | C42×C256 | direct (N × H) |
| 459. C4 | C2×C3×C4×C256 | direct (N × H) |
| 463. C3 | C2×C42×C256 | direct (N × H) |
| 464. C2 | C3×C42×C256 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C42×C128 | 2 |
| C22×C3×C4×C256 | 2 |
| C3×C42×C256 | 2 |
| C2×C42×C256 | 3 |