Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
4252 conjugacy classes of subgroups.
The subgroup lattice has 568 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4×C128 | C24×C3 |
| socle, prime radical | C24×C3 | C4×C128 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×C8×C256 | C2 | direct (N × H) |
| 4. C22×C8×C256 | C3 | direct (N × H) |
| 13. C3×C8×C256 | C22 | direct (N × H) |
| 16. C2×C8×C256 | C2×C3 | direct (N × H) |
| 31. C22×C3×C256 | C8 | direct (N × H) |
| 43. C8×C256 | C22×C3 | direct (N × H) |
| 71. C2×C3×C256 | C2×C8 | direct (N × H) |
| 87. C22×C256 | C3×C8 | direct (N × H) |
| 128. C3×C256 | C22×C8 | direct (N × H) |
| 148. C2×C256 | C2×C3×C8 | direct (N × H) |
| 239. C256 | C22×C3×C8 | direct (N × H) |
| 330. C22×C3×C8 | C256 | direct (N × H) |
| 420. C2×C3×C8 | C2×C256 | direct (N × H) |
| 441. C22×C8 | C3×C256 | direct (N × H) |
| 489. C3×C8 | C22×C256 | direct (N × H) |
| 503. C2×C8 | C2×C3×C256 | direct (N × H) |
| 524. C22×C3 | C8×C256 | direct (N × H) |
| 545. C8 | C22×C3×C256 | direct (N × H) |
| 551. C2×C3 | C2×C8×C256 | direct (N × H) |
| 554. C22 | C3×C8×C256 | direct (N × H) |
| 563. C3 | C22×C8×C256 | direct (N × H) |
| 564. C2 | C2×C3×C8×C256 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C8×C128 | 2 |
| C22×C3×C4×C256 | 2 |
| C2×C3×C8×C256 | 2 |
| C22×C8×C256 | 3 |