Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
8092 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 | C4×C8×C32 | C23×C3 |
| socle, prime radical | C23×C3 | C4×C8×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C8×C16×C64 | C3 | direct (N × H) |
| 30. C3×C16×C64 | C8 | direct (N × H) |
| 62. C3×C8×C64 | C16 | direct (N × H) |
| 78. C16×C64 | C3×C8 | direct (N × H) |
| 133. C8×C64 | C3×C16 | direct (N × H) |
| 167. C3×C8×C16 | C64 | direct (N × H) |
| 247. C3×C64 | C8×C16 | direct (N × H) |
| 278. C8×C16 | C3×C64 | direct (N × H) |
| 358. C64 | C3×C8×C16 | direct (N × H) |
| 380. C3×C16 | C8×C64 | direct (N × H) |
| 426. C3×C8 | C16×C64 | direct (N × H) |
| 452. C16 | C3×C8×C64 | direct (N × H) |
| 474. C8 | C3×C16×C64 | direct (N × H) |
| 491. C3 | C8×C16×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C8×C16×C32 | 2 |
| C3×C82×C64 | 2 |
| C3×C4×C16×C64 | 2 |
| C8×C16×C64 | 3 |