Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
22300 conjugacy classes of subgroups.
The subgroup lattice has 700 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C42×C32 | C24×C3 |
| socle, prime radical | C24×C3 | C42×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C82×C64 | C2 | direct (N × H) |
| 4. C2×C82×C64 | C3 | direct (N × H) |
| 16. C82×C64 | C2×C3 | direct (N × H) |
| 34. C2×C3×C8×C64 | C8 | direct (N × H) |
| 78. C3×C8×C64 | C2×C8 | direct (N × H) |
| 94. C2×C8×C64 | C3×C8 | direct (N × H) |
| 180. C8×C64 | C2×C3×C8 | direct (N × H) |
| 218. C2×C3×C82 | C64 | direct (N × H) |
| 227. C2×C3×C64 | C82 | direct (N × H) |
| 345. C3×C82 | C2×C64 | direct (N × H) |
| 349. C3×C64 | C2×C82 | direct (N × H) |
| 387. C2×C82 | C3×C64 | direct (N × H) |
| 396. C2×C64 | C3×C82 | direct (N × H) |
| 514. C82 | C2×C3×C64 | direct (N × H) |
| 518. C64 | C2×C3×C82 | direct (N × H) |
| 528. C2×C3×C8 | C8×C64 | direct (N × H) |
| 615. C3×C8 | C2×C8×C64 | direct (N × H) |
| 630. C2×C8 | C3×C8×C64 | direct (N × H) |
| 677. C8 | C2×C3×C8×C64 | direct (N × H) |
| 683. C2×C3 | C82×C64 | direct (N × H) |
| 695. C3 | C2×C82×C64 | direct (N × H) |
| 696. C2 | C3×C82×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C82×C32 | 2 |
| C2×C3×C4×C8×C64 | 2 |
| C3×C82×C64 | 2 |
| C2×C82×C64 | 3 |