Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
9372 conjugacy classes of subgroups.
The subgroup lattice has 456 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C16×C32 | C24×C3 |
| socle, prime radical | C24×C3 | C16×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C32×C64 | C2 | direct (N × H) |
| 4. C22×C32×C64 | C3 | direct (N × H) |
| 12. C3×C32×C64 | C22 | direct (N × H) |
| 14. C2×C32×C64 | C2×C3 | direct (N × H) |
| 38. C32×C64 | C22×C3 | direct (N × H) |
| 82. C22×C3×C64 | C32 | direct (N × H) |
| 134. C22×C3×C32 | C64 | direct (N × H) |
| 148. C2×C3×C64 | C2×C32 | direct (N × H) |
| 169. C22×C64 | C3×C32 | direct (N × H) |
| 210. C2×C3×C32 | C2×C64 | direct (N × H) |
| 227. C3×C64 | C22×C32 | direct (N × H) |
| 232. C22×C32 | C3×C64 | direct (N × H) |
| 246. C2×C64 | C2×C3×C32 | direct (N × H) |
| 290. C3×C32 | C22×C64 | direct (N × H) |
| 308. C2×C32 | C2×C3×C64 | direct (N × H) |
| 325. C64 | C22×C3×C32 | direct (N × H) |
| 377. C32 | C22×C3×C64 | direct (N × H) |
| 417. C22×C3 | C32×C64 | direct (N × H) |
| 441. C2×C3 | C2×C32×C64 | direct (N × H) |
| 443. C22 | C3×C32×C64 | direct (N × H) |
| 451. C3 | C22×C32×C64 | direct (N × H) |
| 453. C2 | C2×C3×C32×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C322 | 2 |
| C2×C3×C32×C64 | 2 |
| C22×C3×C16×C64 | 2 |
| C22×C32×C64 | 3 |