Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
33564 conjugacy classes of subgroups.
The subgroup lattice has 624 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C64 | C25×C3 |
| socle, prime radical | C25×C3 | C22×C64 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×C42×C128 | C2 | direct (N × H) |
| 4. C22×C42×C128 | C3 | direct (N × H) |
| 10. C22×C3×C4×C128 | C4 | direct (N × H) |
| 13. C3×C42×C128 | C22 | direct (N × H) |
| 16. C2×C42×C128 | C2×C3 | direct (N × H) |
| 31. C2×C3×C4×C128 | C2×C4 | direct (N × H) |
| 40. C22×C4×C128 | C3×C4 | direct (N × H) |
| 43. C42×C128 | C22×C3 | direct (N × H) |
| 71. C22×C3×C128 | C42 | direct (N × H) |
| 74. C3×C4×C128 | C22×C4 | direct (N × H) |
| 91. C2×C4×C128 | C2×C3×C4 | direct (N × H) |
| 131. C2×C3×C128 | C2×C42 | direct (N × H) |
| 155. C22×C128 | C3×C42 | direct (N × H) |
| 166. C4×C128 | C22×C3×C4 | direct (N × H) |
| 217. C3×C128 | C22×C42 | direct (N × H) |
| 256. C2×C128 | C2×C3×C42 | direct (N × H) |
| 270. C22×C3×C42 | C128 | direct (N × H) |
| 362. C128 | C22×C3×C42 | direct (N × H) |
| 383. C2×C3×C42 | C2×C128 | direct (N × H) |
| 415. C22×C42 | C3×C128 | direct (N × H) |
| 460. C22×C3×C4 | C4×C128 | direct (N × H) |
| 481. C3×C42 | C22×C128 | direct (N × H) |
| 508. C2×C42 | C2×C3×C128 | direct (N × H) |
| 536. C2×C3×C4 | C2×C4×C128 | direct (N × H) |
| 552. C22×C4 | C3×C4×C128 | direct (N × H) |
| 572. C42 | C22×C3×C128 | direct (N × H) |
| 580. C22×C3 | C42×C128 | direct (N × H) |
| 585. C3×C4 | C22×C4×C128 | direct (N × H) |
| 595. C2×C4 | C2×C3×C4×C128 | direct (N × H) |
| 607. C2×C3 | C2×C42×C128 | direct (N × H) |
| 610. C22 | C3×C42×C128 | direct (N × H) |
| 615. C4 | C22×C3×C4×C128 | direct (N × H) |
| 619. C3 | C22×C42×C128 | direct (N × H) |
| 620. C2 | C2×C3×C42×C128 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C42×C64 | 2 |
| C23×C3×C4×C128 | 2 |
| C2×C3×C42×C128 | 2 |
| C22×C42×C128 | 3 |