Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
1811228 conjugacy classes of subgroups.
The subgroup lattice has 304 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C82 | C27×C3 |
| socle, prime radical | C27×C3 | C82 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×C3×C162 | C2 | direct (N × H) |
| 3. C25×C162 | C3 | direct (N × H) |
| 8. C23×C3×C162 | C22 | direct (N × H) |
| 10. C24×C162 | C2×C3 | direct (N × H) |
| 19. C22×C3×C162 | C23 | direct (N × H) |
| 24. C23×C162 | C22×C3 | direct (N × H) |
| 35. C25×C3×C16 | C16 | direct (N × H) |
| 39. C2×C3×C162 | C24 | direct (N × H) |
| 48. C22×C162 | C23×C3 | direct (N × H) |
| 55. C24×C3×C16 | C2×C16 | direct (N × H) |
| 68. C3×C162 | C25 | direct (N × H) |
| 71. C25×C16 | C3×C16 | direct (N × H) |
| 83. C2×C162 | C24×C3 | direct (N × H) |
| 95. C23×C3×C16 | C22×C16 | direct (N × H) |
| 114. C24×C16 | C2×C3×C16 | direct (N × H) |
| 127. C162 | C25×C3 | direct (N × H) |
| 141. C22×C3×C16 | C23×C16 | direct (N × H) |
| 163. C23×C16 | C22×C3×C16 | direct (N × H) |
| 176. C25×C3 | C162 | direct (N × H) |
| 190. C2×C3×C16 | C24×C16 | direct (N × H) |
| 209. C22×C16 | C23×C3×C16 | direct (N × H) |
| 220. C24×C3 | C2×C162 | direct (N × H) |
| 234. C3×C16 | C25×C16 | direct (N × H) |
| 235. C25 | C3×C162 | direct (N × H) |
| 249. C2×C16 | C24×C3×C16 | direct (N × H) |
| 255. C23×C3 | C22×C162 | direct (N × H) |
| 264. C24 | C2×C3×C162 | direct (N × H) |
| 278. C16 | C25×C3×C16 | direct (N × H) |
| 279. C22×C3 | C23×C162 | direct (N × H) |
| 284. C23 | C22×C3×C162 | direct (N × H) |
| 293. C2×C3 | C24×C162 | direct (N × H) |
| 295. C22 | C23×C3×C162 | direct (N × H) |
| 300. C3 | C25×C162 | direct (N × H) |
| 301. C2 | C24×C3×C162 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C3×C8×C16 | 2 |
| C24×C3×C162 | 2 |
| C25×C162 | 3 |