Finite Group DatabasebetaFinite Group DatabasebetaOrder 24576.
23432988 conjugacy classes of subgroups.
The subgroup lattice has 420 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C24×C4 | C27×C3 |
| socle, prime radical | C27×C3 | C24×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×C44×C8 | C2 | direct (N × H) |
| 4. C22×C44×C8 | C3 | direct (N × H) |
| 10. C22×C3×C43×C8 | C4 | direct (N × H) |
| 12. C3×C44×C8 | C22 | direct (N × H) |
| 15. C2×C44×C8 | C2×C3 | direct (N × H) |
| 18. C22×C3×C44 | C8 | direct (N × H) |
| 28. C2×C3×C43×C8 | C2×C4 | direct (N × H) |
| 35. C22×C43×C8 | C3×C4 | direct (N × H) |
| 37. C44×C8 | C22×C3 | direct (N × H) |
| 48. C2×C3×C44 | C2×C8 | direct (N × H) |
| 55. C22×C3×C42×C8 | C42 | direct (N × H) |
| 58. C3×C43×C8 | C22×C4 | direct (N × H) |
| 61. C22×C44 | C3×C8 | direct (N × H) |
| 71. C2×C43×C8 | C2×C3×C4 | direct (N × H) |
| 86. C22×C3×C43 | C4×C8 | direct (N × H) |
| 95. C2×C3×C42×C8 | C2×C42 | direct (N × H) |
| 98. C3×C44 | C22×C8 | direct (N × H) |
| 111. C22×C42×C8 | C3×C42 | direct (N × H) |
| 115. C2×C44 | C2×C3×C8 | direct (N × H) |
| 120. C43×C8 | C22×C3×C4 | direct (N × H) |
| 141. C22×C3×C4×C8 | C43 | direct (N × H) |
| 145. C2×C3×C43 | C2×C4×C8 | direct (N × H) |
| 151. C3×C42×C8 | C22×C42 | direct (N × H) |
| 165. C22×C43 | C3×C4×C8 | direct (N × H) |
| 174. C2×C42×C8 | C2×C3×C42 | direct (N × H) |
| 177. C44 | C22×C3×C8 | direct (N × H) |
| 195. C22×C3×C42 | C42×C8 | direct (N × H) |
| 205. C2×C3×C4×C8 | C2×C43 | direct (N × H) |
| 208. C3×C43 | C22×C4×C8 | direct (N × H) |
| 230. C22×C4×C8 | C3×C43 | direct (N × H) |
| 234. C2×C43 | C2×C3×C4×C8 | direct (N × H) |
| 240. C42×C8 | C22×C3×C42 | direct (N × H) |
| 257. C22×C3×C8 | C44 | direct (N × H) |
| 261. C2×C3×C42 | C2×C42×C8 | direct (N × H) |
| 267. C3×C4×C8 | C22×C43 | direct (N × H) |
| 284. C22×C42 | C3×C42×C8 | direct (N × H) |
| 294. C2×C4×C8 | C2×C3×C43 | direct (N × H) |
| 297. C43 | C22×C3×C4×C8 | direct (N × H) |
| 305. C22×C3×C4 | C43×C8 | direct (N × H) |
| 315. C2×C3×C8 | C2×C44 | direct (N × H) |
| 318. C3×C42 | C22×C42×C8 | direct (N × H) |
| 336. C22×C8 | C3×C44 | direct (N × H) |
| 340. C2×C42 | C2×C3×C42×C8 | direct (N × H) |
| 346. C4×C8 | C22×C3×C43 | direct (N × H) |
| 353. C2×C3×C4 | C2×C43×C8 | direct (N × H) |
| 360. C3×C8 | C22×C44 | direct (N × H) |
| 367. C22×C4 | C3×C43×C8 | direct (N × H) |
| 377. C2×C8 | C2×C3×C44 | direct (N × H) |
| 379. C42 | C22×C3×C42×C8 | direct (N × H) |
| 382. C22×C3 | C44×C8 | direct (N × H) |
| 387. C3×C4 | C22×C43×C8 | direct (N × H) |
| 396. C2×C4 | C2×C3×C43×C8 | direct (N × H) |
| 403. C8 | C22×C3×C44 | direct (N × H) |
| 404. C2×C3 | C2×C44×C8 | direct (N × H) |
| 407. C22 | C3×C44×C8 | direct (N × H) |
| 412. C4 | C22×C3×C43×C8 | direct (N × H) |
| 415. C3 | C22×C44×C8 | direct (N × H) |
| 416. C2 | C2×C3×C44×C8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C45 | 2 |
| C23×C3×C43×C8 | 2 |
| C2×C3×C44×C8 | 2 |
| C22×C44×C8 | 3 |