Finite Group DatabasebetaFinite Group DatabasebetaOrder 28672.
4823286 conjugacy classes of subgroups.
The subgroup lattice has 380 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C23×C4 | C27×C7 |
| socle, prime radical | C27×C7 | C23×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C43×C7×C8 | C2 | direct (N × H) |
| 9. C23×C42×C7×C8 | C4 | direct (N × H) |
| 11. C2×C43×C7×C8 | C22 | direct (N × H) |
| 12. C23×C43×C8 | C7 | direct (N × H) |
| 15. C23×C43×C7 | C8 | direct (N × H) |
| 25. C22×C42×C7×C8 | C2×C4 | direct (N × H) |
| 27. C43×C7×C8 | C23 | direct (N × H) |
| 30. C22×C43×C8 | C2×C7 | direct (N × H) |
| 36. C23×C42×C8 | C4×C7 | direct (N × H) |
| 38. C2×C43×C8 | C22×C7 | direct (N × H) |
| 49. C23×C4×C7×C8 | C42 | direct (N × H) |
| 53. C22×C43×C7 | C2×C8 | direct (N × H) |
| 58. C2×C42×C7×C8 | C22×C4 | direct (N × H) |
| 72. C23×C42×C7 | C4×C8 | direct (N × H) |
| 81. C22×C4×C7×C8 | C2×C42 | direct (N × H) |
| 85. C2×C43×C7 | C22×C8 | direct (N × H) |
| 90. C42×C7×C8 | C23×C4 | direct (N × H) |
| 96. C23×C43 | C7×C8 | direct (N × H) |
| 101. C22×C42×C8 | C2×C4×C7 | direct (N × H) |
| 105. C43×C8 | C23×C7 | direct (N × H) |
| 119. C23×C7×C8 | C43 | direct (N × H) |
| 124. C22×C42×C7 | C2×C4×C8 | direct (N × H) |
| 133. C2×C4×C7×C8 | C22×C42 | direct (N × H) |
| 136. C43×C7 | C23×C8 | direct (N × H) |
| 148. C23×C4×C8 | C42×C7 | direct (N × H) |
| 152. C22×C43 | C2×C7×C8 | direct (N × H) |
| 157. C2×C42×C8 | C22×C4×C7 | direct (N × H) |
| 166. C23×C4×C7 | C42×C8 | direct (N × H) |
| 179. C22×C7×C8 | C2×C43 | direct (N × H) |
| 183. C2×C42×C7 | C22×C4×C8 | direct (N × H) |
| 189. C4×C7×C8 | C23×C42 | direct (N × H) |
| 200. C23×C42 | C4×C7×C8 | direct (N × H) |
| 209. C22×C4×C8 | C2×C42×C7 | direct (N × H) |
| 213. C2×C43 | C22×C7×C8 | direct (N × H) |
| 218. C42×C8 | C23×C4×C7 | direct (N × H) |
| 227. C22×C4×C7 | C2×C42×C8 | direct (N × H) |
| 237. C2×C7×C8 | C22×C43 | direct (N × H) |
| 240. C42×C7 | C23×C4×C8 | direct (N × H) |
| 255. C23×C8 | C43×C7 | direct (N × H) |
| 260. C22×C42 | C2×C4×C7×C8 | direct (N × H) |
| 269. C2×C4×C8 | C22×C42×C7 | direct (N × H) |
| 272. C43 | C23×C7×C8 | direct (N × H) |
| 274. C23×C7 | C43×C8 | direct (N × H) |
| 282. C2×C4×C7 | C22×C42×C8 | direct (N × H) |
| 288. C7×C8 | C23×C43 | direct (N × H) |
| 294. C23×C4 | C42×C7×C8 | direct (N × H) |
| 307. C22×C8 | C2×C43×C7 | direct (N × H) |
| 311. C2×C42 | C22×C4×C7×C8 | direct (N × H) |
| 317. C4×C8 | C23×C42×C7 | direct (N × H) |
| 318. C22×C7 | C2×C43×C8 | direct (N × H) |
| 323. C4×C7 | C23×C42×C8 | direct (N × H) |
| 334. C22×C4 | C2×C42×C7×C8 | direct (N × H) |
| 344. C2×C8 | C22×C43×C7 | direct (N × H) |
| 347. C42 | C23×C4×C7×C8 | direct (N × H) |
| 349. C2×C7 | C22×C43×C8 | direct (N × H) |
| 352. C23 | C43×C7×C8 | direct (N × H) |
| 359. C2×C4 | C22×C42×C7×C8 | direct (N × H) |
| 366. C8 | C23×C43×C7 | direct (N × H) |
| 367. C7 | C23×C43×C8 | direct (N × H) |
| 368. C22 | C2×C43×C7×C8 | direct (N × H) |
| 373. C4 | C23×C42×C7×C8 | direct (N × H) |
| 377. C2 | C22×C43×C7×C8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C44×C7 | 2 |
| C22×C43×C7×C8 | 2 |
| C24×C42×C7×C8 | 2 |
| C23×C43×C8 | 7 |