Finite Group DatabasebetaFinite Group DatabasebetaOrder 31104.
108000 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C9 | C25×C33 |
| socle, prime radical | C25×C33 | C22×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C32×C42×C27 | C2 | direct (N × H) |
| 4. C23×C3×C42×C27 | C3 | direct (N × H) |
| 6. C23×C32×C4×C27 | C4 | direct (N × H) |
| 8. C2×C32×C42×C27 | C22 | direct (N × H) |
| 12. C22×C3×C42×C27 | C2×C3 | direct (N × H) |
| 15. C22×C32×C4×C27 | C2×C4 | direct (N × H) |
| 17. C32×C42×C27 | C23 | direct (N × H) |
| 21. C23×C42×C27 | C32 | direct (N × H) |
| 26. C23×C3×C4×C27 | C3×C4 | direct (N × H) |
| 29. C2×C3×C42×C27 | C22×C3 | direct (N × H) |
| 30. C23×C32×C27 | C42 | direct (N × H) |
| 33. C2×C32×C4×C27 | C22×C4 | direct (N × H) |
| 42. C22×C42×C27 | C2×C32 | direct (N × H) |
| 50. C22×C3×C4×C27 | C2×C3×C4 | direct (N × H) |
| 52. C3×C42×C27 | C23×C3 | direct (N × H) |
| 53. C23×C32×C42 | C27 | direct (N × H) |
| 57. C22×C32×C27 | C2×C42 | direct (N × H) |
| 60. C32×C4×C27 | C23×C4 | direct (N × H) |
| 74. C23×C4×C27 | C32×C4 | direct (N × H) |
| 76. C2×C42×C27 | C22×C32 | direct (N × H) |
| 82. C23×C3×C27 | C3×C42 | direct (N × H) |
| 85. C2×C3×C4×C27 | C22×C3×C4 | direct (N × H) |
| 89. C22×C32×C42 | C2×C27 | direct (N × H) |
| 95. C2×C32×C27 | C22×C42 | direct (N × H) |
| 114. C22×C4×C27 | C2×C32×C4 | direct (N × H) |
| 116. C42×C27 | C23×C32 | direct (N × H) |
| 117. C23×C3×C42 | C3×C27 | direct (N × H) |
| 123. C22×C3×C27 | C2×C3×C42 | direct (N × H) |
| 126. C3×C4×C27 | C23×C3×C4 | direct (N × H) |
| 129. C23×C32×C4 | C4×C27 | direct (N × H) |
| 135. C2×C32×C42 | C22×C27 | direct (N × H) |
| 143. C32×C27 | C23×C42 | direct (N × H) |
| 159. C23×C27 | C32×C42 | direct (N × H) |
| 162. C2×C4×C27 | C22×C32×C4 | direct (N × H) |
| 166. C22×C3×C42 | C2×C3×C27 | direct (N × H) |
| 170. C2×C3×C27 | C22×C3×C42 | direct (N × H) |
| 176. C22×C32×C4 | C2×C4×C27 | direct (N × H) |
| 184. C32×C42 | C23×C27 | direct (N × H) |
| 192. C23×C42 | C32×C27 | direct (N × H) |
| 205. C22×C27 | C2×C32×C42 | direct (N × H) |
| 208. C4×C27 | C23×C32×C4 | direct (N × H) |
| 211. C23×C3×C4 | C3×C4×C27 | direct (N × H) |
| 215. C2×C3×C42 | C22×C3×C27 | direct (N × H) |
| 218. C3×C27 | C23×C3×C42 | direct (N × H) |
| 219. C23×C32 | C42×C27 | direct (N × H) |
| 225. C2×C32×C4 | C22×C4×C27 | direct (N × H) |
| 240. C22×C42 | C2×C32×C27 | direct (N × H) |
| 247. C2×C27 | C22×C32×C42 | direct (N × H) |
| 253. C22×C3×C4 | C2×C3×C4×C27 | direct (N × H) |
| 257. C3×C42 | C23×C3×C27 | direct (N × H) |
| 259. C22×C32 | C2×C42×C27 | direct (N × H) |
| 265. C32×C4 | C23×C4×C27 | direct (N × H) |
| 276. C23×C4 | C32×C4×C27 | direct (N × H) |
| 278. C2×C42 | C22×C32×C27 | direct (N × H) |
| 282. C27 | C23×C32×C42 | direct (N × H) |
| 283. C23×C3 | C3×C42×C27 | direct (N × H) |
| 289. C2×C3×C4 | C22×C3×C4×C27 | direct (N × H) |
| 293. C2×C32 | C22×C42×C27 | direct (N × H) |
| 303. C22×C4 | C2×C32×C4×C27 | direct (N × H) |
| 305. C42 | C23×C32×C27 | direct (N × H) |
| 306. C22×C3 | C2×C3×C42×C27 | direct (N × H) |
| 312. C3×C4 | C23×C3×C4×C27 | direct (N × H) |
| 314. C32 | C23×C42×C27 | direct (N × H) |
| 318. C23 | C32×C42×C27 | direct (N × H) |
| 321. C2×C4 | C22×C32×C4×C27 | direct (N × H) |
| 323. C2×C3 | C22×C3×C42×C27 | direct (N × H) |
| 327. C22 | C2×C32×C42×C27 | direct (N × H) |
| 330. C4 | C23×C32×C4×C27 | direct (N × H) |
| 331. C3 | C23×C3×C42×C27 | direct (N × H) |
| 333. C2 | C22×C32×C42×C27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C32×C4×C27 | 2 |
| C22×C32×C42×C27 | 2 |
| C23×C32×C42×C9 | 3 |
| C23×C3×C42×C27 | 3 |