Finite Group DatabasebetaFinite Group DatabasebetaOrder 4096.
Only the 2335 normal subgroups are shown; non-normal subgroups are omitted.
The subgroup lattice has 331 nodes (normal subgroups only) — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72210025 |
| commutator, Frattini, cosocle | C22×H(5,𝔽2) | C25 |
| LCS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72324336 | C2 | semidirect |
| 2. #72324305 | C2 | semidirect |
| 3. #72324282 | C2 | semidirect |
| 4. #72324281 | C22 | semidirect |
| 5. #72324280 | C22 | semidirect |
| 6. #72324265 | C2 | semidirect |
| 7. #72324264 | C22 | semidirect |
| 8. #72324263 | C22 | semidirect |
| 9. #72306591 | C2 | semidirect |
| 11. #72262724 | C22 | semidirect |
| 12. #72256349 | C2 | semidirect |
| 13. #72256330 | C22 | semidirect |
| 14. #72256325 | C22 | semidirect |
| 15. #72256267 | C22 | semidirect |
| 16. #72256265 | C22 | semidirect |
| 17. #72256144 | C22 | semidirect |
| 18. #72249638 | C22 | semidirect |
| 19. #72237882 | C22 | semidirect |
| 21. #72233051 | C22 | semidirect |
| 22. #72228008 | C22 | semidirect |
| 23. #72226174 | C2 | semidirect |
| 24. #72226081 | C22 | semidirect |
| 25. #72226041 | C22 | semidirect |
| 26. #72226017 | C22 | semidirect |
| 27. #72225996 | C22 | semidirect |
| 28. #72225641 | C22 | semidirect |
| 29. #72219393 | C22 | semidirect |
| 31. C2×#6499874 | C22 | semidirect |
| 32. #72216396 | C22 | semidirect |
| 33. C2×#62817 | D4 | semidirect |
| 35. C2×#60399 | C23 | semidirect |
| 36. #9040971 | D4 | semidirect |
| 37. #9040959 | D4 | semidirect |
| 38. #9040459 | D4 | semidirect |
| 41. #8912605 | D4 | semidirect |
| 42. #8910147 | D4 | semidirect |
| 44. #8902653 | D4 | semidirect |
| 46. #8797921 | D4 | semidirect |
| 47. #8765959 | D4 | semidirect |
| 48. #8093167 | D4 | semidirect |
| 49. #8093108 | D4 | semidirect |
| 50. #7819709 | D4 | semidirect |
| 52. #7670609 | D4 | semidirect |
| 53. #7670203 | D4 | semidirect |
| 55. #7642244 | D4 | semidirect |
| 57. C22×#2679 | D4 | semidirect |
| 58. #7626264 | D4 | semidirect |
| 59. #6628896 | C23 | semidirect |
| 60. #6628876 | C23 | semidirect |
| 61. C2×#21776 | C23 | semidirect |
| 62. #496240 | C23 | semidirect |
| 63. #496219 | C23 | semidirect |
| 64. #496200 | C23 | semidirect |
| 65. #494721 | C23 | semidirect |
| 66. #494700 | C23 | semidirect |
| 67. #493424 | C23 | semidirect |
| 68. #382436 | C23 | semidirect |
| 69. #382434 | C23 | semidirect |
| 70. #379815 | C23 | semidirect |
| 71. #379806 | C23 | semidirect |
| 72. #214599 | C23 | semidirect |
| 73. #213890 | C23 | semidirect |
| 74. #213828 | C23 | semidirect |
| 75. C2×H(7,𝔽2) | C24 | semidirect |
| 83. C2×#2883 | C24 | semidirect |
| 129. #20592 | C24 | semidirect |
| 130. #20585 | C24 | semidirect |
| 132. C27 | #120 | semidirect |
| 134. C22×H(5,𝔽2) | #120 | semidirect |
| 135. C22×H(5,𝔽2) | #120 | semidirect |
| 136. C22×H(5,𝔽2) | #120 | semidirect |
| 139. C2×H(3,𝔽4) | #120 | semidirect |
| 140. C2×H(3,𝔽4) | #120 | semidirect |
| 141. C2×H(3,𝔽4) | #120 | semidirect |
| 142. C2×#560 | #120 | semidirect |
| 143. C2×Q82 | #120 | semidirect |
| 144. C2×#551 | #120 | semidirect |
| 145. C2×D42 | #120 | semidirect |
| 146. C2×D42 | #120 | semidirect |
| 148. C2×#543 | #120 | semidirect |
| 152. C2×#534 | #120 | semidirect |
| 156. C2×#525 | #120 | semidirect |
| 159. C22×#120 | #120 | semidirect |
| 166. C2×#518 | #120 | semidirect |
| 167. C2×#518 | #120 | semidirect |
| 168. C2×#518 | #120 | semidirect |
| 170. C23×C4⋊C4 | #120 | semidirect |
| 172. C23×C22⋊C4 | #120 | semidirect |
| 173. C23×C42 | #120 | semidirect |
| 174. #2840 | #120 | semidirect |
| 175. #2816 | #120 | semidirect |
| 176. #2811 | #120 | semidirect |
| 177. #2748 | #120 | semidirect |
| 178. #2731 | #120 | semidirect |
| 179. #2684 | #120 | semidirect |
| 180. #2684 | #120 | semidirect |
| 181. #2684 | #120 | semidirect |
| 182. #2679 | #120 | semidirect |
| 183. #2679 | #120 | semidirect |
| 184. #2679 | #120 | semidirect |
| 185. #2679 | #120 | semidirect |
| 186. #2679 | #120 | semidirect |
| 187. #2669 | #120 | semidirect |
| 188. #2669 | #120 | semidirect |
| 189. #2669 | #120 | semidirect |
| 190. #2608 | #120 | semidirect |
| 191. #2561 | #120 | semidirect |
| 192. #2539 | #120 | semidirect |
| 199. #2403 | #120 | semidirect |
| 202. #2379 | #120 | semidirect |
| 203. #2379 | #120 | semidirect |
| 204. #2376 | #120 | semidirect |
| 205. #2375 | #120 | semidirect |
| 206. #2375 | #120 | semidirect |
| 207. #2374 | #120 | semidirect |
| 208. #2369 | #120 | semidirect |
| 209. #2366 | #120 | semidirect |
| 210. #2358 | #120 | semidirect |
| 211. #2358 | #120 | semidirect |
| 212. #2333 | #120 | semidirect |
| 213. #2332 | #120 | semidirect |
| 214. #2316 | #120 | semidirect |
| 215. #2315 | #120 | semidirect |
| 216. #2315 | #120 | semidirect |
| 217. #2311 | #120 | semidirect |
| 219. #2308 | #120 | semidirect |
| 220. #2308 | #120 | semidirect |
| 224. #2297 | #120 | semidirect |
| 226. C2×#390 | #120 | semidirect |
| 228. #2285 | #120 | semidirect |
| 229. C2×#379 | #120 | semidirect |
| 230. C2×#379 | #120 | semidirect |
| 231. C2×#379 | #120 | semidirect |
| 232. #2273 | #120 | semidirect |
| 233. #2273 | #120 | semidirect |
| 237. C2×H(5,𝔽2) | B(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72324336 | 2 |
| #72324305 | 2 |
| #72324282 | 2 |
| #72324265 | 2 |
| #72306591 | 2 |
| #72256349 | 2 |
| #72226174 | 2 |