Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
At least 6087 conjugacy classes of subgroups; only the 4038 normal subgroups are shown.
The subgroup lattice has 679 nodes (normal subgroups only) — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C26 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#127 ◁ C2×#3498 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72340519 | C2 | semidirect |
| 2. #72340516 | C2 | semidirect |
| 3. #72338887 | C2 | semidirect |
| 4. #72338861 | C2 | semidirect |
| 5. #72338640 | C2 | semidirect |
| 6. #72318112 | C2 | semidirect |
| 7. #72317502 | C2 | semidirect |
| 8. #72315030 | C2 | semidirect |
| 9. #72311970 | C2 | semidirect |
| 10. #72252010 | C2 | semidirect |
| 11. #72249464 | C2 | semidirect |
| 12. #72236627 | C2 | semidirect |
| 13. #72236574 | C2 | semidirect |
| 14. #72236189 | C2 | semidirect |
| 15. #72234132 | C2 | semidirect |
| 16. D4×#2196 | C2 | semidirect |
| 17. #72210487 | C2 | semidirect |
| 18. #10586617 | C22 | semidirect |
| 19. #8934366 | C22 | semidirect |
| 20. #8698894 | C22 | semidirect |
| 21. #8195808 | C22 | semidirect |
| 22. #7985294 | C22 | semidirect |
| 24. #7737411 | C22 | semidirect |
| 25. D43 | C22 | semidirect |
| 27. #7667013 | C22 | semidirect |
| 28. C22×#2196 | C22 | semidirect |
| 29. C2×#33359 | C22 | semidirect |
| 30. #6596199 | C22 | semidirect |
| 31. #6595772 | C22 | semidirect |
| 32. #6595646 | C22 | semidirect |
| 33. #6595612 | C22 | semidirect |
| 34. #6591514 | C22 | semidirect |
| 35. D4×B(4,𝔽2) | C22 | semidirect |
| 36. #6573693 | C22 | semidirect |
| 37. D4×#453 | C22 | semidirect |
| 38. #6491480 | C22 | semidirect |
| 39. #6490945 | C22 | semidirect |
| 40. #6470231 | C22 | semidirect |
| 41. #6470043 | C22 | semidirect |
| 42. #6454873 | C22 | semidirect |
| 43. #6453831 | C22 | semidirect |
| 44. #6437999 | C22 | semidirect |
| 45. #6437456 | C22 | semidirect |
| 46. #6385998 | C22 | semidirect |
| 47. #6379015 | C22 | semidirect |
| 48. #512122 | C22 | semidirect |
| 49. #512008 | C22 | semidirect |
| 50. #511908 | C22 | semidirect |
| 51. #511771 | C22 | semidirect |
| 52. #508249 | C22 | semidirect |
| 53. #508193 | C22 | semidirect |
| 54. #508095 | C22 | semidirect |
| 55. #503579 | C22 | semidirect |
| 56. #503456 | C22 | semidirect |
| 57. #503444 | C22 | semidirect |
| 58. #503347 | C22 | semidirect |
| 59. #503194 | C22 | semidirect |
| 60. #503182 | C22 | semidirect |
| 61. #502969 | C22 | semidirect |
| 62. #502962 | C22 | semidirect |
| 63. #502798 | C22 | semidirect |
| 64. #502780 | C22 | semidirect |
| 65. #502709 | C22 | semidirect |
| 66. #502709 | C22 | semidirect |
| 67. #502662 | C22 | semidirect |
| 68. #502662 | C22 | semidirect |
| 69. #501156 | C22 | semidirect |
| 70. #501156 | C22 | semidirect |
| 71. #501097 | C22 | semidirect |
| 72. #501096 | C22 | semidirect |
| 73. #501096 | C22 | semidirect |
| 74. #501085 | C22 | semidirect |
| 75. #501085 | C22 | semidirect |
| 76. #501085 | C22 | semidirect |
| 77. #501084 | C22 | semidirect |
| 78. #501084 | C22 | semidirect |
| 79. #500955 | C22 | semidirect |
| 80. #500769 | C22 | semidirect |
| 81. #500757 | C22 | semidirect |
| 82. #500757 | C22 | semidirect |
| 83. #500507 | C22 | semidirect |
| 84. #500507 | C22 | semidirect |
| 85. #500489 | C22 | semidirect |
| 86. #500405 | C22 | semidirect |
| 87. #500218 | C22 | semidirect |
| 88. #500177 | C22 | semidirect |
| 89. #500170 | C22 | semidirect |
| 90. #500057 | C22 | semidirect |
| 91. #499843 | C22 | semidirect |
| 92. #499832 | C22 | semidirect |
| 93. #499787 | C22 | semidirect |
| 94. #499282 | C22 | semidirect |
| 95. #499137 | C22 | semidirect |
| 96. #498927 | C22 | semidirect |
| 97. #497484 | C22 | semidirect |
| 98. #497484 | C22 | semidirect |
| 99. #497307 | C22 | semidirect |
| 100. D4×#353 | C22 | semidirect |
| 101. D4×#353 | C22 | semidirect |
| 102. D4×AΣL(1,𝔽16) | C22 | semidirect |
| 103. D4×AΣL(1,𝔽16) | C22 | semidirect |
| 104. #496520 | C22 | semidirect |
| 105. #496237 | C22 | semidirect |
| 106. C4×#2196 | C22 | semidirect |
| 107. C4×#2196 | C22 | semidirect |
| 108. #495592 | C22 | semidirect |
| 109. #495174 | C22 | semidirect |
| 110. #494722 | C22 | semidirect |
| 111. #494700 | C22 | semidirect |
| 112. #494596 | C22 | semidirect |
| 113. #494370 | C22 | semidirect |
| 114. #493707 | C22 | semidirect |
| 115. C2×#12759 | C22 | semidirect |
| 116. #493431 | C22 | semidirect |
| 117. C4×#2126 | C22 | semidirect |
| 118. #62983 | C23 | semidirect |
| 119. D4×H(5,𝔽2) | C23 | semidirect |
| 120. #62842 | C23 | semidirect |
| 121. C2×#3498 | D4 | semidirect |
| 124. #62781 | C23 | semidirect |
| 125. C22×D42 | D4 | semidirect |
| 127. C22×B(4,𝔽2) | C23 | semidirect |
| 128. C2×#3019 | C23 | semidirect |
| 130. C2×#2883 | C23 | semidirect |
| 144. #39361 | D4 | semidirect |
| 146. #38904 | D4 | semidirect |
| 148. #36702 | D4 | semidirect |
| 150. C2×#2867 | D4 | semidirect |
| 151. #36128 | D4 | semidirect |
| 153. D4×#120 | D4 | semidirect |
| 155. #35527 | D4 | semidirect |
| 158. #35272 | D4 | semidirect |
| 159. #35272 | D4 | semidirect |
| 161. #35237 | D4 | semidirect |
| 163. #34364 | D4 | semidirect |
| 166. C4×#532 | D4 | semidirect |
| 171. C2×#2196 | C23 | semidirect |
| 172. C2×#2196 | C23 | semidirect |
| 173. C2×#2128 | C23 | semidirect |
| 174. C2×#2128 | C23 | semidirect |
| 175. C2×#2127 | C23 | semidirect |
| 181. C22×AΣL(1,𝔽16) | C23 | semidirect |
| 182. C22×AΣL(1,𝔽16) | C23 | semidirect |
| 183. #33020 | C23 | semidirect |
| 184. #32990 | C23 | semidirect |
| 185. #32976 | C23 | semidirect |
| 187. #32920 | C23 | semidirect |
| 188. #32920 | C23 | semidirect |
| 189. #32898 | C23 | semidirect |
| 190. #32888 | C23 | semidirect |
| 192. #25918 | C23 | semidirect |
| 193. #25900 | C23 | semidirect |
| 194. #25853 | C23 | semidirect |
| 196. #25739 | C23 | semidirect |
| 197. #24412 | C23 | semidirect |
| 198. #24412 | C23 | semidirect |
| 199. #24406 | C23 | semidirect |
| 200. #24406 | C23 | semidirect |
| 201. #24406 | C23 | semidirect |
| 202. #24404 | C23 | semidirect |
| 203. #24404 | C23 | semidirect |
| 204. #24350 | C23 | semidirect |
| 205. #24314 | C23 | semidirect |
| 207. #24260 | C23 | semidirect |
| 208. #24224 | C23 | semidirect |
| 210. #24058 | C23 | semidirect |
| 211. #24021 | C23 | semidirect |
| 212. #23913 | C23 | semidirect |
| 213. #23912 | C23 | semidirect |
| 216. #23900 | C23 | semidirect |
| 217. D4×#100 | C23 | semidirect |
| 218. D4×#100 | C23 | semidirect |
| 219. D4×#99 | C23 | semidirect |
| 220. D4×#99 | C23 | semidirect |
| 221. C4×B(4,𝔽2) | C23 | semidirect |
| 222. C4×B(4,𝔽2) | C23 | semidirect |
| 223. C4×#455 | C23 | semidirect |
| 225. C2×#1889 | C23 | semidirect |
| 226. #20953 | C23 | semidirect |
| 228. #20889 | C23 | semidirect |
| 229. #20728 | C23 | semidirect |
| 230. #20632 | C23 | semidirect |
| 231. #20609 | C23 | semidirect |
| 232. #20592 | C23 | semidirect |
| 233. #20586 | C23 | semidirect |
| 234. C4×#410 | C23 | semidirect |
| 235. #13463 | C23 | semidirect |
| 236. #13463 | C23 | semidirect |
| 237. #13424 | C23 | semidirect |
| 238. #13424 | C23 | semidirect |
| 239. #13424 | C23 | semidirect |
| 240. #13422 | C23 | semidirect |
| 241. #13422 | C23 | semidirect |
| 242. #13422 | C23 | semidirect |
| 243. #13422 | C23 | semidirect |
| 246. #13377 | C23 | semidirect |
| 247. #13377 | C23 | semidirect |
| 248. #13348 | C23 | semidirect |
| 249. #13348 | C23 | semidirect |
| 250. #13348 | C23 | semidirect |
| 251. #13346 | C23 | semidirect |
| 252. #13346 | C23 | semidirect |
| 253. #13346 | C23 | semidirect |
| 254. #13346 | C23 | semidirect |
| 257. #13295 | C23 | semidirect |
| 258. #13295 | C23 | semidirect |
| 259. #13292 | C23 | semidirect |
| 260. #13223 | C23 | semidirect |
| 261. #13223 | C23 | semidirect |
| 265. #13088 | C23 | semidirect |
| 266. #13088 | C23 | semidirect |
| 267. #13087 | C23 | semidirect |
| 268. #13087 | C23 | semidirect |
| 269. #13087 | C23 | semidirect |
| 270. #13086 | C23 | semidirect |
| 271. #13086 | C23 | semidirect |
| 272. #13082 | C23 | semidirect |
| 273. #13082 | C23 | semidirect |
| 274. #13049 | C23 | semidirect |
| 275. #13049 | C23 | semidirect |
| 279. #13041 | C23 | semidirect |
| 280. #13041 | C23 | semidirect |
| 281. #13041 | C23 | semidirect |
| 282. #12966 | C23 | semidirect |
| 283. #12966 | C23 | semidirect |
| 287. #12958 | C23 | semidirect |
| 288. #12958 | C23 | semidirect |
| 289. #12958 | C23 | semidirect |
| 290. #12854 | C23 | semidirect |
| 291. #12798 | C23 | semidirect |
| 293. #12793 | C23 | semidirect |
| 294. #12793 | C23 | semidirect |
| 298. #12759 | C23 | semidirect |
| 299. #12759 | C23 | semidirect |
| 300. #12696 | C23 | semidirect |
| 302. #12687 | C23 | semidirect |
| 303. C4×#354 | C23 | semidirect |
| 306. C4×AΣL(1,𝔽16) | C23 | semidirect |
| 307. C4×AΣL(1,𝔽16) | C23 | semidirect |
| 308. #12581 | C23 | semidirect |
| 309. #12581 | C23 | semidirect |
| 310. #12574 | C23 | semidirect |
| 311. #12574 | C23 | semidirect |
| 312. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 315. C24×D4 | C2×D4 | semidirect |
| 321. C2×#550 | C2×D4 | semidirect |
| 323. C2×D42 | C2×D4 | semidirect |
| 329. C2×#535 | C2×D4 | semidirect |
| 330. C2×#535 | C2×D4 | semidirect |
| 331. C2×#535 | C2×D4 | semidirect |
| 336. C2×#532 | C2×D4 | semidirect |
| 340. C22×#127 | C2×D4 | semidirect |
| 341. C22×#127 | C2×D4 | semidirect |
| 342. C22×#127 | C2×D4 | semidirect |
| 344. C22×#121 | C2×D4 | semidirect |
| 345. C22×#120 | C2×D4 | semidirect |
| 346. C22×#120 | C2×D4 | semidirect |
| 350. C2×C4×#41 | C2×D4 | semidirect |
| 351. C2×C4×#41 | C2×D4 | semidirect |
| 353. C22×C4×D4 | C2×D4 | semidirect |
| 368. #2867 | C2×D4 | semidirect |
| 369. #2867 | C2×D4 | semidirect |
| 370. #2856 | C2×D4 | semidirect |
| 371. #2847 | C2×D4 | semidirect |
| 372. #2847 | C2×D4 | semidirect |
| 373. #2846 | C2×D4 | semidirect |
| 378. #2622 | C2×D4 | semidirect |
| 379. #2619 | C2×D4 | semidirect |
| 380. #2548 | C2×D4 | semidirect |
| 381. #2548 | C2×D4 | semidirect |
| 382. #2541 | C2×D4 | semidirect |
| 384. #2504 | C2×D4 | semidirect |
| 385. #2487 | C2×D4 | semidirect |
| 387. #2482 | C2×D4 | semidirect |
| 388. #2433 | C2×D4 | semidirect |
| 389. #2433 | C2×D4 | semidirect |
| 391. #2431 | C2×D4 | semidirect |
| 392. #2431 | C2×D4 | semidirect |
| 393. #2410 | C2×D4 | semidirect |
| 394. #2410 | C2×D4 | semidirect |
| 395. #2408 | C2×D4 | semidirect |
| 397. #2404 | C2×D4 | semidirect |
| 399. #2403 | C2×D4 | semidirect |
| 404. D4×C22⋊C4 | C2×D4 | semidirect |
| 405. D4×C22⋊C4 | C2×D4 | semidirect |
| 406. #2309 | C2×D4 | semidirect |
| 407. #2309 | C2×D4 | semidirect |
| 408. #2308 | C2×D4 | semidirect |
| 409. #2308 | C2×D4 | semidirect |
| 411. C4×C4⋊Q8 | C2×D4 | semidirect |
| 413. C4×#124 | C2×D4 | semidirect |
| 415. C4×#120 | C2×D4 | semidirect |
| 417. #2297 | C2×D4 | semidirect |
| 418. #2297 | C2×D4 | semidirect |
| 420. #2293 | C2×D4 | semidirect |
| 421. #2285 | C2×D4 | semidirect |
| 422. #2285 | C2×D4 | semidirect |
| 423. #2283 | C2×D4 | semidirect |
| 424. C42×D4 | C2×D4 | semidirect |
| 425. C2×C43 | C2×D4 | semidirect |
| 582. #353 | H(5,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72340519 | 2 |
| #72340516 | 2 |
| #72338887 | 2 |
| #72338861 | 2 |
| #72338640 | 2 |
| #72318112 | 2 |
| #72317502 | 2 |
| #72315030 | 2 |
| #72311970 | 2 |
| #72252010 | 2 |
| #72249464 | 2 |
| #72236627 | 2 |
| #72236574 | 2 |
| #72236189 | 2 |
| #72234132 | 2 |
| D4×#2196 | 2 |
| #72210487 | 2 |