Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
Only the 4054 normal subgroups are shown; non-normal subgroups are omitted.
The subgroup lattice has 397 nodes (normal subgroups only) — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#2196 |
| commutator, Frattini, cosocle | #127 | C26 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C22×#41 ◁ #62812 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72338456 | C2 | semidirect |
| 2. #72338452 | C2 | semidirect |
| 3. #72338450 | C2 | semidirect |
| 4. #72338344 | C2 | semidirect |
| 5. #72338314 | C2 | semidirect |
| 6. #72338308 | C2 | semidirect |
| 7. #72317553 | C2 | semidirect |
| 8. #72306575 | C2 | semidirect |
| 9. #72279437 | C2 | semidirect |
| 10. #72279417 | C2 | semidirect |
| 11. C2×#512864 | C2 | semidirect |
| 12. #72237528 | C2 | semidirect |
| 13. #72216138 | C2 | semidirect |
| 14. #10586342 | C22 | semidirect |
| 15. #10576184 | C22 | semidirect |
| 16. #10576183 | C22 | semidirect |
| 17. #10576177 | C22 | semidirect |
| 18. C2×#61110 | C22 | semidirect |
| 21. #7622885 | C22 | semidirect |
| 22. #7622883 | C22 | semidirect |
| 23. #7622882 | C22 | semidirect |
| 24. #7622880 | C22 | semidirect |
| 25. #7622874 | C22 | semidirect |
| 26. C2×#33546 | C22 | semidirect |
| 27. #7622466 | C22 | semidirect |
| 28. #6353352 | C22 | semidirect |
| 29. #6353348 | C22 | semidirect |
| 30. #6353305 | C22 | semidirect |
| 31. #6353301 | C22 | semidirect |
| 32. #6348454 | C22 | semidirect |
| 33. #6348452 | C22 | semidirect |
| 34. #6348073 | C22 | semidirect |
| 35. C2×#18527 | C22 | semidirect |
| 36. #6344271 | C22 | semidirect |
| 37. #512868 | C22 | semidirect |
| 38. #512867 | C22 | semidirect |
| 39. #512866 | C22 | semidirect |
| 40. #512865 | C22 | semidirect |
| 41. #512865 | C22 | semidirect |
| 42. #512864 | C22 | semidirect |
| 43. #512864 | C22 | semidirect |
| 44. #512864 | C22 | semidirect |
| 45. #512864 | C22 | semidirect |
| 46. #512864 | C22 | semidirect |
| 47. #512863 | C22 | semidirect |
| 48. #512862 | C22 | semidirect |
| 49. #512861 | C22 | semidirect |
| 50. #512860 | C22 | semidirect |
| 51. #512858 | C22 | semidirect |
| 52. #512857 | C22 | semidirect |
| 53. #512857 | C22 | semidirect |
| 54. C2×#13675 | C22 | semidirect |
| 55. C2×#13673 | C22 | semidirect |
| 56. #512616 | C22 | semidirect |
| 57. #512613 | C22 | semidirect |
| 58. #512601 | C22 | semidirect |
| 59. C2×#13585 | C22 | semidirect |
| 60. #512566 | C22 | semidirect |
| 61. #512565 | C22 | semidirect |
| 62. #512564 | C22 | semidirect |
| 63. C2×#13562 | C22 | semidirect |
| 64. #63103 | D4 | semidirect |
| 65. #63022 | C23 | semidirect |
| 66. #63021 | C23 | semidirect |
| 67. #63016 | C23 | semidirect |
| 68. #63011 | C23 | semidirect |
| 70. C2×#3498 | D4 | semidirect |
| 71. #62759 | C23 | semidirect |
| 72. C2×#3481 | C23 | semidirect |
| 79. #60647 | D4 | semidirect |
| 81. #60646 | D4 | semidirect |
| 83. #60639 | D4 | semidirect |
| 85. C2×#3151 | D4 | semidirect |
| 88. #60390 | D4 | semidirect |
| 89. #60389 | D4 | semidirect |
| 91. #60387 | D4 | semidirect |
| 93. #60076 | D4 | semidirect |
| 94. C4×#568 | D4 | semidirect |
| 95. #33561 | C23 | semidirect |
| 96. #33559 | C23 | semidirect |
| 97. #33557 | C23 | semidirect |
| 98. #33546 | C23 | semidirect |
| 99. #33546 | C23 | semidirect |
| 100. #33546 | C23 | semidirect |
| 101. #33546 | C23 | semidirect |
| 102. C2×#2198 | C23 | semidirect |
| 103. C2×#2196 | C23 | semidirect |
| 104. #33360 | C23 | semidirect |
| 105. #33359 | C23 | semidirect |
| 106. C2×#2126 | C23 | semidirect |
| 107. #18718 | C23 | semidirect |
| 108. #18714 | C23 | semidirect |
| 109. #18712 | C23 | semidirect |
| 110. #18688 | C23 | semidirect |
| 111. #18686 | C23 | semidirect |
| 112. #18682 | C23 | semidirect |
| 113. #18528 | C23 | semidirect |
| 114. #18528 | C23 | semidirect |
| 115. #18527 | C23 | semidirect |
| 116. #18527 | C23 | semidirect |
| 117. #18527 | C23 | semidirect |
| 118. #18527 | C23 | semidirect |
| 119. #18527 | C23 | semidirect |
| 120. #18526 | C23 | semidirect |
| 121. #18524 | C23 | semidirect |
| 122. #18480 | C23 | semidirect |
| 123. #18480 | C23 | semidirect |
| 124. C2×#1620 | C23 | semidirect |
| 125. C2×#1619 | C23 | semidirect |
| 126. #17773 | C23 | semidirect |
| 127. #17770 | C23 | semidirect |
| 128. #17738 | C23 | semidirect |
| 129. C2×#1560 | C23 | semidirect |
| 130. #13676 | C23 | semidirect |
| 131. #13676 | C23 | semidirect |
| 132. #13675 | C23 | semidirect |
| 133. #13675 | C23 | semidirect |
| 134. #13675 | C23 | semidirect |
| 135. #13674 | C23 | semidirect |
| 136. #13674 | C23 | semidirect |
| 137. #13673 | C23 | semidirect |
| 138. #13673 | C23 | semidirect |
| 139. #13673 | C23 | semidirect |
| 140. #13586 | C23 | semidirect |
| 141. #13585 | C23 | semidirect |
| 142. #13585 | C23 | semidirect |
| 143. #13584 | C23 | semidirect |
| 144. #13583 | C23 | semidirect |
| 145. C2×#1408 | C23 | semidirect |
| 146. #13562 | C23 | semidirect |
| 147. #13562 | C23 | semidirect |
| 148. #13562 | C23 | semidirect |
| 149. #13561 | C23 | semidirect |
| 150. #13560 | C23 | semidirect |
| 151. C2×#1402 | C23 | semidirect |
| 152. 2−1+6 | C2×D4 | semidirect |
| 153. H(7,𝔽2) | C2×D4 | semidirect |
| 155. C2×#585 | C2×D4 | semidirect |
| 156. #3550 | C24 | semidirect |
| 157. #3549 | C24 | semidirect |
| 158. #3543 | C24 | semidirect |
| 164. #3498 | C2×D4 | semidirect |
| 165. #3498 | C2×D4 | semidirect |
| 169. #3481 | C24 | semidirect |
| 170. #3481 | C24 | semidirect |
| 171. C2×#559 | C24 | semidirect |
| 172. C2×D42 | C24 | semidirect |
| 175. C2×#532 | C2×D4 | semidirect |
| 177. C22×#127 | C2×D4 | semidirect |
| 181. #3152 | C2×D4 | semidirect |
| 182. #3152 | C2×D4 | semidirect |
| 183. #3152 | C2×D4 | semidirect |
| 185. #3151 | C2×D4 | semidirect |
| 186. #3151 | C2×D4 | semidirect |
| 187. #3151 | C2×D4 | semidirect |
| 188. #3151 | C2×D4 | semidirect |
| 189. #3151 | C2×D4 | semidirect |
| 193. #3147 | C2×D4 | semidirect |
| 194. #3147 | C2×D4 | semidirect |
| 196. C2×#493 | C2×D4 | semidirect |
| 198. C2×#492 | C2×D4 | semidirect |
| 199. C2×#492 | C2×D4 | semidirect |
| 200. #3022 | C2×D4 | semidirect |
| 201. #3021 | C2×D4 | semidirect |
| 202. #3019 | C2×D4 | semidirect |
| 203. #3018 | C2×D4 | semidirect |
| 204. #3018 | C2×D4 | semidirect |
| 207. C2×#455 | C2×D4 | semidirect |
| 209. #2889 | C2×D4 | semidirect |
| 210. #2888 | C2×D4 | semidirect |
| 211. C2×#413 | C2×D4 | semidirect |
| 212. C4×#131 | C2×D4 | semidirect |
| 213. C4×#131 | C2×D4 | semidirect |
| 214. C2×C4×M16 | C2×D4 | semidirect |
| 215. C2×C4×M16 | C2×D4 | semidirect |
| 216. #2198 | C24 | semidirect |
| 217. #2198 | C24 | semidirect |
| 218. #2196 | C24 | semidirect |
| 219. #2196 | C24 | semidirect |
| 221. #2126 | C24 | semidirect |
| 222. C2×#353 | C24 | semidirect |
| 223. #1630 | C24 | semidirect |
| 224. #1630 | C24 | semidirect |
| 225. #1629 | C24 | semidirect |
| 226. #1629 | C24 | semidirect |
| 227. #1620 | C24 | semidirect |
| 228. #1620 | C24 | semidirect |
| 229. #1620 | C24 | semidirect |
| 230. #1619 | C24 | semidirect |
| 231. #1619 | C24 | semidirect |
| 232. #1619 | C24 | semidirect |
| 235. #1560 | C24 | semidirect |
| 236. #1555 | C24 | semidirect |
| 237. #1543 | C24 | semidirect |
| 238. C2×#331 | C24 | semidirect |
| 239. #1409 | C24 | semidirect |
| 240. #1408 | C24 | semidirect |
| 241. #1402 | C24 | semidirect |
| 242. #1402 | C24 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72338456 | 2 |
| #72338452 | 2 |
| #72338450 | 2 |
| #72338344 | 2 |
| #72338314 | 2 |
| #72338308 | 2 |
| #72317553 | 2 |
| #72306575 | 2 |
| #72279437 | 2 |
| #72279417 | 2 |
| C2×#512864 | 2 |
| #72237528 | 2 |
| #72216138 | 2 |