Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3395 conjugacy classes of subgroups.
The subgroup lattice has 431 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3568 | C2 | semidirect |
| 2. #3561 | C2 | semidirect |
| 3. #3558 | C2 | semidirect |
| 4. #3545 | C2 | semidirect |
| 5. #3545 | C2 | semidirect |
| 6. #3540 | C2 | semidirect |
| 7. #3532 | C2 | semidirect |
| 8. #3529 | C2 | semidirect |
| 9. C2×H(3,𝔽4) | C2 | semidirect |
| 10. #3514 | C2 | semidirect |
| 11. #3510 | C2 | semidirect |
| 12. #3490 | C2 | semidirect |
| 13. #563 | C22 | semidirect |
| 14. #563 | C22 | semidirect |
| 15. #563 | C22 | semidirect |
| 16. #562 | C22 | semidirect |
| 17. #562 | C22 | semidirect |
| 18. #562 | C22 | semidirect |
| 19. H(3,𝔽4) | C22 | semidirect |
| 20. H(3,𝔽4) | C22 | semidirect |
| 21. H(3,𝔽4) | C22 | semidirect |
| 22. H(3,𝔽4) | C22 | semidirect |
| 23. H(3,𝔽4) | C22 | semidirect |
| 24. H(3,𝔽4) | C22 | semidirect |
| 25. #560 | C22 | semidirect |
| 26. #560 | C22 | semidirect |
| 27. #559 | C22 | semidirect |
| 28. #559 | C22 | semidirect |
| 29. #556 | C22 | semidirect |
| 30. #556 | C22 | semidirect |
| 31. #556 | C22 | semidirect |
| 32. #555 | C22 | semidirect |
| 33. #555 | C22 | semidirect |
| 34. #551 | C22 | semidirect |
| 35. #551 | C22 | semidirect |
| 36. #551 | C22 | semidirect |
| 37. #551 | C22 | semidirect |
| 38. #551 | C22 | semidirect |
| 39. #551 | C22 | semidirect |
| 40. #551 | C22 | semidirect |
| 41. #551 | C22 | semidirect |
| 42. D4×Q8 | C22 | semidirect |
| 43. #548 | C22 | semidirect |
| 44. #548 | C22 | semidirect |
| 45. #548 | C22 | semidirect |
| 46. #546 | C22 | semidirect |
| 47. #546 | C22 | semidirect |
| 48. #546 | C22 | semidirect |
| 49. #546 | C22 | semidirect |
| 50. #546 | C22 | semidirect |
| 51. #546 | C22 | semidirect |
| 52. #546 | C22 | semidirect |
| 53. D42 | C22 | semidirect |
| 54. D42 | C22 | semidirect |
| 55. #543 | C22 | semidirect |
| 56. #542 | C22 | semidirect |
| 57. #542 | C22 | semidirect |
| 58. #542 | C22 | semidirect |
| 59. #542 | C22 | semidirect |
| 60. #542 | C22 | semidirect |
| 61. #542 | C22 | semidirect |
| 62. #542 | C22 | semidirect |
| 63. #538 | C22 | semidirect |
| 64. #538 | C22 | semidirect |
| 65. #538 | C22 | semidirect |
| 66. #538 | C22 | semidirect |
| 67. #538 | C22 | semidirect |
| 68. #538 | C22 | semidirect |
| 69. #538 | C22 | semidirect |
| 70. #536 | C22 | semidirect |
| 71. #536 | C22 | semidirect |
| 73. #535 | C22 | semidirect |
| 74. #534 | C22 | semidirect |
| 75. #534 | C22 | semidirect |
| 76. #533 | C22 | semidirect |
| 77. #532 | C22 | semidirect |
| 78. #529 | C22 | semidirect |
| 79. #529 | C22 | semidirect |
| 80. #529 | C22 | semidirect |
| 81. #529 | C22 | semidirect |
| 82. #529 | C22 | semidirect |
| 83. C2×#124 | C22 | semidirect |
| 84. C2×#124 | C22 | semidirect |
| 85. C2×#124 | C22 | semidirect |
| 86. #525 | C22 | semidirect |
| 87. #525 | C22 | semidirect |
| 88. C2×#120 | C22 | semidirect |
| 89. C2×#120 | C22 | semidirect |
| 90. #519 | C22 | semidirect |
| 91. #518 | C22 | semidirect |
| 92. #518 | C22 | semidirect |
| 93. C4×#41 | C22 | semidirect |
| 94. C22×C22⋊C4 | C22 | semidirect |
| 99. C2×#41 | C23 | semidirect |
| 102. C22×Q8 | C23 | semidirect |
| 107. C22×D4 | C23 | semidirect |
| 108. C23×C4 | C23 | semidirect |
| 109. C4⋊Q8 | C23 | semidirect |
| 110. C4⋊Q8 | C23 | semidirect |
| 111. C4⋊Q8 | C23 | semidirect |
| 112. C4⋊Q8 | C23 | semidirect |
| 115. #126 | C23 | semidirect |
| 116. #126 | C23 | semidirect |
| 117. #126 | C23 | semidirect |
| 118. #126 | C23 | semidirect |
| 119. #126 | C23 | semidirect |
| 120. #126 | C23 | semidirect |
| 121. #126 | C23 | semidirect |
| 122. #126 | C23 | semidirect |
| 123. #126 | C23 | semidirect |
| 124. #125 | C23 | semidirect |
| 125. #125 | C23 | semidirect |
| 140. #124 | C23 | semidirect |
| 141. #124 | C23 | semidirect |
| 142. #124 | C23 | semidirect |
| 143. #124 | C23 | semidirect |
| 144. #124 | C23 | semidirect |
| 145. #124 | C23 | semidirect |
| 146. #124 | C23 | semidirect |
| 147. #124 | C23 | semidirect |
| 148. #124 | C23 | semidirect |
| 155. #123 | C23 | semidirect |
| 156. #123 | C23 | semidirect |
| 157. #123 | C23 | semidirect |
| 158. #123 | C23 | semidirect |
| 159. #123 | C23 | semidirect |
| 160. #123 | C23 | semidirect |
| 161. #123 | C23 | semidirect |
| 162. #123 | C23 | semidirect |
| 163. #123 | C23 | semidirect |
| 166. C22⋊Q8 | C23 | semidirect |
| 167. C22⋊Q8 | C23 | semidirect |
| 168. C22⋊Q8 | C23 | semidirect |
| 169. C22⋊Q8 | C23 | semidirect |
| 170. C22⋊Q8 | C23 | semidirect |
| 171. C22⋊Q8 | C23 | semidirect |
| 172. C22⋊Q8 | C23 | semidirect |
| 173. C22⋊Q8 | C23 | semidirect |
| 174. C22⋊Q8 | C23 | semidirect |
| 175. C22⋊Q8 | C23 | semidirect |
| 176. C22⋊Q8 | C23 | semidirect |
| 177. C22⋊Q8 | C23 | semidirect |
| 178. C22⋊Q8 | C23 | semidirect |
| 191. #121 | C23 | semidirect |
| 192. #121 | C23 | semidirect |
| 193. #121 | C23 | semidirect |
| 194. #121 | C23 | semidirect |
| 206. #120 | C23 | semidirect |
| 207. #120 | C23 | semidirect |
| 208. #120 | C23 | semidirect |
| 209. C4×Q8 | C23 | semidirect |
| 210. C4×Q8 | C23 | semidirect |
| 211. C4×Q8 | C23 | semidirect |
| 212. C4×Q8 | C23 | semidirect |
| 213. C4×Q8 | C23 | semidirect |
| 214. C4×Q8 | C23 | semidirect |
| 215. C4×Q8 | C23 | semidirect |
| 225. C4×D4 | C23 | semidirect |
| 226. C4×D4 | C23 | semidirect |
| 227. C4×D4 | C23 | semidirect |
| 228. C4×D4 | C23 | semidirect |
| 229. C4×D4 | C23 | semidirect |
| 230. C4×D4 | C23 | semidirect |
| 231. C4×D4 | C23 | semidirect |
| 232. C4×D4 | C23 | semidirect |
| 236. #117 | C23 | semidirect |
| 237. #117 | C23 | semidirect |
| 238. #117 | C23 | semidirect |
| 239. #117 | C23 | semidirect |
| 240. #117 | C23 | semidirect |
| 241. #117 | C23 | semidirect |
| 247. C2×C22⋊C4 | C23 | semidirect |
| 248. C2×C22⋊C4 | C23 | semidirect |
| 249. C2×C22⋊C4 | C23 | semidirect |
| 250. C2×C22⋊C4 | C23 | semidirect |
| 252. C2×C42 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3568 | 2 |
| #3561 | 2 |
| #3558 | 2 |
| #3545 | 2 |
| #3540 | 2 |
| #3532 | 2 |
| #3529 | 2 |
| C2×H(3,𝔽4) | 2 |
| #3514 | 2 |
| #3510 | 2 |
| #3490 | 2 |