Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3733 conjugacy classes of subgroups.
The subgroup lattice has 399 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. #3560 | C2 | semidirect |
| 2. #3558 | C2 | semidirect |
| 3. #3537 | C2 | semidirect |
| 4. #3534 | C2 | semidirect |
| 5. #3533 | C2 | semidirect |
| 6. #3532 | C2 | semidirect |
| 7. #3505 | C2 | semidirect |
| 8. #3496 | C2 | semidirect |
| 9. #3484 | C2 | semidirect |
| 10. C2×#538 | C2 | semidirect |
| 11. C2×#534 | C2 | semidirect |
| 12. C2×H(5,𝔽2) | C22 | semidirect |
| 13. C23×D4 | C22 | semidirect |
| 14. #563 | C22 | semidirect |
| 15. #562 | C22 | semidirect |
| 16. H(3,𝔽4) | C22 | semidirect |
| 17. #560 | C22 | semidirect |
| 18. #560 | C22 | semidirect |
| 19. #560 | C22 | semidirect |
| 20. #559 | C22 | semidirect |
| 21. #553 | C22 | semidirect |
| 22. #553 | C22 | semidirect |
| 23. #553 | C22 | semidirect |
| 24. #552 | C22 | semidirect |
| 25. #551 | C22 | semidirect |
| 26. #551 | C22 | semidirect |
| 27. #551 | C22 | semidirect |
| 28. #548 | C22 | semidirect |
| 29. #547 | C22 | semidirect |
| 30. #547 | C22 | semidirect |
| 31. #546 | C22 | semidirect |
| 32. #546 | C22 | semidirect |
| 33. #546 | C22 | semidirect |
| 34. #546 | C22 | semidirect |
| 35. #546 | C22 | semidirect |
| 36. #546 | C22 | semidirect |
| 37. D42 | C22 | semidirect |
| 38. D42 | C22 | semidirect |
| 39. D42 | C22 | semidirect |
| 40. D42 | C22 | semidirect |
| 41. D42 | C22 | semidirect |
| 42. #542 | C22 | semidirect |
| 43. #540 | C22 | semidirect |
| 44. #539 | C22 | semidirect |
| 45. #539 | C22 | semidirect |
| 46. #538 | C22 | semidirect |
| 47. #538 | C22 | semidirect |
| 48. #538 | C22 | semidirect |
| 49. #538 | C22 | semidirect |
| 50. #538 | C22 | semidirect |
| 51. #538 | C22 | semidirect |
| 52. #538 | C22 | semidirect |
| 53. #537 | C22 | semidirect |
| 54. #535 | C22 | semidirect |
| 55. #535 | C22 | semidirect |
| 56. #534 | C22 | semidirect |
| 57. #534 | C22 | semidirect |
| 58. #534 | C22 | semidirect |
| 59. #534 | C22 | semidirect |
| 60. #534 | C22 | semidirect |
| 61. #534 | C22 | semidirect |
| 62. #532 | C22 | semidirect |
| 63. #529 | C22 | semidirect |
| 64. C2×#126 | C22 | semidirect |
| 65. C2×#124 | C22 | semidirect |
| 66. #525 | C22 | semidirect |
| 67. #525 | C22 | semidirect |
| 68. C2×#123 | C22 | semidirect |
| 69. C2×#123 | C22 | semidirect |
| 70. C2×C22⋊Q8 | C22 | semidirect |
| 71. C2×#121 | C22 | semidirect |
| 72. C2×#121 | C22 | semidirect |
| 73. C2×#121 | C22 | semidirect |
| 74. C2×#121 | C22 | semidirect |
| 75. C2×#120 | C22 | semidirect |
| 76. C2×#120 | C22 | semidirect |
| 77. C2×#120 | C22 | semidirect |
| 78. #518 | C22 | semidirect |
| 79. C2×C4×D4 | C22 | semidirect |
| 80. C22×C22⋊C4 | C22 | semidirect |
| 85. C2×#41 | C23 | semidirect |
| 86. C2×#41 | C23 | semidirect |
| 87. C22×Q8 | C23 | semidirect |
| 97. C22×D4 | C23 | semidirect |
| 98. C22×D4 | C23 | semidirect |
| 99. C22×D4 | C23 | semidirect |
| 100. C22×D4 | C23 | semidirect |
| 103. C23×C4 | C23 | semidirect |
| 104. C4⋊Q8 | C23 | semidirect |
| 107. #127 | C23 | semidirect |
| 108. #126 | C23 | semidirect |
| 109. #126 | C23 | semidirect |
| 110. #126 | C23 | semidirect |
| 111. #126 | C23 | semidirect |
| 112. #126 | C23 | semidirect |
| 113. #125 | C23 | semidirect |
| 114. #125 | C23 | semidirect |
| 117. #124 | C23 | semidirect |
| 118. #124 | C23 | semidirect |
| 119. #124 | C23 | semidirect |
| 120. #124 | C23 | semidirect |
| 121. #124 | C23 | semidirect |
| 128. #123 | C23 | semidirect |
| 129. #123 | C23 | semidirect |
| 130. #123 | C23 | semidirect |
| 131. #123 | C23 | semidirect |
| 132. #123 | C23 | semidirect |
| 133. #123 | C23 | semidirect |
| 134. #123 | C23 | semidirect |
| 135. #123 | C23 | semidirect |
| 137. C22⋊Q8 | C23 | semidirect |
| 138. C22⋊Q8 | C23 | semidirect |
| 139. C22⋊Q8 | C23 | semidirect |
| 140. C22⋊Q8 | C23 | semidirect |
| 141. C22⋊Q8 | C23 | semidirect |
| 142. C22⋊Q8 | C23 | semidirect |
| 152. #121 | C23 | semidirect |
| 153. #121 | C23 | semidirect |
| 154. #121 | C23 | semidirect |
| 155. #121 | C23 | semidirect |
| 156. #121 | C23 | semidirect |
| 157. #121 | C23 | semidirect |
| 158. #121 | C23 | semidirect |
| 159. #121 | C23 | semidirect |
| 160. #121 | C23 | semidirect |
| 161. #121 | C23 | semidirect |
| 162. #121 | C23 | semidirect |
| 163. #121 | C23 | semidirect |
| 164. #121 | C23 | semidirect |
| 173. #120 | C23 | semidirect |
| 174. #120 | C23 | semidirect |
| 175. #120 | C23 | semidirect |
| 176. #120 | C23 | semidirect |
| 177. #120 | C23 | semidirect |
| 178. C4×Q8 | C23 | semidirect |
| 181. C4×D4 | C23 | semidirect |
| 182. C4×D4 | C23 | semidirect |
| 183. C4×D4 | C23 | semidirect |
| 184. C4×D4 | C23 | semidirect |
| 185. C4×D4 | C23 | semidirect |
| 186. C4×D4 | C23 | semidirect |
| 187. C4×D4 | C23 | semidirect |
| 188. C4×D4 | C23 | semidirect |
| 189. C4×D4 | C23 | semidirect |
| 190. C4×D4 | C23 | semidirect |
| 191. C4×D4 | C23 | semidirect |
| 193. #117 | C23 | semidirect |
| 194. #117 | C23 | semidirect |
| 196. C2×C4⋊C4 | C23 | semidirect |
| 197. C2×C4⋊C4 | C23 | semidirect |
| 198. C2×C4⋊C4 | C23 | semidirect |
| 199. C2×C4⋊C4 | C23 | semidirect |
| 200. C2×C4⋊C4 | C23 | semidirect |
| 209. C2×C22⋊C4 | C23 | semidirect |
| 210. C2×C22⋊C4 | C23 | semidirect |
| 211. C2×C22⋊C4 | C23 | semidirect |
| 212. C2×C22⋊C4 | C23 | semidirect |
| 213. C2×C42 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3560 | 2 |
| #3558 | 2 |
| #3537 | 2 |
| #3534 | 2 |
| #3533 | 2 |
| #3532 | 2 |
| #3505 | 2 |
| #3496 | 2 |
| #3484 | 2 |
| C2×#538 | 2 |
| C2×#534 | 2 |