Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
At least 6116 conjugacy classes of subgroups; only the 1060 normal subgroups are shown.
The subgroup lattice has 275 nodes (normal subgroups only) — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#2846 |
| commutator, Frattini, cosocle | C26 | C25 |
| LCS | G ▷ C26 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C27 ◁ G | |
| LECS | G ▷ C26 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72266460 | C2 | semidirect |
| 2. #72266458 | C2 | semidirect |
| 3. #72266457 | C2 | semidirect |
| 4. #72235057 | C2 | semidirect |
| 5. #72235054 | C2 | semidirect |
| 6. #72235053 | C2 | semidirect |
| 7. #72232440 | C2 | semidirect |
| 8. #72217730 | C2 | semidirect |
| 9. #72217546 | C2 | semidirect |
| 10. C2×#449475 | C2 | semidirect |
| 11. #72216432 | C2 | semidirect |
| 12. #9040958 | C22 | semidirect |
| 13. #8932885 | C22 | semidirect |
| 14. #8932883 | C22 | semidirect |
| 15. #8932872 | C22 | semidirect |
| 18. C22×#2846 | C22 | semidirect |
| 19. C22×#2846 | C22 | semidirect |
| 21. #6342371 | C22 | semidirect |
| 22. #449521 | C22 | semidirect |
| 23. #449475 | C22 | semidirect |
| 24. #449475 | C22 | semidirect |
| 25. #449475 | C22 | semidirect |
| 26. #449475 | C22 | semidirect |
| 27. #449475 | C22 | semidirect |
| 28. #417744 | C22 | semidirect |
| 29. #417493 | C22 | semidirect |
| 30. #417493 | C22 | semidirect |
| 31. #382422 | C22 | semidirect |
| 32. #379723 | C22 | semidirect |
| 33. #374022 | C22 | semidirect |
| 34. #359428 | C22 | semidirect |
| 35. C2×#10965 | C22 | semidirect |
| 36. C2×#10965 | C22 | semidirect |
| 37. #97555 | C22 | semidirect |
| 38. #97552 | C22 | semidirect |
| 39. #97551 | C22 | semidirect |
| 40. #97551 | C22 | semidirect |
| 41. #97550 | C22 | semidirect |
| 42. #97550 | C22 | semidirect |
| 43. #97550 | C22 | semidirect |
| 44. #97550 | C22 | semidirect |
| 45. #97550 | C22 | semidirect |
| 46. C28 | C23 | semidirect |
| 47. C23×H(5,𝔽2) | D4 | semidirect |
| 48. C2×#3484 | D4 | semidirect |
| 49. C2×#3484 | D4 | semidirect |
| 54. C2×#2846 | C23 | semidirect |
| 55. C2×#2846 | C23 | semidirect |
| 56. C2×#2846 | C23 | semidirect |
| 57. C2×#2846 | C23 | semidirect |
| 59. #35350 | D4 | semidirect |
| 61. #35348 | D4 | semidirect |
| 64. #35346 | D4 | semidirect |
| 65. #35346 | D4 | semidirect |
| 66. #35346 | D4 | semidirect |
| 67. #35346 | D4 | semidirect |
| 73. #34992 | D4 | semidirect |
| 74. #34981 | D4 | semidirect |
| 75. #34980 | D4 | semidirect |
| 76. #34198 | D4 | semidirect |
| 77. #34171 | D4 | semidirect |
| 78. C2×#2308 | D4 | semidirect |
| 79. C2×#2308 | D4 | semidirect |
| 80. C22×#379 | C23 | semidirect |
| 81. C22×#379 | C23 | semidirect |
| 82. #14679 | C23 | semidirect |
| 83. #14679 | C23 | semidirect |
| 84. #11196 | C23 | semidirect |
| 85. #11196 | C23 | semidirect |
| 88. #10965 | C23 | semidirect |
| 89. #10965 | C23 | semidirect |
| 90. #10965 | C23 | semidirect |
| 91. #10965 | C23 | semidirect |
| 92. #10965 | C23 | semidirect |
| 93. #10965 | C23 | semidirect |
| 94. #10965 | C23 | semidirect |
| 95. #10965 | C23 | semidirect |
| 98. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 99. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 100. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 101. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 102. C22×H(5,𝔽2) | C2×D4 | semidirect |
| 107. C2×#534 | C2×D4 | semidirect |
| 108. C2×#534 | C2×D4 | semidirect |
| 109. C2×#534 | C2×D4 | semidirect |
| 110. C2×#534 | C2×D4 | semidirect |
| 127. C2×#518 | C2×D4 | semidirect |
| 128. C2×#518 | C2×D4 | semidirect |
| 129. C2×#518 | C2×D4 | semidirect |
| 134. #2315 | C2×D4 | semidirect |
| 135. #2315 | C2×D4 | semidirect |
| 136. #2315 | C2×D4 | semidirect |
| 137. #2308 | C2×D4 | semidirect |
| 138. #2308 | C2×D4 | semidirect |
| 139. #2308 | C2×D4 | semidirect |
| 140. #2308 | C2×D4 | semidirect |
| 141. #2308 | C2×D4 | semidirect |
| 142. #2308 | C2×D4 | semidirect |
| 143. #2308 | C2×D4 | semidirect |
| 144. #2308 | C2×D4 | semidirect |
| 145. #2308 | C2×D4 | semidirect |
| 146. #2308 | C2×D4 | semidirect |
| 147. #2308 | C2×D4 | semidirect |
| 148. #2308 | C2×D4 | semidirect |
| 149. #2308 | C2×D4 | semidirect |
| 150. #2308 | C2×D4 | semidirect |
| 158. #2273 | C2×D4 | semidirect |
| 214. C25 | B(4,𝔽2) | semidirect |
| 231. C22×D4 | B(4,𝔽2) | semidirect |
| 251. C24 | C2×B(4,𝔽2) | semidirect |
| 257. C22×C4 | C2×B(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72266460 | 2 |
| #72266458 | 2 |
| #72266457 | 2 |
| #72235057 | 2 |
| #72235054 | 2 |
| #72235053 | 2 |
| #72232440 | 2 |
| #72217730 | 2 |
| #72217546 | 2 |
| C2×#449475 | 2 |
| #72216432 | 2 |