Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1342 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3585 | C2 | semidirect |
| 2. #3292 | C2 | semidirect |
| 3. #3291 | C2 | semidirect |
| 4. #3288 | C2 | semidirect |
| 5. #3151 | C2 | semidirect |
| 6. #3068 | C2 | semidirect |
| 7. #3018 | C2 | semidirect |
| 8. C2×#453 | C2 | semidirect |
| 9. #2956 | C2 | semidirect |
| 10. #585 | C22 | semidirect |
| 11. C2×H(5,𝔽2) | C22 | semidirect |
| 13. #577 | C22 | semidirect |
| 15. #576 | C22 | semidirect |
| 18. #575 | C22 | semidirect |
| 20. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 21. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 22. #568 | C22 | semidirect |
| 23. #532 | C22 | semidirect |
| 24. C2×#127 | C22 | semidirect |
| 25. #496 | C22 | semidirect |
| 26. #493 | C22 | semidirect |
| 27. #492 | C22 | semidirect |
| 28. #472 | C22 | semidirect |
| 29. #471 | C22 | semidirect |
| 30. #456 | C22 | semidirect |
| 31. #455 | C22 | semidirect |
| 32. #454 | C22 | semidirect |
| 33. #453 | C22 | semidirect |
| 34. #453 | C22 | semidirect |
| 35. #453 | C22 | semidirect |
| 36. #444 | C22 | semidirect |
| 37. #443 | C22 | semidirect |
| 38. #430 | C22 | semidirect |
| 39. C2×#104 | C22 | semidirect |
| 40. #413 | C22 | semidirect |
| 41. C2×#100 | C22 | semidirect |
| 42. C4×M16 | C22 | semidirect |
| 50. H(5,𝔽2) | C23 | semidirect |
| 62. #137 | D4 | semidirect |
| 63. #137 | C23 | semidirect |
| 71. AGL(1,ℤ/8) | D4 | semidirect |
| 73. AGL(1,ℤ/8) | C23 | semidirect |
| 80. C2×SD16 | C23 | semidirect |
| 83. C2×D8 | C23 | semidirect |
| 85. #131 | C23 | semidirect |
| 89. C4⋊Q8 | C23 | semidirect |
| 93. #127 | C23 | semidirect |
| 94. #124 | C23 | semidirect |
| 98. #108 | C23 | semidirect |
| 99. #104 | C23 | semidirect |
| 100. #104 | C23 | semidirect |
| 101. #101 | C23 | semidirect |
| 102. #100 | C23 | semidirect |
| 103. #100 | C23 | semidirect |
| 104. #97 | C23 | semidirect |
| 105. C4×C8 | C23 | semidirect |
| 154. Q16 | C2×D4 | semidirect |
| 159. SD16 | C2×D4 | semidirect |
| 163. D8 | C2×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3585 | 2 |
| #3292 | 2 |
| #3291 | 2 |
| #3288 | 2 |
| #3151 | 2 |
| #3068 | 2 |
| #3018 | 2 |
| C2×#453 | 2 |
| #2956 | 2 |