Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2437 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2437 conjugacy classes of subgroups — too many to draw. Only its 557 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 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(7,𝔽2) | C2 | semidirect |
| 2. #3585 | C2 | semidirect |
| 3. #3498 | C2 | semidirect |
| 4. #3021 | C2 | semidirect |
| 5. #3019 | C2 | semidirect |
| 6. #3018 | C2 | semidirect |
| 7. C2×#453 | C2 | semidirect |
| 8. #2901 | C2 | semidirect |
| 9. #2888 | C2 | semidirect |
| 13. #585 | C22 | semidirect |
| 17. C2×H(5,𝔽2) | C22 | semidirect |
| 18. C2×H(5,𝔽2) | C22 | semidirect |
| 21. #575 | C22 | semidirect |
| 22. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 23. #568 | C22 | semidirect |
| 26. #535 | C22 | semidirect |
| 27. #535 | C22 | semidirect |
| 28. #532 | C22 | semidirect |
| 29. C2×#127 | C22 | semidirect |
| 30. C4×#41 | C22 | semidirect |
| 31. #456 | C22 | semidirect |
| 32. #455 | C22 | semidirect |
| 33. #454 | C22 | semidirect |
| 34. #453 | C22 | semidirect |
| 35. #453 | C22 | semidirect |
| 36. #421 | C22 | semidirect |
| 37. C2×#104 | C22 | semidirect |
| 38. #413 | C22 | semidirect |
| 39. C2×#100 | C22 | semidirect |
| 43. 2−1+4 | D4 | semidirect |
| 54. H(5,𝔽2) | D4 | semidirect |
| 67. C2×#41 | C23 | semidirect |
| 75. C22×D4 | C23 | semidirect |
| 77. #137 | C23 | semidirect |
| 80. AGL(1,ℤ/8) | C23 | semidirect |
| 85. C2×M16 | C23 | semidirect |
| 86. C4⋊Q8 | C23 | semidirect |
| 88. #127 | C23 | semidirect |
| 89. #127 | C23 | semidirect |
| 90. #124 | C23 | semidirect |
| 91. #124 | C23 | semidirect |
| 100. #117 | C23 | semidirect |
| 101. C2×C42 | C23 | semidirect |
| 102. #104 | C23 | semidirect |
| 103. #104 | C23 | semidirect |
| 104. #101 | C23 | semidirect |
| 105. #100 | C23 | semidirect |
| 125. #41 | C2×D4 | semidirect |
| 131. C2×Q8 | C2×D4 | semidirect |
| 155. C2×D4 | C2×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(7,𝔽2) | 2 |
| #3585 | 2 |
| #3498 | 2 |
| #3021 | 2 |
| #3019 | 2 |
| #3018 | 2 |
| C2×#453 | 2 |
| #2901 | 2 |
| #2888 | 2 |