Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1341 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1341 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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. C2×#577 | C2 | semidirect |
| 2. #3291 | C2 | direct (N × H) |
| 3. C2×#492 | C2 | semidirect |
| 4. C2×#471 | C2 | semidirect |
| 5. C2×#456 | C2 | semidirect |
| 6. C2×#453 | C2 | semidirect |
| 7. C2×#443 | C2 | semidirect |
| 9. C2×H(5,𝔽2) | C22 | semidirect |
| 11. #577 | C22 | semidirect |
| 13. C2×#137 | C22 | semidirect |
| 19. C22×SD16 | C22 | semidirect |
| 20. C2×#131 | C22 | semidirect |
| 21. C2×C4⋊Q8 | C22 | semidirect |
| 23. #492 | C22 | semidirect |
| 24. #471 | C22 | semidirect |
| 25. #456 | C22 | semidirect |
| 26. #453 | C22 | semidirect |
| 27. #443 | C22 | semidirect |
| 28. C2×#108 | C22 | semidirect |
| 29. C2×#104 | C22 | semidirect |
| 30. C2×#101 | C22 | semidirect |
| 31. C2×#100 | C22 | semidirect |
| 32. C2×C4×C8 | C22 | semidirect |
| 38. H(5,𝔽2) | C23 | semidirect |
| 51. #137 | C23 | semidirect |
| 56. #135 | D4 | semidirect |
| 59. C2×Q16 | D4 | semidirect |
| 66. C2×SD16 | D4 | semidirect |
| 67. C2×SD16 | C23 | semidirect |
| 69. C2×D8 | D4 | semidirect |
| 71. #131 | C23 | semidirect |
| 77. C4⋊Q8 | C23 | semidirect |
| 82. #108 | C23 | semidirect |
| 83. #104 | C23 | semidirect |
| 84. #101 | C23 | semidirect |
| 85. #100 | C23 | semidirect |
| 86. C4×C8 | C23 | semidirect |
| 128. Q16 | C2×D4 | semidirect |
| 133. SD16 | C2×D4 | semidirect |
| 135. D8 | C2×D4 | semidirect |
| 207. C2 | #3291 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#577 | 2 |
| #3291 | 2 |
| C2×#492 | 2 |
| C2×#471 | 2 |
| C2×#456 | 2 |
| C2×#453 | 2 |
| C2×#443 | 2 |