Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
7615 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 7615 conjugacy classes of subgroups — too many to draw. Only its 3183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D4 | C2 | semidirect |
| 2. C2×#534 | C2 | direct (N × H) |
| 3. C22×#123 | C2 | semidirect |
| 4. C22×#121 | C2 | semidirect |
| 5. C22×#120 | C2 | semidirect |
| 6. C23×C22⋊C4 | C2 | semidirect |
| 8. C26 | C22 | semidirect |
| 11. C23×D4 | C22 | semidirect |
| 12. C23×D4 | C22 | semidirect |
| 13. C23×D4 | C22 | semidirect |
| 14. C24×C4 | C22 | semidirect |
| 15. C24×C4 | C22 | semidirect |
| 16. #534 | C22 | direct (N × H) |
| 17. C2×#123 | C22 | semidirect |
| 18. C2×#121 | C22 | semidirect |
| 19. C2×#120 | C22 | semidirect |
| 20. C22×C4⋊C4 | C22 | semidirect |
| 21. C22×C22⋊C4 | C22 | semidirect |
| 22. C22×C22⋊C4 | C22 | semidirect |
| 23. C22×C22⋊C4 | C22 | semidirect |
| 27. C25 | D4 | semidirect |
| 32. C25 | C23 | semidirect |
| 38. C22×D4 | C23 | semidirect |
| 39. C22×D4 | C23 | semidirect |
| 40. C22×D4 | C23 | semidirect |
| 45. C23×C4 | C23 | semidirect |
| 46. C23×C4 | C23 | semidirect |
| 47. #123 | C23 | semidirect |
| 48. #121 | C23 | semidirect |
| 49. #120 | C23 | semidirect |
| 50. C2×C4⋊C4 | C23 | semidirect |
| 51. C2×C22⋊C4 | C23 | semidirect |
| 52. C2×C22⋊C4 | C23 | semidirect |
| 53. C2×C22⋊C4 | C23 | semidirect |
| 65. C24 | C2×D4 | semidirect |
| 71. C24 | C24 | semidirect |
| 77. C2×D4 | C24 | semidirect |
| 78. C2×D4 | C24 | semidirect |
| 85. C22×C4 | C24 | semidirect |
| 86. C22×C4 | C24 | semidirect |
| 87. C4⋊C4 | C24 | semidirect |
| 88. C22⋊C4 | C24 | semidirect |
| 89. C22⋊C4 | C24 | semidirect |
| 105. C23 | C22×D4 | semidirect |
| 129. C22 | #534 | direct (N × H) |
| 140. C2 | C2×#534 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D4 | 2 |
| C2×#534 | 2 |
| C22×#123 | 2 |
| C22×#121 | 2 |
| C22×#120 | 2 |
| C23×C22⋊C4 | 2 |