Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
4803 conjugacy classes of subgroups; only the 2887 normal subgroups are shown.
This group has 4803 conjugacy classes of subgroups — too many to draw. Only its 2887 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. C2×#560 | C2 | direct (N × H) |
| 2. C22×#127 | C2 | semidirect |
| 3. C22×#126 | C2 | semidirect |
| 4. C22×#123 | C2 | semidirect |
| 5. C22×#121 | C2 | semidirect |
| 6. C22×#120 | C2 | semidirect |
| 10. C23×D4 | C22 | semidirect |
| 11. C24×C4 | C22 | semidirect |
| 12. #560 | C22 | direct (N × H) |
| 13. C2×#127 | C22 | semidirect |
| 14. C2×#126 | C22 | semidirect |
| 15. C2×#123 | C22 | semidirect |
| 16. C2×#121 | C22 | semidirect |
| 17. C2×#120 | C22 | semidirect |
| 18. C22×C4⋊C4 | C22 | semidirect |
| 21. C22×C22⋊C4 | C22 | semidirect |
| 22. C22×C42 | C22 | semidirect |
| 33. C22×D4 | C23 | semidirect |
| 37. C23×C4 | C23 | semidirect |
| 38. #127 | C23 | semidirect |
| 39. #126 | C23 | semidirect |
| 40. #123 | C23 | semidirect |
| 41. #121 | C23 | semidirect |
| 42. #120 | C23 | semidirect |
| 43. C2×C4⋊C4 | C23 | semidirect |
| 46. C2×C22⋊C4 | C23 | semidirect |
| 47. C2×C42 | C23 | semidirect |
| 64. C2×D4 | C24 | semidirect |
| 70. C22×C4 | C24 | semidirect |
| 71. C4⋊C4 | C24 | semidirect |
| 74. C22⋊C4 | C24 | semidirect |
| 75. C42 | C24 | semidirect |
| 106. C22 | #560 | direct (N × H) |
| 114. C2 | C2×#560 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#560 | 2 |
| C22×#127 | 2 |
| C22×#126 | 2 |
| C22×#123 | 2 |
| C22×#121 | 2 |
| C22×#120 | 2 |