Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
7295 conjugacy classes of subgroups; only the 5327 normal subgroups are shown.
This group has 7295 conjugacy classes of subgroups — too many to draw. Only its 5327 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 | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#518 | C2 | direct (N × H) |
| 3. C22×C4×D4 | C2 | semidirect |
| 4. C22×#117 | C2 | semidirect |
| 5. C23×C22⋊C4 | C2 | semidirect |
| 7. C23×D4 | C4 | semidirect |
| 11. C24×C4 | C22 | semidirect |
| 12. #518 | C22 | direct (N × H) |
| 13. C2×C4×D4 | C22 | semidirect |
| 14. C2×#117 | C22 | semidirect |
| 15. C22×C4⋊C4 | C22 | semidirect |
| 17. C22×C22⋊C4 | C22 | semidirect |
| 18. C22×C22⋊C4 | C22 | semidirect |
| 19. C22×C42 | C22 | semidirect |
| 21. C25 | C2×C4 | semidirect |
| 25. C22×D4 | C2×C4 | semidirect |
| 26. C22×D4 | C2×C4 | semidirect |
| 30. C23×C4 | C2×C4 | semidirect |
| 34. C23×C4 | C23 | semidirect |
| 35. C4×D4 | C23 | semidirect |
| 36. #117 | C23 | semidirect |
| 37. C2×C4⋊C4 | C23 | semidirect |
| 39. C2×C22⋊C4 | C23 | semidirect |
| 40. C2×C22⋊C4 | C23 | semidirect |
| 41. C2×C42 | C23 | semidirect |
| 48. C24 | C22×C4 | semidirect |
| 53. C2×D4 | C22×C4 | semidirect |
| 54. C2×D4 | C22×C4 | semidirect |
| 59. C22×C4 | C22×C4 | semidirect |
| 63. C22×C4 | C24 | semidirect |
| 64. C4⋊C4 | C24 | semidirect |
| 66. C22⋊C4 | C24 | semidirect |
| 67. C42 | C24 | semidirect |
| 77. C23 | C23×C4 | semidirect |
| 82. D4 | C23×C4 | semidirect |
| 86. C2×C4 | C23×C4 | semidirect |
| 94. C22 | #518 | direct (N × H) |
| 104. C2 | C2×#518 | 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×#518 | 2 |
| C22×C4×D4 | 2 |
| C22×#117 | 2 |
| C23×C22⋊C4 | 2 |