Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2123 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 2123 conjugacy classes of subgroups — too many to draw. Only its 743 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 | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2375 | C2 | direct (N × H) |
| 3. C2×#390 | C2 | semidirect |
| 4. C2×#388 | C2 | semidirect |
| 5. C2×#386 | C2 | semidirect |
| 6. C2×#379 | C2 | semidirect |
| 9. C24×C4 | C22 | semidirect |
| 10. C2×#120 | C4 | semidirect |
| 12. C22×C4⋊C4 | C22 | semidirect |
| 16. C22×C22⋊C4 | C22 | semidirect |
| 17. C22×C42 | C22 | semidirect |
| 18. #390 | C22 | semidirect |
| 19. #388 | C22 | semidirect |
| 20. #386 | C22 | semidirect |
| 21. #379 | C22 | semidirect |
| 22. C2×#95 | C22 | semidirect |
| 25. C25 | C2×C4 | semidirect |
| 37. C23×C4 | C23 | semidirect |
| 38. #120 | C2×C4 | semidirect |
| 40. C2×C4⋊C4 | C23 | semidirect |
| 44. C2×C22⋊C4 | C2×C4 | semidirect |
| 48. C2×C22⋊C4 | C23 | semidirect |
| 50. C2×C42 | C23 | semidirect |
| 51. #95 | C23 | semidirect |
| 65. C24 | C22×C4 | semidirect |
| 89. C22⋊C4 | C22×C4 | semidirect |
| 149. C2 | #2375 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#120 | 2 |
| #2375 | 2 |
| C2×#390 | 2 |
| C2×#388 | 2 |
| C2×#386 | 2 |
| C2×#379 | 2 |