Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
4071 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 4071 conjugacy classes of subgroups — too many to draw. Only its 1039 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 | C24×D4 | 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. C23×C22⋊C4 | C2 | semidirect |
| 3. #2308 | C2 | direct (N × H) |
| 4. C2×#386 | C2 | semidirect |
| 5. C2×#381 | C2 | semidirect |
| 6. C2×#379 | C2 | semidirect |
| 10. C23×D4 | C4 | semidirect |
| 14. C24×C4 | C22 | semidirect |
| 17. C22×C22⋊C4 | C22 | semidirect |
| 18. C22×C22⋊C4 | C22 | semidirect |
| 19. C22×C22⋊C4 | C22 | semidirect |
| 20. #386 | C22 | semidirect |
| 21. #381 | C22 | semidirect |
| 22. #379 | C22 | semidirect |
| 23. C2×#95 | C22 | semidirect |
| 28. C25 | C2×C4 | semidirect |
| 39. C22×D4 | C2×C4 | semidirect |
| 40. C22×D4 | C2×C4 | semidirect |
| 45. C23×C4 | C2×C4 | semidirect |
| 49. C23×C4 | C23 | semidirect |
| 55. C2×C22⋊C4 | C23 | semidirect |
| 56. C2×C22⋊C4 | C23 | semidirect |
| 57. #95 | C23 | semidirect |
| 70. C24 | C22⋊C4 | semidirect |
| 75. C24 | C22×C4 | semidirect |
| 88. C2×D4 | C22×C4 | semidirect |
| 98. C22×C4 | C22×C4 | semidirect |
| 125. C23 | C2×C22⋊C4 | semidirect |
| 171. C2 | #2308 | 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 |
| C23×C22⋊C4 | 2 |
| #2308 | 2 |
| C2×#386 | 2 |
| C2×#381 | 2 |
| C2×#379 | 2 |