Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3751 conjugacy classes of subgroups; only the 751 normal subgroups are shown.
This group has 3751 conjugacy classes of subgroups — too many to draw. Only its 751 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 | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ 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. C22×C22⋊Q8 | C2 | semidirect |
| 3. #2848 | C2 | direct (N × H) |
| 4. C2×#385 | C2 | semidirect |
| 5. C2×#380 | C2 | semidirect |
| 8. C23×Q8 | C22 | semidirect |
| 14. C2×C22⋊Q8 | C22 | semidirect |
| 15. C22×C4⋊C4 | C22 | semidirect |
| 16. C22×C4⋊C4 | C22 | semidirect |
| 19. #385 | C22 | semidirect |
| 20. #380 | C22 | semidirect |
| 22. C2×#95 | C22 | semidirect |
| 25. C25 | Q8 | semidirect |
| 29. C22×Q8 | C23 | semidirect |
| 45. C2×C4⋊C4 | C23 | semidirect |
| 46. C2×C4⋊C4 | C23 | semidirect |
| 51. #95 | C23 | semidirect |
| 61. C24 | C2×Q8 | semidirect |
| 98. C23 | C22⋊Q8 | semidirect |
| 127. C22 | C2×C22⋊Q8 | semidirect |
| 137. C2 | #2848 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C4 | 2 |
| C22×C22⋊Q8 | 2 |
| #2848 | 2 |
| C2×#385 | 2 |
| C2×#380 | 2 |
| C2×#379 | 2 |