Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3703 conjugacy classes of subgroups; only the 3145 normal subgroups are shown.
This group has 3703 conjugacy classes of subgroups — too many to draw. Only its 3145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3503 | C2 | semidirect |
| 2. #3501 | C2 | semidirect |
| 3. Q8×#41 | C2 | semidirect |
| 4. C2×D4×Q8 | C2 | semidirect |
| 7. C23×Q8 | C22 | semidirect |
| 8. Q82 | C22 | semidirect |
| 9. Q82 | C22 | semidirect |
| 10. #557 | C22 | semidirect |
| 11. #554 | C22 | semidirect |
| 12. #554 | C22 | semidirect |
| 13. D4×Q8 | C22 | semidirect |
| 14. D4×Q8 | C22 | semidirect |
| 15. #544 | C22 | semidirect |
| 16. #543 | C22 | semidirect |
| 17. #533 | C22 | semidirect |
| 18. C2×C4⋊Q8 | C22 | semidirect |
| 19. C2×C22⋊Q8 | C22 | semidirect |
| 23. C2×C4×Q8 | C22 | semidirect |
| 25. H(5,𝔽2) | Q8 | direct (N × H) |
| 54. #41 | C2×Q8 | semidirect |
| 62. C2×D4 | C2×Q8 | semidirect |
| 79. C23 | C22×Q8 | semidirect |
| 84. Q8 | C22×Q8 | semidirect |
| 85. Q8 | H(5,𝔽2) | direct (N × H) |
| 86. D4 | C22×Q8 | semidirect |
| 90. C2×C4 | C22×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3503 | 2 |
| #3501 | 2 |
| Q8×#41 | 2 |
| C2×D4×Q8 | 2 |
| C4×H(5,𝔽2) | 2 |