Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1693 conjugacy classes of subgroups; only the 563 normal subgroups are shown.
This group has 1693 conjugacy classes of subgroups — too many to draw. Only its 563 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#576 | C2 | semidirect |
| 2. #3498 | C2 | semidirect |
| 3. #3155 | C2 | semidirect |
| 4. #3151 | C2 | semidirect |
| 5. #3147 | C2 | semidirect |
| 6. C2×#493 | C2 | semidirect |
| 7. C4×#131 | C2 | semidirect |
| 10. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 11. C2×#135 | C22 | semidirect |
| 12. C22×D8 | C22 | semidirect |
| 16. #535 | C22 | semidirect |
| 17. #532 | C22 | semidirect |
| 18. C2×#127 | C22 | semidirect |
| 20. #496 | C22 | semidirect |
| 21. #495 | C22 | semidirect |
| 22. #494 | C22 | semidirect |
| 23. #493 | C22 | semidirect |
| 24. #493 | C22 | semidirect |
| 25. #493 | C22 | semidirect |
| 26. #492 | C22 | semidirect |
| 27. #405 | C22 | semidirect |
| 28. C4×M16 | C22 | semidirect |
| 29. C2×C4×C8 | C22 | semidirect |
| 44. #131 | D4 | semidirect |
| 86. M16 | C2×D4 | semidirect |
| 87. C2×C8 | C2×D4 | semidirect |
| 128. C4 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#576 | 2 |
| #3498 | 2 |
| #3155 | 2 |
| #3151 | 2 |
| #3147 | 2 |
| C2×#493 | 2 |
| C4×#131 | 2 |