Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
383 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3103 | C2 | semidirect |
| 3. C2×#460 | C2 | semidirect |
| 4. Q8×M16 | C2 | semidirect |
| 5. #1688 | C2 | semidirect |
| 6. #1686 | C2 | semidirect |
| 7. #1668 | C2 | semidirect |
| 8. #1666 | C2 | semidirect |
| 9. #1565 | C2 | semidirect |
| 10. #1547 | C2 | semidirect |
| 11. #1545 | C2 | semidirect |
| 22. #475 | C22 | semidirect |
| 23. #475 | C22 | semidirect |
| 28. #460 | C22 | semidirect |
| 29. #460 | C22 | semidirect |
| 32. #446 | C22 | semidirect |
| 33. C8×Q8 | C22 | semidirect |
| 43. #335 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #332 | C22 | semidirect |
| 48. #331 | C22 | semidirect |
| 152. M16 | SD16 | semidirect |
| 191. Q8 | AGL(1,ℤ/8) | semidirect |
| 230. C8 | C2×SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3151 | 2 |
| #3103 | 2 |
| C2×#460 | 2 |
| Q8×M16 | 2 |
| #1688 | 2 |
| #1686 | 2 |
| #1668 | 2 |
| #1666 | 2 |
| #1565 | 2 |
| #1547 | 2 |
| #1545 | 2 |