Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
347 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 109 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 | #3205 | 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. #3112 | C2 | semidirect |
| 3. C2×#460 | C2 | semidirect |
| 4. #2984 | C2 | semidirect |
| 5. #1698 | C2 | semidirect |
| 6. #1688 | C2 | semidirect |
| 7. #1678 | C2 | semidirect |
| 8. #1668 | C2 | semidirect |
| 9. #1573 | C2 | semidirect |
| 10. #1555 | C2 | semidirect |
| 11. #1551 | C2 | semidirect |
| 19. #477 | C22 | semidirect |
| 20. #477 | C22 | semidirect |
| 25. #460 | C22 | semidirect |
| 26. #460 | C22 | semidirect |
| 32. #446 | C22 | semidirect |
| 33. C8×Q8 | C22 | semidirect |
| 43. #335 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #333 | C22 | semidirect |
| 48. #331 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3205 | 2 |
| #3112 | 2 |
| C2×#460 | 2 |
| #2984 | 2 |
| #1698 | 2 |
| #1688 | 2 |
| #1678 | 2 |
| #1668 | 2 |
| #1573 | 2 |
| #1555 | 2 |
| #1551 | 2 |