Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 372 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 | #3036 | 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 |
|---|---|---|
| 1. #3167 | C2 | semidirect |
| 2. #3077 | C2 | semidirect |
| 4. C2×C8×D4 | C2 | semidirect |
| 5. #1679 | C2 | semidirect |
| 6. #1677 | C2 | semidirect |
| 7. #1671 | C2 | semidirect |
| 8. #1665 | C2 | semidirect |
| 9. #1566 | C2 | semidirect |
| 10. #1550 | C2 | semidirect |
| 11. #1547 | C2 | semidirect |
| 17. #498 | C22 | semidirect |
| 18. #492 | C22 | semidirect |
| 19. #487 | C22 | semidirect |
| 20. #486 | C22 | semidirect |
| 43. #334 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #333 | C22 | semidirect |
| 46. #332 | C22 | semidirect |
| 47. #332 | C22 | semidirect |
| 48. #331 | C22 | semidirect |
| 123. C2×D4 | SD16 | semidirect |
| 218. C8 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3167 | 2 |
| #3077 | 2 |
| #3036 | 2 |
| C2×C8×D4 | 2 |
| #1679 | 2 |
| #1677 | 2 |
| #1671 | 2 |
| #1665 | 2 |
| #1566 | 2 |
| #1550 | 2 |
| #1547 | 2 |