Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
689 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2948 | C2 | semidirect |
| 3. #2883 | C2 | semidirect |
| 4. #1976 | C2 | semidirect |
| 5. #1904 | C2 | semidirect |
| 6. #1812 | C2 | semidirect |
| 7. #1795 | C2 | semidirect |
| 8. #1757 | C2 | semidirect |
| 12. C2×AGL(1,ℤ/8) | C4 | semidirect |
| 14. C2×#135 | C4 | semidirect |
| 16. C22×D8 | C4 | semidirect |
| 19. #518 | C22 | semidirect |
| 22. #427 | C22 | semidirect |
| 23. #421 | C22 | semidirect |
| 24. #418 | C22 | semidirect |
| 28. #410 | C22 | semidirect |
| 30. #405 | C22 | semidirect |
| 31. #344 | C22 | semidirect |
| 32. H(3,ℤ/4) | C22 | semidirect |
| 36. C2×#41 | C2×C4 | semidirect |
| 41. C22×D4 | C2×C4 | semidirect |
| 49. C2×Q16 | C2×C4 | semidirect |
| 50. C2×SD16 | C2×C4 | semidirect |
| 51. C2×SD16 | C2×C4 | semidirect |
| 56. C2×D8 | C2×C4 | semidirect |
| 57. C2×D8 | C2×C4 | semidirect |
| 62. C2×M16 | C2×C4 | semidirect |
| 65. C22×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#576 | 2 |
| #2948 | 2 |
| #2883 | 2 |
| #1976 | 2 |
| #1904 | 2 |
| #1812 | 2 |
| #1795 | 2 |
| #1757 | 2 |