Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
313 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C2×#127 | C22 |
| 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. #3255 | C2 | semidirect |
| 4. #2986 | C2 | semidirect |
| 5. #1702 | C2 | semidirect |
| 6. #1686 | C2 | semidirect |
| 7. #1682 | C2 | semidirect |
| 8. #1666 | C2 | semidirect |
| 9. #1574 | C2 | semidirect |
| 11. #1544 | C2 | semidirect |
| 20. #475 | C22 | semidirect |
| 23. #462 | C22 | semidirect |
| 31. #446 | C22 | semidirect |
| 33. C8×Q8 | C22 | semidirect |
| 39. #335 | C22 | semidirect |
| 40. #334 | C22 | semidirect |
| 41. #332 | C22 | semidirect |
| 42. #332 | C22 | semidirect |
| 43. #332 | C22 | semidirect |
| 44. #332 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3255 | 2 |
| #3208 | 2 |
| #2986 | 2 |
| #1702 | 2 |
| #1686 | 2 |
| #1682 | 2 |
| #1666 | 2 |
| #1574 | 2 |
| #1545 | 2 |
| #1544 | 2 |