Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
352 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 121 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 | #3038 | 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. #3208 | C2 | semidirect |
| 2. #3076 | C2 | semidirect |
| 4. #2988 | C2 | semidirect |
| 5. #1678 | C2 | semidirect |
| 6. #1677 | C2 | semidirect |
| 7. #1666 | C2 | semidirect |
| 8. #1665 | C2 | semidirect |
| 9. #1570 | C2 | semidirect |
| 10. #1550 | C2 | semidirect |
| 11. C2×#331 | C2 | semidirect |
| 20. #477 | C22 | semidirect |
| 21. #476 | C22 | semidirect |
| 22. #475 | C22 | semidirect |
| 23. #474 | C22 | semidirect |
| 31. C8×Q8 | C22 | semidirect |
| 33. C8×D4 | C22 | semidirect |
| 43. #334 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #333 | C22 | semidirect |
| 46. #332 | C22 | semidirect |
| 112. #41 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3208 | 2 |
| #3076 | 2 |
| #3038 | 2 |
| #2988 | 2 |
| #1678 | 2 |
| #1677 | 2 |
| #1666 | 2 |
| #1665 | 2 |
| #1570 | 2 |
| #1550 | 2 |
| C2×#331 | 2 |