Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
324 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 324 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 | #3217 | 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. #3238 | C2 | semidirect |
| 3. C2×#486 | C2 | semidirect |
| 4. #2966 | C2 | semidirect |
| 5. #1698 | C2 | semidirect |
| 6. #1697 | C2 | semidirect |
| 7. #1678 | C2 | semidirect |
| 8. #1677 | C2 | semidirect |
| 9. #1568 | C2 | semidirect |
| 10. #1558 | C2 | semidirect |
| 11. #1555 | C2 | semidirect |
| 25. #477 | C22 | semidirect |
| 26. #476 | C22 | semidirect |
| 31. #446 | C22 | semidirect |
| 32. C8×Q8 | C22 | semidirect |
| 33. #436 | C22 | semidirect |
| 36. C8×D4 | C22 | semidirect |
| 45. #335 | C22 | semidirect |
| 46. #334 | C22 | semidirect |
| 47. #333 | C22 | semidirect |
| 48. #333 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3238 | 2 |
| #3217 | 2 |
| C2×#486 | 2 |
| #2966 | 2 |
| #1698 | 2 |
| #1697 | 2 |
| #1678 | 2 |
| #1677 | 2 |
| #1568 | 2 |
| #1558 | 2 |
| #1555 | 2 |