Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
356 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 111 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. #3239 | C2 | semidirect |
| 3. C2×#492 | C2 | semidirect |
| 4. #2966 | C2 | semidirect |
| 5. #1686 | C2 | semidirect |
| 6. #1685 | C2 | semidirect |
| 7. #1666 | C2 | semidirect |
| 8. #1665 | C2 | semidirect |
| 9. #1568 | C2 | semidirect |
| 10. #1556 | C2 | semidirect |
| 11. #1555 | C2 | semidirect |
| 27. #475 | C22 | semidirect |
| 28. #474 | C22 | semidirect |
| 33. #446 | C22 | semidirect |
| 34. C8×Q8 | C22 | semidirect |
| 35. #436 | C22 | semidirect |
| 38. C8×D4 | C22 | semidirect |
| 45. #335 | C22 | semidirect |
| 46. #334 | C22 | semidirect |
| 47. #332 | C22 | semidirect |
| 48. #332 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3239 | 2 |
| #3217 | 2 |
| C2×#492 | 2 |
| #2966 | 2 |
| #1686 | 2 |
| #1685 | 2 |
| #1666 | 2 |
| #1665 | 2 |
| #1568 | 2 |
| #1556 | 2 |
| #1555 | 2 |