Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
361 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 361 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 | #3151 | 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. #3246 | C2 | semidirect |
| 2. #3208 | C2 | semidirect |
| 4. #2962 | C2 | semidirect |
| 5. #1688 | C2 | semidirect |
| 6. #1686 | C2 | semidirect |
| 7. #1668 | C2 | semidirect |
| 8. #1666 | C2 | semidirect |
| 9. #1565 | C2 | semidirect |
| 10. #1540 | C2 | semidirect |
| 11. C2×#331 | C2 | semidirect |
| 23. #475 | C22 | semidirect |
| 28. #460 | C22 | semidirect |
| 31. #446 | C22 | semidirect |
| 32. C8×Q8 | C22 | semidirect |
| 39. #335 | C22 | semidirect |
| 40. #334 | C22 | semidirect |
| 41. #332 | C22 | semidirect |
| 42. #332 | C22 | semidirect |
| 45. #331 | C22 | semidirect |
| 46. #331 | C22 | semidirect |
| 137. M16 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3246 | 2 |
| #3208 | 2 |
| #3151 | 2 |
| #2962 | 2 |
| #1688 | 2 |
| #1686 | 2 |
| #1668 | 2 |
| #1666 | 2 |
| #1565 | 2 |
| #1540 | 2 |
| C2×#331 | 2 |