Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
337 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C22×D4 | D4 |
| 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. #3165 | C2 | semidirect |
| 5. #1695 | C2 | semidirect |
| 6. #1691 | C2 | semidirect |
| 7. #1675 | C2 | semidirect |
| 8. #1671 | C2 | semidirect |
| 9. #1565 | C2 | semidirect |
| 10. #1550 | C2 | semidirect |
| 11. #1548 | C2 | semidirect |
| 17. #500 | C22 | semidirect |
| 18. #498 | C22 | semidirect |
| 43. #335 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #333 | C22 | semidirect |
| 46. #333 | C22 | semidirect |
| 47. #333 | C22 | semidirect |
| 48. #332 | 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 |
|---|---|
| #3165 | 2 |
| #3101 | 2 |
| C2×#476 | 2 |
| D4×M16 | 2 |
| #1695 | 2 |
| #1691 | 2 |
| #1675 | 2 |
| #1671 | 2 |
| #1565 | 2 |
| #1550 | 2 |
| #1548 | 2 |