Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
782 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 139 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 | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D8 | C2 | semidirect |
| 2. C2×#472 | C2 | semidirect |
| 3. #3003 | C2 | semidirect |
| 4. #2929 | C2 | semidirect |
| 5. #2045 | C2 | semidirect |
| 6. #2018 | C2 | semidirect |
| 7. #1978 | C2 | semidirect |
| 8. #1810 | C2 | semidirect |
| 9. #1783 | C2 | semidirect |
| 11. C2×AGL(1,ℤ/8) | C22 | semidirect |
| 17. C2×#131 | C22 | semidirect |
| 25. C2×#108 | C22 | semidirect |
| 26. #424 | C22 | semidirect |
| 27. #418 | C22 | semidirect |
| 28. C2×#100 | C22 | semidirect |
| 29. C2×#100 | C22 | semidirect |
| 30. #408 | C22 | semidirect |
| 33. #405 | C22 | semidirect |
| 34. #341 | C22 | semidirect |
| 55. C2×D8 | D4 | semidirect |
| 191. C22 | #472 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C2×#472 | 2 |
| #3003 | 2 |
| #2929 | 2 |
| #2045 | 2 |
| #2018 | 2 |
| #1978 | 2 |
| #1810 | 2 |
| #1783 | 2 |