Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
725 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 725 conjugacy classes of subgroups — too many to draw. Only its 137 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 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| 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 |
|---|---|---|
| 2. C2×#471 | C2 | semidirect |
| 3. #3007 | C2 | semidirect |
| 4. C2×#450 | C2 | semidirect |
| 5. #2880 | C2 | semidirect |
| 6. #2051 | C2 | semidirect |
| 7. #2019 | C2 | semidirect |
| 8. #1907 | C2 | semidirect |
| 9. #1901 | C2 | semidirect |
| 10. #1855 | C2 | semidirect |
| 11. #1783 | C2 | semidirect |
| 14. C2×#137 | C22 | semidirect |
| 15. C2×#137 | C22 | semidirect |
| 22. C22×SD16 | C22 | semidirect |
| 24. C2×#131 | C22 | semidirect |
| 35. C2×#108 | C22 | semidirect |
| 36. #417 | C22 | semidirect |
| 37. #417 | C22 | semidirect |
| 38. C2×#102 | C22 | semidirect |
| 39. C2×#101 | C22 | semidirect |
| 40. C2×#100 | C22 | semidirect |
| 41. C2×#100 | C22 | semidirect |
| 43. #408 | C22 | semidirect |
| 49. #343 | C22 | semidirect |
| 50. #341 | C22 | semidirect |
| 334. C22 | #471 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AGL(1,ℤ/8) | 2 |
| C2×#471 | 2 |
| #3007 | 2 |
| C2×#450 | 2 |
| #2880 | 2 |
| #2051 | 2 |
| #2019 | 2 |
| #1907 | 2 |
| #1901 | 2 |
| #1855 | 2 |
| #1783 | 2 |