Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
501 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 117 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 | C2×#447 | 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 |
|---|---|---|
| 1. #3094 | C2 | semidirect |
| 3. #2924 | C2 | semidirect |
| 4. C2×#418 | C2 | semidirect |
| 5. #2063 | C2 | semidirect |
| 6. #2045 | C2 | semidirect |
| 7. #1978 | C2 | semidirect |
| 8. #1931 | C2 | semidirect |
| 9. #1900 | C2 | semidirect |
| 10. #1855 | C2 | semidirect |
| 11. #1783 | C2 | semidirect |
| 27. C2×#108 | C22 | semidirect |
| 28. #428 | C22 | semidirect |
| 29. #428 | C22 | semidirect |
| 30. #424 | C22 | semidirect |
| 35. #418 | C22 | semidirect |
| 36. #418 | C22 | semidirect |
| 42. C2×#100 | C22 | semidirect |
| 46. #405 | C22 | semidirect |
| 47. #343 | C22 | semidirect |
| 48. #341 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3094 | 2 |
| C2×#447 | 2 |
| #2924 | 2 |
| C2×#418 | 2 |
| #2063 | 2 |
| #2045 | 2 |
| #1978 | 2 |
| #1931 | 2 |
| #1900 | 2 |
| #1855 | 2 |
| #1783 | 2 |