Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
512 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 119 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 | #3003 | 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. #3092 | C2 | semidirect |
| 3. #2929 | C2 | semidirect |
| 4. C22×#102 | C2 | semidirect |
| 5. #2063 | C2 | semidirect |
| 6. #2051 | C2 | semidirect |
| 7. #1931 | C2 | semidirect |
| 8. #1907 | C2 | semidirect |
| 9. #1821 | C2 | semidirect |
| 10. #1810 | C2 | semidirect |
| 11. #1783 | C2 | semidirect |
| 14. C2×#131 | C22 | semidirect |
| 30. C2×#108 | C22 | semidirect |
| 31. #428 | C22 | semidirect |
| 32. #424 | C22 | semidirect |
| 34. #417 | C22 | semidirect |
| 43. #408 | C22 | semidirect |
| 47. #405 | C22 | semidirect |
| 48. #343 | C22 | semidirect |
| 49. #343 | C22 | semidirect |
| 50. #341 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3092 | 2 |
| #3003 | 2 |
| #2929 | 2 |
| C22×#102 | 2 |
| #2063 | 2 |
| #2051 | 2 |
| #1931 | 2 |
| #1907 | 2 |
| #1821 | 2 |
| #1810 | 2 |
| #1783 | 2 |