Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
854 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C8 | C2 | semidirect |
| 2. C2×#435 | C2 | semidirect |
| 4. #1869 | C2 | semidirect |
| 5. #1850 | C2 | semidirect |
| 6. #1779 | C2 | semidirect |
| 7. #1752 | C2 | semidirect |
| 9. C22×M16 | C22 | semidirect |
| 16. C2×C4⋊C8 | C22 | semidirect |
| 19. C2×C22⋊C8 | C22 | semidirect |
| 20. C2×#97 | C22 | semidirect |
| 25. #336 | C22 | semidirect |
| 38. C22×C8 | D4 | semidirect |
| 60. C24 | M16 | semidirect |
| 129. C8 | #120 | semidirect |
| 138. C22 | #435 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8 | 2 |
| C2×#435 | 2 |
| C4×#120 | 2 |
| #1869 | 2 |
| #1850 | 2 |
| #1779 | 2 |
| #1752 | 2 |