Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
513 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 153 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 |
|---|---|---|
| 2. #1955 | C2 | semidirect |
| 3. #1924 | C2 | semidirect |
| 4. #1870 | C2 | semidirect |
| 5. #1779 | C2 | semidirect |
| 7. C22×M16 | C22 | semidirect |
| 10. C2×C4⋊C8 | C22 | semidirect |
| 13. C2×C22⋊C8 | C22 | semidirect |
| 14. C2×#97 | C22 | semidirect |
| 19. #336 | C22 | semidirect |
| 47. C24 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#120 | 2 |
| #1955 | 2 |
| #1924 | 2 |
| #1870 | 2 |
| #1779 | 2 |