Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
597 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1854 | C2 | semidirect |
| 5. #1853 | C2 | semidirect |
| 6. #1852 | C2 | semidirect |
| 7. #1808 | C2 | semidirect |
| 8. #1807 | C2 | semidirect |
| 14. C2×#107 | C22 | semidirect |
| 15. C2×#106 | C22 | semidirect |
| 16. C2×C22⋊C8 | C4 | semidirect |
| 32. #340 | C22 | semidirect |
| 33. #340 | C22 | semidirect |
| 46. C22×C8 | C2×C4 | semidirect |
| 67. C22⋊C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊C8 | 2 |
| #2280 | 2 |
| C2×#380 | 2 |
| #1854 | 2 |
| #1853 | 2 |
| #1852 | 2 |
| #1808 | 2 |
| #1807 | 2 |