Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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 | C42 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3160 | C2 | semidirect |
| 4. #1676 | C2 | semidirect |
| 5. #1675 | C2 | semidirect |
| 6. #1674 | C2 | semidirect |
| 7. #1673 | C2 | semidirect |
| 8. #1672 | C2 | semidirect |
| 9. #1671 | C2 | semidirect |
| 10. #1568 | C2 | semidirect |
| 11. #1554 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3160 | 2 |
| #3116 | 2 |
| C8×#41 | 2 |
| #1676 | 2 |
| #1675 | 2 |
| #1674 | 2 |
| #1673 | 2 |
| #1672 | 2 |
| #1671 | 2 |
| #1568 | 2 |
| #1554 | 2 |