Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
570 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 131 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, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#555 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#475 | C2 | semidirect |
| 3. C2×#440 | C2 | semidirect |
| 4. #2536 | C2 | semidirect |
| 5. #2034 | C2 | semidirect |
| 6. #2000 | C2 | semidirect |
| 7. #1993 | C2 | semidirect |
| 8. #1955 | C2 | semidirect |
| 9. #1873 | C2 | semidirect |
| 10. #1854 | C2 | semidirect |
| 11. #1804 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#555 | 2 |
| C2×#475 | 2 |
| C2×#440 | 2 |
| #2536 | 2 |
| #2034 | 2 |
| #2000 | 2 |
| #1993 | 2 |
| #1955 | 2 |
| #1873 | 2 |
| #1854 | 2 |
| #1804 | 2 |