Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
566 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 566 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×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#441 | C2 | semidirect |
| 4. #2581 | C2 | semidirect |
| 5. #2042 | C2 | semidirect |
| 6. #2015 | C2 | semidirect |
| 7. #1985 | C2 | semidirect |
| 9. #1877 | C2 | semidirect |
| 10. #1864 | C2 | semidirect |
| 11. #1805 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#559 | 2 |
| C2×#460 | 2 |
| C2×#441 | 2 |
| #2581 | 2 |
| #2042 | 2 |
| #2015 | 2 |
| #1985 | 2 |
| #1961 | 2 |
| #1877 | 2 |
| #1864 | 2 |
| #1805 | 2 |