Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
690 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 135 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×#550 | C2 |
| 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 |
|---|---|---|
| 2. C2×#460 | C2 | semidirect |
| 3. C2×#440 | C2 | semidirect |
| 4. #2581 | C2 | semidirect |
| 5. #2041 | C2 | semidirect |
| 6. #2014 | C2 | semidirect |
| 7. #1985 | C2 | semidirect |
| 8. #1961 | C2 | semidirect |
| 9. #1877 | C2 | semidirect |
| 10. #1862 | C2 | semidirect |
| 11. #1804 | C2 | semidirect |
| 116. C2×Q8 | #41 | semidirect |
| 163. C2×C8 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#550 | 2 |
| C2×#460 | 2 |
| C2×#440 | 2 |
| #2581 | 2 |
| #2041 | 2 |
| #2014 | 2 |
| #1985 | 2 |
| #1961 | 2 |
| #1877 | 2 |
| #1862 | 2 |
| #1804 | 2 |