Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
573 conjugacy classes of subgroups; only the 323 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 323 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1770 | C2 | semidirect |
| 4. #1767 | C2 | semidirect |
| 5. #1752 | C2 | semidirect |
| 6. #1749 | C2 | semidirect |
| 7. #1744 | C2 | semidirect |
| 8. C2×#336 | C2 | semidirect |
| 13. C8×D4 | C4 | semidirect |
| 34. C22×C8 | C2×C4 | semidirect |
| 52. C4⋊C8 | C2×C4 | semidirect |
| 53. C22⋊C8 | C2×C4 | semidirect |
| 56. C4×C8 | C2×C4 | semidirect |
| 104. D4 | #97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D4 | 2 |
| C42×D4 | 2 |
| #1770 | 2 |
| #1767 | 2 |
| #1752 | 2 |
| #1749 | 2 |
| #1744 | 2 |
| C2×#336 | 2 |