Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
891 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 891 conjugacy classes of subgroups — too many to draw. Only its 223 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#432 | C2 | semidirect |
| 3. #2285 | C2 | semidirect |
| 4. #1824 | C2 | semidirect |
| 5. #1802 | C2 | semidirect |
| 7. #1762 | C2 | semidirect |
| 13. C2×C4×D4 | C4 | semidirect |
| 21. C2×C4⋊C8 | C22 | semidirect |
| 24. C2×#97 | C22 | semidirect |
| 25. #384 | C22 | semidirect |
| 30. C4×C4⋊C4 | C22 | semidirect |
| 31. #340 | C22 | semidirect |
| 153. D4 | #117 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| C2×#432 | 2 |
| #2285 | 2 |
| #1824 | 2 |
| #1802 | 2 |
| #1786 | 2 |
| #1762 | 2 |