Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
721 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 721 conjugacy classes of subgroups — too many to draw. Only its 191 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 | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#433 | C2 | semidirect |
| 3. #2314 | C2 | semidirect |
| 4. #1983 | C2 | semidirect |
| 5. #1982 | C2 | semidirect |
| 6. #1961 | C2 | semidirect |
| 7. #1959 | C2 | semidirect |
| 8. #1958 | C2 | semidirect |
| 9. #1957 | C2 | semidirect |
| 10. #1788 | C2 | semidirect |
| 11. #1786 | C2 | semidirect |
| 15. #535 | C4 | semidirect |
| 24. C2×C4⋊C8 | C22 | semidirect |
| 25. C2×#103 | C22 | semidirect |
| 26. C2×#103 | C22 | semidirect |
| 31. C2×#102 | C22 | semidirect |
| 32. C2×#102 | C22 | semidirect |
| 34. C2×#97 | C22 | semidirect |
| 35. C2×C4×C8 | C22 | semidirect |
| 38. #384 | C22 | semidirect |
| 39. #382 | C22 | semidirect |
| 41. C4×C4⋊C4 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#535 | 2 |
| C2×#433 | 2 |
| #2314 | 2 |
| #1983 | 2 |
| #1982 | 2 |
| #1961 | 2 |
| #1959 | 2 |
| #1958 | 2 |
| #1957 | 2 |
| #1788 | 2 |
| #1786 | 2 |