Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
5605 conjugacy classes of subgroups; only the 5295 normal subgroups are shown.
This group has 5605 conjugacy classes of subgroups — too many to draw. Only its 5295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C22 | C26 |
| socle | C23 | C25 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×2−1+4 | C2 | direct (N × H) |
| 3. C2×#520 | C2 | semidirect |
| 4. C2×#519 | C2 | semidirect |
| 5. C2×C4×#41 | C2 | semidirect |
| 6. C22×C4×Q8 | C2 | semidirect |
| 7. C2×2−1+4 | C4 | direct (N × H) |
| 11. #520 | C22 | semidirect |
| 12. #519 | C22 | semidirect |
| 13. C4×#41 | C22 | semidirect |
| 17. C2×C4×Q8 | C22 | semidirect |
| 23. 2−1+4 | C2×C4 | direct (N × H) |
| 24. C2×#41 | C2×C4 | semidirect |
| 26. C22×Q8 | C2×C4 | semidirect |
| 47. #41 | C22×C4 | semidirect |
| 49. C2×Q8 | C22×C4 | semidirect |
| 81. C2×C4 | 2−1+4 | direct (N × H) |
| 97. C4 | C2×2−1+4 | direct (N × H) |
| 99. C2 | C4×2−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×2−1+4 | 2 |
| C4×2−1+4 | 2 |
| C2×#520 | 2 |
| C2×#519 | 2 |
| C2×C4×#41 | 2 |
| C22×C4×Q8 | 2 |