Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1251 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 743 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2383 | C2 | direct (N × H) |
| 3. C2×#389 | C2 | semidirect |
| 5. C2×#387 | C2 | semidirect |
| 7. C2×#384 | C2 | semidirect |
| 8. C2×#383 | C2 | semidirect |
| 10. C2×#126 | C4 | semidirect |
| 20. #389 | C22 | semidirect |
| 22. #387 | C22 | semidirect |
| 24. #384 | C22 | semidirect |
| 25. #383 | C22 | semidirect |
| 39. #126 | C2×C4 | semidirect |
| 43. C2×C4⋊C4 | C2×C4 | semidirect |
| 53. C2×C42 | C2×C4 | semidirect |
| 89. C4⋊C4 | C22×C4 | semidirect |
| 93. C42 | C22×C4 | semidirect |
| 141. C2 | #2383 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#126 | 2 |
| #2383 | 2 |
| C2×#389 | 2 |
| C2×#388 | 2 |
| C2×#387 | 2 |
| C2×#385 | 2 |
| C2×#384 | 2 |
| C2×#383 | 2 |