Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1307 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1307 conjugacy classes of subgroups — too many to draw. Only its 567 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, Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#125 | C2 | semidirect |
| 3. #2629 | C2 | direct (N × H) |
| 4. C2×#398 | C2 | semidirect |
| 6. C2×#389 | C2 | semidirect |
| 7. C2×#384 | C2 | semidirect |
| 11. C2×#125 | C22 | semidirect |
| 24. #398 | C22 | semidirect |
| 26. #389 | C22 | semidirect |
| 27. #384 | C22 | semidirect |
| 175. C2 | #2629 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#125 | 2 |
| C22×#117 | 2 |
| #2629 | 2 |
| C2×#398 | 2 |
| C2×#396 | 2 |
| C2×#389 | 2 |
| C2×#384 | 2 |
| C2×#380 | 2 |