Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1459 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1459 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 |
|---|---|---|
| 3. #2503 | C2 | direct (N × H) |
| 4. C2×#400 | C2 | semidirect |
| 5. C2×#391 | C2 | semidirect |
| 6. C2×#382 | C2 | semidirect |
| 23. #400 | C22 | semidirect |
| 24. #391 | C22 | semidirect |
| 25. #382 | C22 | semidirect |
| 162. C2 | #2503 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| C23×C4⋊C4 | 2 |
| #2503 | 2 |
| C2×#400 | 2 |
| C2×#391 | 2 |
| C2×#382 | 2 |
| C2×#381 | 2 |
| C2×C4×C22⋊C4 | 2 |