Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
313 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C22 | #323 |
| prime radical | C25 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#349 | C2 | semidirect |
| 3. #1320 | C2 | direct (N × H) |
| 13. #349 | C22 | semidirect |
| 176. C2 | #1320 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#407 | 2 |
| C2×#349 | 2 |
| #1320 | 2 |