Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
201 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×C16 | C2 | semidirect |
| 3. #1564 | C2 | semidirect |
| 4. #1332 | C2 | semidirect |
| 5. #1331 | C2 | semidirect |
| 25. #135 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#135 | 2 |
| C2×C4×C16 | 2 |
| #1564 | 2 |
| #1332 | 2 |
| #1331 | 2 |