Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
487 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | #95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1276 | C2 | direct (N × H) |
| 9. #431 | C4 | semidirect |
| 11. C2×C4⋊C8 | C4 | semidirect |
| 26. C2×C42 | C8 | semidirect |
| 38. C4⋊C8 | C2×C4 | semidirect |
| 75. C42 | C2×C8 | semidirect |
| 131. C2 | #1276 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#431 | 2 |
| C2×C43 | 2 |
| #1276 | 2 |