Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
471 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1306 | C2 | direct (N × H) |
| 9. #407 | C4 | semidirect |
| 12. C2×C22⋊C8 | C4 | semidirect |
| 40. C22⋊C8 | C2×C4 | semidirect |
| 166. C2 | #1306 | 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×#381 | 2 |
| #1306 | 2 |