Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
751 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 191 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 | #99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#354 | C2 | direct (N × H) |
| 4. C23×Q8 | C4 | semidirect |
| 7. C2×#124 | C4 | semidirect |
| 11. C22×C42 | C4 | semidirect |
| 14. #354 | C22 | direct (N × H) |
| 19. C22×Q8 | C2×C4 | semidirect |
| 31. #124 | C2×C4 | semidirect |
| 38. C2×C42 | C2×C4 | semidirect |
| 52. C2×Q8 | C22×C4 | semidirect |
| 79. C42 | C22×C4 | semidirect |
| 123. C22 | #354 | direct (N × H) |
| 136. C2 | C2×#354 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#124 | 2 |
| C22×#99 | 2 |
| C2×#354 | 2 |