Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
391 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C22 | #342 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1391 | C2 | direct (N × H) |
| 5. C22×M16 | C4 | semidirect |
| 6. C23×C8 | C4 | semidirect |
| 10. #410 | C4 | semidirect |
| 13. C2×#99 | C4 | semidirect |
| 15. #408 | C4 | semidirect |
| 36. C2×M16 | C2×C4 | semidirect |
| 42. C22×C8 | C2×C4 | semidirect |
| 53. #99 | C2×C4 | semidirect |
| 208. C2 | #1391 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#410 | 2 |
| C2×#408 | 2 |
| #1391 | 2 |