Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
227 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 227 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 | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#106 | C2 | semidirect |
| 4. #1566 | C2 | semidirect |
| 5. #1303 | C2 | semidirect |
| 7. C8×Q8 | C4 | semidirect |
| 8. C8×Q8 | C4 | semidirect |
| 11. C8×D4 | C4 | semidirect |
| 12. C8×D4 | C4 | semidirect |
| 108. Q8 | #106 | semidirect |
| 111. D4 | #106 | semidirect |
| 139. C8 | #104 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#41 | 2 |
| #1806 | 2 |
| C4×#106 | 2 |
| #1566 | 2 |
| #1303 | 2 |