Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
408 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 121 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, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | #2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C2×C4×D8 | C2 | semidirect |
| 3. #2922 | C2 | semidirect |
| 4. #2089 | C2 | semidirect |
| 5. #2078 | C2 | semidirect |
| 6. #2063 | C2 | semidirect |
| 7. #2051 | C2 | semidirect |
| 8. #2045 | C2 | semidirect |
| 10. C4×#108 | C2 | semidirect |
| 11. C4×#100 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×AGL(1,ℤ/8) | 2 |
| C2×C4×D8 | 2 |
| #2922 | 2 |
| #2089 | 2 |
| #2078 | 2 |
| #2063 | 2 |
| #2051 | 2 |
| #2045 | 2 |
| #2018 | 2 |
| C4×#108 | 2 |
| C4×#100 | 2 |