Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
376 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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. #3079 | C2 | semidirect |
| 2. C2×#474 | C2 | semidirect |
| 4. #2906 | C2 | semidirect |
| 5. #2089 | C2 | semidirect |
| 6. #2063 | C2 | semidirect |
| 7. #1997 | C2 | semidirect |
| 8. #1950 | C2 | semidirect |
| 10. #1816 | C2 | semidirect |
| 11. #1800 | C2 | semidirect |
| 54. AGL(1,ℤ/8) | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3079 | 2 |
| C2×#474 | 2 |
| C4×AGL(1,ℤ/8) | 2 |
| #2906 | 2 |
| #2089 | 2 |
| #2063 | 2 |
| #1997 | 2 |
| #1950 | 2 |
| #1907 | 2 |
| #1816 | 2 |
| #1800 | 2 |