Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
343 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 111 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. C2×#489 | C2 | semidirect |
| 2. #3079 | C2 | semidirect |
| 4. #2923 | C2 | semidirect |
| 5. #2064 | C2 | semidirect |
| 6. #2063 | C2 | semidirect |
| 7. #2052 | C2 | semidirect |
| 9. #1997 | C2 | semidirect |
| 10. #1967 | C2 | semidirect |
| 11. #1849 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#489 | 2 |
| #3079 | 2 |
| #3040 | 2 |
| #2923 | 2 |
| #2064 | 2 |
| #2063 | 2 |
| #2052 | 2 |
| #2051 | 2 |
| #1997 | 2 |
| #1967 | 2 |
| #1849 | 2 |