Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
437 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 117 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×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2924 | C2 | semidirect |
| 4. C2×#419 | C2 | semidirect |
| 5. #2064 | C2 | semidirect |
| 6. #2046 | C2 | semidirect |
| 7. #1977 | C2 | semidirect |
| 8. #1930 | C2 | semidirect |
| 10. #1855 | C2 | semidirect |
| 11. #1784 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3094 | 2 |
| C2×#448 | 2 |
| #2924 | 2 |
| C2×#419 | 2 |
| #2064 | 2 |
| #2046 | 2 |
| #1977 | 2 |
| #1930 | 2 |
| #1900 | 2 |
| #1855 | 2 |
| #1784 | 2 |