Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
343 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 115 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3164 | C2 | semidirect |
| 5. #2028 | C2 | semidirect |
| 6. #1979 | C2 | semidirect |
| 7. #1969 | C2 | semidirect |
| 8. #1930 | C2 | semidirect |
| 10. #1772 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3164 | 2 |
| #3137 | 2 |
| #3059 | 2 |
| #2038 | 2 |
| #2028 | 2 |
| #1979 | 2 |
| #1969 | 2 |
| #1930 | 2 |
| #1816 | 2 |
| #1772 | 2 |
| #1764 | 2 |