Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
292 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 292 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#427 | C2 | semidirect |
| 3. #2237 | C2 | semidirect |
| 4. #2234 | C2 | semidirect |
| 5. #2159 | C2 | semidirect |
| 6. #2139 | C2 | semidirect |
| 7. #2112 | C2 | semidirect |
| 22. #367 | C22 | semidirect |
| 23. #366 | C22 | semidirect |
| 24. #358 | C22 | semidirect |
| 25. #357 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3050 | 2 |
| C2×#427 | 2 |
| #2237 | 2 |
| #2234 | 2 |
| #2159 | 2 |
| #2139 | 2 |
| #2112 | 2 |