Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
273 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3160 | C2 | semidirect |
| 3. #2229 | C2 | semidirect |
| 4. #2228 | C2 | semidirect |
| 5. #2227 | C2 | semidirect |
| 6. #2226 | C2 | semidirect |
| 7. #2225 | C2 | semidirect |
| 8. #2224 | C2 | semidirect |
| 9. #2158 | C2 | semidirect |
| 10. C2×C4⋊C16 | C2 | semidirect |
| 45. #135 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3160 | 2 |
| C4×#135 | 2 |
| #2229 | 2 |
| #2228 | 2 |
| #2227 | 2 |
| #2226 | 2 |
| #2225 | 2 |
| #2224 | 2 |
| #2158 | 2 |
| C2×C4⋊C16 | 2 |
| #2138 | 2 |