Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
931 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 931 conjugacy classes of subgroups — too many to draw. Only its 209 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×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. D4×M16 | C2 | semidirect |
| 3. C22×#100 | C2 | semidirect |
| 4. #1965 | C2 | semidirect |
| 5. #1900 | C2 | semidirect |
| 6. #1800 | C2 | semidirect |
| 7. #1783 | C2 | semidirect |
| 8. C4×#100 | C2 | semidirect |
| 11. C22×M16 | C22 | semidirect |
| 24. #423 | C22 | semidirect |
| 29. C2×#100 | C22 | semidirect |
| 30. C2×#100 | C22 | semidirect |
| 31. #407 | C22 | semidirect |
| 32. C4×M16 | C22 | semidirect |
| 33. #343 | C22 | semidirect |
| 34. #343 | C22 | semidirect |
| 86. #100 | D4 | direct (N × H) |
| 149. M16 | C2×D4 | semidirect |
| 200. D4 | #100 | direct (N × H) |
| 245. C22 | C2×#100 | semidirect |
| 253. C4 | C2×#100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D42 | 2 |
| D4×M16 | 2 |
| C22×#100 | 2 |
| #1965 | 2 |
| #1900 | 2 |
| #1800 | 2 |
| #1783 | 2 |
| C4×#100 | 2 |