Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
711 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 711 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×D8 | C2 | semidirect |
| 3. C4×#127 | C2 | semidirect |
| 4. #1970 | C2 | semidirect |
| 5. #1957 | C2 | semidirect |
| 6. #1845 | C2 | semidirect |
| 7. C42×C8 | C2 | semidirect |
| 12. #493 | C4 | direct (N × H) |
| 29. C2×D8 | C2×C4 | semidirect |
| 33. #127 | C2×C4 | semidirect |
| 49. C4×C8 | C2×C4 | semidirect |
| 50. C4×C8 | D4 | semidirect |
| 80. C42 | D8 | semidirect |
| 111. C8 | C4×D4 | semidirect |
| 125. C4 | C4×D8 | semidirect |
| 126. C4 | #493 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#493 | 2 |
| C2×C4×D8 | 2 |
| C4×#127 | 2 |
| #1970 | 2 |
| #1957 | 2 |
| #1845 | 2 |
| C42×C8 | 2 |