Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1503 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 1503 conjugacy classes of subgroups — too many to draw. Only its 1039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C2×D4 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#427 | C2 | direct (N × H) |
| 4. C22×#107 | C2 | semidirect |
| 5. C22×#106 | C2 | semidirect |
| 7. C23×C8 | C4 | semidirect |
| 16. #427 | C22 | direct (N × H) |
| 17. C2×#107 | C22 | semidirect |
| 18. C2×#106 | C22 | semidirect |
| 29. C22×C8 | C2×C4 | semidirect |
| 39. #107 | C23 | semidirect |
| 40. #106 | C23 | semidirect |
| 60. C2×C8 | C22×C4 | semidirect |
| 90. C22 | #427 | direct (N × H) |
| 101. C2 | C2×#427 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8 | 2 |
| C22×#117 | 2 |
| C2×#427 | 2 |
| C22×#107 | 2 |
| C22×#106 | 2 |