Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2859 conjugacy classes of subgroups; only the 1055 normal subgroups are shown.
This group has 2859 conjugacy classes of subgroups — too many to draw. Only its 1055 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×Q8 | C2 | semidirect |
| 3. #2341 | C2 | direct (N × H) |
| 4. C2×#390 | C2 | semidirect |
| 5. C2×#380 | C2 | semidirect |
| 6. C2×C4×C22⋊C4 | C2 | semidirect |
| 8. C22×#41 | C4 | semidirect |
| 14. C2×C4×Q8 | C22 | semidirect |
| 19. #390 | C22 | semidirect |
| 20. #380 | C22 | semidirect |
| 21. C4×C22⋊C4 | C22 | semidirect |
| 27. C2×#41 | C2×C4 | semidirect |
| 60. C2×Q8 | C22⋊C4 | semidirect |
| 78. C22×C4 | #41 | semidirect |
| 103. Q8 | C2×C22⋊C4 | semidirect |
| 119. C2×C4 | C2×#41 | semidirect |
| 141. C2 | #2341 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C22×C4×Q8 | 2 |
| #2341 | 2 |
| C2×#390 | 2 |
| C2×#380 | 2 |
| C2×C4×C22⋊C4 | 2 |