Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
971 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 311 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 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4⋊C8 | C2 | semidirect |
| 4. C2×#382 | C2 | semidirect |
| 5. #1804 | C2 | direct (N × H) |
| 6. C2×#340 | C2 | semidirect |
| 12. C2×C4×D4 | C4 | semidirect |
| 19. C2×C4⋊C8 | C22 | semidirect |
| 23. #382 | C22 | semidirect |
| 26. #340 | C22 | semidirect |
| 50. C4×D4 | C2×C4 | semidirect |
| 202. C2 | #1804 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| C22×C4⋊C8 | 2 |
| C22×#102 | 2 |
| C2×#382 | 2 |
| #1804 | 2 |
| C2×#340 | 2 |