Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
487 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 185 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #121 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2984 | C2 | semidirect |
| 3. #2276 | C2 | semidirect |
| 5. #1506 | C2 | semidirect |
| 6. #1503 | C2 | semidirect |
| 8. #557 | C4 | semidirect |
| 9. #557 | C4 | semidirect |
| 10. #556 | C4 | semidirect |
| 13. D4×Q8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3523 | 2 |
| #2984 | 2 |
| #2276 | 2 |
| #1509 | 2 |
| #1506 | 2 |
| #1503 | 2 |
| #1500 | 2 |