Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
278 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 103 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 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×Q16 | C2 | semidirect |
| 4. #1710 | C2 | semidirect |
| 5. #1704 | C2 | semidirect |
| 7. #1674 | C2 | semidirect |
| 8. #1645 | C2 | semidirect |
| 10. #1591 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16 | 2 |
| #3369 | 2 |
| #3342 | 2 |
| #1710 | 2 |
| #1704 | 2 |
| #1675 | 2 |
| #1674 | 2 |
| #1645 | 2 |
| #1638 | 2 |
| #1591 | 2 |
| C8×SD16 | 2 |