Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
341 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#475 | C2 | semidirect |
| 4. Q8×M16 | C2 | semidirect |
| 5. #1690 | C2 | semidirect |
| 6. #1686 | C2 | semidirect |
| 7. #1670 | C2 | semidirect |
| 8. #1666 | C2 | semidirect |
| 9. #1565 | C2 | semidirect |
| 11. #1546 | C2 | semidirect |
| 142. M16 | SD16 | semidirect |
| 184. Q8 | #137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3152 | 2 |
| #3103 | 2 |
| C2×#475 | 2 |
| Q8×M16 | 2 |
| #1690 | 2 |
| #1686 | 2 |
| #1670 | 2 |
| #1666 | 2 |
| #1565 | 2 |
| #1547 | 2 |
| #1546 | 2 |