Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
336 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #3283 | C2 |
| 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×D8 | C2 | semidirect |
| 2. #3298 | C2 | semidirect |
| 4. #1707 | C2 | semidirect |
| 5. #1679 | C2 | semidirect |
| 6. #1673 | C2 | semidirect |
| 7. #1670 | C2 | semidirect |
| 8. #1654 | C2 | semidirect |
| 9. #1651 | C2 | semidirect |
| 10. C8×SD16 | C2 | semidirect |
| 11. C8×D8 | C2 | semidirect |
| 115. D8 | SD16 | semidirect |
| 150. Q8 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D8 | 2 |
| #3298 | 2 |
| #3283 | 2 |
| #1707 | 2 |
| #1679 | 2 |
| #1673 | 2 |
| #1670 | 2 |
| #1654 | 2 |
| #1651 | 2 |
| C8×SD16 | 2 |
| C8×D8 | 2 |