Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
963 conjugacy classes of subgroups; only the 483 normal subgroups are shown.
This group has 963 conjugacy classes of subgroups — too many to draw. Only its 483 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 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3502 | C2 | semidirect |
| 3. #3392 | C2 | semidirect |
| 4. Q8×SD16 | C2 | semidirect |
| 5. #3362 | C2 | semidirect |
| 6. #3359 | C2 | semidirect |
| 7. #3246 | C2 | semidirect |
| 8. #3167 | C2 | semidirect |
| 9. C2×C8×Q8 | C2 | semidirect |
| 87. C2×Q8 | SD16 | semidirect |
| 146. C8 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3502 | 2 |
| #3497 | 2 |
| #3392 | 2 |
| Q8×SD16 | 2 |
| #3362 | 2 |
| #3359 | 2 |
| #3246 | 2 |
| #3167 | 2 |
| C2×C8×Q8 | 2 |