Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
5387 conjugacy classes of subgroups; only the 3143 normal subgroups are shown.
This group has 5387 conjugacy classes of subgroups — too many to draw. Only its 3143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C26 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×2−1+4 | C2 | semidirect |
| 2. C2×#578 | C2 | direct (N × H) |
| 3. C22×#137 | C2 | semidirect |
| 4. C22×#135 | C2 | semidirect |
| 5. C23×Q16 | C2 | semidirect |
| 6. C22×#131 | C2 | semidirect |
| 8. C2×2−1+4 | C22 | semidirect |
| 14. #578 | C22 | direct (N × H) |
| 15. C2×#137 | C22 | semidirect |
| 16. C2×#135 | C22 | semidirect |
| 19. C22×Q16 | C22 | semidirect |
| 22. C2×#131 | C22 | semidirect |
| 26. 2−1+4 | C23 | semidirect |
| 44. #137 | C23 | semidirect |
| 45. #135 | C23 | semidirect |
| 48. C2×Q16 | C23 | semidirect |
| 51. #131 | C23 | semidirect |
| 115. C22 | #578 | direct (N × H) |
| 124. C2 | C2×#578 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×2−1+4 | 2 |
| C2×#578 | 2 |
| C22×#137 | 2 |
| C22×#135 | 2 |
| C23×Q16 | 2 |
| C22×#131 | 2 |