Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
399 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 399 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 |
| 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 |
|---|---|---|
| 1. #3169 | C2 | semidirect |
| 2. #3036 | C2 | semidirect |
| 3. C2×C8×D4 | C2 | semidirect |
| 4. #1681 | C2 | semidirect |
| 5. #1673 | C2 | semidirect |
| 6. #1667 | C2 | semidirect |
| 7. #1567 | C2 | semidirect |
| 8. #1547 | C2 | semidirect |
| 13. #500 | C22 | semidirect |
| 14. #493 | C22 | semidirect |
| 15. #486 | C22 | semidirect |
| 32. #335 | C22 | semidirect |
| 33. #335 | C22 | semidirect |
| 34. #332 | C22 | semidirect |
| 35. #331 | C22 | semidirect |
| 91. C2×D4 | D8 | semidirect |
| 170. C8 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3169 | 2 |
| #3036 | 2 |
| C2×C8×D4 | 2 |
| #1681 | 2 |
| #1673 | 2 |
| #1667 | 2 |
| #1567 | 2 |
| #1547 | 2 |