Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
360 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 127 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×#127 |
| 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 |
|---|---|---|
| 2. #3209 | C2 | semidirect |
| 4. C2×#498 | C2 | semidirect |
| 6. #3139 | C2 | semidirect |
| 7. #1720 | C2 | semidirect |
| 8. #1718 | C2 | semidirect |
| 9. #1708 | C2 | semidirect |
| 10. #1707 | C2 | semidirect |
| 11. C8×M16 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3226 | 2 |
| #3209 | 2 |
| #3167 | 2 |
| C2×#498 | 2 |
| #3141 | 2 |
| #3139 | 2 |
| #1720 | 2 |
| #1718 | 2 |
| #1708 | 2 |
| #1707 | 2 |
| C8×M16 | 2 |