Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
837 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 179 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 | C22×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#423 | C2 | semidirect |
| 3. C2×#417 | C2 | semidirect |
| 4. C2×#418 | C2 | semidirect |
| 5. #2297 | C2 | semidirect |
| 6. C2×#390 | C2 | semidirect |
| 7. #2055 | C2 | semidirect |
| 8. #2054 | C2 | semidirect |
| 9. #1736 | C2 | semidirect |
| 24. C2×C4⋊C8 | C22 | semidirect |
| 27. C2×#103 | C22 | semidirect |
| 29. C2×#102 | C22 | semidirect |
| 30. #391 | C22 | semidirect |
| 34. #390 | C22 | semidirect |
| 38. #340 | C22 | semidirect |
| 245. C2×C4 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AGL(1,ℤ/8) | 2 |
| C2×#423 | 2 |
| C2×#417 | 2 |
| C2×#418 | 2 |
| #2297 | 2 |
| C2×#390 | 2 |
| #2055 | 2 |
| #2054 | 2 |
| #1736 | 2 |