Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
570 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 129 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#510 | C2 | semidirect |
| 3. C2×#467 | C2 | semidirect |
| 4. #2189 | C2 | semidirect |
| 5. #2188 | C2 | semidirect |
| 6. #2187 | C2 | semidirect |
| 7. #2139 | C2 | semidirect |
| 8. #2112 | C2 | semidirect |
| 16. C22×Q16 | C22 | semidirect |
| 22. C2×Q32 | C22 | semidirect |
| 23. C2×SD32 | C22 | semidirect |
| 24. C2×M32 | C22 | semidirect |
| 28. #467 | C22 | semidirect |
| 29. C2×#107 | C22 | semidirect |
| 31. #358 | C22 | semidirect |
| 32. #357 | C22 | semidirect |
| 34. C22⋊C16 | C22 | semidirect |
| 210. C22 | #510 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C2×#510 | 2 |
| C2×#467 | 2 |
| #2189 | 2 |
| #2188 | 2 |
| #2187 | 2 |
| #2139 | 2 |
| #2112 | 2 |