Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
572 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 572 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| 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 |
|---|---|---|
| 1. C22×#135 | C2 | semidirect |
| 2. C2×#510 | C2 | semidirect |
| 3. C2×#509 | C2 | semidirect |
| 4. #3050 | C2 | semidirect |
| 5. #2189 | C2 | semidirect |
| 6. #2186 | C2 | semidirect |
| 7. #2141 | C2 | semidirect |
| 8. #2140 | C2 | semidirect |
| 9. #2112 | C2 | semidirect |
| 23. C2×Q32 | C22 | semidirect |
| 24. C2×SD32 | C22 | semidirect |
| 25. C2×SD32 | C22 | semidirect |
| 26. C2×D16 | C22 | semidirect |
| 29. #467 | C22 | semidirect |
| 30. #466 | C22 | semidirect |
| 34. #358 | C22 | semidirect |
| 35. #357 | C22 | semidirect |
| 36. C22⋊C16 | C22 | semidirect |
| 54. C2×Q16 | D4 | semidirect |
| 62. C2×D8 | D4 | 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×#509 | 2 |
| #3050 | 2 |
| #2189 | 2 |
| #2186 | 2 |
| #2141 | 2 |
| #2140 | 2 |
| #2112 | 2 |