Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
306 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #2184 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×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. #3168 | C2 | semidirect |
| 2. C2×C4×D8 | C2 | semidirect |
| 3. #2235 | C2 | semidirect |
| 4. #2232 | C2 | semidirect |
| 5. #2228 | C2 | semidirect |
| 6. #2210 | C2 | semidirect |
| 7. #2187 | C2 | semidirect |
| 9. #2177 | C2 | semidirect |
| 10. #2175 | C2 | semidirect |
| 11. #2163 | C2 | semidirect |
| 17. #499 | C22 | semidirect |
| 18. #495 | C22 | semidirect |
| 23. C4×Q16 | C22 | semidirect |
| 29. C4×D8 | C22 | semidirect |
| 33. #367 | C22 | semidirect |
| 34. #366 | C22 | semidirect |
| 35. C4⋊C16 | C22 | semidirect |
| 36. #358 | C22 | semidirect |
| 39. #357 | C22 | semidirect |
| 42. #346 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3168 | 2 |
| C2×C4×D8 | 2 |
| #2235 | 2 |
| #2232 | 2 |
| #2228 | 2 |
| #2210 | 2 |
| #2187 | 2 |
| #2184 | 2 |
| #2177 | 2 |
| #2175 | 2 |
| #2163 | 2 |