Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
845 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 151 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 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#486 | C2 | semidirect |
| 3. C2×#474 | C2 | semidirect |
| 4. C2×#461 | C2 | semidirect |
| 5. #1639 | C2 | direct (N × H) |
| 6. C2×#331 | C2 | semidirect |
| 7. C2×D4⋊C8 | C2 | semidirect |
| 8. C2×D4⋊C8 | C2 | semidirect |
| 16. C2×C4⋊Q8 | C22 | semidirect |
| 24. #486 | C22 | semidirect |
| 26. #474 | C22 | semidirect |
| 28. #461 | C22 | semidirect |
| 30. C2×C4⋊C8 | C22 | semidirect |
| 31. C2×C4⋊C8 | C22 | semidirect |
| 35. C2×C4×C8 | C22 | semidirect |
| 36. #331 | C22 | semidirect |
| 37. D4⋊C8 | C22 | semidirect |
| 38. D4⋊C8 | C22 | semidirect |
| 66. C4⋊Q8 | C23 | semidirect |
| 96. C4⋊C8 | C23 | semidirect |
| 97. C4⋊C8 | C23 | semidirect |
| 102. C4×C8 | C23 | semidirect |
| 139. C2×D4 | SD16 | semidirect |
| 216. D4 | C2×SD16 | semidirect |
| 274. C2 | #1639 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D42 | 2 |
| C2×#486 | 2 |
| C2×#474 | 2 |
| C2×#461 | 2 |
| #1639 | 2 |
| C2×#331 | 2 |
| C2×D4⋊C8 | 2 |