Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
755 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 755 conjugacy classes of subgroups — too many to draw. Only its 183 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 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#99 | C2 | semidirect |
| 3. #2127 | C2 | direct (N × H) |
| 4. C2×#354 | C2 | semidirect |
| 5. C2×#353 | C2 | semidirect |
| 6. C22×#41 | C4 | semidirect |
| 9. #535 | C4 | semidirect |
| 11. C2×#127 | C4 | semidirect |
| 13. C2×#124 | C4 | semidirect |
| 17. C2×#117 | C4 | semidirect |
| 22. C2×#99 | C22 | semidirect |
| 23. #354 | C22 | semidirect |
| 24. #353 | C22 | semidirect |
| 29. C2×#41 | C2×C4 | semidirect |
| 30. C22×Q8 | C2×C4 | semidirect |
| 38. C22×D4 | C2×C4 | semidirect |
| 48. #127 | C2×C4 | semidirect |
| 49. #127 | C2×C4 | semidirect |
| 53. #124 | C2×C4 | semidirect |
| 54. #124 | C2×C4 | semidirect |
| 61. #117 | C2×C4 | semidirect |
| 70. C2×C42 | C2×C4 | semidirect |
| 71. C2×C42 | C2×C4 | semidirect |
| 90. C2×Q8 | C22×C4 | semidirect |
| 116. C2×D4 | C22×C4 | semidirect |
| 145. C42 | C22×C4 | semidirect |
| 146. C42 | C22×C4 | semidirect |
| 249. C2 | #2127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#535 | 2 |
| C22×#99 | 2 |
| #2127 | 2 |
| C2×#354 | 2 |
| C2×#353 | 2 |