Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
369 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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 |
|---|---|---|
| 2. C4×#135 | C2 | semidirect |
| 3. C2×C4×D8 | C2 | semidirect |
| 4. #2177 | C2 | semidirect |
| 5. #2175 | C2 | semidirect |
| 6. C4×SD32 | C2 | semidirect |
| 7. C4×D16 | C2 | semidirect |
| 8. #2159 | C2 | semidirect |
| 9. #2140 | C2 | semidirect |
| 10. #2139 | C2 | semidirect |
| 11. C4×M32 | C2 | semidirect |
| 16. #509 | C4 | direct (N × H) |
| 21. C4×Q16 | C22 | semidirect |
| 22. C4×Q16 | C22 | semidirect |
| 29. C4×D8 | C22 | semidirect |
| 30. C4×D8 | C22 | semidirect |
| 36. #367 | C22 | semidirect |
| 37. #366 | C22 | semidirect |
| 38. #358 | C22 | semidirect |
| 41. #357 | C22 | semidirect |
| 42. #346 | C22 | semidirect |
| 43. C4×C16 | C22 | semidirect |
| 48. #135 | C2×C4 | semidirect |
| 53. C2×D8 | C2×C4 | semidirect |
| 74. SD32 | C2×C4 | semidirect |
| 75. D16 | C2×C4 | semidirect |
| 77. M32 | C2×C4 | semidirect |
| 116. Q16 | C22×C4 | semidirect |
| 122. D8 | C22×C4 | semidirect |
| 149. C16 | C22×C4 | semidirect |
| 208. C4 | #509 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#509 | 2 |
| C4×#135 | 2 |
| C2×C4×D8 | 2 |
| #2177 | 2 |
| #2175 | 2 |
| C4×SD32 | 2 |
| C4×D16 | 2 |
| #2159 | 2 |
| #2140 | 2 |
| #2139 | 2 |
| C4×M32 | 2 |