Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
303 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 99 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 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3175 | C2 | semidirect |
| 2. #3155 | C2 | semidirect |
| 3. #2250 | C2 | semidirect |
| 4. #2249 | C2 | semidirect |
| 5. #2243 | C2 | semidirect |
| 6. #2240 | C2 | semidirect |
| 7. #2211 | C2 | semidirect |
| 8. #2210 | C2 | semidirect |
| 9. #2207 | C2 | semidirect |
| 10. #2206 | C2 | semidirect |
| 11. #2189 | C2 | semidirect |
| 12. #2187 | C2 | semidirect |
| 13. #2186 | C2 | semidirect |
| 14. #2184 | C2 | semidirect |
| 15. #2164 | C2 | semidirect |
| 21. C2×Q32 | C22 | semidirect |
| 24. C2×SD32 | C22 | semidirect |
| 25. C2×SD32 | C22 | semidirect |
| 28. C2×D16 | C22 | semidirect |
| 29. #500 | C22 | semidirect |
| 34. #495 | C22 | semidirect |
| 42. C4×Q16 | C22 | semidirect |
| 43. C4×D8 | C22 | semidirect |
| 45. C4⋊C16 | C22 | semidirect |
| 46. C4⋊C16 | C22 | semidirect |
| 47. #358 | C22 | semidirect |
| 48. #358 | C22 | semidirect |
| 53. #357 | C22 | semidirect |
| 54. #357 | C22 | semidirect |
| 57. #346 | C22 | semidirect |
| 58. C4×C16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3175 | 2 |
| #3155 | 2 |
| #2250 | 2 |
| #2249 | 2 |
| #2243 | 2 |
| #2240 | 2 |
| #2211 | 2 |
| #2210 | 2 |
| #2207 | 2 |
| #2206 | 2 |
| #2189 | 2 |
| #2187 | 2 |
| #2186 | 2 |
| #2184 | 2 |
| #2164 | 2 |