Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
287 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 287 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 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3341 | C2 | semidirect |
| 2. #3280 | C2 | semidirect |
| 3. #2243 | C2 | semidirect |
| 4. #2237 | C2 | semidirect |
| 5. #2234 | C2 | semidirect |
| 6. #2224 | C2 | semidirect |
| 7. #2220 | C2 | semidirect |
| 8. #2215 | C2 | semidirect |
| 9. #2210 | C2 | semidirect |
| 10. #2186 | C2 | semidirect |
| 11. #2184 | C2 | semidirect |
| 12. #2177 | C2 | semidirect |
| 13. #2168 | C2 | semidirect |
| 14. #2140 | C2 | semidirect |
| 15. #2138 | C2 | semidirect |
| 20. C2×SD32 | C22 | semidirect |
| 24. C22×C16 | C22 | semidirect |
| 25. #500 | C22 | semidirect |
| 33. #466 | C22 | semidirect |
| 38. C4×D8 | C22 | semidirect |
| 41. #427 | C22 | semidirect |
| 42. #367 | C22 | semidirect |
| 43. #366 | C22 | semidirect |
| 44. #366 | C22 | semidirect |
| 45. C4⋊C16 | C22 | semidirect |
| 46. #358 | C22 | semidirect |
| 47. #358 | C22 | semidirect |
| 48. #358 | C22 | semidirect |
| 55. #357 | C22 | semidirect |
| 57. C22⋊C16 | C22 | semidirect |
| 58. #346 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3341 | 2 |
| #3280 | 2 |
| #2243 | 2 |
| #2237 | 2 |
| #2234 | 2 |
| #2224 | 2 |
| #2220 | 2 |
| #2215 | 2 |
| #2210 | 2 |
| #2186 | 2 |
| #2184 | 2 |
| #2177 | 2 |
| #2168 | 2 |
| #2140 | 2 |
| #2138 | 2 |