Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
58050 conjugacy classes of subgroups; only the 31970 normal subgroups are shown.
This group has 58050 conjugacy classes of subgroups — too many to draw. Only its 31970 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C27 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C25×Q8 | C2 | semidirect |
| 3. C23×#137 | C2 | direct (N × H) |
| 4. C24×Q16 | C2 | semidirect |
| 5. C24×SD16 | C2 | semidirect |
| 6. C24×M16 | C2 | semidirect |
| 13. C24×Q8 | C22 | semidirect |
| 19. C22×#137 | C22 | direct (N × H) |
| 20. C23×Q16 | C22 | semidirect |
| 21. C23×SD16 | C22 | semidirect |
| 22. C23×M16 | C22 | semidirect |
| 34. C23×Q8 | C23 | semidirect |
| 46. C2×#137 | C23 | direct (N × H) |
| 47. C22×Q16 | C23 | semidirect |
| 48. C22×SD16 | C23 | semidirect |
| 49. C22×M16 | C23 | semidirect |
| 64. C22×Q8 | C24 | semidirect |
| 77. #137 | C24 | direct (N × H) |
| 78. C2×Q16 | C24 | semidirect |
| 79. C2×SD16 | C24 | semidirect |
| 80. C2×M16 | C24 | semidirect |
| 85. C24 | #137 | direct (N × H) |
| 96. C2×Q8 | C25 | semidirect |
| 109. Q16 | C25 | semidirect |
| 110. SD16 | C25 | semidirect |
| 111. M16 | C25 | semidirect |
| 116. C23 | C2×#137 | direct (N × H) |
| 140. C22 | C22×#137 | direct (N × H) |
| 150. C2 | C23×#137 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#41 | 2 |
| C25×Q8 | 2 |
| C23×#137 | 2 |
| C24×Q16 | 2 |
| C24×SD16 | 2 |
| C24×M16 | 2 |