Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
24546 conjugacy classes of subgroups; only the 4514 normal subgroups are shown.
This group has 24546 conjugacy classes of subgroups — too many to draw. Only its 4514 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 | C2×C4 | C26 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×Q8 | C2 | semidirect |
| 2. C24×SD16 | C2 | semidirect |
| 4. C22×#448 | C2 | direct (N × H) |
| 5. C23×#103 | C2 | semidirect |
| 6. C23×C22⋊C8 | C2 | semidirect |
| 11. C24×Q8 | C22 | semidirect |
| 17. C23×SD16 | C22 | semidirect |
| 23. C2×#448 | C22 | direct (N × H) |
| 24. C22×#103 | C22 | semidirect |
| 25. C22×C22⋊C8 | C22 | semidirect |
| 35. C23×Q8 | D4 | semidirect |
| 37. C23×Q8 | C23 | semidirect |
| 53. C22×SD16 | C23 | semidirect |
| 61. #448 | C23 | direct (N × H) |
| 62. C2×#103 | C23 | semidirect |
| 63. C2×C22⋊C8 | C23 | semidirect |
| 70. C25 | SD16 | semidirect |
| 81. C22×Q8 | C2×D4 | semidirect |
| 83. C22×Q8 | C24 | semidirect |
| 106. C2×SD16 | C24 | semidirect |
| 114. #103 | C24 | semidirect |
| 115. C22⋊C8 | C24 | semidirect |
| 126. C24 | C2×SD16 | semidirect |
| 137. C2×Q8 | C22×D4 | semidirect |
| 175. C23 | #448 | direct (N × H) |
| 178. C23 | C22×SD16 | semidirect |
| 185. Q8 | C23×D4 | semidirect |
| 210. C22 | C2×#448 | direct (N × H) |
| 213. C22 | C23×SD16 | semidirect |
| 223. C2 | C22×#448 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×Q8 | 2 |
| C24×SD16 | 2 |
| C23×#121 | 2 |
| C22×#448 | 2 |
| C23×#103 | 2 |
| C23×C22⋊C8 | 2 |