Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6786 conjugacy classes of subgroups; only the 1602 normal subgroups are shown.
This group has 6786 conjugacy classes of subgroups — too many to draw. Only its 1602 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 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×#328 | C2 | direct (N × H) |
| 5. C24×Q8 | C4 | semidirect |
| 11. C22×C22⋊Q8 | C4 | semidirect |
| 14. C23×C4⋊C4 | C4 | semidirect |
| 20. C2×#328 | C22 | direct (N × H) |
| 24. C23×Q8 | C2×C4 | semidirect |
| 43. C2×C22⋊Q8 | C2×C4 | semidirect |
| 48. C22×C4⋊C4 | C2×C4 | semidirect |
| 55. #328 | C23 | direct (N × H) |
| 66. C22×Q8 | C22×C4 | semidirect |
| 96. C22⋊Q8 | C22×C4 | semidirect |
| 101. C2×C4⋊C4 | C22×C4 | semidirect |
| 126. C2×Q8 | C23×C4 | semidirect |
| 159. C4⋊C4 | C23×C4 | semidirect |
| 169. C23 | #328 | direct (N × H) |
| 210. C22 | C2×#328 | direct (N × H) |
| 228. C2 | C22×#328 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊Q8 | 2 |
| C23×C22⋊C8 | 2 |
| C23×#95 | 2 |
| C22×#328 | 2 |