Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7410 conjugacy classes of subgroups; only the 3906 normal subgroups are shown.
This group has 7410 conjugacy classes of subgroups — too many to draw. Only its 3906 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2863 | C2 | direct (N × H) |
| 13. #2863 | C22 | direct (N × H) |
| 35. C22×C42 | Q8 | semidirect |
| 72. C2×C42 | C2×Q8 | semidirect |
| 109. C42 | C22×Q8 | semidirect |
| 133. C22 | #2863 | direct (N × H) |
| 145. C2 | C2×#2863 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2863 | 2 |
| C22×#391 | 2 |
| C22×#389 | 2 |
| C22×#387 | 2 |
| C22×#383 | 2 |
| C22×C43 | 2 |