Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6091 conjugacy classes of subgroups; only the 3204 normal subgroups are shown.
This group has 6091 conjugacy classes of subgroups — too many to draw. Only its 3204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#564 | C2 | semidirect |
| 2. C4⋊#564 | C2 | semidirect |
| 3. C22⋊#564 | C2 | semidirect |
| 5. C4×#564 | C2 | semidirect |
| 24. #564 | D4 | direct (N × H) |
| 98. D4 | #564 | direct (N × H) |
| 111. C22 | C2×#564 | semidirect |
| 118. C4 | C2×#564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#564 | 2 |
| C4⋊#564 | 2 |
| C22⋊#564 | 2 |
| D4×#125 | 2 |
| C4×#564 | 2 |