Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
414 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q64 | C2 | semidirect |
| 4. #13701 | C2 | direct (N × H) |
| 5. C2×#1425 | C2 | semidirect |
| 6. C2×#1420 | C2 | semidirect |
| 13. C2×Q64 | C22 | semidirect |
| 24. #1425 | C22 | semidirect |
| 25. #1420 | C22 | semidirect |
| 193. C2 | #13701 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q64 | 2 |
| C2×#2258 | 2 |
| C2×#2222 | 2 |
| #13701 | 2 |
| C2×#1425 | 2 |
| C2×#1420 | 2 |