Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
342 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 342 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 | #2226 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ 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. C2×#2253 | C2 | semidirect |
| 3. #13709 | C2 | direct (N × H) |
| 4. C2×#1426 | C2 | semidirect |
| 5. C2×#1422 | C2 | semidirect |
| 12. #2253 | C22 | semidirect |
| 20. #1426 | C22 | semidirect |
| 21. #1422 | C22 | semidirect |
| 185. C2 | #13709 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2253 | 2 |
| C2×#2178 | 2 |
| #13709 | 2 |
| C2×#1426 | 2 |
| C2×#1422 | 2 |
| C2×#1418 | 2 |