Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
498 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2247 | C2 | semidirect |
| 4. #12437 | C2 | direct (N × H) |
| 5. C2×#1373 | C2 | semidirect |
| 6. C2×#1358 | C2 | semidirect |
| 14. #2247 | C22 | semidirect |
| 24. #1373 | C22 | semidirect |
| 25. #1358 | C22 | semidirect |
| 245. C2 | #12437 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2247 | 2 |
| C2×#2214 | 2 |
| C2×#2170 | 2 |
| #12437 | 2 |
| C2×#1373 | 2 |
| C2×#1358 | 2 |