Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1458 conjugacy classes of subgroups; only the 738 normal subgroups are shown.
This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 738 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 | C24 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#2255 | C2 | direct (N × H) |
| 20. #2255 | C22 | direct (N × H) |
| 38. C22×C16 | Q8 | semidirect |
| 85. C2×C16 | C2×Q8 | semidirect |
| 131. C16 | C22×Q8 | semidirect |
| 155. C22 | #2255 | direct (N × H) |
| 168. C2 | C2×#2255 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#500 | 2 |
| C22×#499 | 2 |
| C2×#2255 | 2 |
| C22×#367 | 2 |
| C22×#366 | 2 |
| C22×#346 | 2 |