Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
362 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C23 |
| commutator | C22 | C2×C82 |
| socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. #346 | C8 | semidirect |
| 30. C4×C16 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#346 | 2 |
| C42×C16 | 2 |
| #7841 | 2 |
| #7136 | 2 |