Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10349 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #453 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ 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 |
|---|---|---|
| 38. #498 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28823 | 2 |
| #13271 | 2 |
| #9755 | 2 |
| #7271 | 2 |
| #7181 | 2 |
| #7180 | 2 |