Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
788 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×#127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 22. #1710 | C4 | direct (N × H) |
| 172. C4 | #1710 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#500 | 2 |
| C4×#487 | 2 |
| C2×#1710 | 2 |
| #9823 | 2 |
| #9821 | 2 |
| #8816 | 2 |
| C4×C82 | 2 |