Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
658 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13128 | C2 | semidirect |
| 3. #12788 | C2 | semidirect |
| 5. #12605 | C2 | semidirect |
| 11. C2×#364 | C22 | semidirect |
| 15. C2×#347 | C22 | semidirect |
| 20. #1376 | C22 | semidirect |
| 21. #1375 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1753 | 2 |
| #13128 | 2 |
| #12788 | 2 |
| #12681 | 2 |
| #12605 | 2 |