Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
398 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. #1328 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20427 | 2 |
| #12607 | 2 |
| C2×#1328 | 2 |
| #7194 | 2 |