Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
190 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1744 | C22 |
| prime radical | #3169 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ #1744 ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #7142 | C2 | semidirect |
| 3. #7018 | C2 | semidirect |
| 12. #500 | C8 | semidirect |
| 13. #493 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #9847 | 2 |
| #7142 | 2 |
| #7018 | 2 |