Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
612 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #2184 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1420 | C2 | semidirect |
| 3. C2×#1417 | C2 | semidirect |
| 4. #13616 | C2 | semidirect |
| 5. C2×#1416 | C2 | semidirect |
| 6. C2×#1401 | C2 | semidirect |
| 7. #12842 | C2 | semidirect |
| 12. #3417 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3417 | 2 |
| C2×#1420 | 2 |
| C2×#1417 | 2 |
| #13616 | 2 |
| C2×#1416 | 2 |
| C2×#1401 | 2 |
| #12842 | 2 |