Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
184 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 184 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 | #1667 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #12515 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12425 | C2 | semidirect |
| 3. #12054 | C2 | semidirect |
| 4. #7101 | C2 | semidirect |
| 5. #7096 | C2 | semidirect |
| 87. C16 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12515 | 2 |
| #12425 | 2 |
| #12054 | 2 |
| #7101 | 2 |
| #7096 | 2 |