Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
347 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | C22×D8 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12012 | C2 | semidirect |
| 3. #11629 | C2 | semidirect |
| 4. #11487 | C2 | semidirect |
| 5. #11411 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D8 | 2 |
| #12012 | 2 |
| #11629 | 2 |
| #11487 | 2 |
| #11411 | 2 |