Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10431 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28249 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ 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. #12824 | C2 | semidirect |
| 3. #12820 | C2 | semidirect |
| 4. #11604 | C2 | semidirect |
| 5. #11604 | C2 | semidirect |
| 6. #8793 | C2 | semidirect |
| 7. #7270 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28249 | 2 |
| #12824 | 2 |
| #12820 | 2 |
| #11604 | 2 |
| #8793 | 2 |
| #7270 | 2 |