Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
353 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 76 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 | #10536 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33717 | 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. #13261 | C2 | semidirect |
| 3. #13259 | C2 | semidirect |
| 4. #13164 | C2 | semidirect |
| 5. #11920 | C2 | semidirect |
| 6. C2×#1357 | C2 | semidirect |
| 7. #7162 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33717 | 2 |
| #13261 | 2 |
| #13259 | 2 |
| #13164 | 2 |
| #11920 | 2 |
| C2×#1357 | 2 |
| #7162 | 2 |