Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
337 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #3106 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ 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 |
|---|---|---|
| 1. #33505 | C2 | semidirect |
| 3. #13191 | C2 | semidirect |
| 4. #11630 | C2 | semidirect |
| 5. #11613 | C2 | semidirect |
| 6. #7212 | C2 | semidirect |
| 7. #7163 | C2 | semidirect |
| 229. Q8 | #350 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33505 | 2 |
| #17912 | 2 |
| #13191 | 2 |
| #11630 | 2 |
| #11613 | 2 |
| #7212 | 2 |
| #7163 | 2 |