Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
600 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #33171 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10797 | C2 | semidirect |
| 3. #9712 | C2 | semidirect |
| 4. #8596 | C2 | semidirect |
| 5. #7664 | C2 | semidirect |
| 6. C2×#493 | C4 | semidirect |
| 13. #1957 | C4 | semidirect |
| 14. #1845 | C4 | semidirect |
| 138. #95 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33171 | 2 |
| #10797 | 2 |
| #9712 | 2 |
| #8596 | 2 |
| #7664 | 2 |