Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
555 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 555 conjugacy classes of subgroups — too many to draw. Only its 104 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 | #386 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| prime radical | #28156 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22260 | C2 | semidirect |
| 3. #9644 | C2 | semidirect |
| 4. #8588 | C2 | semidirect |
| 5. #7651 | C2 | semidirect |
| 6. #7040 | C2 | semidirect |
| 7. #7018 | C2 | semidirect |
| 156. C4×C8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28156 | 2 |
| #22260 | 2 |
| #9644 | 2 |
| #8588 | 2 |
| #7651 | 2 |
| #7040 | 2 |
| #7018 | 2 |