Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
537 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 100 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 | C2×#493 | C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ 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. #22264 | C2 | semidirect |
| 3. #9642 | C2 | semidirect |
| 4. #7587 | C2 | semidirect |
| 5. #7587 | C2 | semidirect |
| 6. #7018 | C2 | semidirect |
| 7. #7014 | C2 | semidirect |
| 105. #127 | M16 | semidirect |
| 154. C4×C8 | M16 | semidirect |
| 155. C4×C8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22269 | 2 |
| #22264 | 2 |
| #9642 | 2 |
| #7587 | 2 |
| #7018 | 2 |
| #7014 | 2 |