Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
517 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 102 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 | C22×D4 | C22⋊C4 |
| 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 |
|---|---|---|
| 3. #9686 | C2 | semidirect |
| 5. #7680 | C2 | semidirect |
| 6. #7049 | C2 | semidirect |
| 7. #7030 | C2 | semidirect |
| 276. D4 | #330 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33066 | 2 |
| D4×#97 | 2 |
| #9686 | 2 |
| #8174 | 2 |
| #7680 | 2 |
| #7049 | 2 |
| #7030 | 2 |