Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 96 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×#127 | C2×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 |
|---|---|---|
| 2. #22242 | C2 | semidirect |
| 4. #8589 | C2 | semidirect |
| 6. #7040 | C2 | semidirect |
| 7. #7018 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28201 | 2 |
| #22242 | 2 |
| #9644 | 2 |
| #8589 | 2 |
| #7651 | 2 |
| #7040 | 2 |
| #7018 | 2 |