Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
766 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 170 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 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28252 | C2 | semidirect |
| 3. #22238 | C2 | semidirect |
| 5. #10474 | C2 | semidirect |
| 6. #8810 | C2 | semidirect |
| 7. #8649 | C2 | semidirect |
| 8. #8588 | C2 | semidirect |
| 89. C4⋊Q8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28252 | 2 |
| #22269 | 2 |
| #22238 | 2 |
| C2×#1668 | 2 |
| #10474 | 2 |
| #8810 | 2 |
| #8649 | 2 |
| #8588 | 2 |