Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1248 conjugacy classes of subgroups; only the 330 normal subgroups are shown.
This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 330 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×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2867 | 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. #33312 | C2 | semidirect |
| 2. #33310 | C2 | semidirect |
| 5. #33225 | C2 | semidirect |
| 7. C2×#1714 | C2 | semidirect |
| 8. C4×#322 | C2 | semidirect |
| 67. #322 | D4 | semidirect |
| 111. C4×C8 | SD16 | semidirect |
| 188. C8 | #492 | semidirect |
| 200. C4 | #1714 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33312 | 2 |
| #33310 | 2 |
| #33300 | 2 |
| #33225 | 2 |
| #33224 | 2 |
| C2×#1714 | 2 |
| C4×#322 | 2 |