Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
592 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33434 | C2 | semidirect |
| 3. #13192 | C2 | semidirect |
| 4. #12980 | C2 | semidirect |
| 5. #12884 | C2 | semidirect |
| 6. #12592 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33434 | 2 |
| #21708 | 2 |
| #13192 | 2 |
| #12980 | 2 |
| #12884 | 2 |
| #12592 | 2 |