Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1581 conjugacy classes of subgroups; only the 864 normal subgroups are shown.
This group has 1581 conjugacy classes of subgroups — too many to draw. Only its 864 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#106 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C22×#340 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#1384 | C2 | semidirect |
| 7. C2×#504 | C4 | semidirect |
| 10. #2912 | C4 | semidirect |
| 28. C2×M32 | C2×C4 | semidirect |
| 30. C22×C16 | C2×C4 | semidirect |
| 34. #427 | C2×C4 | semidirect |
| 37. C2×#106 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3407 | 2 |
| #60108 | 2 |
| C2×#1384 | 2 |