Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
370 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C4⋊C4 | C22 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C4⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 19. C2×C4⋊C8 | C8 | semidirect |
| 24. C2×#97 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10835 | 2 |
| #8558 | 2 |