Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C2×C4×C4⋊C4 | C22 |
| socle | C23 | #342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ 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 |
|---|---|---|
| 18. C2×C4⋊C8 | C8 | semidirect |
| 24. C2×C4×C8 | C8 | semidirect |
| 32. #324 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10834 | 2 |
| C4×#324 | 2 |