Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
921 conjugacy classes of subgroups; only the 348 normal subgroups are shown.
This group has 921 conjugacy classes of subgroups — too many to draw. Only its 348 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 132. Q8 | #333 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36593 | 2 |
| Q8×C4⋊C8 | 2 |
| #9691 | 2 |
| #8587 | 2 |
| C4×#333 | 2 |