Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
834 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 834 conjugacy classes of subgroups — too many to draw. Only its 226 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 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 86. C4⋊Q8 | Q16 | semidirect |
| 210. C4 | #1670 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33302 | 2 |
| #28512 | 2 |
| C8×C4⋊Q8 | 2 |
| C2×#1670 | 2 |
| #10276 | 2 |
| #9692 | 2 |
| C4×#335 | 2 |