Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
356 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2331 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22311 |
| prime radical | C23 | #568 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 53. #346 | Q8 | semidirect |
| 55. C4×C16 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21736 | 2 |
| #13269 | 2 |
| #13164 | 2 |
| #13108 | 2 |
| #12882 | 2 |
| #12667 | 2 |
| #12594 | 2 |