Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
448 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×D4 | C2×D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33962 | C2 | semidirect |
| 4. #12516 | C2 | semidirect |
| 5. #12494 | C2 | semidirect |
| 6. #12440 | C2 | semidirect |
| 8. #12334 | C2 | semidirect |
| 10. #12083 | C2 | semidirect |
| 184. SD16 | Q32 | semidirect |
| 228. Q8 | C2×Q32 | semidirect |
| 250. C8 | C2×Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33962 | 2 |
| D4×Q32 | 2 |
| #19928 | 2 |
| #12516 | 2 |
| #12494 | 2 |
| #12440 | 2 |
| #12431 | 2 |
| #12334 | 2 |
| #12322 | 2 |
| #12083 | 2 |
| C16×SD16 | 2 |