Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
165 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×D8 | C2×C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C16×Q16 | C2 | semidirect |
| 4. #7101 | C2 | semidirect |
| 5. #7069 | C2 | semidirect |
| 7. C162 | C2 | semidirect |
| 52. SD32 | C16 | direct (N × H) |
| 79. Q16 | C2×C16 | semidirect |
| 104. C16 | C2×C16 | semidirect |
| 105. C16 | SD32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C16×Q16 | 2 |
| C8×SD32 | 2 |
| C16×D8 | 2 |
| #7101 | 2 |
| #7069 | 2 |
| #7068 | 2 |
| C162 | 2 |