Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1714 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 1714 conjugacy classes of subgroups — too many to draw. Only its 562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3391 | C2 | semidirect |
| 2. #33958 | C2 | direct (N × H) |
| 3. C2×#2246 | C2 | semidirect |
| 4. C2×#2206 | C2 | semidirect |
| 5. C2×Q8×C16 | C2 | semidirect |
| 6. C2×C4×D16 | C2 | semidirect |
| 11. #3391 | C22 | semidirect |
| 16. #2246 | C22 | semidirect |
| 17. #2206 | C22 | semidirect |
| 18. Q8×C16 | C22 | semidirect |
| 19. C4×D16 | C22 | semidirect |
| 83. C2×C16 | #41 | semidirect |
| 94. C2×Q8 | D16 | semidirect |
| 127. C16 | C2×#41 | semidirect |
| 133. Q8 | C2×D16 | semidirect |
| 169. C2 | #33958 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3391 | 2 |
| #33958 | 2 |
| C2×#2246 | 2 |
| C2×#2206 | 2 |
| C2×Q8×C16 | 2 |
| C2×C4×D16 | 2 |