Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
402 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 104 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 | #1879 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #17854 | C2 | semidirect |
| 3. #11599 | C2 | semidirect |
| 4. #11593 | C2 | semidirect |
| 5. #9668 | C2 | semidirect |
| 6. #7157 | C2 | semidirect |
| 7. #7140 | C2 | semidirect |
| 35. C2×D16 | C8 | semidirect |
| 89. C2×D8 | C2×C8 | semidirect |
| 116. C2×C16 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2177 | 2 |
| #17854 | 2 |
| #11599 | 2 |
| #11593 | 2 |
| #9668 | 2 |
| #7157 | 2 |
| #7140 | 2 |