Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
354 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #1667 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ 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. #13262 | C2 | semidirect |
| 4. #13161 | C2 | semidirect |
| 5. C2×#1358 | C2 | semidirect |
| 7. #7160 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2248 | 2 |
| #13262 | 2 |
| #13261 | 2 |
| #13161 | 2 |
| C2×#1358 | 2 |
| C2×#1357 | 2 |
| #7160 | 2 |