Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
461 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 96 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 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ 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. #22689 | C2 | semidirect |
| 3. #13261 | C2 | semidirect |
| 4. C2×#1336 | C2 | semidirect |
| 5. #9837 | C2 | semidirect |
| 6. #7268 | C2 | semidirect |
| 7. #7182 | C2 | semidirect |
| 14. #2213 | C4 | semidirect |
| 43. #493 | C2×C4 | semidirect |
| 70. #364 | C2×C4 | semidirect |
| 180. Q16 | #102 | semidirect |
| 192. D8 | #102 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2213 | 2 |
| #22689 | 2 |
| #13261 | 2 |
| C2×#1336 | 2 |
| #9837 | 2 |
| #7268 | 2 |
| #7182 | 2 |