Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
989 conjugacy classes of subgroups; only the 352 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 352 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#102 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C22×D4⋊C8 |
| prime radical | #3584 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#131 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33401 | C2 | semidirect |
| 3. #17316 | C2 | semidirect |
| 4. #11604 | C2 | semidirect |
| 5. C2×#1336 | C2 | semidirect |
| 18. C2×#364 | C22 | semidirect |
| 19. #1450 | C22 | semidirect |
| 21. C2×#322 | C22 | semidirect |
| 22. #1336 | C22 | semidirect |
| 23. #1336 | C22 | semidirect |
| 25. #578 | C8 | semidirect |
| 27. #576 | C8 | semidirect |
| 74. #135 | C2×C8 | semidirect |
| 76. C2×Q16 | C2×C8 | semidirect |
| 78. C2×D8 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60183 | 2 |
| #33401 | 2 |
| #17316 | 2 |
| #11604 | 2 |
| C2×#1336 | 2 |