Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
355 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8921 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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. #13188 | C2 | semidirect |
| 3. #13182 | C2 | semidirect |
| 6. #7181 | C2 | semidirect |
| 7. #7180 | C2 | semidirect |
| 27. #1359 | C4 | semidirect |
| 65. #364 | C2×C4 | semidirect |
| 66. #364 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23198 | 2 |
| #13188 | 2 |
| #13182 | 2 |
| C2×#1359 | 2 |
| #9768 | 2 |
| #7181 | 2 |
| #7180 | 2 |