Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
339 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13409 | C2 | semidirect |
| 3. #13409 | C2 | semidirect |
| 5. #7121 | C2 | semidirect |
| 6. #7120 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #32462 | 2 |
| #13409 | 2 |
| #10274 | 2 |
| #7121 | 2 |
| #7120 | 2 |
| #7102 | 2 |