Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
319 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ 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. C4×#364 | C2 | semidirect |
| 3. #8133 | C2 | semidirect |
| 4. #7119 | C2 | semidirect |
| 26. C4×Q16 | C8 | semidirect |
| 27. C4×Q16 | C8 | semidirect |
| 30. C4×D8 | C8 | semidirect |
| 31. C4×D8 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21292 | 2 |
| C4×#364 | 2 |
| #8133 | 2 |
| #7119 | 2 |
| #7114 | 2 |