Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
556 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 98 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 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×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. #12844 | C2 | semidirect |
| 3. #7874 | C2 | semidirect |
| 4. #7113 | C2 | semidirect |
| 5. #7112 | C2 | semidirect |
| 82. C2×Q16 | M16 | semidirect |
| 89. C2×D8 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21224 | 2 |
| #12844 | 2 |
| #7874 | 2 |
| #7113 | 2 |
| #7112 | 2 |