Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
462 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#127 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×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. #12653 | C2 | semidirect |
| 3. C4×C4⋊C16 | C2 | semidirect |
| 4. C2×#1330 | C2 | semidirect |
| 52. C4⋊C16 | D4 | semidirect |
| 239. C4 | #1330 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21681 | 2 |
| #12653 | 2 |
| C4×C4⋊C16 | 2 |
| C2×#1330 | 2 |