Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1026 conjugacy classes of subgroups; only the 690 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 690 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12898 | C2 | direct (N × H) |
| 39. C2×C42 | C16 | semidirect |
| 74. C42 | C2×C16 | semidirect |
| 84. C2×C4 | C4⋊C16 | semidirect |
| 106. C4 | C2×C4⋊C16 | semidirect |
| 109. C2 | #12898 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C16 | 2 |
| C2×C42×C8 | 2 |
| #12898 | 2 |