Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
8238 conjugacy classes of subgroups; only the 1626 normal subgroups are shown.
This group has 8238 conjugacy classes of subgroups — too many to draw. Only its 1626 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #15190 | C2 | direct (N × H) |
| 42. C2×#95 | Q8 | semidirect |
| 84. #95 | C2×Q8 | semidirect |
| 118. C22×C4 | C4⋊Q8 | semidirect |
| 158. C2×C4 | C2×C4⋊Q8 | semidirect |
| 175. C2 | #15190 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4⋊Q8 | 2 |
| C22×#391 | 2 |
| #15190 | 2 |
| C2×#1444 | 2 |
| C2×C4×#95 | 2 |