Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
672 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. #1847 | C4 | semidirect |
| 14. #1846 | C4 | semidirect |
| 101. #97 | C4⋊C4 | semidirect |
| 102. C4×C8 | C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20801 | 2 |
| C2×#1841 | 2 |
| #11214 | 2 |
| #11213 | 2 |
| #11212 | 2 |
| #11211 | 2 |