Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
698 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 282 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 | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 65. #322 | Q8 | semidirect |
| 66. C82 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33316 | 2 |
| #33289 | 2 |
| #33256 | 2 |
| #30305 | 2 |
| #8819 | 2 |
| #8815 | 2 |
| #8814 | 2 |
| C8×#97 | 2 |