Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
301 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 100 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 | #1853 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #448 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ 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 |
|---|---|---|
| 24. Q8⋊C16 | C4 | semidirect |
| 25. Q8⋊C16 | C4 | semidirect |
| 196. Q8 | #365 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#108 | 2 |
| #13183 | 2 |
| #12903 | 2 |
| #11607 | 2 |
| #8643 | 2 |
| #7280 | 2 |
| #7173 | 2 |