Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
383 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 124 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 | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 103. Q32 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33954 | 2 |
| #20008 | 2 |
| #12522 | 2 |
| #12462 | 2 |
| #12450 | 2 |
| #12410 | 2 |
| #12405 | 2 |
| #12054 | 2 |
| C8×Q32 | 2 |