Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
339 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 104 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 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ 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 |
|---|---|---|
| 53. C4⋊C16 | Q8 | semidirect |
| 55. C4×C16 | Q8 | semidirect |
| 61. #335 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28813 | 2 |
| #13176 | 2 |
| #12905 | 2 |
| #9614 | 2 |
| #7274 | 2 |
| #7171 | 2 |
| #7167 | 2 |