Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
346 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ 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 |
|---|---|---|
| 16. #1579 | C4 | semidirect |
| 22. #1334 | C4 | semidirect |
| 47. #358 | C8 | semidirect |
| 121. Q16 | C4×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1579 | 2 |
| #12638 | 2 |
| C2×#1334 | 2 |
| C2×#1313 | 2 |
| #8284 | 2 |
| #7178 | 2 |
| #7156 | 2 |