Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
420 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 3. C2×#1334 | C2 | semidirect |
| 5. #9658 | C2 | semidirect |
| 6. #7156 | C2 | semidirect |
| 7. #7137 | C2 | semidirect |
| 36. C2×SD32 | C8 | semidirect |
| 84. C2×Q16 | C2×C8 | semidirect |
| 122. C2×C16 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2175 | 2 |
| C2×#1580 | 2 |
| C2×#1334 | 2 |
| C2×#1333 | 2 |
| #9658 | 2 |
| #7156 | 2 |
| #7137 | 2 |