Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
592 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 196 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 | #2380 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #9090 | C2 | semidirect |
| 3. #9087 | C2 | semidirect |
| 5. #8958 | C2 | semidirect |
| 6. #8800 | C2 | semidirect |
| 32. #492 | C8 | semidirect |
| 66. C2×SD16 | C2×C8 | semidirect |
| 71. C4⋊Q8 | C2×C8 | semidirect |
| 108. C4×C8 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21692 | 2 |
| #9090 | 2 |
| #9087 | 2 |
| #8959 | 2 |
| #8958 | 2 |
| #8800 | 2 |