Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
351 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ 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 |
|---|---|---|
| 4. #11402 | C2 | semidirect |
| 5. C8×#108 | C2 | semidirect |
| 6. #7176 | C2 | semidirect |
| 7. #7159 | C2 | semidirect |
| 98. #108 | M16 | semidirect |
| 207. C8 | #361 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1589 | 2 |
| C4×#361 | 2 |
| #11599 | 2 |
| #11402 | 2 |
| C8×#108 | 2 |
| #7176 | 2 |
| #7159 | 2 |