Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
167 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 48 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 | #1667 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #12424 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #12509 | C2 | semidirect |
| 2. #12447 | C2 | semidirect |
| 4. #12039 | C2 | semidirect |
| 5. #7100 | C2 | semidirect |
| 6. #7098 | C2 | semidirect |
| 7. #7096 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12509 | 2 |
| #12447 | 2 |
| #12424 | 2 |
| #12039 | 2 |
| #7100 | 2 |
| #7098 | 2 |
| #7096 | 2 |