Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
374 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8174 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ #1726 ◁ 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 |
|---|---|---|
| 2. #12660 | C2 | semidirect |
| 4. #8138 | C2 | semidirect |
| 5. #7180 | C2 | semidirect |
| 21. #1335 | C4 | semidirect |
| 58. #364 | C2×C4 | semidirect |
| 65. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20514 | 2 |
| #12660 | 2 |
| #11602 | 2 |
| #8138 | 2 |
| #7180 | 2 |