Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
672 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 162 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 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ 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 |
|---|---|---|
| 1. #28731 | C2 | semidirect |
| 3. C2×#1713 | C2 | semidirect |
| 4. #19193 | C2 | semidirect |
| 6. #10164 | C2 | semidirect |
| 7. #9798 | C2 | semidirect |
| 8. C8×#108 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28731 | 2 |
| #19639 | 2 |
| C2×#1713 | 2 |
| #19193 | 2 |
| #17956 | 2 |
| #10164 | 2 |
| #9798 | 2 |
| C8×#108 | 2 |