Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1543 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #23847 | C2 | 1 | 24 | yes | maximal |
| 2 | #20355 | C2 | 1 | 7 | yes | maximal |
| 3 | C2×#519 | C22 | 1 | 7 | yes | — |
| 4 | #2104 | C22 | 1 | 64 | yes | — |
| 5 | #1740 | C22 | 1 | 84 | yes | — |
| 6 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 7 | #519 | — | 2 | 56 | no | — |
| 8 | C2×C4×Q8 | — | 2 | 28 | no | — |
| 9 | C2×#117 | C23 | 1 | 42 | yes | — |
| 10 | #342 | C23 | 1 | 112 | yes | — |
| 11 | C22×Q8 | #41 | 1 | 28 | yes | — |
| 12 | C23×C4 | C24 | 1 | 3 | yes | — |
| 13 | C4×Q8 | — | 4 | 112 | no | — |
| 14 | #117 | — | 2 | 168 | no | — |
| 15 | C2×C4⋊C4 | — | 2 | 42 | no | — |
| 16 | C2×C22⋊C4 | C24 | 1 | 28 | yes | — |
| 17 | C2×C42 | — | 2 | 42 | no | — |
| 18 | C24 | C25 | 1 | 1 | yes | characteristic |
| 19 | C2×Q8 | #126 | 1 | 112 | yes | — |
| 20 | C22×C4 | — | 2 | 28 | no | — |
| 21 | C22×C4 | C2×#41 | 1 | 42 | yes | — |
| 22 | C4⋊C4 | — | 4 | 84 | no | — |
| 23 | C22⋊C4 | — | 2 | 56 | no | — |
| 24 | C42 | — | 4 | 84 | no | — |
| 25 | C23 | — | 8 | 1 | no | — |
| 26 | C23 | C22×#41 | 1 | 7 | yes | — |
| 27 | Q8 | #401 | 1 | 64 | yes | — |
| 28 | C2×C4 | — | 4 | 28 | no | — |
| 29 | C2×C4 | — | 8 | 28 | no | — |
| 30 | C2×C4 | C2×#126 | 1 | 84 | yes | — |
| 31 | C22 | — | 4 | 7 | no | — |
| 32 | C22 | C22×#126 | 1 | 7 | yes | — |
| 33 | C4 | — | 8 | 28 | no | — |
| 34 | C4 | C2×#401 | 1 | 24 | yes | — |
| 35 | C2 | — | 2 | 7 | no | — |
| 36 | C2 | C22×#401 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#401 |
| commutator, Frattini, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23847 | 2 |
| #20355 | 2 |