Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4870 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C4×#95 | C2 | 1 | 15 | yes | maximal |
| 2 | C23×C42 | C22 | 1 | 15 | yes | — |
| 3 | C22×#95 | C22 | 1 | 20 | yes | — |
| 4 | C4×#95 | C4 | 1 | 120 | yes | complemented |
| 5 | C24×C4 | C23 | 1 | 15 | yes | — |
| 6 | C22×C42 | — | 4 | 60 | no | — |
| 7 | C22×C42 | C2×C4 | 1 | 180 | yes | — |
| 8 | C2×#95 | C2×C4 | 1 | 240 | yes | — |
| 9 | C25 | C24 | 1 | 1 | yes | characteristic |
| 10 | C23×C4 | — | 2 | 90 | no | — |
| 11 | C23×C4 | — | 2 | 30 | no | — |
| 12 | C23×C4 | C22×C4 | 1 | 180 | yes | — |
| 13 | C23×C4 | C22×C4 | 1 | 30 | yes | — |
| 14 | C2×C42 | — | 4 | 360 | no | — |
| 15 | C2×C42 | C42 | 1 | 240 | yes | complemented |
| 16 | #95 | C42 | 1 | 320 | yes | complemented |
| 17 | C24 | C23×C4 | 1 | 15 | yes | — |
| 18 | C24 | H(5,𝔽2) | 1 | 10 | yes | — |
| 19 | C24 | 2−1+4 | 1 | 6 | yes | — |
| 20 | C22×C4 | — | 2 | 90 | no | — |
| 21 | C22×C4 | — | 2 | 360 | no | — |
| 22 | C22×C4 | — | 2 | 30 | no | — |
| 23 | C22×C4 | — | 2 | 360 | no | — |
| 24 | C22×C4 | C2×C42 | 1 | 240 | yes | — |
| 25 | C22×C4 | C2×C42 | 1 | 180 | yes | — |
| 26 | C42 | — | 4 | 240 | no | — |
| 27 | C23 | C22×C42 | 1 | 15 | yes | — |
| 28 | C23 | C22×C42 | 1 | 20 | yes | — |
| 29 | C23 | #518 | 1 | 45 | yes | — |
| 30 | C23 | #519 | 1 | 15 | yes | — |
| 31 | C23 | #520 | 1 | 60 | yes | — |
| 32 | C2×C4 | — | 2 | 180 | no | — |
| 33 | C2×C4 | — | 2 | 180 | no | — |
| 34 | C2×C4 | — | 2 | 480 | no | — |
| 35 | C2×C4 | C43 | 1 | 120 | yes | — |
| 36 | C22 | C2×C43 | 1 | 15 | yes | — |
| 37 | C22 | #2273 | 1 | 15 | yes | — |
| 38 | C22 | #2274 | 1 | 45 | yes | — |
| 39 | C22 | #2275 | 1 | 60 | yes | — |
| 40 | C22 | #2276 | 1 | 20 | yes | — |
| 41 | C4 | — | 2 | 120 | no | — |
| 42 | C2 | C44 | 1 | 1 | yes | characteristic |
| 43 | C2 | #13749 | 1 | 15 | yes | — |
| 44 | C2 | #13750 | 1 | 15 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C2 | C44 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C4×#95 | C4 | semidirect |
| 15. C2×C42 | C42 | semidirect |
| 16. #95 | C42 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#95 | 2 |