Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
51 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×Q128 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | M256 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #7547 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×Q64 | — | 2 | 1 | no | — |
| 5 | Q128 | — | 2 | 1 | no | — |
| 6 | Q128 | C4 | 1 | 2 | yes | — |
| 7 | C2×C64 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1425 | — | 2 | 1 | no | — |
| 9 | C128 | — | 2 | 1 | no | — |
| 10 | C2×Q32 | — | 4 | 1 | no | — |
| 11 | Q64 | — | 2 | 2 | no | — |
| 12 | Q64 | — | 4 | 1 | no | — |
| 13 | C2×C32 | D4 | 1 | 1 | yes | characteristic |
| 14 | #368 | — | 4 | 1 | no | — |
| 15 | C64 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C64 | D4 | 1 | 1 | yes | characteristic |
| 17 | C2×Q16 | — | 8 | 1 | no | — |
| 18 | Q32 | — | 4 | 2 | no | — |
| 19 | Q32 | — | 8 | 1 | no | — |
| 20 | C2×C16 | D8 | 1 | 1 | yes | characteristic |
| 21 | #108 | — | 8 | 1 | no | — |
| 22 | C32 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 23 | C32 | SD16 | 1 | 1 | yes | characteristic |
| 24 | C2×Q8 | — | 16 | 1 | no | — |
| 25 | Q16 | — | 8 | 2 | no | — |
| 26 | Q16 | — | 16 | 1 | no | — |
| 27 | M16 | — | 16 | 1 | no | — |
| 28 | C2×C8 | D16 | 1 | 1 | yes | characteristic |
| 29 | C16 | #102 | 1 | 1 | yes | characteristic |
| 30 | C16 | SD32 | 1 | 1 | yes | characteristic |
| 31 | Q8 | — | 16 | 2 | no | — |
| 32 | Q8 | — | 32 | 1 | no | — |
| 33 | C2×C4 | — | 32 | 1 | no | — |
| 34 | C2×C4 | D32 | 1 | 1 | yes | characteristic |
| 35 | C8 | — | 32 | 1 | no | — |
| 36 | C8 | #357 | 1 | 1 | yes | characteristic |
| 37 | C8 | SD64 | 1 | 1 | yes | characteristic |
| 38 | C22 | SD128 | 1 | 1 | yes | characteristic |
| 39 | C4 | — | 32 | 2 | no | — |
| 40 | C4 | #1415 | 1 | 1 | yes | characteristic |
| 41 | C4 | D64 | 1 | 1 | yes | characteristic |
| 42 | C2 | — | 2 | 1 | no | — |
| 43 | C2 | #7537 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7537 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| prime radical | C22 | SD128 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q128 | 2 |
| M256 | 2 |
| #7547 | 2 |