Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
204 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #8136 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #7019 | C2 | 1 | 3 | yes | maximal |
| 3 | #7019 | C2 | 1 | 3 | yes | maximal |
| 4 | #1839 | C22 | 1 | 1 | yes | characteristic |
| 5 | #1839 | C22 | 1 | 3 | yes | — |
| 6 | C4×#97 | C22 | 1 | 3 | yes | — |
| 7 | #1273 | C4 | 1 | 1 | yes | characteristic |
| 8 | #1273 | C4 | 1 | 3 | yes | — |
| 9 | C2×#97 | — | 2 | 3 | no | — |
| 10 | C2×#97 | — | 2 | 3 | no | — |
| 11 | C2×C4×C8 | C2×C4 | 1 | 3 | yes | — |
| 12 | C2×C4×C8 | C2×C4 | 1 | 3 | yes | — |
| 13 | C43 | C23 | 1 | 1 | yes | characteristic |
| 14 | #336 | — | 2 | 3 | no | — |
| 15 | #336 | — | 2 | 3 | no | — |
| 16 | #336 | — | 2 | 6 | no | — |
| 17 | C22×C8 | — | 2 | 3 | no | — |
| 18 | C22×C8 | — | 2 | 3 | no | — |
| 19 | C22×C8 | — | 4 | 1 | no | — |
| 20 | C22×C8 | — | 4 | 3 | no | — |
| 21 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C42 | #41 | 1 | 3 | yes | — |
| 23 | C2×C42 | #41 | 1 | 3 | yes | — |
| 24 | #97 | — | 4 | 6 | no | — |
| 25 | #97 | — | 4 | 6 | no | — |
| 26 | C4×C8 | — | 2 | 6 | no | — |
| 27 | C4×C8 | — | 2 | 3 | no | — |
| 28 | C4×C8 | — | 2 | 3 | no | — |
| 29 | C22×C4 | #117 | 1 | 3 | yes | — |
| 30 | C22×C4 | #126 | 1 | 1 | yes | characteristic |
| 31 | C22×C4 | #126 | 1 | 3 | yes | — |
| 32 | C2×C8 | — | 2 | 6 | no | — |
| 33 | C2×C8 | — | 2 | 6 | no | — |
| 34 | C2×C8 | — | 4 | 3 | no | — |
| 35 | C2×C8 | — | 4 | 3 | no | — |
| 36 | C2×C8 | — | 4 | 3 | no | — |
| 37 | C2×C8 | — | 4 | 3 | no | — |
| 38 | C2×C8 | — | 8 | 3 | no | — |
| 39 | C2×C8 | — | 8 | 3 | no | — |
| 40 | C2×C8 | — | 8 | 6 | no | — |
| 41 | C42 | — | 4 | 3 | no | — |
| 42 | C42 | — | 4 | 3 | no | — |
| 43 | C42 | #131 | 1 | 3 | yes | — |
| 44 | C42 | #131 | 1 | 1 | yes | characteristic |
| 45 | C23 | #383 | 1 | 1 | yes | characteristic |
| 46 | C2×C4 | — | 2 | 3 | no | — |
| 47 | C2×C4 | — | 2 | 3 | no | — |
| 48 | C2×C4 | — | 2 | 3 | no | — |
| 49 | C2×C4 | — | 2 | 3 | no | — |
| 50 | C2×C4 | — | 2 | 6 | no | — |
| 51 | C2×C4 | #433 | 1 | 3 | yes | — |
| 52 | C2×C4 | #433 | 1 | 3 | yes | — |
| 53 | C8 | — | 4 | 6 | no | — |
| 54 | C8 | — | 4 | 6 | no | — |
| 55 | C8 | — | 16 | 1 | no | — |
| 56 | C8 | — | 16 | 3 | no | — |
| 57 | C22 | #1840 | 1 | 1 | yes | characteristic |
| 58 | C22 | #1841 | 1 | 3 | yes | — |
| 59 | C22 | #1841 | 1 | 3 | yes | — |
| 60 | C4 | — | 2 | 3 | no | — |
| 61 | C4 | — | 2 | 3 | no | — |
| 62 | C4 | — | 4 | 1 | no | — |
| 63 | C4 | — | 4 | 3 | no | — |
| 64 | C2 | #8802 | 1 | 3 | yes | — |
| 65 | C2 | #8803 | 1 | 1 | yes | characteristic |
| 66 | C2 | #8804 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #383 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #8136 | 2 |
| #7019 | 2 |