Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
655 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #23847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11319 | C2 | 1 | 7 | yes | maximal |
| 3 | #10711 | C2 | 1 | 7 | yes | maximal |
| 4 | C2×#405 | C22 | 1 | 7 | yes | — |
| 5 | #2104 | C4 | 1 | 8 | yes | — |
| 6 | C2×#344 | C22 | 1 | 7 | yes | — |
| 7 | C2×#344 | C22 | 1 | 7 | yes | — |
| 8 | #1742 | C22 | 1 | 7 | yes | — |
| 9 | #1740 | C22 | 1 | 7 | yes | — |
| 10 | C22×M16 | C23 | 1 | 7 | yes | — |
| 11 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×#117 | C23 | 1 | 7 | yes | — |
| 13 | #405 | — | 2 | 7 | no | — |
| 14 | #405 | — | 2 | 7 | no | — |
| 15 | #405 | — | 2 | 7 | no | — |
| 16 | #405 | — | 2 | 7 | no | — |
| 17 | C2×#97 | — | 2 | 7 | no | — |
| 18 | C2×C4×C8 | — | 2 | 7 | no | — |
| 19 | #344 | — | 2 | 14 | no | — |
| 20 | #344 | — | 2 | 14 | no | — |
| 21 | #343 | — | 2 | 14 | no | — |
| 22 | #342 | C2×C4 | 1 | 28 | yes | — |
| 23 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 24 | C2×M16 | — | 2 | 7 | no | — |
| 25 | C2×M16 | — | 2 | 7 | no | — |
| 26 | C2×M16 | — | 2 | 7 | no | — |
| 27 | C2×M16 | — | 2 | 7 | no | — |
| 28 | C2×M16 | — | 2 | 7 | no | — |
| 29 | C2×M16 | #41 | 1 | 14 | yes | — |
| 30 | C22×C8 | — | 2 | 7 | no | — |
| 31 | C22×C8 | #41 | 1 | 7 | yes | — |
| 32 | C22×C8 | #41 | 1 | 7 | yes | — |
| 33 | #117 | — | 2 | 7 | no | — |
| 34 | #117 | — | 2 | 7 | no | — |
| 35 | #117 | — | 2 | 7 | no | — |
| 36 | #117 | — | 2 | 7 | no | — |
| 37 | C2×C4⋊C4 | — | 2 | 7 | no | — |
| 38 | C2×C22⋊C4 | C22×C4 | 1 | 14 | yes | — |
| 39 | C2×C42 | — | 2 | 7 | no | — |
| 40 | #97 | — | 4 | 7 | no | — |
| 41 | #97 | — | 4 | 7 | no | — |
| 42 | C4×C8 | — | 4 | 7 | no | — |
| 43 | C4×C8 | — | 4 | 7 | no | — |
| 44 | C24 | C23×C4 | 1 | 1 | yes | characteristic |
| 45 | C22×C4 | — | 2 | 14 | no | — |
| 46 | C22×C4 | C2×#41 | 1 | 7 | yes | — |
| 47 | C22×C4 | C2×#41 | 1 | 7 | yes | — |
| 48 | M16 | — | 2 | 14 | no | — |
| 49 | M16 | — | 2 | 14 | no | — |
| 50 | M16 | — | 4 | 7 | no | — |
| 51 | M16 | — | 4 | 7 | no | — |
| 52 | C2×C8 | — | 2 | 14 | no | — |
| 53 | C2×C8 | — | 4 | 7 | no | — |
| 54 | C2×C8 | — | 4 | 7 | no | — |
| 55 | C2×C8 | #126 | 1 | 7 | yes | — |
| 56 | C2×C8 | #126 | 1 | 7 | yes | — |
| 57 | C2×C8 | #126 | 1 | 7 | yes | — |
| 58 | C2×C8 | #126 | 1 | 7 | yes | — |
| 59 | C4⋊C4 | — | 4 | 7 | no | — |
| 60 | C4⋊C4 | — | 4 | 7 | no | — |
| 61 | C22⋊C4 | — | 2 | 14 | no | — |
| 62 | C22⋊C4 | — | 2 | 14 | no | — |
| 63 | C42 | — | 4 | 7 | no | — |
| 64 | C42 | — | 4 | 7 | no | — |
| 65 | C23 | — | 8 | 1 | no | — |
| 66 | C23 | C4×#41 | 1 | 7 | yes | — |
| 67 | C2×C4 | — | 4 | 14 | no | — |
| 68 | C2×C4 | — | 8 | 14 | no | — |
| 69 | C2×C4 | C2×#126 | 1 | 7 | yes | — |
| 70 | C2×C4 | #529 | 1 | 7 | yes | — |
| 71 | C2×C4 | #529 | 1 | 7 | yes | — |
| 72 | C2×C4 | #529 | 1 | 7 | yes | — |
| 73 | C8 | — | 4 | 14 | no | — |
| 74 | C8 | #401 | 1 | 7 | yes | — |
| 75 | C8 | #401 | 1 | 1 | yes | characteristic |
| 76 | C22 | — | 4 | 7 | no | — |
| 77 | C22 | C4×#126 | 1 | 7 | yes | — |
| 78 | C4 | — | 8 | 14 | no | — |
| 79 | C4 | C2×#401 | 1 | 1 | yes | characteristic |
| 80 | C4 | #2401 | 1 | 7 | yes | — |
| 81 | C2 | — | 2 | 7 | no | — |
| 82 | C2 | C4×#401 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #401 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#401 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23847 | 2 |
| #11319 | 2 |
| #10711 | 2 |