Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3428 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #56845 | C2 | 1 | 63 | yes | maximal |
| 2 | #2833 | C22 | 1 | 63 | yes | — |
| 3 | #2811 | C22 | 1 | 21 | yes | — |
| 4 | #2789 | C22 | 1 | 63 | yes | — |
| 5 | #2774 | C22 | 1 | 63 | yes | — |
| 6 | #2762 | C22 | 1 | 63 | yes | — |
| 7 | #2762 | C22 | 1 | 63 | yes | — |
| 8 | #2758 | C22 | 1 | 63 | yes | — |
| 9 | #2666 | C22 | 1 | 63 | yes | — |
| 10 | #2588 | C22 | 1 | 63 | yes | — |
| 11 | #2520 | C22 | 1 | 63 | yes | — |
| 12 | #2514 | C22 | 1 | 63 | yes | — |
| 13 | #401 | C23 | 1 | 63 | yes | — |
| 14 | #401 | C23 | 1 | 9 | yes | — |
| 15 | #400 | C23 | 1 | 63 | yes | — |
| 16 | #400 | C23 | 1 | 63 | yes | — |
| 17 | #400 | C23 | 1 | 63 | yes | — |
| 18 | #400 | C23 | 1 | 63 | yes | — |
| 19 | #398 | C23 | 1 | 63 | yes | — |
| 20 | #398 | C23 | 1 | 63 | yes | — |
| 21 | #395 | C23 | 1 | 63 | yes | — |
| 22 | #395 | C23 | 1 | 63 | yes | — |
| 23 | #391 | C23 | 1 | 63 | yes | — |
| 24 | #391 | C23 | 1 | 63 | yes | — |
| 25 | #391 | C23 | 1 | 63 | yes | — |
| 26 | #389 | C23 | 1 | 63 | yes | — |
| 27 | #387 | C23 | 1 | 63 | yes | — |
| 28 | #387 | C23 | 1 | 63 | yes | — |
| 29 | #387 | C23 | 1 | 63 | yes | — |
| 30 | #387 | C23 | 1 | 63 | yes | — |
| 31 | #387 | C23 | 1 | 63 | yes | — |
| 32 | #387 | C23 | 1 | 63 | yes | — |
| 33 | #383 | C23 | 1 | 63 | yes | — |
| 34 | #382 | C23 | 1 | 63 | yes | — |
| 35 | C4×C4⋊C4 | C23 | 1 | 63 | yes | — |
| 36 | C22×Q8 | C24 | 1 | 21 | yes | — |
| 37 | C2×C4⋊C4 | C24 | 1 | 63 | yes | — |
| 38 | C2×C4⋊C4 | C24 | 1 | 63 | yes | — |
| 39 | C2×C4⋊C4 | C24 | 1 | 63 | yes | — |
| 40 | C2×C42 | C24 | 1 | 63 | yes | — |
| 41 | #95 | C24 | 1 | 63 | yes | — |
| 42 | #95 | C24 | 1 | 63 | yes | — |
| 43 | #95 | C24 | 1 | 63 | yes | — |
| 44 | #95 | C24 | 1 | 63 | yes | — |
| 45 | #95 | C24 | 1 | 63 | yes | — |
| 46 | #95 | C24 | 1 | 63 | yes | — |
| 47 | C2×Q8 | — | 4 | 21 | no | — |
| 48 | C2×Q8 | — | 4 | 21 | no | — |
| 49 | C2×Q8 | — | 4 | 21 | no | — |
| 50 | C22×C4 | C25 | 1 | 63 | yes | — |
| 51 | C4⋊C4 | — | 4 | 63 | no | — |
| 52 | C4⋊C4 | — | 4 | 63 | no | — |
| 53 | C4⋊C4 | — | 4 | 63 | no | — |
| 54 | C42 | — | 4 | 63 | no | — |
| 55 | C23 | C26 | 1 | 1 | yes | characteristic |
| 56 | Q8 | — | 16 | 21 | no | — |
| 57 | C2×C4 | — | 2 | 63 | no | — |
| 58 | C2×C4 | — | 2 | 63 | no | — |
| 59 | C2×C4 | — | 2 | 63 | no | — |
| 60 | C22 | 2−1+6 | 1 | 7 | yes | — |
| 61 | C4 | — | 4 | 63 | no | — |
| 62 | C2 | #62980 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #56845 | 2 |