← #149507
Subgroups — #149507
Order 512.
183 conjugacy classes of subgroups.
The subgroup lattice has 133 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #13639 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×M64 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C16×M16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7331 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7331 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C8×C32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×M64 | C4 | 1 | 2 | yes | — |
| 9 | #2162 | C4 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C8×M16 | C4 | 1 | 1 | yes | characteristic |
| 12 | C4⋊C32 | C4 | 1 | 2 | yes | — |
| 13 | C22⋊C32 | C4 | 1 | 2 | yes | — |
| 14 | #1397 | C4 | 1 | 2 | yes | — |
| 15 | C4×C32 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×C32 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×C32 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×C32 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1312 | C4 | 1 | 1 | yes | characteristic |
| 20 | #1312 | C4 | 1 | 1 | yes | characteristic |
| 21 | C8×C16 | C22 | 1 | 1 | yes | characteristic |
| 22 | C8×C16 | C22 | 1 | 1 | yes | characteristic |
| 23 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | #431 | C8 | 1 | 1 | yes | characteristic |
| 26 | C4×M16 | C8 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | M64 | — | 2 | 4 | no | — |
| 29 | C2×C32 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×C32 | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×C32 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×C32 | C2×C4 | 1 | 2 | yes | — |
| 33 | C4⋊C16 | C8 | 1 | 2 | yes | — |
| 34 | C22⋊C16 | C8 | 1 | 2 | yes | — |
| 35 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 41 | #322 | C8 | 1 | 1 | yes | characteristic |
| 42 | #322 | C8 | 1 | 1 | yes | characteristic |
| 43 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C2×M16 | C16 | 1 | 2 | yes | — |
| 46 | C22×C8 | C42 | 1 | 1 | yes | characteristic |
| 47 | C22×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 48 | C2×C42 | C2×C8 | 1 | 1 | yes | characteristic |
| 49 | C2×C16 | — | 2 | 2 | no | — |
| 50 | C2×C16 | — | 2 | 2 | no | — |
| 51 | C2×C16 | C42 | 1 | 1 | yes | characteristic |
| 52 | C2×C16 | C42 | 1 | 1 | yes | characteristic |
| 53 | C2×C16 | C2×C8 | 1 | 2 | yes | — |
| 54 | C2×C16 | C2×C8 | 1 | 2 | yes | — |
| 55 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 56 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 57 | C4⋊C8 | C16 | 1 | 2 | yes | — |
| 58 | C22⋊C8 | C16 | 1 | 2 | yes | — |
| 59 | #97 | C16 | 1 | 2 | yes | — |
| 60 | C4×C8 | C42 | 1 | 1 | yes | characteristic |
| 61 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 62 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 63 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 64 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 65 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 66 | C4×C8 | C2×C8 | 1 | 1 | yes | characteristic |
| 67 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C32 | — | 2 | 2 | no | — |
| 69 | C32 | — | 2 | 2 | no | — |
| 70 | C32 | — | 2 | 2 | no | — |
| 71 | C32 | M16 | 1 | 4 | yes | complemented |
| 72 | C22×C4 | C4×C8 | 1 | 1 | yes | characteristic |
| 73 | C22×C4 | C2×C16 | 1 | 1 | yes | characteristic |
| 74 | C22×C4 | C2×C16 | 1 | 1 | yes | characteristic |
| 75 | M16 | — | 2 | 4 | no | — |
| 76 | C2×C8 | — | 2 | 1 | no | — |
| 77 | C2×C8 | — | 2 | 1 | no | — |
| 78 | C2×C8 | — | 2 | 2 | no | — |
| 79 | C2×C8 | C4×C8 | 1 | 1 | yes | characteristic |
| 80 | C2×C8 | C4×C8 | 1 | 1 | yes | characteristic |
| 81 | C2×C8 | C2×C16 | 1 | 2 | yes | — |
| 82 | C2×C8 | C2×C16 | 1 | 2 | yes | — |
| 83 | C2×C8 | C2×C16 | 1 | 2 | yes | — |
| 84 | C2×C8 | C2×C16 | 1 | 2 | yes | — |
| 85 | C2×C8 | C2×C42 | 1 | 1 | yes | characteristic |
| 86 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 87 | C42 | C4×C8 | 1 | 1 | yes | characteristic |
| 88 | C42 | C2×C16 | 1 | 1 | yes | characteristic |
| 89 | C42 | C2×C16 | 1 | 1 | yes | characteristic |
| 90 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 91 | C16 | — | 2 | 1 | no | — |
| 92 | C16 | — | 2 | 1 | no | — |
| 93 | C16 | — | 2 | 2 | no | — |
| 94 | C16 | — | 2 | 2 | no | — |
| 95 | C16 | C2×M16 | 1 | 2 | yes | — |
| 96 | C16 | #131 | 1 | 2 | yes | — |
| 97 | C23 | C4×C16 | 1 | 1 | yes | characteristic |
| 98 | C2×C4 | — | 2 | 1 | no | — |
| 99 | C2×C4 | — | 2 | 1 | no | — |
| 100 | C2×C4 | — | 2 | 2 | no | — |
| 101 | C2×C4 | — | 2 | 2 | no | — |
| 102 | C2×C4 | C4×C16 | 1 | 1 | yes | characteristic |
| 103 | C2×C4 | C4×C16 | 1 | 1 | yes | characteristic |
| 104 | C2×C4 | C4×C16 | 1 | 1 | yes | characteristic |
| 105 | C2×C4 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 106 | C2×C4 | C22×C16 | 1 | 1 | yes | characteristic |
| 107 | C2×C4 | C22×C16 | 1 | 1 | yes | characteristic |
| 108 | C8 | — | 2 | 1 | no | — |
| 109 | C8 | — | 2 | 1 | no | — |
| 110 | C8 | — | 2 | 2 | no | — |
| 111 | C8 | — | 2 | 2 | no | — |
| 112 | C8 | — | 2 | 2 | no | — |
| 113 | C8 | M64 | 1 | 4 | yes | complemented |
| 114 | C8 | C4×M16 | 1 | 1 | yes | characteristic |
| 115 | C8 | #405 | 1 | 1 | yes | characteristic |
| 116 | C8 | #504 | 1 | 2 | yes | — |
| 117 | C22 | — | 2 | 1 | no | — |
| 118 | C22 | — | 2 | 1 | no | — |
| 119 | C22 | — | 2 | 1 | no | — |
| 120 | C22 | C2×C4×C16 | 1 | 1 | yes | characteristic |
| 121 | C4 | — | 2 | 1 | no | — |
| 122 | C4 | — | 2 | 1 | no | — |
| 123 | C4 | — | 2 | 1 | no | — |
| 124 | C4 | C8×M16 | 1 | 1 | yes | characteristic |
| 125 | C4 | #2108 | 1 | 1 | yes | characteristic |
| 126 | C4 | C2×M64 | 1 | 2 | yes | — |
| 127 | C4 | #2258 | 1 | 2 | yes | — |
| 128 | C2 | — | 4 | 1 | no | — |
| 129 | C2 | C16×M16 | 1 | 1 | yes | characteristic |
| 130 | C2 | C4×M64 | 1 | 1 | yes | characteristic |
| 131 | C2 | #13549 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta