← #512749
Subgroups — #512749
Order 512.
418 conjugacy classes of subgroups.
The subgroup lattice has 166 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#2258 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33441 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13648 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #13647 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #13630 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13558 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #13549 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#504 | C22 | 1 | 2 | yes | — |
| 9 | #2955 | C4 | 1 | 1 | yes | characteristic |
| 10 | #2258 | — | 2 | 2 | no | — |
| 11 | #2258 | — | 2 | 2 | no | — |
| 12 | #2258 | C22 | 1 | 4 | yes | complemented |
| 13 | #2258 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×M64 | — | 2 | 2 | no | — |
| 15 | C2×M64 | C22 | 1 | 2 | yes | — |
| 16 | C2×M64 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×C32 | — | 2 | 2 | no | — |
| 18 | C22×C32 | C22 | 1 | 1 | yes | characteristic |
| 19 | #2171 | C4 | 1 | 4 | yes | — |
| 20 | #2170 | C22 | 1 | 4 | yes | — |
| 21 | #2153 | C4 | 1 | 1 | yes | characteristic |
| 22 | #2117 | C4 | 1 | 2 | yes | — |
| 23 | #2108 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1422 | C22 | 1 | 4 | yes | complemented |
| 25 | #1401 | C22 | 1 | 8 | yes | complemented |
| 26 | #1397 | C22 | 1 | 2 | yes | complemented |
| 27 | C4×C32 | C22 | 1 | 2 | yes | complemented |
| 28 | #575 | C8 | 1 | 1 | yes | characteristic |
| 29 | C2×#131 | C2×C4 | 1 | 2 | yes | — |
| 30 | #504 | — | 2 | 2 | no | — |
| 31 | #504 | — | 2 | 2 | no | — |
| 32 | #504 | C2×C4 | 1 | 4 | yes | — |
| 33 | #504 | C23 | 1 | 4 | yes | — |
| 34 | C2×M32 | — | 2 | 2 | no | — |
| 35 | C2×M32 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×M32 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22×C16 | — | 2 | 2 | no | — |
| 38 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 39 | #444 | C8 | 1 | 4 | yes | — |
| 40 | #443 | C2×C4 | 1 | 4 | yes | — |
| 41 | #430 | C8 | 1 | 1 | yes | characteristic |
| 42 | #421 | C8 | 1 | 2 | yes | — |
| 43 | #405 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | M64 | — | 2 | 2 | no | — |
| 45 | M64 | — | 2 | 2 | no | — |
| 46 | M64 | — | 2 | 4 | no | — |
| 47 | M64 | — | 2 | 4 | no | — |
| 48 | M64 | C23 | 1 | 4 | yes | — |
| 49 | C2×C32 | — | 2 | 2 | no | — |
| 50 | C2×C32 | — | 2 | 4 | no | — |
| 51 | C2×C32 | — | 2 | 4 | no | — |
| 52 | C2×C32 | D4 | 1 | 2 | yes | — |
| 53 | C2×C32 | D4 | 1 | 2 | yes | — |
| 54 | C2×C32 | C23 | 1 | 2 | yes | — |
| 55 | C2×C32 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C32 | C23 | 1 | 1 | yes | characteristic |
| 57 | #364 | C2×C4 | 1 | 4 | yes | — |
| 58 | #350 | C2×C4 | 1 | 8 | yes | — |
| 59 | #346 | C2×C4 | 1 | 2 | yes | — |
| 60 | C4×C16 | C23 | 1 | 2 | yes | — |
| 61 | C2×#41 | C2×C8 | 1 | 2 | yes | — |
| 62 | #137 | C16 | 1 | 4 | yes | — |
| 63 | AGL(1,ℤ/8) | C16 | 1 | 4 | yes | — |
| 64 | #135 | C2×C8 | 1 | 4 | yes | — |
| 65 | #131 | — | 2 | 2 | no | — |
| 66 | #131 | — | 2 | 2 | no | — |
| 67 | #131 | C2×C8 | 1 | 4 | yes | — |
| 68 | #131 | C22×C4 | 1 | 4 | yes | — |
| 69 | C2×M16 | — | 2 | 2 | no | — |
| 70 | C2×M16 | C2×C8 | 1 | 2 | yes | — |
| 71 | C2×M16 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C22×C8 | — | 2 | 2 | no | — |
| 73 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | #117 | C2×C8 | 1 | 1 | yes | characteristic |
| 75 | M32 | — | 2 | 2 | no | — |
| 76 | M32 | — | 2 | 2 | no | — |
| 77 | M32 | — | 2 | 4 | no | — |
| 78 | M32 | — | 2 | 4 | no | — |
| 79 | M32 | C22×C4 | 1 | 4 | yes | — |
| 80 | C2×C16 | — | 2 | 2 | no | — |
| 81 | C2×C16 | — | 2 | 4 | no | — |
| 82 | C2×C16 | — | 2 | 4 | no | — |
| 83 | C2×C16 | C22×C4 | 1 | 2 | yes | — |
| 84 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 86 | C2×C16 | #41 | 1 | 2 | yes | — |
| 87 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 88 | #108 | C2×C8 | 1 | 4 | yes | — |
| 89 | #104 | C2×C8 | 1 | 8 | yes | — |
| 90 | #97 | C2×C8 | 1 | 2 | yes | — |
| 91 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 92 | C32 | — | 2 | 4 | no | — |
| 93 | C32 | — | 2 | 2 | no | — |
| 94 | C32 | C2×D4 | 1 | 2 | yes | — |
| 95 | C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | #41 | — | 2 | 2 | no | — |
| 98 | #41 | — | 2 | 2 | no | — |
| 99 | #41 | C2×C16 | 1 | 4 | yes | — |
| 100 | #41 | C22×C8 | 1 | 4 | yes | — |
| 101 | C2×Q8 | C2×C16 | 1 | 2 | yes | — |
| 102 | C2×D4 | — | 2 | 2 | no | — |
| 103 | C2×D4 | C2×C16 | 1 | 2 | yes | — |
| 104 | C22×C4 | — | 2 | 2 | no | — |
| 105 | C22×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 106 | Q16 | C2×C16 | 1 | 4 | yes | — |
| 107 | SD16 | C2×C16 | 1 | 8 | yes | — |
| 108 | D8 | C2×C16 | 1 | 4 | yes | — |
| 109 | M16 | — | 2 | 2 | no | — |
| 110 | M16 | — | 2 | 4 | no | — |
| 111 | M16 | — | 2 | 4 | no | — |
| 112 | M16 | C2×C16 | 1 | 4 | yes | — |
| 113 | M16 | C22×C8 | 1 | 4 | yes | — |
| 114 | C2×C8 | — | 2 | 2 | no | — |
| 115 | C2×C8 | — | 2 | 4 | no | — |
| 116 | C2×C8 | — | 2 | 4 | no | — |
| 117 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 118 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 119 | C2×C8 | C22×C8 | 1 | 2 | yes | — |
| 120 | C2×C8 | #131 | 1 | 2 | yes | — |
| 121 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 122 | C4⋊C4 | — | 2 | 1 | no | — |
| 123 | C22⋊C4 | — | 2 | 1 | no | — |
| 124 | C42 | C22×C8 | 1 | 2 | yes | — |
| 125 | C16 | — | 2 | 4 | no | — |
| 126 | C16 | — | 2 | 2 | no | — |
| 127 | C16 | C4×D4 | 1 | 2 | yes | — |
| 128 | C16 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C16 | C2×#41 | 1 | 1 | yes | characteristic |
| 130 | C23 | — | 2 | 2 | no | — |
| 131 | C23 | #504 | 1 | 1 | yes | characteristic |
| 132 | Q8 | — | 2 | 2 | no | — |
| 133 | Q8 | C22×C16 | 1 | 4 | yes | — |
| 134 | D4 | — | 2 | 4 | no | — |
| 135 | D4 | — | 2 | 4 | no | — |
| 136 | D4 | — | 2 | 2 | no | — |
| 137 | D4 | C22×C16 | 1 | 4 | yes | — |
| 138 | C2×C4 | — | 2 | 2 | no | — |
| 139 | C2×C4 | — | 2 | 4 | no | — |
| 140 | C2×C4 | — | 2 | 4 | no | — |
| 141 | C2×C4 | — | 2 | 2 | no | — |
| 142 | C2×C4 | C8×D4 | 1 | 2 | yes | — |
| 143 | C2×C4 | C22×C16 | 1 | 2 | yes | — |
| 144 | C2×C4 | #504 | 1 | 1 | yes | characteristic |
| 145 | C2×C4 | C23×C8 | 1 | 1 | yes | characteristic |
| 146 | C8 | — | 2 | 4 | no | — |
| 147 | C8 | C8×D4 | 1 | 2 | yes | — |
| 148 | C8 | C22×C16 | 1 | 4 | yes | — |
| 149 | C8 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 150 | C8 | C2×#131 | 1 | 1 | yes | characteristic |
| 151 | C22 | — | 2 | 4 | no | — |
| 152 | C22 | — | 4 | 1 | no | — |
| 153 | C22 | — | 4 | 4 | no | — |
| 154 | C22 | D4×C16 | 1 | 2 | yes | — |
| 155 | C22 | C2×#504 | 1 | 1 | yes | characteristic |
| 156 | C4 | — | 2 | 4 | no | — |
| 157 | C4 | — | 4 | 2 | no | — |
| 158 | C4 | D4×C16 | 1 | 2 | yes | — |
| 159 | C4 | C2×C8×D4 | 1 | 1 | yes | characteristic |
| 160 | C4 | C23×C16 | 1 | 1 | yes | characteristic |
| 161 | C2 | — | 2 | 1 | no | — |
| 162 | C2 | — | 2 | 2 | no | — |
| 163 | C2 | — | 4 | 4 | no | — |
| 164 | C2 | C2×D4×C16 | 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