← #213236
Subgroups — #213236
Order 512.
762 conjugacy classes of subgroups.
The subgroup lattice has 239 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #21732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21705 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19596 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #17957 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #9856 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9855 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #9843 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #8822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C4×C82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3167 | C22 | 1 | 4 | yes | — |
| 11 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 12 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2960 | C22 | 1 | 2 | yes | — |
| 14 | C2×#443 | — | 2 | 2 | no | — |
| 15 | #2931 | C22 | 1 | 2 | yes | — |
| 16 | #1994 | C22 | 1 | 2 | yes | — |
| 17 | #1974 | — | 2 | 2 | no | — |
| 18 | #1962 | C22 | 1 | 2 | yes | — |
| 19 | #1945 | — | 2 | 2 | no | — |
| 20 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1772 | — | 2 | 2 | no | — |
| 23 | C42×C8 | C22 | 1 | 2 | yes | — |
| 24 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1708 | C4 | 1 | 2 | yes | complemented |
| 26 | #1707 | C4 | 1 | 2 | yes | complemented |
| 27 | #1706 | C4 | 1 | 2 | yes | complemented |
| 28 | #1705 | C4 | 1 | 2 | yes | complemented |
| 29 | #1590 | — | 2 | 8 | no | — |
| 30 | C8×SD16 | — | 2 | 8 | no | — |
| 31 | #1566 | C22 | 1 | 4 | yes | — |
| 32 | #1493 | C22 | 1 | 4 | yes | — |
| 33 | #1492 | C22 | 1 | 4 | yes | — |
| 34 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 36 | C2×C82 | C22 | 1 | 2 | yes | — |
| 37 | C2×#135 | — | 2 | 2 | no | — |
| 38 | C2×#131 | — | 2 | 2 | no | — |
| 39 | #533 | C23 | 1 | 2 | yes | — |
| 40 | #532 | C23 | 1 | 2 | yes | — |
| 41 | #500 | C2×C4 | 1 | 2 | yes | — |
| 42 | #499 | — | 2 | 2 | no | — |
| 43 | #498 | C2×C4 | 1 | 2 | yes | — |
| 44 | #498 | C2×C4 | 1 | 4 | yes | — |
| 45 | #495 | — | 2 | 2 | no | — |
| 46 | #494 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #492 | C2×C4 | 1 | 2 | yes | — |
| 49 | #492 | C2×C4 | 1 | 4 | yes | — |
| 50 | #487 | C2×C4 | 1 | 4 | yes | — |
| 51 | #486 | C2×C4 | 1 | 4 | yes | — |
| 52 | #465 | — | 2 | 8 | no | — |
| 53 | #446 | — | 2 | 4 | no | — |
| 54 | C8×Q8 | — | 2 | 4 | no | — |
| 55 | #443 | — | 4 | 8 | no | — |
| 56 | C4×SD16 | — | 2 | 8 | no | — |
| 57 | #436 | — | 2 | 4 | no | — |
| 58 | C8×D4 | — | 2 | 4 | no | — |
| 59 | C2×#108 | — | 2 | 2 | no | — |
| 60 | C2×#108 | — | 2 | 2 | no | — |
| 61 | #427 | — | 2 | 2 | no | — |
| 62 | #424 | — | 2 | 2 | no | — |
| 63 | #423 | C23 | 1 | 2 | yes | — |
| 64 | #423 | C23 | 1 | 2 | yes | — |
| 65 | C2×#104 | — | 2 | 4 | no | — |
| 66 | #408 | — | 2 | 2 | no | — |
| 67 | #405 | — | 2 | 2 | no | — |
| 68 | C2×C4×C8 | — | 2 | 4 | no | — |
| 69 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 70 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 71 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 72 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 73 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 75 | C43 | C23 | 1 | 1 | yes | characteristic |
| 76 | #339 | — | 2 | 4 | no | — |
| 77 | #334 | — | 2 | 8 | no | — |
| 78 | Q8⋊C8 | — | 2 | 8 | no | — |
| 79 | D4⋊C8 | — | 2 | 8 | no | — |
| 80 | C82 | — | 2 | 2 | no | — |
| 81 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 82 | C82 | C2×C4 | 1 | 2 | yes | — |
| 83 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 84 | C82 | D4 | 1 | 4 | yes | — |
| 85 | C82 | D4 | 1 | 4 | yes | — |
| 86 | C2×#41 | — | 2 | 2 | no | — |
| 87 | #135 | — | 4 | 4 | no | — |
| 88 | #135 | — | 4 | 4 | no | — |
| 89 | C2×Q16 | — | 2 | 2 | no | — |
| 90 | C2×SD16 | — | 2 | 4 | no | — |
| 91 | C2×SD16 | — | 2 | 8 | no | — |
| 92 | C2×D8 | — | 2 | 2 | no | — |
| 93 | #131 | — | 4 | 8 | no | — |
| 94 | C2×M16 | — | 2 | 2 | no | — |
| 95 | C2×M16 | — | 2 | 2 | no | — |
| 96 | C2×M16 | — | 4 | 2 | no | — |
| 97 | C22×C8 | — | 2 | 2 | no | — |
| 98 | C22×C8 | — | 4 | 2 | no | — |
| 99 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 101 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C22×C8 | #41 | 1 | 4 | yes | — |
| 103 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 104 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 105 | #127 | C22×C4 | 1 | 2 | yes | — |
| 106 | #125 | — | 2 | 2 | no | — |
| 107 | #124 | — | 2 | 2 | no | — |
| 108 | C22⋊Q8 | — | 2 | 4 | no | — |
| 109 | #121 | — | 2 | 4 | no | — |
| 110 | C4×Q8 | — | 2 | 4 | no | — |
| 111 | C4×D4 | — | 2 | 4 | no | — |
| 112 | #117 | — | 2 | 2 | no | — |
| 113 | C2×C42 | — | 2 | 2 | no | — |
| 114 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 115 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 116 | #108 | — | 4 | 4 | no | — |
| 117 | #108 | — | 4 | 4 | no | — |
| 118 | #107 | — | 2 | 4 | no | — |
| 119 | #106 | — | 2 | 4 | no | — |
| 120 | #106 | — | 2 | 8 | no | — |
| 121 | C4⋊C8 | — | 2 | 4 | no | — |
| 122 | C4⋊C8 | — | 2 | 4 | no | — |
| 123 | C4⋊C8 | — | 4 | 4 | no | — |
| 124 | #104 | — | 4 | 8 | no | — |
| 125 | #103 | — | 2 | 8 | no | — |
| 126 | #102 | — | 2 | 8 | no | — |
| 127 | C22⋊C8 | — | 4 | 4 | no | — |
| 128 | #97 | — | 4 | 2 | no | — |
| 129 | C4×C8 | — | 2 | 16 | no | — |
| 130 | C4×C8 | — | 2 | 4 | no | — |
| 131 | C4×C8 | — | 2 | 2 | no | — |
| 132 | C4×C8 | — | 4 | 2 | no | — |
| 133 | C4×C8 | SD16 | 1 | 8 | yes | complemented |
| 134 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 135 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 136 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 137 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 138 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 139 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 140 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 141 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 142 | C4×C8 | #41 | 1 | 2 | yes | — |
| 143 | #41 | — | 4 | 4 | no | — |
| 144 | #41 | — | 4 | 4 | no | — |
| 145 | C2×Q8 | — | 2 | 4 | no | — |
| 146 | C2×Q8 | — | 2 | 2 | no | — |
| 147 | C2×D4 | — | 2 | 4 | no | — |
| 148 | C2×D4 | — | 2 | 2 | no | — |
| 149 | C2×D4 | — | 4 | 2 | no | — |
| 150 | C22×C4 | — | 2 | 2 | no | — |
| 151 | C22×C4 | — | 4 | 2 | no | — |
| 152 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 153 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 154 | Q16 | — | 4 | 4 | no | — |
| 155 | SD16 | — | 4 | 8 | no | — |
| 156 | SD16 | — | 4 | 16 | no | — |
| 157 | D8 | — | 4 | 4 | no | — |
| 158 | M16 | — | 4 | 4 | no | — |
| 159 | M16 | — | 4 | 4 | no | — |
| 160 | M16 | — | 8 | 4 | no | — |
| 161 | C2×C8 | — | 2 | 8 | no | — |
| 162 | C2×C8 | — | 2 | 4 | no | — |
| 163 | C2×C8 | — | 2 | 2 | no | — |
| 164 | C2×C8 | — | 2 | 8 | no | — |
| 165 | C2×C8 | — | 4 | 2 | no | — |
| 166 | C2×C8 | — | 4 | 2 | no | — |
| 167 | C2×C8 | — | 8 | 4 | no | — |
| 168 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 169 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 170 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 171 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 172 | C2×C8 | #127 | 1 | 2 | yes | — |
| 173 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | C2×SD16 | 1 | 4 | yes | — |
| 176 | C2×C8 | C2×SD16 | 1 | 4 | yes | — |
| 177 | C2×C8 | #135 | 1 | 4 | yes | — |
| 178 | C4⋊C4 | — | 2 | 4 | no | — |
| 179 | C4⋊C4 | — | 2 | 4 | no | — |
| 180 | C4⋊C4 | — | 2 | 4 | no | — |
| 181 | C4⋊C4 | — | 4 | 4 | no | — |
| 182 | C22⋊C4 | — | 4 | 2 | no | — |
| 183 | C22⋊C4 | — | 4 | 4 | no | — |
| 184 | C42 | — | 2 | 8 | no | — |
| 185 | C42 | — | 2 | 2 | no | — |
| 186 | C42 | — | 4 | 2 | no | — |
| 187 | C42 | C2×SD16 | 1 | 4 | yes | — |
| 188 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 189 | C42 | C22×D4 | 1 | 2 | yes | — |
| 190 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 191 | C23 | — | 4 | 2 | no | — |
| 192 | C23 | #532 | 1 | 1 | yes | characteristic |
| 193 | Q8 | — | 4 | 4 | no | — |
| 194 | Q8 | — | 4 | 8 | no | — |
| 195 | D4 | — | 4 | 4 | no | — |
| 196 | D4 | — | 4 | 8 | no | — |
| 197 | D4 | — | 8 | 4 | no | — |
| 198 | C2×C4 | — | 2 | 8 | no | — |
| 199 | C2×C4 | — | 2 | 4 | no | — |
| 200 | C2×C4 | — | 2 | 4 | no | — |
| 201 | C2×C4 | — | 4 | 4 | no | — |
| 202 | C2×C4 | — | 4 | 2 | no | — |
| 203 | C2×C4 | — | 8 | 2 | no | — |
| 204 | C2×C4 | — | 8 | 2 | no | — |
| 205 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 207 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | #532 | 1 | 2 | yes | — |
| 209 | C2×C4 | C22×SD16 | 1 | 2 | yes | — |
| 210 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 211 | C8 | — | 2 | 8 | no | — |
| 212 | C8 | — | 2 | 2 | no | — |
| 213 | C8 | — | 2 | 4 | no | — |
| 214 | C8 | — | 8 | 2 | no | — |
| 215 | C8 | — | 8 | 2 | no | — |
| 216 | C8 | C4×SD16 | 1 | 4 | yes | — |
| 217 | C8 | C4×SD16 | 1 | 4 | yes | — |
| 218 | C8 | #443 | 1 | 8 | yes | complemented |
| 219 | C8 | #492 | 1 | 2 | yes | — |
| 220 | C8 | #492 | 1 | 2 | yes | — |
| 221 | C22 | — | 2 | 2 | no | — |
| 222 | C22 | — | 8 | 2 | no | — |
| 223 | C22 | — | 8 | 4 | no | — |
| 224 | C22 | C4×#127 | 1 | 1 | yes | characteristic |
| 225 | C22 | #3167 | 1 | 2 | yes | — |
| 226 | C4 | — | 2 | 8 | no | — |
| 227 | C4 | — | 2 | 2 | no | — |
| 228 | C4 | — | 8 | 2 | no | — |
| 229 | C4 | — | 8 | 4 | no | — |
| 230 | C4 | C2×C4×SD16 | 1 | 2 | yes | — |
| 231 | C4 | C2×#443 | 1 | 4 | yes | — |
| 232 | C4 | C2×#492 | 1 | 1 | yes | characteristic |
| 233 | C4 | #3147 | 1 | 1 | yes | characteristic |
| 234 | C2 | — | 2 | 2 | no | — |
| 235 | C2 | — | 16 | 2 | no | — |
| 236 | C2 | C4×#492 | 1 | 1 | yes | characteristic |
| 237 | C2 | #21705 | 1 | 2 | yes | — |
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