← #17951927
Subgroups — #17951927
Order 4352.
420 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #14544075 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14544075 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14543987 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#11763651 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #14543617 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #14543617 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #14537424 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #11763825 | C22 | 1 | 1 | yes | characteristic |
| 9 | #11763825 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×#3624 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×#3624 | C22 | 1 | 1 | yes | characteristic |
| 12 | Q8⋊#3624 | C4 | 1 | 2 | yes | — |
| 13 | Q8⋊#3624 | C4 | 1 | 2 | yes | — |
| 14 | D4⋊#3624 | C4 | 1 | 2 | yes | — |
| 15 | D4⋊#3624 | C4 | 1 | 2 | yes | — |
| 16 | #11763651 | — | 2 | 1 | no | — |
| 17 | #11763651 | — | 2 | 1 | no | — |
| 18 | #11763650 | — | 2 | 1 | no | — |
| 19 | C8×#3624 | — | 2 | 1 | no | — |
| 20 | #41×Dic17 | C22 | 1 | 1 | yes | characteristic |
| 21 | #11762867 | — | 2 | 1 | no | — |
| 22 | #11762867 | — | 2 | 1 | no | — |
| 23 | #11762859 | — | 2 | 1 | no | — |
| 24 | #11762859 | — | 2 | 1 | no | — |
| 25 | #11762837 | C4 | 1 | 1 | yes | characteristic |
| 26 | #11762835 | C4 | 1 | 1 | yes | characteristic |
| 27 | #11762776 | C4 | 1 | 1 | yes | characteristic |
| 28 | M16×Dic17 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C8×Dic17 | C22 | 1 | 1 | yes | characteristic |
| 30 | C17×#408 | C4 | 1 | 1 | yes | characteristic |
| 31 | C2×#41×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C22×#3624 | — | 2 | 1 | no | — |
| 33 | C22×#3624 | — | 2 | 1 | no | — |
| 34 | C2×M16×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C22×C8×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | Q8×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | Q8×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | D4×Dic17 | — | 2 | 1 | no | — |
| 39 | D4×Dic17 | — | 2 | 1 | no | — |
| 40 | D4×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | D4×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #10587612 | — | 2 | 1 | no | — |
| 43 | #10587612 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C2×C4×Dic17 | — | 2 | 1 | no | — |
| 45 | C2×C4×Dic17 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×#63223 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C22×#3622 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C22⋊#3624 | — | 2 | 1 | no | — |
| 49 | C22⋊#3624 | — | 2 | 1 | no | — |
| 50 | C4⋊#3624 | C2×C4 | 1 | 2 | yes | — |
| 51 | C4⋊#3624 | C2×C4 | 1 | 2 | yes | — |
| 52 | C4×#3624 | — | 2 | 1 | no | — |
| 53 | C4×#3624 | — | 2 | 1 | no | — |
| 54 | C4×#3624 | — | 2 | 1 | no | — |
| 55 | C4×#3624 | — | 2 | 1 | no | — |
| 56 | C4×#3624 | C2×C4 | 1 | 2 | yes | — |
| 57 | C4×#3624 | C2×C4 | 1 | 2 | yes | — |
| 58 | C17×C22⋊C8 | — | 2 | 1 | no | — |
| 59 | C17×C22⋊C8 | — | 2 | 1 | no | — |
| 60 | C22⋊#3622 | — | 2 | 1 | no | — |
| 61 | C22⋊#3622 | — | 2 | 1 | no | — |
| 62 | #10587486 | — | 2 | 1 | no | — |
| 63 | #10587485 | — | 2 | 1 | no | — |
| 64 | #10587485 | — | 2 | 1 | no | — |
| 65 | #10587485 | — | 2 | 1 | no | — |
| 66 | #10587485 | — | 2 | 1 | no | — |
| 67 | C8×Dic17 | — | 2 | 1 | no | — |
| 68 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 70 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 71 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 72 | #41×C17 | — | 2 | 2 | no | — |
| 73 | #41×C17 | — | 2 | 2 | no | — |
| 74 | C2×Q8×C17 | C42 | 1 | 1 | yes | characteristic |
| 75 | C2×D4×C17 | — | 2 | 1 | no | — |
| 76 | C2×D4×C17 | C42 | 1 | 1 | yes | characteristic |
| 77 | C22×C4×C17 | — | 2 | 1 | no | — |
| 78 | C22×C4×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C22×Dic17 | — | 2 | 1 | no | — |
| 80 | C22×Dic17 | — | 2 | 1 | no | — |
| 81 | C22×Dic17 | #41 | 1 | 1 | yes | characteristic |
| 82 | C22×Dic17 | #41 | 1 | 1 | yes | characteristic |
| 83 | C2×#3624 | — | 2 | 4 | no | — |
| 84 | C2×#3624 | — | 2 | 2 | no | — |
| 85 | C2×#3624 | — | 2 | 2 | no | — |
| 86 | C2×#3624 | — | 2 | 2 | no | — |
| 87 | C2×#3624 | — | 2 | 2 | no | — |
| 88 | C2×#3624 | — | 4 | 2 | no | — |
| 89 | M16×C17 | — | 2 | 2 | no | — |
| 90 | C2×C8×C17 | — | 2 | 2 | no | — |
| 91 | C2×C8×C17 | — | 2 | 1 | no | — |
| 92 | C2×C8×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 94 | C22⋊Dic17 | — | 2 | 1 | no | — |
| 95 | C22⋊Dic17 | — | 2 | 1 | no | — |
| 96 | C22⋊Dic17 | — | 2 | 1 | no | — |
| 97 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 98 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 99 | C4⋊Dic17 | C42 | 1 | 1 | yes | characteristic |
| 100 | C4⋊Dic17 | C42 | 1 | 1 | yes | characteristic |
| 101 | C4×Dic17 | — | 2 | 1 | no | — |
| 102 | C4×Dic17 | — | 2 | 1 | no | — |
| 103 | C4×Dic17 | — | 2 | 1 | no | — |
| 104 | C4×Dic17 | C22×C4 | 1 | 1 | yes | characteristic |
| 105 | C4×Dic17 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | C4×Dic17 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C4×Dic17 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | #63223 | — | 2 | 2 | no | — |
| 109 | C2×#3622 | — | 2 | 2 | no | — |
| 110 | C2×#3622 | — | 2 | 1 | no | — |
| 111 | C2×#3622 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 112 | C2×#3622 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 113 | #8189 | — | 17 | 1 | no | maximal, Sylow |
| 114 | C23×C17 | — | 2 | 1 | no | — |
| 115 | C23×C17 | #117 | 1 | 1 | yes | characteristic |
| 116 | Q8×C17 | — | 2 | 2 | no | — |
| 117 | D4×C17 | — | 2 | 2 | no | — |
| 118 | D4×C17 | — | 4 | 2 | no | — |
| 119 | C2×C4×C17 | — | 2 | 1 | no | — |
| 120 | C2×C4×C17 | — | 4 | 1 | no | — |
| 121 | C2×C4×C17 | — | 4 | 1 | no | — |
| 122 | C2×C4×C17 | C2×C42 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C17 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C17 | #131 | 1 | 2 | yes | — |
| 125 | C2×C4×C17 | #131 | 1 | 2 | yes | — |
| 126 | C2×Dic17 | — | 2 | 2 | no | — |
| 127 | C2×Dic17 | — | 2 | 2 | no | — |
| 128 | C2×Dic17 | — | 2 | 1 | no | — |
| 129 | C2×Dic17 | — | 2 | 1 | no | — |
| 130 | C2×Dic17 | — | 4 | 2 | no | — |
| 131 | C2×Dic17 | — | 4 | 2 | no | — |
| 132 | C2×Dic17 | C4×D4 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic17 | C4×D4 | 1 | 1 | yes | characteristic |
| 134 | C2×Dic17 | C4×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×Dic17 | C4×D4 | 1 | 1 | yes | characteristic |
| 136 | #3624 | — | 2 | 4 | no | — |
| 137 | #3624 | — | 4 | 2 | no | — |
| 138 | #3624 | — | 4 | 2 | no | — |
| 139 | #3624 | — | 4 | 2 | no | — |
| 140 | C8×C17 | — | 2 | 1 | no | — |
| 141 | C8×C17 | — | 2 | 1 | no | — |
| 142 | C8×C17 | — | 4 | 1 | no | — |
| 143 | #3622 | — | 2 | 1 | no | — |
| 144 | #3622 | — | 2 | 1 | no | — |
| 145 | #3622 | — | 4 | 1 | no | — |
| 146 | #2922 | — | 17 | 1 | no | — |
| 147 | #1476 | — | 17 | 1 | no | — |
| 148 | #1476 | — | 17 | 1 | no | — |
| 149 | #1450 | — | 17 | 1 | no | — |
| 150 | C2×#322 | — | 17 | 1 | no | — |
| 151 | #1278 | — | 17 | 1 | no | — |
| 152 | #1278 | — | 17 | 1 | no | — |
| 153 | C22×C17 | — | 2 | 1 | no | — |
| 154 | C22×C17 | — | 2 | 1 | no | — |
| 155 | C22×C17 | — | 4 | 1 | no | — |
| 156 | C22×C17 | — | 4 | 1 | no | — |
| 157 | C22×C17 | — | 4 | 1 | no | — |
| 158 | C22×C17 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 159 | C22×C17 | #433 | 1 | 2 | yes | — |
| 160 | C4×C17 | — | 4 | 1 | no | — |
| 161 | C4×C17 | #405 | 1 | 2 | yes | — |
| 162 | C4×C17 | #408 | 1 | 1 | yes | characteristic |
| 163 | C4×C17 | #408 | 1 | 1 | yes | characteristic |
| 164 | Dic17 | — | 2 | 1 | no | — |
| 165 | Dic17 | — | 2 | 1 | no | — |
| 166 | Dic17 | — | 4 | 1 | no | — |
| 167 | Dic17 | — | 4 | 1 | no | — |
| 168 | Dic17 | #443 | 1 | 2 | yes | — |
| 169 | Dic17 | #444 | 1 | 2 | yes | — |
| 170 | C4×#41 | — | 17 | 1 | no | — |
| 171 | #433 | — | 34 | 1 | no | — |
| 172 | #433 | — | 34 | 1 | no | — |
| 173 | #431 | — | 17 | 1 | no | — |
| 174 | #431 | — | 17 | 1 | no | — |
| 175 | #431 | — | 34 | 1 | no | — |
| 176 | #431 | — | 34 | 1 | no | — |
| 177 | #424 | — | 17 | 1 | no | — |
| 178 | #424 | — | 17 | 1 | no | — |
| 179 | #408 | — | 17 | 1 | no | — |
| 180 | #408 | #595 | 1 | 1 | yes | characteristic |
| 181 | C4×M16 | — | 17 | 1 | no | — |
| 182 | C2×C4×C8 | — | 17 | 1 | no | — |
| 183 | C2×C4×C8 | — | 17 | 1 | no | — |
| 184 | C2×C4×C8 | — | 17 | 1 | no | — |
| 185 | Q8⋊C8 | — | 17 | 2 | no | — |
| 186 | Q8⋊C8 | — | 17 | 2 | no | — |
| 187 | D4⋊C8 | — | 17 | 2 | no | — |
| 188 | D4⋊C8 | — | 17 | 2 | no | — |
| 189 | #322 | — | 34 | 1 | no | — |
| 190 | #322 | — | 34 | 1 | no | — |
| 191 | #322 | — | 34 | 1 | no | — |
| 192 | C82 | — | 34 | 1 | no | — |
| 193 | C2×C17 | — | 2 | 2 | no | — |
| 194 | C2×C17 | — | 8 | 1 | no | — |
| 195 | C2×C17 | #1750 | 1 | 1 | yes | characteristic |
| 196 | C2×C17 | #1764 | 1 | 1 | yes | characteristic |
| 197 | C2×C17 | #1765 | 1 | 1 | yes | characteristic |
| 198 | C2×#41 | C2×#595 | 1 | 1 | yes | characteristic |
| 199 | C2×M16 | — | 17 | 1 | no | — |
| 200 | C2×M16 | C2×#595 | 1 | 1 | yes | characteristic |
| 201 | C22×C8 | — | 17 | 1 | no | — |
| 202 | C22×C8 | — | 34 | 1 | no | — |
| 203 | C22×C8 | — | 34 | 1 | no | — |
| 204 | C22×C8 | C2×#595 | 1 | 1 | yes | characteristic |
| 205 | C4×Q8 | — | 17 | 1 | no | — |
| 206 | C4×Q8 | — | 17 | 1 | no | — |
| 207 | C4×D4 | — | 17 | 1 | no | — |
| 208 | C4×D4 | — | 17 | 1 | no | — |
| 209 | C4×D4 | — | 34 | 1 | no | — |
| 210 | C4×D4 | — | 34 | 1 | no | — |
| 211 | #117 | — | 17 | 1 | no | — |
| 212 | #117 | — | 34 | 1 | no | — |
| 213 | C2×C42 | — | 17 | 1 | no | — |
| 214 | C2×C42 | — | 34 | 1 | no | — |
| 215 | C4⋊C8 | — | 17 | 2 | no | — |
| 216 | C4⋊C8 | — | 17 | 2 | no | — |
| 217 | C4⋊C8 | — | 34 | 1 | no | — |
| 218 | C4⋊C8 | — | 34 | 1 | no | — |
| 219 | C4⋊C8 | — | 34 | 1 | no | — |
| 220 | C4⋊C8 | — | 34 | 1 | no | — |
| 221 | C22⋊C8 | — | 2 | 1 | no | — |
| 222 | C22⋊C8 | — | 2 | 1 | no | — |
| 223 | C22⋊C8 | — | 34 | 1 | no | — |
| 224 | C22⋊C8 | — | 34 | 1 | no | — |
| 225 | C22⋊C8 | — | 34 | 1 | no | — |
| 226 | C22⋊C8 | — | 34 | 1 | no | — |
| 227 | #97 | — | 34 | 1 | no | — |
| 228 | C4×C8 | — | 17 | 1 | no | — |
| 229 | C4×C8 | — | 17 | 1 | no | — |
| 230 | C4×C8 | — | 17 | 1 | no | — |
| 231 | C4×C8 | — | 17 | 1 | no | — |
| 232 | C4×C8 | — | 17 | 2 | no | — |
| 233 | C4×C8 | — | 17 | 2 | no | — |
| 234 | C4×C8 | — | 34 | 1 | no | — |
| 235 | C4×C8 | — | 34 | 1 | no | — |
| 236 | C4×C8 | — | 34 | 1 | no | — |
| 237 | C4×C8 | — | 34 | 1 | no | — |
| 238 | C4×C8 | — | 34 | 1 | no | — |
| 239 | C17 | #8189 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 240 | #41 | — | 2 | 2 | no | — |
| 241 | #41 | — | 2 | 2 | no | — |
| 242 | C2×Q8 | C4×#595 | 1 | 1 | yes | characteristic |
| 243 | C2×D4 | — | 2 | 1 | no | — |
| 244 | C2×D4 | C4×#595 | 1 | 1 | yes | characteristic |
| 245 | C22×C4 | — | 2 | 1 | no | — |
| 246 | C22×C4 | — | 17 | 1 | no | — |
| 247 | C22×C4 | — | 17 | 1 | no | — |
| 248 | C22×C4 | — | 34 | 1 | no | — |
| 249 | C22×C4 | — | 34 | 1 | no | — |
| 250 | C22×C4 | C22×#595 | 1 | 1 | yes | characteristic |
| 251 | M16 | — | 2 | 2 | no | — |
| 252 | M16 | — | 34 | 2 | no | — |
| 253 | C2×C8 | — | 2 | 2 | no | — |
| 254 | C2×C8 | — | 2 | 1 | no | — |
| 255 | C2×C8 | — | 17 | 1 | no | — |
| 256 | C2×C8 | — | 17 | 1 | no | — |
| 257 | C2×C8 | — | 34 | 2 | no | — |
| 258 | C2×C8 | — | 34 | 4 | no | — |
| 259 | C2×C8 | — | 34 | 2 | no | — |
| 260 | C2×C8 | — | 34 | 2 | no | — |
| 261 | C2×C8 | — | 34 | 2 | no | — |
| 262 | C2×C8 | — | 34 | 2 | no | — |
| 263 | C2×C8 | — | 34 | 1 | no | — |
| 264 | C2×C8 | — | 68 | 2 | no | — |
| 265 | C2×C8 | C22⋊#595 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | C22⋊#595 | 1 | 1 | yes | characteristic |
| 267 | C4⋊C4 | — | 17 | 1 | no | — |
| 268 | C4⋊C4 | — | 17 | 1 | no | — |
| 269 | C4⋊C4 | — | 34 | 1 | no | — |
| 270 | C4⋊C4 | — | 34 | 1 | no | — |
| 271 | C22⋊C4 | — | 34 | 1 | no | — |
| 272 | C22⋊C4 | — | 34 | 1 | no | — |
| 273 | C22⋊C4 | — | 34 | 1 | no | — |
| 274 | C42 | — | 17 | 1 | no | — |
| 275 | C42 | — | 17 | 1 | no | — |
| 276 | C42 | — | 17 | 1 | no | — |
| 277 | C42 | — | 17 | 1 | no | — |
| 278 | C42 | — | 34 | 1 | no | — |
| 279 | C42 | — | 34 | 1 | no | — |
| 280 | C42 | — | 34 | 1 | no | — |
| 281 | C23 | — | 2 | 1 | no | — |
| 282 | C23 | #10587677 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 2 | 2 | no | — |
| 284 | D4 | — | 2 | 2 | no | — |
| 285 | D4 | — | 4 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 4 | 1 | no | — |
| 288 | C2×C4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 17 | 1 | no | — |
| 290 | C2×C4 | — | 17 | 1 | no | — |
| 291 | C2×C4 | — | 17 | 1 | no | — |
| 292 | C2×C4 | — | 17 | 1 | no | — |
| 293 | C2×C4 | — | 34 | 2 | no | — |
| 294 | C2×C4 | — | 34 | 2 | no | — |
| 295 | C2×C4 | — | 34 | 1 | no | — |
| 296 | C2×C4 | — | 34 | 1 | no | — |
| 297 | C2×C4 | — | 68 | 2 | no | — |
| 298 | C2×C4 | — | 68 | 2 | no | — |
| 299 | C2×C4 | C2×C4×#595 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #10587678 | 1 | 2 | yes | — |
| 301 | C2×C4 | #10587678 | 1 | 2 | yes | — |
| 302 | C2×C4 | C2×C22⋊#595 | 1 | 1 | yes | characteristic |
| 303 | C8 | — | 2 | 1 | no | — |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 4 | 1 | no | — |
| 306 | C8 | — | 34 | 4 | no | — |
| 307 | C8 | — | 34 | 1 | no | — |
| 308 | C8 | — | 34 | 1 | no | — |
| 309 | C8 | — | 68 | 2 | no | — |
| 310 | C8 | — | 68 | 2 | no | — |
| 311 | C8 | — | 68 | 2 | no | — |
| 312 | C8 | — | 68 | 1 | no | — |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 4 | 1 | no | — |
| 316 | C22 | — | 4 | 1 | no | — |
| 317 | C22 | — | 4 | 1 | no | — |
| 318 | C22 | C4×C22⋊#595 | 1 | 1 | yes | characteristic |
| 319 | C22 | #11763833 | 1 | 2 | yes | — |
| 320 | C4 | — | 4 | 1 | no | — |
| 321 | C4 | — | 17 | 2 | no | — |
| 322 | C4 | — | 17 | 2 | no | — |
| 323 | C4 | — | 34 | 1 | no | — |
| 324 | C4 | — | 34 | 1 | no | — |
| 325 | C4 | — | 68 | 1 | no | — |
| 326 | C4 | — | 68 | 1 | no | — |
| 327 | C4 | #11763773 | 1 | 2 | yes | — |
| 328 | C4 | #11763782 | 1 | 1 | yes | characteristic |
| 329 | C4 | #11763783 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 2 | no | — |
| 331 | C2 | — | 8 | 1 | no | — |
| 332 | C2 | #14544514 | 1 | 1 | yes | characteristic |
| 333 | C2 | #14544558 | 1 | 1 | yes | characteristic |
| 334 | C2 | #14544562 | 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