← #17026186
Subgroups — #17026186
Order 4352.
606 conjugacy classes of subgroups.
The subgroup lattice has 332 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #14541681 | C2 | 1 | 2 | yes | maximal |
| 2 | #14540661 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #14528381 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #14528291 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #14528235 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #14528036 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #14527920 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #14527682 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C8×D8×C17 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #11763516 | C22 | 1 | 2 | yes | — |
| 11 | D4×Dic34 | C22 | 1 | 2 | yes | — |
| 12 | #11763217 | — | 2 | 2 | no | — |
| 13 | #11763214 | C22 | 1 | 1 | yes | characteristic |
| 14 | #11763207 | C22 | 1 | 2 | yes | — |
| 15 | #11763189 | C22 | 1 | 1 | yes | characteristic |
| 16 | #11763160 | C22 | 1 | 2 | yes | — |
| 17 | #11763145 | — | 2 | 2 | no | — |
| 18 | #11763111 | — | 2 | 2 | no | — |
| 19 | #11763105 | C22 | 1 | 2 | yes | — |
| 20 | #11763099 | C22 | 1 | 2 | yes | — |
| 21 | #11763097 | — | 2 | 2 | no | — |
| 22 | #11763089 | C22 | 1 | 2 | yes | — |
| 23 | #11763050 | — | 2 | 2 | no | — |
| 24 | #11763020 | C22 | 1 | 2 | yes | — |
| 25 | #11762967 | — | 2 | 2 | no | — |
| 26 | #11762939 | C22 | 1 | 2 | yes | — |
| 27 | #11762921 | — | 2 | 2 | no | — |
| 28 | C4×Dic68 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#10587487 | — | 2 | 2 | no | — |
| 30 | #11762562 | C22 | 1 | 1 | yes | characteristic |
| 31 | #11762557 | C22 | 1 | 2 | yes | — |
| 32 | #11762556 | C22 | 1 | 2 | yes | — |
| 33 | #11762541 | C22 | 1 | 2 | yes | — |
| 34 | C4×D8×C17 | C22 | 1 | 1 | yes | characteristic |
| 35 | C8×D4×C17 | C22 | 1 | 2 | yes | — |
| 36 | C17×#335 | C22 | 1 | 1 | yes | characteristic |
| 37 | C17×D4⋊C8 | C22 | 1 | 2 | yes | — |
| 38 | C82×C17 | C22 | 1 | 1 | yes | characteristic |
| 39 | C22×Dic34 | — | 2 | 2 | no | — |
| 40 | C2×D8×C17 | D4 | 1 | 1 | yes | characteristic |
| 41 | C22×C8×C17 | — | 2 | 2 | no | — |
| 42 | C4×D4×C17 | C23 | 1 | 2 | yes | — |
| 43 | D4×Dic17 | — | 2 | 2 | no | — |
| 44 | D4×Dic17 | — | 2 | 2 | no | — |
| 45 | C2×C4⋊Dic17 | — | 2 | 2 | no | — |
| 46 | #10587610 | — | 2 | 2 | no | — |
| 47 | #10587610 | — | 2 | 2 | no | — |
| 48 | C2×Dic68 | — | 2 | 2 | no | — |
| 49 | C2×Dic68 | — | 4 | 1 | no | — |
| 50 | C2×Dic68 | D4 | 1 | 1 | yes | characteristic |
| 51 | #10587576 | — | 2 | 2 | no | — |
| 52 | #10587574 | — | 2 | 2 | no | — |
| 53 | #10587564 | — | 4 | 2 | no | — |
| 54 | #10587564 | — | 4 | 2 | no | — |
| 55 | #10587555 | C23 | 1 | 2 | yes | — |
| 56 | #10587555 | C23 | 1 | 2 | yes | — |
| 57 | C4×Dic34 | C23 | 1 | 2 | yes | — |
| 58 | C17×#107 | D4 | 1 | 1 | yes | characteristic |
| 59 | C17×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 60 | C17×#102 | D4 | 1 | 2 | yes | — |
| 61 | C17×C22⋊C8 | — | 2 | 2 | no | — |
| 62 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 63 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 64 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 65 | #10587503 | — | 2 | 2 | no | — |
| 66 | #10587503 | — | 2 | 2 | no | — |
| 67 | #10587489 | — | 2 | 2 | no | — |
| 68 | #10587489 | — | 4 | 1 | no | — |
| 69 | #10587489 | D4 | 1 | 1 | yes | characteristic |
| 70 | #10587488 | — | 2 | 2 | no | — |
| 71 | #10587488 | — | 2 | 2 | no | — |
| 72 | #10587487 | — | 2 | 2 | no | — |
| 73 | #10587487 | — | 2 | 2 | no | — |
| 74 | #10587487 | — | 2 | 2 | no | — |
| 75 | #10587487 | — | 2 | 2 | no | — |
| 76 | #10587487 | D4 | 1 | 2 | yes | — |
| 77 | #10587486 | — | 2 | 2 | no | — |
| 78 | #10587485 | — | 2 | 2 | no | — |
| 79 | #10587479 | — | 2 | 2 | no | — |
| 80 | #10587478 | — | 2 | 2 | no | — |
| 81 | C4⋊#3622 | C23 | 1 | 2 | yes | — |
| 82 | C2×D4×C17 | C2×D4 | 1 | 2 | yes | — |
| 83 | C22×C4×C17 | — | 2 | 2 | no | — |
| 84 | C22×Dic17 | — | 4 | 2 | no | — |
| 85 | C22×Dic17 | — | 4 | 2 | no | — |
| 86 | C2×Dic34 | — | 2 | 2 | no | — |
| 87 | C2×Dic34 | — | 2 | 2 | no | — |
| 88 | C2×Dic34 | — | 4 | 4 | no | — |
| 89 | C2×Dic34 | — | 4 | 2 | no | — |
| 90 | C2×Dic34 | — | 8 | 2 | no | — |
| 91 | C2×Dic34 | C2×D4 | 1 | 2 | yes | — |
| 92 | D8×C17 | Q16 | 1 | 4 | yes | complemented |
| 93 | C2×C8×C17 | — | 2 | 1 | no | — |
| 94 | C2×C8×C17 | — | 2 | 2 | no | — |
| 95 | C2×C8×C17 | — | 8 | 2 | no | — |
| 96 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C4⋊C4×C17 | C2×D4 | 1 | 2 | yes | — |
| 101 | C22⋊C4×C17 | — | 2 | 2 | no | — |
| 102 | C42×C17 | C24 | 1 | 1 | yes | characteristic |
| 103 | C22⋊Dic17 | — | 4 | 2 | no | — |
| 104 | C22⋊Dic17 | — | 4 | 2 | no | — |
| 105 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 106 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 107 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 108 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 109 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 110 | C4⋊Dic17 | C2×D4 | 1 | 2 | yes | — |
| 111 | #63225 | — | 2 | 2 | no | — |
| 112 | #63225 | — | 4 | 2 | no | — |
| 113 | #63225 | — | 4 | 2 | no | — |
| 114 | C4×Dic17 | — | 2 | 2 | no | — |
| 115 | C2×#3622 | — | 2 | 2 | no | — |
| 116 | Dic68 | — | 4 | 1 | no | — |
| 117 | Dic68 | — | 8 | 2 | no | — |
| 118 | Dic68 | — | 8 | 2 | no | — |
| 119 | #19927 | — | 17 | 1 | no | maximal, Sylow |
| 120 | C23×C17 | — | 2 | 2 | no | — |
| 121 | D4×C17 | — | 2 | 2 | no | — |
| 122 | D4×C17 | C2×Q16 | 1 | 4 | yes | — |
| 123 | C2×C4×C17 | — | 2 | 2 | no | — |
| 124 | C2×C4×C17 | — | 4 | 2 | no | — |
| 125 | C2×C4×C17 | — | 4 | 2 | no | — |
| 126 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C17 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | C2×Dic17 | — | 2 | 2 | no | — |
| 130 | C2×Dic17 | — | 2 | 2 | no | — |
| 131 | C2×Dic17 | — | 4 | 2 | no | — |
| 132 | C2×Dic17 | — | 4 | 2 | no | — |
| 133 | C2×Dic17 | — | 8 | 4 | no | — |
| 134 | C2×Dic17 | — | 16 | 2 | no | — |
| 135 | Dic34 | — | 2 | 2 | no | — |
| 136 | Dic34 | — | 2 | 2 | no | — |
| 137 | Dic34 | — | 4 | 2 | no | — |
| 138 | Dic34 | — | 4 | 2 | no | — |
| 139 | Dic34 | — | 8 | 4 | no | — |
| 140 | Dic34 | — | 8 | 2 | no | — |
| 141 | C8×C17 | — | 2 | 2 | no | — |
| 142 | C8×C17 | — | 2 | 1 | no | — |
| 143 | C8×C17 | — | 2 | 1 | no | — |
| 144 | C8×C17 | — | 2 | 1 | no | — |
| 145 | C8×C17 | — | 4 | 2 | no | — |
| 146 | C8×C17 | C2×Q16 | 1 | 2 | yes | — |
| 147 | #3622 | — | 4 | 2 | no | — |
| 148 | #3384 | — | 17 | 2 | no | — |
| 149 | #3299 | — | 17 | 2 | no | — |
| 150 | #1707 | — | 17 | 1 | no | — |
| 151 | #1692 | — | 17 | 2 | no | — |
| 152 | #1683 | — | 17 | 2 | no | — |
| 153 | #1649 | — | 17 | 2 | no | — |
| 154 | #1632 | — | 17 | 2 | no | — |
| 155 | #1591 | — | 17 | 1 | no | — |
| 156 | C8×D8 | D17 | 1 | 1 | yes | characteristic |
| 157 | C22×C17 | — | 2 | 4 | no | — |
| 158 | C22×C17 | — | 4 | 2 | no | — |
| 159 | C22×C17 | — | 4 | 2 | no | — |
| 160 | C22×C17 | D42 | 1 | 1 | yes | characteristic |
| 161 | C4×C17 | — | 2 | 1 | no | — |
| 162 | C4×C17 | — | 4 | 2 | no | — |
| 163 | C4×C17 | C22×Q16 | 1 | 1 | yes | characteristic |
| 164 | C4×C17 | #575 | 1 | 1 | yes | characteristic |
| 165 | C4×C17 | #576 | 1 | 1 | yes | characteristic |
| 166 | C4×C17 | #578 | 1 | 1 | yes | characteristic |
| 167 | Dic17 | — | 4 | 2 | no | — |
| 168 | Dic17 | — | 4 | 2 | no | — |
| 169 | Dic17 | — | 8 | 2 | no | — |
| 170 | Dic17 | — | 8 | 2 | no | — |
| 171 | #554 | — | 17 | 2 | no | — |
| 172 | D4×Q8 | — | 17 | 2 | no | — |
| 173 | #501 | — | 34 | 2 | no | — |
| 174 | #500 | — | 17 | 1 | no | — |
| 175 | #498 | — | 17 | 2 | no | — |
| 176 | #494 | — | 17 | 1 | no | — |
| 177 | #487 | — | 17 | 2 | no | — |
| 178 | #484 | — | 34 | 2 | no | — |
| 179 | #478 | — | 34 | 2 | no | — |
| 180 | #477 | — | 17 | 2 | no | — |
| 181 | #476 | — | 17 | 2 | no | — |
| 182 | #476 | — | 34 | 2 | no | — |
| 183 | #474 | — | 17 | 2 | no | — |
| 184 | #467 | — | 34 | 2 | no | — |
| 185 | #462 | — | 17 | 2 | no | — |
| 186 | #451 | — | 34 | 2 | no | — |
| 187 | #446 | — | 17 | 2 | no | — |
| 188 | #442 | — | 34 | 2 | no | — |
| 189 | C4×Q16 | — | 17 | 1 | no | — |
| 190 | C4×D8 | C2×D17 | 1 | 1 | yes | characteristic |
| 191 | C8×D4 | C2×D17 | 1 | 2 | yes | — |
| 192 | C2×#103 | — | 34 | 2 | no | — |
| 193 | #335 | — | 17 | 1 | no | — |
| 194 | #335 | C2×D17 | 1 | 1 | yes | characteristic |
| 195 | #333 | — | 17 | 2 | no | — |
| 196 | #332 | — | 17 | 2 | no | — |
| 197 | Q8⋊C8 | — | 17 | 2 | no | — |
| 198 | D4⋊C8 | C2×D17 | 1 | 2 | yes | — |
| 199 | C82 | C2×D17 | 1 | 1 | yes | characteristic |
| 200 | C2×C17 | — | 4 | 4 | no | — |
| 201 | C2×C17 | #3273 | 1 | 1 | yes | characteristic |
| 202 | C2×C17 | D4×Q16 | 1 | 1 | yes | characteristic |
| 203 | C2×C17 | #3291 | 1 | 1 | yes | characteristic |
| 204 | C22×Q8 | — | 34 | 2 | no | — |
| 205 | C2×Q16 | — | 17 | 1 | no | — |
| 206 | C2×Q16 | — | 34 | 2 | no | — |
| 207 | C2×Q16 | — | 68 | 1 | no | — |
| 208 | C2×D8 | D68 | 1 | 1 | yes | characteristic |
| 209 | C22×C8 | — | 2 | 2 | no | — |
| 210 | C4⋊Q8 | — | 17 | 2 | no | — |
| 211 | C4⋊Q8 | — | 17 | 2 | no | — |
| 212 | C4⋊Q8 | — | 34 | 2 | no | — |
| 213 | #125 | — | 34 | 2 | no | — |
| 214 | C22⋊Q8 | — | 34 | 2 | no | — |
| 215 | C22⋊Q8 | — | 34 | 2 | no | — |
| 216 | C22⋊Q8 | — | 68 | 2 | no | — |
| 217 | C22⋊Q8 | — | 68 | 2 | no | — |
| 218 | C4×Q8 | — | 17 | 2 | no | — |
| 219 | C4×D4 | — | 34 | 2 | no | — |
| 220 | C4×D4 | — | 34 | 2 | no | — |
| 221 | C4×D4 | C22×D17 | 1 | 2 | yes | — |
| 222 | C2×C4⋊C4 | — | 34 | 2 | no | — |
| 223 | #107 | — | 17 | 1 | no | — |
| 224 | #107 | — | 34 | 2 | no | — |
| 225 | #107 | — | 34 | 2 | no | — |
| 226 | #107 | — | 68 | 1 | no | — |
| 227 | #107 | D68 | 1 | 1 | yes | characteristic |
| 228 | #106 | — | 34 | 2 | no | — |
| 229 | #106 | — | 34 | 2 | no | — |
| 230 | #106 | — | 34 | 2 | no | — |
| 231 | C4⋊C8 | — | 17 | 2 | no | — |
| 232 | C4⋊C8 | — | 34 | 2 | no | — |
| 233 | C4⋊C8 | C22×D17 | 1 | 2 | yes | — |
| 234 | #103 | — | 17 | 2 | no | — |
| 235 | #103 | — | 34 | 2 | no | — |
| 236 | #103 | — | 34 | 2 | no | — |
| 237 | #103 | — | 34 | 2 | no | — |
| 238 | #103 | — | 34 | 2 | no | — |
| 239 | #102 | — | 34 | 2 | no | — |
| 240 | #102 | — | 34 | 2 | no | — |
| 241 | #102 | D68 | 1 | 2 | yes | — |
| 242 | C22⋊C8 | — | 2 | 2 | no | — |
| 243 | #97 | — | 34 | 2 | no | — |
| 244 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 245 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 246 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 247 | C17 | #19927 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 248 | C2×Q8 | — | 17 | 2 | no | — |
| 249 | C2×Q8 | — | 34 | 2 | no | — |
| 250 | C2×Q8 | — | 34 | 2 | no | — |
| 251 | C2×Q8 | — | 68 | 2 | no | — |
| 252 | C2×Q8 | — | 68 | 4 | no | — |
| 253 | C2×Q8 | — | 136 | 2 | no | — |
| 254 | C2×D4 | C2×D68 | 1 | 2 | yes | — |
| 255 | C22×C4 | — | 2 | 2 | no | — |
| 256 | C22×C4 | — | 68 | 2 | no | — |
| 257 | C22×C4 | — | 68 | 2 | no | — |
| 258 | Q16 | — | 68 | 1 | no | — |
| 259 | Q16 | — | 136 | 2 | no | — |
| 260 | Q16 | — | 136 | 2 | no | — |
| 261 | D8 | Dic68 | 1 | 4 | yes | complemented |
| 262 | C2×C8 | — | 2 | 1 | no | — |
| 263 | C2×C8 | — | 2 | 2 | no | — |
| 264 | C2×C8 | — | 8 | 2 | no | — |
| 265 | C2×C8 | — | 34 | 2 | no | — |
| 266 | C2×C8 | C2×D68 | 1 | 1 | yes | characteristic |
| 267 | C2×C8 | C2×D68 | 1 | 1 | yes | characteristic |
| 268 | C2×C8 | D4×D17 | 1 | 1 | yes | characteristic |
| 269 | C2×C8 | D4×D17 | 1 | 1 | yes | characteristic |
| 270 | C4⋊C4 | — | 17 | 2 | no | — |
| 271 | C4⋊C4 | — | 34 | 2 | no | — |
| 272 | C4⋊C4 | — | 34 | 2 | no | — |
| 273 | C4⋊C4 | — | 34 | 2 | no | — |
| 274 | C4⋊C4 | — | 34 | 2 | no | — |
| 275 | C4⋊C4 | — | 34 | 2 | no | — |
| 276 | C4⋊C4 | — | 68 | 2 | no | — |
| 277 | C4⋊C4 | — | 68 | 2 | no | — |
| 278 | C4⋊C4 | — | 68 | 2 | no | — |
| 279 | C4⋊C4 | C2×D68 | 1 | 2 | yes | — |
| 280 | C22⋊C4 | — | 2 | 2 | no | — |
| 281 | C22⋊C4 | — | 68 | 2 | no | — |
| 282 | C22⋊C4 | — | 68 | 2 | no | — |
| 283 | C42 | — | 34 | 2 | no | — |
| 284 | C42 | C23×D17 | 1 | 1 | yes | characteristic |
| 285 | C23 | — | 2 | 2 | no | — |
| 286 | Q8 | — | 34 | 2 | no | — |
| 287 | Q8 | — | 34 | 2 | no | — |
| 288 | Q8 | — | 68 | 2 | no | — |
| 289 | Q8 | — | 68 | 2 | no | — |
| 290 | Q8 | — | 136 | 2 | no | — |
| 291 | Q8 | — | 136 | 4 | no | — |
| 292 | D4 | — | 2 | 2 | no | — |
| 293 | D4 | C2×Dic68 | 1 | 4 | yes | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 4 | 2 | no | — |
| 296 | C2×C4 | — | 4 | 2 | no | — |
| 297 | C2×C4 | — | 272 | 2 | no | — |
| 298 | C2×C4 | — | 34 | 2 | no | — |
| 299 | C2×C4 | — | 34 | 2 | no | — |
| 300 | C2×C4 | — | 68 | 2 | no | — |
| 301 | C2×C4 | — | 68 | 2 | no | — |
| 302 | C2×C4 | — | 136 | 4 | no | — |
| 303 | C2×C4 | C22×D68 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×D4×D17 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #10587696 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 2 | 2 | no | — |
| 307 | C8 | — | 2 | 1 | no | — |
| 308 | C8 | — | 2 | 1 | no | — |
| 309 | C8 | — | 2 | 1 | no | — |
| 310 | C8 | — | 4 | 2 | no | — |
| 311 | C8 | — | 68 | 2 | no | — |
| 312 | C8 | C2×Dic68 | 1 | 2 | yes | — |
| 313 | C22 | — | 2 | 4 | no | — |
| 314 | C22 | — | 4 | 2 | no | — |
| 315 | C22 | — | 4 | 2 | no | — |
| 316 | C22 | D4×D68 | 1 | 1 | yes | characteristic |
| 317 | C4 | — | 2 | 1 | no | — |
| 318 | C4 | — | 4 | 2 | no | — |
| 319 | C4 | — | 68 | 2 | no | — |
| 320 | C4 | — | 68 | 2 | no | — |
| 321 | C4 | — | 136 | 2 | no | — |
| 322 | C4 | — | 136 | 2 | no | — |
| 323 | C4 | C22×Dic68 | 1 | 1 | yes | characteristic |
| 324 | C4 | #11763606 | 1 | 1 | yes | characteristic |
| 325 | C4 | #11763610 | 1 | 1 | yes | characteristic |
| 326 | C4 | #11763626 | 1 | 1 | yes | characteristic |
| 327 | C2 | — | 4 | 4 | no | — |
| 328 | C2 | #14540394 | 1 | 1 | yes | characteristic |
| 329 | C2 | D4×Dic68 | 1 | 1 | yes | characteristic |
| 330 | C2 | #14540578 | 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