Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
666 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22626228 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#17989498 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22625845 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22623818 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#14556652 | C22 | 1 | 4 | yes | — |
| 6 | #17989498 | — | 2 | 2 | no | — |
| 7 | #17989498 | — | 2 | 2 | no | — |
| 8 | #17989498 | — | 2 | 2 | no | — |
| 9 | #17989496 | — | 2 | 2 | no | — |
| 10 | #17989468 | — | 2 | 4 | no | — |
| 11 | #17989316 | — | 2 | 2 | no | — |
| 12 | #17989310 | — | 2 | 2 | no | — |
| 13 | #17989279 | — | 2 | 4 | no | — |
| 14 | #17989200 | — | 2 | 2 | no | — |
| 15 | #17989163 | C4 | 1 | 1 | yes | characteristic |
| 16 | #17989163 | C22 | 1 | 1 | yes | characteristic |
| 17 | #17989142 | — | 2 | 2 | no | — |
| 18 | C2×#14556591 | C22 | 1 | 1 | yes | characteristic |
| 19 | #17989064 | C4 | 1 | 1 | yes | characteristic |
| 20 | #17989064 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×#11924028 | C4 | 1 | 1 | yes | characteristic |
| 22 | #533×C73 | C4 | 1 | 1 | yes | characteristic |
| 23 | C2×#11924037 | — | 2 | 4 | no | — |
| 24 | #14556652 | — | 2 | 4 | no | — |
| 25 | #14556652 | — | 2 | 4 | no | — |
| 26 | C2×#11924028 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C22×Dic146 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×Dic146 | C23 | 1 | 1 | yes | characteristic |
| 29 | #14556591 | — | 2 | 1 | no | — |
| 30 | #14556591 | — | 2 | 2 | no | — |
| 31 | #14556591 | D4 | 1 | 2 | yes | — |
| 32 | #14556591 | D4 | 1 | 2 | yes | — |
| 33 | #14556591 | D4 | 1 | 2 | yes | — |
| 34 | #14556586 | — | 2 | 4 | no | — |
| 35 | #14556586 | — | 2 | 4 | no | — |
| 36 | #14556582 | — | 2 | 4 | no | — |
| 37 | #14556582 | — | 2 | 4 | no | — |
| 38 | #14556582 | — | 2 | 4 | no | — |
| 39 | #14556579 | — | 2 | 1 | no | — |
| 40 | #14556579 | D4 | 1 | 2 | yes | — |
| 41 | #14556575 | — | 2 | 2 | no | — |
| 42 | #14556575 | — | 2 | 2 | no | — |
| 43 | #14556571 | — | 2 | 2 | no | — |
| 44 | #14556571 | — | 2 | 2 | no | — |
| 45 | #14556567 | — | 2 | 4 | no | — |
| 46 | Q8×Dic73 | — | 2 | 4 | no | — |
| 47 | C4×Dic146 | — | 2 | 1 | no | — |
| 48 | C4×Dic146 | — | 2 | 2 | no | — |
| 49 | C4×Dic146 | — | 2 | 2 | no | — |
| 50 | #14556556 | — | 2 | 2 | no | — |
| 51 | #14556556 | — | 2 | 2 | no | — |
| 52 | C4×D292 | — | 2 | 1 | no | — |
| 53 | C4×C22⋊D73 | — | 2 | 2 | no | — |
| 54 | #14556550 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #14556549 | — | 2 | 4 | no | — |
| 56 | #14556548 | — | 2 | 1 | no | — |
| 57 | #14556547 | — | 2 | 2 | no | — |
| 58 | #14556547 | D4 | 1 | 4 | yes | — |
| 59 | C2×C4⋊Dic73 | — | 2 | 2 | no | — |
| 60 | C42×D73 | — | 2 | 1 | no | — |
| 61 | C4⋊Q8×C73 | — | 2 | 1 | no | — |
| 62 | #125×C73 | — | 2 | 1 | no | — |
| 63 | C22⋊Q8×C73 | — | 2 | 2 | no | — |
| 64 | C4×Q8×C73 | — | 2 | 2 | no | — |
| 65 | #117×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | #117×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C42×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | #11924037 | — | 4 | 4 | no | — |
| 69 | #11924037 | — | 4 | 4 | no | — |
| 70 | C2×#10596833 | — | 4 | 4 | no | — |
| 71 | #11924028 | — | 4 | 2 | no | — |
| 72 | C2×Dic146 | — | 2 | 1 | no | — |
| 73 | C2×Dic146 | — | 4 | 2 | no | — |
| 74 | C2×Dic146 | — | 4 | 2 | no | — |
| 75 | C2×Dic146 | C22⋊C4 | 1 | 2 | yes | — |
| 76 | C2×Dic146 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C2×Dic146 | C2×D4 | 1 | 2 | yes | — |
| 78 | C2×Dic146 | C2×D4 | 1 | 2 | yes | — |
| 79 | C2×D292 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 80 | C2×C22⋊D73 | — | 2 | 1 | no | — |
| 81 | C22×Dic73 | — | 2 | 1 | no | — |
| 82 | C22×Dic73 | C2×D4 | 1 | 2 | yes | — |
| 83 | C2×C4×D73 | — | 2 | 2 | no | — |
| 84 | C2×C4×D73 | — | 2 | 1 | no | — |
| 85 | C4⋊Dic73 | — | 2 | 4 | no | — |
| 86 | C4⋊Dic73 | — | 2 | 1 | no | — |
| 87 | C4⋊Dic73 | — | 2 | 4 | no | — |
| 88 | C4⋊Dic73 | — | 2 | 4 | no | — |
| 89 | C4⋊Dic73 | — | 2 | 4 | no | — |
| 90 | #11924007 | — | 2 | 4 | no | — |
| 91 | #11924007 | — | 2 | 2 | no | — |
| 92 | #11924007 | — | 2 | 4 | no | — |
| 93 | C22⋊Dic73 | C22⋊C4 | 1 | 2 | yes | — |
| 94 | #11924005 | — | 2 | 2 | no | — |
| 95 | #11924005 | — | 2 | 4 | no | — |
| 96 | #11924005 | — | 2 | 4 | no | — |
| 97 | C4×Dic73 | — | 2 | 2 | no | — |
| 98 | C4×Dic73 | — | 2 | 4 | no | — |
| 99 | C4×Dic73 | — | 2 | 1 | no | — |
| 100 | C4×Dic73 | — | 2 | 4 | no | — |
| 101 | C2×Q8×C73 | — | 2 | 2 | no | — |
| 102 | C22×C4×C73 | — | 2 | 1 | no | — |
| 103 | C22×C4×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C4⋊C4×C73 | — | 2 | 2 | no | — |
| 105 | C4⋊C4×C73 | — | 2 | 2 | no | — |
| 106 | C4⋊C4×C73 | — | 2 | 2 | no | — |
| 107 | C4⋊C4×C73 | — | 2 | 1 | no | — |
| 108 | C4⋊C4×C73 | — | 2 | 1 | no | — |
| 109 | C22⋊C4×C73 | — | 2 | 1 | no | — |
| 110 | C22⋊C4×C73 | C22⋊C4 | 1 | 2 | yes | — |
| 111 | C42×C73 | — | 2 | 1 | no | — |
| 112 | C42×C73 | — | 2 | 1 | no | — |
| 113 | C42×C73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C42×C73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C42×C73 | C22⋊C4 | 1 | 2 | yes | — |
| 116 | C23×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C22×D73 | — | 2 | 1 | no | — |
| 118 | Q8×C73 | — | 4 | 4 | no | — |
| 119 | C2×C4×C73 | — | 2 | 1 | no | — |
| 120 | C2×C4×C73 | — | 2 | 2 | no | — |
| 121 | C2×C4×C73 | — | 2 | 1 | no | — |
| 122 | C2×C4×C73 | — | 2 | 2 | no | — |
| 123 | C2×C4×C73 | — | 2 | 2 | no | — |
| 124 | C2×C4×C73 | — | 4 | 2 | no | — |
| 125 | C2×C4×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 127 | C22⋊D73 | — | 4 | 2 | no | — |
| 128 | C2×Dic73 | — | 2 | 4 | no | — |
| 129 | C2×Dic73 | — | 2 | 2 | no | — |
| 130 | C2×Dic73 | — | 2 | 2 | no | — |
| 131 | C2×Dic73 | — | 2 | 1 | no | — |
| 132 | C2×Dic73 | — | 4 | 2 | no | — |
| 133 | C2×Dic73 | #120 | 1 | 4 | yes | — |
| 134 | D292 | — | 4 | 1 | no | — |
| 135 | C4×D73 | — | 4 | 4 | no | — |
| 136 | C4×D73 | — | 4 | 2 | no | — |
| 137 | Dic146 | — | 4 | 2 | no | — |
| 138 | Dic146 | — | 4 | 2 | no | — |
| 139 | Dic146 | — | 4 | 2 | no | — |
| 140 | Dic146 | — | 4 | 2 | no | — |
| 141 | Dic146 | — | 4 | 1 | no | — |
| 142 | #10596833 | — | 8 | 4 | no | — |
| 143 | C22×C73 | — | 4 | 1 | no | — |
| 144 | C22×C73 | #379 | 1 | 1 | yes | characteristic |
| 145 | C22×C73 | #410 | 1 | 2 | yes | — |
| 146 | C2×D73 | — | 4 | 2 | no | — |
| 147 | C2×D73 | — | 4 | 1 | no | — |
| 148 | C4×C73 | — | 2 | 2 | no | — |
| 149 | C4×C73 | — | 2 | 1 | no | — |
| 150 | C4×C73 | — | 2 | 1 | no | — |
| 151 | C4×C73 | — | 4 | 1 | no | — |
| 152 | C4×C73 | — | 4 | 2 | no | — |
| 153 | C4×C73 | — | 4 | 2 | no | — |
| 154 | Dic73 | — | 2 | 4 | no | — |
| 155 | Dic73 | — | 4 | 2 | no | — |
| 156 | Dic73 | — | 4 | 2 | no | — |
| 157 | Dic73 | — | 4 | 1 | no | — |
| 158 | #12730 | — | 73 | 1 | no | maximal, Sylow |
| 159 | C2×C73 | — | 2 | 1 | no | — |
| 160 | C2×C73 | — | 2 | 2 | no | — |
| 161 | C2×C73 | #1782 | 1 | 1 | yes | characteristic |
| 162 | D73 | — | 8 | 1 | no | — |
| 163 | #3509 | — | 73 | 1 | no | — |
| 164 | #2133 | — | 73 | 2 | no | — |
| 165 | C2×#356 | — | 73 | 2 | no | — |
| 166 | #1830 | — | 73 | 2 | no | — |
| 167 | C73 | #12730 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 168 | #557 | — | 146 | 2 | no | — |
| 169 | #556 | — | 146 | 2 | no | — |
| 170 | #552 | — | 146 | 4 | no | — |
| 171 | #541 | — | 146 | 2 | no | — |
| 172 | #536 | — | 73 | 1 | no | — |
| 173 | #536 | — | 73 | 1 | no | — |
| 174 | #533 | — | 146 | 2 | no | — |
| 175 | #533 | #64408 | 1 | 1 | yes | characteristic |
| 176 | C2×C4⋊Q8 | — | 73 | 1 | no | — |
| 177 | #519 | — | 73 | 1 | no | — |
| 178 | #519 | — | 73 | 1 | no | — |
| 179 | C4×#41 | — | 73 | 1 | no | — |
| 180 | C2×#101 | — | 73 | 4 | no | — |
| 181 | #356 | — | 146 | 2 | no | — |
| 182 | #356 | — | 146 | 2 | no | — |
| 183 | #356 | — | 146 | 2 | no | — |
| 184 | #355 | — | 146 | 2 | no | — |
| 185 | #344 | — | 146 | 4 | no | — |
| 186 | C2×#41 | — | 73 | 1 | no | — |
| 187 | C22×Q8 | — | 73 | 1 | no | — |
| 188 | C22×Q8 | — | 73 | 1 | no | — |
| 189 | C2×M16 | — | 146 | 4 | no | — |
| 190 | C4⋊Q8 | — | 2 | 1 | no | — |
| 191 | C4⋊Q8 | — | 73 | 2 | no | — |
| 192 | C4⋊Q8 | — | 73 | 2 | no | — |
| 193 | C4⋊Q8 | — | 73 | 2 | no | — |
| 194 | C4⋊Q8 | — | 146 | 2 | no | — |
| 195 | C4⋊Q8 | — | 146 | 1 | no | — |
| 196 | #126 | — | 146 | 4 | no | — |
| 197 | #126 | — | 146 | 4 | no | — |
| 198 | #125 | — | 2 | 1 | no | — |
| 199 | #125 | — | 146 | 4 | no | — |
| 200 | #125 | — | 146 | 4 | no | — |
| 201 | #125 | — | 146 | 4 | no | — |
| 202 | #124 | — | 73 | 2 | no | — |
| 203 | #124 | — | 146 | 1 | no | — |
| 204 | #123 | — | 146 | 2 | no | — |
| 205 | #123 | — | 146 | 2 | no | — |
| 206 | C22⋊Q8 | — | 2 | 2 | no | — |
| 207 | C22⋊Q8 | — | 146 | 2 | no | — |
| 208 | C22⋊Q8 | — | 146 | 2 | no | — |
| 209 | C22⋊Q8 | — | 146 | 4 | no | — |
| 210 | C4×Q8 | — | 2 | 2 | no | — |
| 211 | C4×Q8 | — | 146 | 1 | no | — |
| 212 | C4×Q8 | — | 146 | 2 | no | — |
| 213 | C4×Q8 | — | 146 | 2 | no | — |
| 214 | C4×Q8 | — | 146 | 2 | no | — |
| 215 | C4×Q8 | — | 146 | 4 | no | — |
| 216 | C4×Q8 | — | 146 | 2 | no | — |
| 217 | C4×D4 | — | 146 | 1 | no | — |
| 218 | C4×D4 | — | 146 | 2 | no | — |
| 219 | #117 | — | 73 | 1 | no | — |
| 220 | #117 | — | 73 | 4 | no | — |
| 221 | #117 | — | 146 | 2 | no | — |
| 222 | #117 | — | 146 | 4 | no | — |
| 223 | #117 | — | 146 | 1 | no | — |
| 224 | #117 | C2×#64408 | 1 | 1 | yes | characteristic |
| 225 | #117 | C2×#64408 | 1 | 1 | yes | characteristic |
| 226 | C2×C4⋊C4 | — | 146 | 2 | no | — |
| 227 | C2×C42 | — | 146 | 1 | no | — |
| 228 | C2×C42 | C2×#64408 | 1 | 1 | yes | characteristic |
| 229 | #101 | — | 146 | 4 | no | — |
| 230 | #101 | — | 146 | 4 | no | — |
| 231 | #41 | — | 292 | 2 | no | — |
| 232 | C2×Q8 | — | 2 | 2 | no | — |
| 233 | C2×Q8 | — | 292 | 2 | no | — |
| 234 | C2×Q8 | — | 292 | 2 | no | — |
| 235 | C2×Q8 | — | 73 | 2 | no | — |
| 236 | C2×Q8 | — | 73 | 2 | no | — |
| 237 | C2×Q8 | — | 73 | 2 | no | — |
| 238 | C2×Q8 | — | 73 | 1 | no | — |
| 239 | C2×Q8 | — | 146 | 1 | no | — |
| 240 | C2×D4 | — | 73 | 1 | no | — |
| 241 | C2×D4 | — | 146 | 1 | no | — |
| 242 | C22×C4 | — | 2 | 1 | no | — |
| 243 | C22×C4 | — | 73 | 2 | no | — |
| 244 | C22×C4 | — | 146 | 2 | no | — |
| 245 | C22×C4 | — | 146 | 1 | no | — |
| 246 | C22×C4 | — | 146 | 1 | no | — |
| 247 | C22×C4 | C22×#64408 | 1 | 1 | yes | characteristic |
| 248 | M16 | — | 292 | 4 | no | — |
| 249 | M16 | — | 292 | 4 | no | — |
| 250 | C2×C8 | — | 292 | 4 | no | — |
| 251 | C4⋊C4 | — | 2 | 2 | no | — |
| 252 | C4⋊C4 | — | 2 | 2 | no | — |
| 253 | C4⋊C4 | — | 2 | 2 | no | — |
| 254 | C4⋊C4 | — | 2 | 1 | no | — |
| 255 | C4⋊C4 | — | 2 | 1 | no | — |
| 256 | C4⋊C4 | — | 146 | 4 | no | — |
| 257 | C4⋊C4 | — | 146 | 4 | no | — |
| 258 | C4⋊C4 | — | 146 | 4 | no | — |
| 259 | C4⋊C4 | — | 146 | 4 | no | — |
| 260 | C4⋊C4 | — | 146 | 2 | no | — |
| 261 | C4⋊C4 | — | 146 | 4 | no | — |
| 262 | C4⋊C4 | — | 146 | 4 | no | — |
| 263 | C4⋊C4 | — | 146 | 1 | no | — |
| 264 | C22⋊C4 | — | 2 | 1 | no | — |
| 265 | C22⋊C4 | — | 73 | 2 | no | — |
| 266 | C22⋊C4 | — | 146 | 4 | no | — |
| 267 | C22⋊C4 | — | 146 | 2 | no | — |
| 268 | C22⋊C4 | — | 146 | 4 | no | — |
| 269 | C22⋊C4 | C22⋊#64408 | 1 | 2 | yes | — |
| 270 | C42 | — | 2 | 1 | no | — |
| 271 | C42 | — | 2 | 1 | no | — |
| 272 | C42 | — | 146 | 4 | no | — |
| 273 | C42 | — | 146 | 2 | no | — |
| 274 | C42 | — | 146 | 4 | no | — |
| 275 | C42 | — | 146 | 1 | no | — |
| 276 | C42 | C22⋊#64408 | 1 | 1 | yes | characteristic |
| 277 | C42 | C22⋊#64408 | 1 | 1 | yes | characteristic |
| 278 | C42 | C22⋊#64408 | 1 | 2 | yes | — |
| 279 | C23 | — | 146 | 1 | no | — |
| 280 | C23 | C2×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 4 | 4 | no | — |
| 282 | Q8 | — | 292 | 2 | no | — |
| 283 | Q8 | — | 292 | 1 | no | — |
| 284 | Q8 | — | 292 | 2 | no | — |
| 285 | Q8 | — | 292 | 2 | no | — |
| 286 | Q8 | — | 292 | 2 | no | — |
| 287 | D4 | — | 292 | 1 | no | — |
| 288 | D4 | — | 292 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 1 | no | — |
| 290 | C2×C4 | — | 2 | 2 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 2 | 1 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 292 | 2 | no | — |
| 296 | C2×C4 | — | 292 | 4 | no | — |
| 297 | C2×C4 | — | 292 | 2 | no | — |
| 298 | C2×C4 | — | 73 | 4 | no | — |
| 299 | C2×C4 | — | 146 | 2 | no | — |
| 300 | C2×C4 | — | 146 | 2 | no | — |
| 301 | C2×C4 | — | 146 | 1 | no | — |
| 302 | C2×C4 | — | 146 | 4 | no | — |
| 303 | C2×C4 | C2×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 305 | C8 | — | 584 | 4 | no | — |
| 306 | C22 | — | 4 | 1 | no | — |
| 307 | C22 | — | 292 | 2 | no | — |
| 308 | C22 | — | 292 | 1 | no | — |
| 309 | C22 | #17989528 | 1 | 1 | yes | characteristic |
| 310 | C22 | #17989568 | 1 | 2 | yes | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 4 | 1 | no | — |
| 315 | C4 | — | 4 | 2 | no | — |
| 316 | C4 | — | 4 | 2 | no | — |
| 317 | C4 | — | 292 | 2 | no | — |
| 318 | C4 | — | 292 | 1 | no | — |
| 319 | C4 | — | 292 | 2 | no | — |
| 320 | C4 | — | 146 | 4 | no | — |
| 321 | C2 | — | 2 | 1 | no | — |
| 322 | C2 | — | 2 | 2 | no | — |
| 323 | C2 | — | 584 | 1 | no | — |
| 324 | C2 | #22625762 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#17989498 | C2 | semidirect |
| 3. #22625845 | C2 | semidirect |
| 167. C73 | #12730 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626228 | 2 |
| C2×#17989498 | 2 |
| #22625845 | 2 |
| #22623818 | 2 |
| #12730 | 73 |