Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
1630 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C2×#17989222 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22620305 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22620031 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22609111 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #22608373 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C73×#1555 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C22×D4×D73 | — | 2 | 2 | no | — |
| 8 | #17989222 | — | 2 | 4 | no | — |
| 9 | #17989222 | — | 2 | 2 | no | — |
| 10 | #17989222 | C22 | 1 | 4 | yes | — |
| 11 | #17989144 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×#14556562 | — | 2 | 2 | no | — |
| 13 | C2×#14556559 | — | 2 | 2 | no | — |
| 14 | C2×C4×D292 | C22 | 1 | 2 | yes | — |
| 15 | #17988951 | — | 2 | 2 | no | — |
| 16 | #17988933 | C22 | 1 | 4 | yes | complemented |
| 17 | #17988923 | — | 2 | 2 | no | — |
| 18 | #17988868 | C22 | 1 | 4 | yes | complemented |
| 19 | #17988793 | C22 | 1 | 4 | yes | complemented |
| 20 | #17988740 | — | 2 | 2 | no | — |
| 21 | #17988689 | — | 2 | 4 | no | — |
| 22 | #17988592 | — | 2 | 2 | no | — |
| 23 | C4×#11924012 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #17988314 | C22 | 1 | 8 | yes | complemented |
| 25 | C2×#127×C73 | C22 | 1 | 1 | yes | characteristic |
| 26 | #432×C73 | C22 | 1 | 2 | yes | complemented |
| 27 | #331×C73 | C22 | 1 | 4 | yes | complemented |
| 28 | C24×D73 | — | 2 | 2 | no | — |
| 29 | C22×D292 | D4 | 1 | 2 | yes | — |
| 30 | C22×C22⋊D73 | — | 2 | 4 | no | — |
| 31 | C2×D4×D73 | — | 2 | 8 | no | — |
| 32 | C2×D4×D73 | — | 2 | 8 | no | — |
| 33 | C2×D4×D73 | — | 4 | 16 | no | — |
| 34 | C22×C4×D73 | — | 2 | 2 | no | — |
| 35 | C2×#11924017 | — | 2 | 4 | no | — |
| 36 | C2×#11924012 | — | 2 | 1 | no | — |
| 37 | #14556591 | C23 | 1 | 2 | yes | complemented |
| 38 | #14556583 | — | 2 | 1 | no | — |
| 39 | #14556570 | — | 2 | 2 | no | — |
| 40 | #14556562 | — | 2 | 8 | no | — |
| 41 | #14556562 | — | 2 | 8 | no | — |
| 42 | #14556559 | — | 2 | 8 | no | — |
| 43 | #14556559 | — | 2 | 8 | no | — |
| 44 | C4×Dic146 | — | 2 | 2 | no | — |
| 45 | C4×D292 | — | 2 | 2 | no | — |
| 46 | C4×D292 | — | 2 | 2 | no | — |
| 47 | C4×D292 | D4 | 1 | 4 | yes | — |
| 48 | C4×D292 | C23 | 1 | 4 | yes | — |
| 49 | #14556550 | — | 2 | 1 | no | — |
| 50 | C2×C4⋊Dic73 | D4 | 1 | 2 | yes | — |
| 51 | C2×C22⋊Dic73 | — | 2 | 2 | no | — |
| 52 | C2×#11924005 | — | 2 | 2 | no | — |
| 53 | #14556525 | — | 2 | 4 | no | — |
| 54 | #14556516 | — | 2 | 4 | no | — |
| 55 | #14556513 | — | 2 | 4 | no | — |
| 56 | #14556512 | — | 2 | 8 | no | — |
| 57 | #14556502 | — | 2 | 1 | no | — |
| 58 | C4×#10596831 | C23 | 1 | 2 | yes | complemented |
| 59 | C22×D4×C73 | — | 2 | 2 | no | — |
| 60 | C22×D4×C73 | D4 | 1 | 2 | yes | — |
| 61 | #127×C73 | — | 2 | 2 | no | — |
| 62 | #127×C73 | D4 | 1 | 2 | yes | — |
| 63 | #127×C73 | C23 | 1 | 2 | yes | — |
| 64 | C2×C42×C73 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4⋊C8×C73 | C23 | 1 | 4 | yes | complemented |
| 66 | C22⋊C8×C73 | — | 2 | 2 | no | — |
| 67 | #97×C73 | — | 2 | 2 | no | — |
| 68 | C23×D73 | — | 2 | 2 | no | — |
| 69 | C23×D73 | — | 2 | 8 | no | — |
| 70 | C23×D73 | — | 4 | 16 | no | — |
| 71 | C23×D73 | — | 8 | 4 | no | — |
| 72 | C2×Dic146 | — | 2 | 2 | no | — |
| 73 | C2×D292 | — | 2 | 2 | no | — |
| 74 | C2×D292 | — | 2 | 8 | no | — |
| 75 | C2×D292 | C2×D4 | 1 | 4 | yes | — |
| 76 | C2×C22⋊D73 | — | 2 | 16 | no | — |
| 77 | C2×C22⋊D73 | — | 4 | 16 | no | — |
| 78 | D4×D73 | — | 4 | 32 | no | — |
| 79 | D4×D73 | — | 4 | 16 | no | — |
| 80 | D4×D73 | — | 4 | 16 | no | — |
| 81 | C22×Dic73 | — | 2 | 2 | no | — |
| 82 | C2×C4×D73 | — | 2 | 4 | no | — |
| 83 | C2×C4×D73 | — | 2 | 4 | no | — |
| 84 | C2×C4×D73 | — | 4 | 8 | no | — |
| 85 | #11924017 | — | 4 | 8 | no | — |
| 86 | #11924012 | — | 4 | 2 | no | — |
| 87 | C2×#10596831 | — | 2 | 2 | no | — |
| 88 | C4⋊Dic73 | — | 2 | 2 | no | — |
| 89 | C4⋊Dic73 | — | 2 | 2 | no | — |
| 90 | C4⋊Dic73 | — | 2 | 2 | no | — |
| 91 | C4⋊Dic73 | C2×D4 | 1 | 4 | yes | — |
| 92 | #11924007 | — | 4 | 2 | no | — |
| 93 | C22⋊Dic73 | — | 2 | 8 | no | — |
| 94 | C22⋊Dic73 | — | 4 | 1 | no | — |
| 95 | #11924005 | — | 2 | 4 | no | — |
| 96 | #11924005 | — | 2 | 4 | no | — |
| 97 | C4×Dic73 | — | 4 | 1 | no | — |
| 98 | C24×C73 | — | 2 | 2 | no | — |
| 99 | C2×D4×C73 | — | 2 | 4 | no | — |
| 100 | C2×D4×C73 | — | 2 | 2 | no | — |
| 101 | C2×D4×C73 | — | 2 | 4 | no | — |
| 102 | C2×D4×C73 | — | 4 | 8 | no | — |
| 103 | C2×D4×C73 | — | 4 | 4 | no | — |
| 104 | C2×D4×C73 | C2×D4 | 1 | 4 | yes | — |
| 105 | C22×C4×C73 | C2×D4 | 1 | 2 | yes | — |
| 106 | C22×C4×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C2×C8×C73 | — | 2 | 4 | no | — |
| 108 | C42×C73 | C2×D4 | 1 | 2 | yes | — |
| 109 | C42×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C73 | C24 | 1 | 1 | yes | characteristic |
| 111 | C23×C73 | — | 2 | 4 | no | — |
| 112 | C23×C73 | — | 4 | 8 | no | — |
| 113 | C23×C73 | — | 4 | 4 | no | — |
| 114 | C23×C73 | #120 | 1 | 1 | yes | characteristic |
| 115 | C22×D73 | — | 2 | 8 | no | — |
| 116 | C22×D73 | — | 2 | 4 | no | — |
| 117 | C22×D73 | — | 4 | 32 | no | — |
| 118 | C22×D73 | — | 4 | 4 | no | — |
| 119 | C22×D73 | — | 4 | 4 | no | — |
| 120 | C22×D73 | — | 8 | 8 | no | — |
| 121 | C22×D73 | — | 8 | 32 | no | — |
| 122 | D4×C73 | — | 4 | 4 | no | — |
| 123 | D4×C73 | — | 4 | 8 | no | — |
| 124 | D4×C73 | — | 4 | 8 | no | — |
| 125 | D4×C73 | — | 8 | 2 | no | — |
| 126 | C2×C4×C73 | — | 2 | 2 | no | — |
| 127 | C2×C4×C73 | — | 2 | 4 | no | — |
| 128 | C2×C4×C73 | #120 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C73 | #120 | 1 | 2 | yes | — |
| 130 | C2×C4×C73 | C22×D4 | 1 | 2 | yes | — |
| 131 | C2×C4×C73 | C22×D4 | 1 | 1 | yes | characteristic |
| 132 | C22⋊D73 | — | 4 | 32 | no | — |
| 133 | C2×Dic73 | — | 2 | 4 | no | — |
| 134 | C2×Dic73 | — | 4 | 4 | no | — |
| 135 | C2×Dic73 | — | 4 | 2 | no | — |
| 136 | D292 | — | 2 | 8 | no | — |
| 137 | D292 | — | 2 | 4 | no | — |
| 138 | D292 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 139 | C4×D73 | — | 4 | 8 | no | — |
| 140 | C4×D73 | — | 4 | 8 | no | — |
| 141 | Dic146 | — | 4 | 4 | no | — |
| 142 | C8×C73 | — | 4 | 4 | no | — |
| 143 | #10596831 | — | 4 | 2 | no | — |
| 144 | C22×C73 | — | 2 | 2 | no | — |
| 145 | C22×C73 | — | 4 | 8 | no | — |
| 146 | C22×C73 | — | 4 | 4 | no | — |
| 147 | C22×C73 | — | 8 | 8 | no | — |
| 148 | C22×C73 | #455 | 1 | 2 | yes | — |
| 149 | C22×C73 | C2×#120 | 1 | 1 | yes | characteristic |
| 150 | C2×D73 | — | 2 | 8 | no | — |
| 151 | C2×D73 | — | 4 | 16 | no | — |
| 152 | C2×D73 | — | 4 | 4 | no | — |
| 153 | C2×D73 | — | 4 | 4 | no | — |
| 154 | C2×D73 | — | 8 | 32 | no | — |
| 155 | C4×C73 | — | 2 | 2 | no | — |
| 156 | C4×C73 | — | 2 | 1 | no | — |
| 157 | C4×C73 | — | 2 | 1 | no | — |
| 158 | C4×C73 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 159 | Dic73 | — | 4 | 4 | no | — |
| 160 | Dic73 | — | 8 | 2 | no | — |
| 161 | #18529 | — | 73 | 1 | no | maximal, Sylow |
| 162 | C2×C73 | — | 2 | 2 | no | — |
| 163 | C2×C73 | — | 8 | 4 | no | — |
| 164 | C2×C73 | #3003 | 1 | 2 | yes | — |
| 165 | C2×C73 | C2×#455 | 1 | 1 | yes | characteristic |
| 166 | D73 | — | 4 | 8 | no | — |
| 167 | C2×D42 | — | 73 | 1 | no | — |
| 168 | #3138 | — | 73 | 1 | no | — |
| 169 | #3110 | — | 73 | 2 | no | — |
| 170 | #1639 | — | 73 | 8 | no | — |
| 171 | #1555 | D73 | 1 | 1 | yes | characteristic, complemented |
| 172 | #1490 | — | 73 | 2 | no | — |
| 173 | C73 | #18529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C23×D4 | — | 146 | 2 | no | — |
| 175 | D42 | — | 73 | 4 | no | — |
| 176 | D42 | — | 146 | 4 | no | — |
| 177 | D42 | — | 146 | 2 | no | — |
| 178 | #533 | — | 73 | 1 | no | — |
| 179 | C2×#127 | C2×D73 | 1 | 1 | yes | characteristic |
| 180 | C2×#121 | — | 146 | 2 | no | — |
| 181 | C2×#120 | — | 146 | 2 | no | — |
| 182 | C2×C4×D4 | — | 73 | 2 | no | — |
| 183 | #490 | — | 146 | 2 | no | — |
| 184 | #486 | — | 73 | 4 | no | — |
| 185 | #484 | — | 146 | 2 | no | — |
| 186 | #474 | — | 73 | 4 | no | — |
| 187 | #461 | — | 73 | 4 | no | — |
| 188 | #450 | — | 146 | 2 | no | — |
| 189 | #440 | — | 146 | 4 | no | — |
| 190 | #432 | C2×D73 | 1 | 2 | yes | complemented |
| 191 | #418 | — | 146 | 2 | no | — |
| 192 | C4×M16 | — | 73 | 1 | no | — |
| 193 | #331 | C2×D73 | 1 | 4 | yes | complemented |
| 194 | D4⋊C8 | — | 73 | 8 | no | — |
| 195 | C25 | — | 146 | 2 | no | — |
| 196 | C22×D4 | — | 2 | 2 | no | — |
| 197 | C22×D4 | — | 292 | 16 | no | — |
| 198 | C22×D4 | — | 73 | 2 | no | — |
| 199 | C22×D4 | — | 146 | 8 | no | — |
| 200 | C22×D4 | — | 146 | 8 | no | — |
| 201 | C22×D4 | — | 146 | 4 | no | — |
| 202 | C22×D4 | D292 | 1 | 2 | yes | — |
| 203 | C23×C4 | — | 146 | 2 | no | — |
| 204 | C2×SD16 | — | 146 | 4 | no | — |
| 205 | C2×M16 | — | 146 | 1 | no | — |
| 206 | C4⋊Q8 | — | 73 | 2 | no | — |
| 207 | #127 | — | 2 | 2 | no | — |
| 208 | #127 | D292 | 1 | 2 | yes | — |
| 209 | #127 | C22×D73 | 1 | 2 | yes | — |
| 210 | #125 | — | 146 | 1 | no | — |
| 211 | C22⋊Q8 | — | 146 | 2 | no | — |
| 212 | #121 | — | 146 | 8 | no | — |
| 213 | #121 | — | 146 | 8 | no | — |
| 214 | #120 | — | 146 | 8 | no | — |
| 215 | #120 | — | 146 | 8 | no | — |
| 216 | C4×Q8 | — | 146 | 2 | no | — |
| 217 | C4×D4 | — | 73 | 4 | no | — |
| 218 | C4×D4 | — | 73 | 4 | no | — |
| 219 | C4×D4 | — | 146 | 2 | no | — |
| 220 | C4×D4 | — | 146 | 2 | no | — |
| 221 | #117 | — | 146 | 1 | no | — |
| 222 | C2×C4⋊C4 | — | 73 | 2 | no | — |
| 223 | C2×C22⋊C4 | — | 146 | 2 | no | — |
| 224 | C2×C22⋊C4 | — | 146 | 2 | no | — |
| 225 | C2×C42 | C22×D73 | 1 | 1 | yes | characteristic |
| 226 | #107 | — | 146 | 4 | no | — |
| 227 | C4⋊C8 | C22×D73 | 1 | 4 | yes | complemented |
| 228 | #103 | — | 146 | 4 | no | — |
| 229 | #102 | — | 146 | 4 | no | — |
| 230 | #102 | — | 146 | 8 | no | — |
| 231 | C22⋊C8 | — | 2 | 2 | no | — |
| 232 | #97 | — | 2 | 2 | no | — |
| 233 | #97 | — | 146 | 1 | no | — |
| 234 | C4×C8 | — | 73 | 2 | no | — |
| 235 | C24 | — | 2 | 2 | no | — |
| 236 | C24 | — | 292 | 16 | no | — |
| 237 | C24 | — | 584 | 4 | no | — |
| 238 | C24 | — | 146 | 8 | no | — |
| 239 | C24 | — | 146 | 2 | no | — |
| 240 | C2×Q8 | — | 146 | 2 | no | — |
| 241 | C2×D4 | — | 2 | 4 | no | — |
| 242 | C2×D4 | — | 2 | 2 | no | — |
| 243 | C2×D4 | — | 2 | 4 | no | — |
| 244 | C2×D4 | — | 4 | 8 | no | — |
| 245 | C2×D4 | — | 4 | 4 | no | — |
| 246 | C2×D4 | — | 292 | 16 | no | — |
| 247 | C2×D4 | — | 292 | 32 | no | — |
| 248 | C2×D4 | — | 292 | 16 | no | — |
| 249 | C2×D4 | — | 292 | 16 | no | — |
| 250 | C2×D4 | — | 73 | 4 | no | — |
| 251 | C2×D4 | — | 146 | 8 | no | — |
| 252 | C2×D4 | — | 146 | 16 | no | — |
| 253 | C2×D4 | — | 146 | 2 | no | — |
| 254 | C2×D4 | C2×D292 | 1 | 4 | yes | — |
| 255 | C22×C4 | — | 292 | 8 | no | — |
| 256 | C22×C4 | — | 146 | 2 | no | — |
| 257 | C22×C4 | — | 146 | 4 | no | — |
| 258 | C22×C4 | — | 146 | 4 | no | — |
| 259 | C22×C4 | D4×D73 | 1 | 2 | yes | — |
| 260 | C22×C4 | C2×D292 | 1 | 1 | yes | characteristic |
| 261 | SD16 | — | 292 | 8 | no | — |
| 262 | M16 | — | 292 | 2 | no | — |
| 263 | C2×C8 | — | 2 | 4 | no | — |
| 264 | C2×C8 | — | 146 | 2 | no | — |
| 265 | C4⋊C4 | — | 292 | 2 | no | — |
| 266 | C4⋊C4 | — | 73 | 4 | no | — |
| 267 | C4⋊C4 | — | 146 | 2 | no | — |
| 268 | C4⋊C4 | — | 146 | 2 | no | — |
| 269 | C4⋊C4 | — | 146 | 2 | no | — |
| 270 | C22⋊C4 | — | 292 | 1 | no | — |
| 271 | C22⋊C4 | — | 146 | 8 | no | — |
| 272 | C22⋊C4 | — | 146 | 4 | no | — |
| 273 | C22⋊C4 | — | 146 | 4 | no | — |
| 274 | C42 | — | 292 | 1 | no | — |
| 275 | C42 | D4×D73 | 1 | 2 | yes | — |
| 276 | C42 | C2×D292 | 1 | 1 | yes | characteristic |
| 277 | C42 | C23×D73 | 1 | 1 | yes | characteristic |
| 278 | C23 | — | 2 | 4 | no | — |
| 279 | C23 | — | 4 | 8 | no | — |
| 280 | C23 | — | 4 | 4 | no | — |
| 281 | C23 | — | 292 | 4 | no | — |
| 282 | C23 | — | 292 | 32 | no | — |
| 283 | C23 | — | 292 | 4 | no | — |
| 284 | C23 | — | 584 | 8 | no | — |
| 285 | C23 | — | 584 | 32 | no | — |
| 286 | C23 | — | 146 | 8 | no | — |
| 287 | C23 | — | 146 | 4 | no | — |
| 288 | C23 | #14556561 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 292 | 4 | no | — |
| 290 | D4 | — | 4 | 4 | no | — |
| 291 | D4 | — | 4 | 8 | no | — |
| 292 | D4 | — | 4 | 8 | no | — |
| 293 | D4 | — | 8 | 2 | no | — |
| 294 | D4 | — | 292 | 32 | no | — |
| 295 | D4 | — | 73 | 8 | no | — |
| 296 | D4 | — | 146 | 8 | no | — |
| 297 | D4 | — | 146 | 4 | no | — |
| 298 | C2×C4 | — | 2 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 4 | no | — |
| 300 | C2×C4 | — | 292 | 8 | no | — |
| 301 | C2×C4 | — | 292 | 4 | no | — |
| 302 | C2×C4 | — | 292 | 8 | no | — |
| 303 | C2×C4 | — | 292 | 2 | no | — |
| 304 | C2×C4 | — | 146 | 4 | no | — |
| 305 | C2×C4 | #14556561 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | #14556561 | 1 | 2 | yes | — |
| 307 | C2×C4 | C2×D4×D73 | 1 | 2 | yes | — |
| 308 | C2×C4 | C22×D292 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 4 | 4 | no | — |
| 310 | C8 | — | 292 | 2 | no | — |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 4 | 8 | no | — |
| 313 | C22 | — | 4 | 4 | no | — |
| 314 | C22 | — | 8 | 8 | no | — |
| 315 | C22 | — | 292 | 16 | no | — |
| 316 | C22 | — | 292 | 4 | no | — |
| 317 | C22 | — | 292 | 4 | no | — |
| 318 | C22 | — | 584 | 32 | no | — |
| 319 | C22 | — | 146 | 8 | no | — |
| 320 | C22 | #17988767 | 1 | 2 | yes | — |
| 321 | C22 | C2×#14556561 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 292 | 4 | no | — |
| 326 | C4 | — | 584 | 2 | no | — |
| 327 | C4 | AGL(1,ℤ/8)×D73 | 1 | 4 | yes | — |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 8 | 4 | no | — |
| 330 | C2 | — | 292 | 8 | no | — |
| 331 | C2 | #22619160 | 1 | 2 | yes | — |
| 332 | C2 | C2×#17988767 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22620305 | C2 | semidirect |
| 3. #22620031 | C2 | semidirect |
| 4. #22609111 | C2 | semidirect |
| 5. #22608373 | C2 | semidirect |
| 6. C73×#1555 | C2 | semidirect |
| 11. #17989144 | C22 | semidirect |
| 16. #17988933 | C22 | semidirect |
| 18. #17988868 | C22 | semidirect |
| 19. #17988793 | C22 | semidirect |
| 23. C4×#11924012 | C22 | semidirect |
| 24. #17988314 | C22 | semidirect |
| 26. #432×C73 | C22 | semidirect |
| 27. #331×C73 | C22 | semidirect |
| 37. #14556591 | C23 | semidirect |
| 58. C4×#10596831 | C23 | semidirect |
| 65. C4⋊C8×C73 | C23 | semidirect |
| 171. #1555 | D73 | semidirect |
| 173. C73 | #18529 | semidirect |
| 190. #432 | C2×D73 | semidirect |
| 193. #331 | C2×D73 | semidirect |
| 227. C4⋊C8 | C22×D73 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17989222 | 2 |
| #22620305 | 2 |
| #22620031 | 2 |
| #22609111 | 2 |
| #22608373 | 2 |
| C73×#1555 | 2 |
| #18529 | 73 |