Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
392 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #22620826 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22611884 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22611035 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22607926 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C73×#1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #17989387 | C22 | 1 | 1 | yes | characteristic |
| 7 | #17989385 | — | 2 | 1 | no | — |
| 8 | #17989158 | — | 2 | 1 | no | — |
| 9 | #17989030 | — | 2 | 2 | no | — |
| 10 | #17988929 | — | 2 | 2 | no | — |
| 11 | #17988845 | C22 | 1 | 2 | yes | — |
| 12 | #17988828 | — | 2 | 1 | no | — |
| 13 | #17988745 | — | 2 | 2 | no | — |
| 14 | #17988553 | — | 2 | 1 | no | — |
| 15 | #17988398 | — | 2 | 2 | no | — |
| 16 | #17988345 | C22 | 1 | 1 | yes | characteristic |
| 17 | #408×C73 | C22 | 1 | 1 | yes | characteristic |
| 18 | #349×C73 | C22 | 1 | 2 | yes | — |
| 19 | C2×#11924028 | — | 2 | 1 | no | — |
| 20 | C22×Dic146 | — | 2 | 1 | no | — |
| 21 | #14556624 | — | 4 | 1 | no | — |
| 22 | #14556622 | — | 4 | 1 | no | — |
| 23 | #14556621 | — | 4 | 1 | no | — |
| 24 | #14556619 | — | 4 | 1 | no | — |
| 25 | #14556617 | — | 4 | 1 | no | — |
| 26 | #14556616 | — | 4 | 1 | no | — |
| 27 | #14556613 | — | 2 | 2 | no | — |
| 28 | #14556611 | — | 4 | 2 | no | — |
| 29 | C2×Dic292 | — | 2 | 1 | no | — |
| 30 | C2×Dic292 | D4 | 1 | 2 | yes | — |
| 31 | C2×#11924012 | — | 2 | 1 | no | — |
| 32 | #14556589 | — | 4 | 1 | no | — |
| 33 | #14556578 | — | 4 | 1 | no | — |
| 34 | #14556572 | — | 4 | 1 | no | — |
| 35 | #14556572 | — | 4 | 1 | no | — |
| 36 | #14556570 | — | 2 | 2 | no | — |
| 37 | #14556570 | — | 2 | 2 | no | — |
| 38 | #14556550 | — | 2 | 1 | no | — |
| 39 | Dic584 | — | 4 | 2 | no | — |
| 40 | #14556536 | — | 4 | 2 | no | — |
| 41 | #14556525 | — | 2 | 1 | no | — |
| 42 | #14556523 | D4 | 1 | 2 | yes | — |
| 43 | #14556516 | — | 2 | 2 | no | — |
| 44 | #14556516 | — | 2 | 2 | no | — |
| 45 | #14556503 | — | 4 | 2 | no | — |
| 46 | C2×#41×C73 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×M16×C73 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C73 | D4 | 1 | 1 | yes | characteristic |
| 49 | M32×C73 | — | 2 | 2 | no | — |
| 50 | C22⋊C8×C73 | — | 2 | 2 | no | — |
| 51 | #11924028 | — | 4 | 1 | no | — |
| 52 | #11924028 | — | 4 | 1 | no | — |
| 53 | #11924028 | — | 4 | 1 | no | — |
| 54 | #11924028 | — | 4 | 1 | no | — |
| 55 | C2×Dic146 | — | 2 | 1 | no | — |
| 56 | C2×Dic146 | — | 2 | 2 | no | — |
| 57 | C2×Dic146 | — | 4 | 1 | no | — |
| 58 | C2×Dic146 | — | 8 | 1 | no | — |
| 59 | C2×Dic146 | — | 8 | 1 | no | — |
| 60 | C2×D292 | — | 2 | 1 | no | — |
| 61 | C2×C22⋊D73 | — | 4 | 1 | no | — |
| 62 | C22×Dic73 | — | 4 | 1 | no | — |
| 63 | C2×C4×D73 | — | 4 | 1 | no | — |
| 64 | Dic292 | — | 2 | 2 | no | — |
| 65 | Dic292 | — | 4 | 2 | no | — |
| 66 | Dic292 | — | 8 | 1 | no | — |
| 67 | #11924018 | — | 4 | 2 | no | — |
| 68 | #11924017 | — | 4 | 2 | no | — |
| 69 | #11924016 | — | 4 | 2 | no | — |
| 70 | #11924015 | — | 4 | 2 | no | — |
| 71 | D584 | — | 2 | 2 | no | — |
| 72 | #11924013 | — | 4 | 2 | no | — |
| 73 | #11924012 | — | 4 | 1 | no | — |
| 74 | #11924012 | — | 4 | 1 | no | — |
| 75 | C2×#10596831 | — | 2 | 2 | no | — |
| 76 | C4⋊Dic73 | — | 2 | 2 | no | — |
| 77 | #11924007 | — | 4 | 2 | no | — |
| 78 | #11924007 | — | 4 | 2 | no | — |
| 79 | C22⋊Dic73 | — | 4 | 1 | no | — |
| 80 | C22⋊Dic73 | — | 4 | 2 | no | — |
| 81 | C22⋊Dic73 | — | 4 | 2 | no | — |
| 82 | C4×Dic73 | — | 4 | 1 | no | — |
| 83 | #41×C73 | — | 4 | 1 | no | — |
| 84 | #41×C73 | — | 4 | 1 | no | — |
| 85 | C2×Q8×C73 | D8 | 1 | 1 | yes | characteristic |
| 86 | C2×D4×C73 | — | 2 | 1 | no | — |
| 87 | C2×D4×C73 | D8 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C73 | — | 2 | 1 | no | — |
| 89 | C22×C4×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | M16×C73 | — | 4 | 1 | no | — |
| 91 | C2×C8×C73 | — | 2 | 1 | no | — |
| 92 | C2×C8×C73 | — | 4 | 1 | no | — |
| 93 | C2×C8×C73 | C2×D4 | 1 | 2 | yes | — |
| 94 | C16×C73 | — | 4 | 2 | no | — |
| 95 | C23×C73 | — | 2 | 1 | no | — |
| 96 | C23×C73 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 97 | C22×D73 | — | 4 | 1 | no | — |
| 98 | Q8×C73 | — | 4 | 1 | no | — |
| 99 | D4×C73 | — | 4 | 1 | no | — |
| 100 | D4×C73 | — | 8 | 1 | no | — |
| 101 | C2×C4×C73 | — | 2 | 1 | no | — |
| 102 | C2×C4×C73 | — | 2 | 1 | no | — |
| 103 | C2×C4×C73 | — | 2 | 1 | no | — |
| 104 | C2×C4×C73 | — | 4 | 1 | no | — |
| 105 | C2×C4×C73 | — | 4 | 1 | no | — |
| 106 | C2×C4×C73 | #120 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C73 | C2×D8 | 1 | 1 | yes | characteristic |
| 108 | C22⋊D73 | — | 8 | 1 | no | — |
| 109 | C22⋊D73 | — | 8 | 1 | no | — |
| 110 | C2×Dic73 | — | 4 | 1 | no | — |
| 111 | C2×Dic73 | — | 4 | 2 | no | — |
| 112 | C2×Dic73 | — | 4 | 2 | no | — |
| 113 | C2×Dic73 | — | 8 | 2 | no | — |
| 114 | D292 | — | 4 | 1 | no | — |
| 115 | D292 | — | 4 | 1 | no | — |
| 116 | C4×D73 | — | 8 | 1 | no | — |
| 117 | C4×D73 | — | 8 | 1 | no | — |
| 118 | Dic146 | — | 4 | 1 | no | — |
| 119 | Dic146 | — | 4 | 2 | no | — |
| 120 | Dic146 | — | 4 | 1 | no | — |
| 121 | Dic146 | — | 4 | 2 | no | — |
| 122 | Dic146 | — | 8 | 2 | no | — |
| 123 | C8×C73 | — | 2 | 2 | no | — |
| 124 | C8×C73 | — | 4 | 1 | no | — |
| 125 | #10596831 | — | 4 | 2 | no | — |
| 126 | C22×C73 | — | 2 | 1 | no | — |
| 127 | C22×C73 | — | 4 | 1 | no | — |
| 128 | C22×C73 | — | 4 | 2 | no | — |
| 129 | C22×C73 | — | 4 | 1 | no | — |
| 130 | C22×C73 | #447 | 1 | 1 | yes | characteristic |
| 131 | C2×D73 | — | 8 | 2 | no | — |
| 132 | C2×D73 | — | 8 | 1 | no | — |
| 133 | C4×C73 | — | 2 | 1 | no | — |
| 134 | C4×C73 | — | 4 | 1 | no | — |
| 135 | C4×C73 | #453 | 1 | 1 | yes | characteristic |
| 136 | C4×C73 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 137 | Dic73 | — | 8 | 2 | no | — |
| 138 | Dic73 | — | 8 | 2 | no | — |
| 139 | Dic73 | — | 8 | 1 | no | — |
| 140 | #12118 | — | 73 | 1 | no | maximal, Sylow |
| 141 | C2×C73 | — | 2 | 1 | no | — |
| 142 | C2×C73 | — | 4 | 1 | no | — |
| 143 | C2×C73 | — | 8 | 1 | no | — |
| 144 | C2×C73 | #1595 | 1 | 1 | yes | characteristic |
| 145 | D73 | — | 16 | 1 | no | — |
| 146 | #3202 | — | 73 | 1 | no | — |
| 147 | #2191 | — | 73 | 2 | no | — |
| 148 | #2017 | — | 73 | 1 | no | — |
| 149 | #1342 | — | 73 | 2 | no | — |
| 150 | #1321 | D73 | 1 | 1 | yes | characteristic, complemented |
| 151 | C73 | #12118 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 152 | #575 | — | 73 | 1 | no | — |
| 153 | #575 | — | 146 | 1 | no | — |
| 154 | #536 | — | 146 | 1 | no | — |
| 155 | #510 | — | 146 | 2 | no | — |
| 156 | #485 | — | 146 | 2 | no | — |
| 157 | #470 | — | 73 | 2 | no | — |
| 158 | #467 | — | 146 | 1 | no | — |
| 159 | #451 | — | 146 | 2 | no | — |
| 160 | #408 | — | 146 | 1 | no | — |
| 161 | #408 | C2×D73 | 1 | 1 | yes | characteristic |
| 162 | #360 | — | 146 | 2 | no | — |
| 163 | #349 | C2×D73 | 1 | 2 | yes | — |
| 164 | #338 | — | 73 | 1 | no | — |
| 165 | C2×#41 | — | 146 | 1 | no | — |
| 166 | C2×#41 | D292 | 1 | 1 | yes | characteristic |
| 167 | C22×Q8 | — | 146 | 1 | no | — |
| 168 | #137 | — | 292 | 1 | no | — |
| 169 | #137 | — | 292 | 1 | no | — |
| 170 | #137 | — | 292 | 1 | no | — |
| 171 | AGL(1,ℤ/8) | — | 292 | 1 | no | — |
| 172 | AGL(1,ℤ/8) | — | 292 | 1 | no | — |
| 173 | AGL(1,ℤ/8) | — | 292 | 1 | no | — |
| 174 | #135 | — | 292 | 2 | no | — |
| 175 | #135 | — | 146 | 2 | no | — |
| 176 | C2×Q16 | — | 73 | 2 | no | — |
| 177 | C2×Q16 | — | 146 | 1 | no | — |
| 178 | C2×M16 | — | 146 | 1 | no | — |
| 179 | C2×M16 | C22×D73 | 1 | 1 | yes | characteristic |
| 180 | C22×C8 | D292 | 1 | 1 | yes | characteristic |
| 181 | C4⋊Q8 | — | 292 | 1 | no | — |
| 182 | #124 | — | 292 | 1 | no | — |
| 183 | #123 | — | 292 | 1 | no | — |
| 184 | #123 | — | 292 | 1 | no | — |
| 185 | C22⋊Q8 | — | 146 | 2 | no | — |
| 186 | C22⋊Q8 | — | 146 | 2 | no | — |
| 187 | #117 | — | 146 | 1 | no | — |
| 188 | Q32 | — | 292 | 2 | no | — |
| 189 | SD32 | — | 292 | 2 | no | — |
| 190 | M32 | — | 2 | 2 | no | — |
| 191 | #107 | — | 146 | 1 | no | — |
| 192 | #106 | — | 73 | 2 | no | — |
| 193 | #103 | — | 146 | 2 | no | — |
| 194 | #103 | — | 146 | 2 | no | — |
| 195 | C22⋊C8 | — | 2 | 2 | no | — |
| 196 | C22⋊C8 | — | 292 | 2 | no | — |
| 197 | #41 | — | 4 | 1 | no | — |
| 198 | #41 | — | 4 | 1 | no | — |
| 199 | #41 | — | 292 | 1 | no | — |
| 200 | #41 | — | 292 | 1 | no | — |
| 201 | #41 | — | 292 | 1 | no | — |
| 202 | #41 | — | 292 | 1 | no | — |
| 203 | C2×Q8 | — | 292 | 1 | no | — |
| 204 | C2×Q8 | — | 584 | 1 | no | — |
| 205 | C2×Q8 | — | 584 | 1 | no | — |
| 206 | C2×Q8 | — | 146 | 1 | no | — |
| 207 | C2×Q8 | — | 146 | 2 | no | — |
| 208 | C2×Q8 | D584 | 1 | 1 | yes | characteristic |
| 209 | C2×D4 | — | 2 | 1 | no | — |
| 210 | C2×D4 | — | 292 | 1 | no | — |
| 211 | C2×D4 | — | 146 | 1 | no | — |
| 212 | C2×D4 | D584 | 1 | 1 | yes | characteristic |
| 213 | C22×C4 | — | 2 | 1 | no | — |
| 214 | C22×C4 | — | 292 | 1 | no | — |
| 215 | C22×C4 | — | 292 | 1 | no | — |
| 216 | C22×C4 | C2×D292 | 1 | 1 | yes | characteristic |
| 217 | Q16 | — | 292 | 2 | no | — |
| 218 | Q16 | — | 292 | 2 | no | — |
| 219 | Q16 | — | 584 | 1 | no | — |
| 220 | Q16 | — | 146 | 2 | no | — |
| 221 | SD16 | — | 292 | 2 | no | — |
| 222 | SD16 | — | 292 | 2 | no | — |
| 223 | SD16 | — | 292 | 2 | no | — |
| 224 | D8 | — | 292 | 2 | no | — |
| 225 | D8 | — | 146 | 2 | no | — |
| 226 | M16 | — | 4 | 1 | no | — |
| 227 | M16 | — | 292 | 1 | no | — |
| 228 | M16 | — | 292 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 1 | no | — |
| 230 | C2×C8 | — | 4 | 1 | no | — |
| 231 | C2×C8 | — | 146 | 2 | no | — |
| 232 | C2×C8 | D4×D73 | 1 | 2 | yes | — |
| 233 | C4⋊C4 | — | 292 | 2 | no | — |
| 234 | C4⋊C4 | — | 292 | 2 | no | — |
| 235 | C4⋊C4 | — | 146 | 2 | no | — |
| 236 | C22⋊C4 | — | 292 | 1 | no | — |
| 237 | C22⋊C4 | — | 292 | 2 | no | — |
| 238 | C22⋊C4 | — | 292 | 2 | no | — |
| 239 | C42 | — | 292 | 1 | no | — |
| 240 | C16 | — | 4 | 2 | no | — |
| 241 | C23 | — | 2 | 1 | no | — |
| 242 | C23 | — | 292 | 1 | no | — |
| 243 | C23 | #14556619 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 4 | 1 | no | — |
| 245 | Q8 | — | 292 | 1 | no | — |
| 246 | Q8 | — | 292 | 2 | no | — |
| 247 | Q8 | — | 292 | 1 | no | — |
| 248 | Q8 | — | 292 | 2 | no | — |
| 249 | Q8 | — | 584 | 2 | no | — |
| 250 | D4 | — | 4 | 1 | no | — |
| 251 | D4 | — | 8 | 1 | no | — |
| 252 | D4 | — | 292 | 1 | no | — |
| 253 | D4 | — | 292 | 1 | no | — |
| 254 | D4 | — | 584 | 1 | no | — |
| 255 | D4 | — | 584 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 292 | 1 | no | — |
| 262 | C2×C4 | — | 292 | 2 | no | — |
| 263 | C2×C4 | — | 292 | 2 | no | — |
| 264 | C2×C4 | — | 584 | 1 | no | — |
| 265 | C2×C4 | — | 584 | 2 | no | — |
| 266 | C2×C4 | — | 584 | 1 | no | — |
| 267 | C2×C4 | #14556561 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | C2×D584 | 1 | 1 | yes | characteristic |
| 269 | C8 | — | 2 | 2 | no | — |
| 270 | C8 | — | 4 | 1 | no | — |
| 271 | C8 | — | 292 | 2 | no | — |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 4 | 1 | no | — |
| 274 | C22 | — | 4 | 2 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 584 | 2 | no | — |
| 277 | C22 | — | 584 | 1 | no | — |
| 278 | C22 | #17988723 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 1 | no | — |
| 280 | C4 | — | 4 | 1 | no | — |
| 281 | C4 | — | 584 | 2 | no | — |
| 282 | C4 | — | 584 | 1 | no | — |
| 283 | C4 | — | 584 | 2 | no | — |
| 284 | C4 | #17988757 | 1 | 1 | yes | characteristic |
| 285 | C4 | #17988775 | 1 | 1 | yes | characteristic |
| 286 | C2 | — | 2 | 1 | no | — |
| 287 | C2 | — | 4 | 1 | no | — |
| 288 | C2 | — | 8 | 1 | no | — |
| 289 | C2 | — | 1168 | 1 | no | — |
| 290 | C2 | #22608851 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22620826 | C2 | semidirect |
| 2. #22611884 | C2 | semidirect |
| 4. #22607926 | C2 | semidirect |
| 5. C73×#1321 | C2 | semidirect |
| 150. #1321 | D73 | semidirect |
| 151. C73 | #12118 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22620826 | 2 |
| #22611884 | 2 |
| #22611035 | 2 |
| #22607926 | 2 |
| C73×#1321 | 2 |
| #12118 | 73 |