Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
492 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #22626467 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22625939 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22625887 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22625027 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #355×D73 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #17989655 | C22 | 1 | 1 | yes | characteristic |
| 7 | #17989598 | — | 2 | 2 | no | — |
| 8 | #17989579 | — | 2 | 1 | no | — |
| 9 | D73⋊#101 | C22 | 1 | 1 | yes | characteristic |
| 10 | (C2×#64408)⋊Q8 | C22 | 1 | 2 | yes | — |
| 11 | #17989537 | — | 2 | 2 | no | — |
| 12 | #17989535 | — | 2 | 1 | no | — |
| 13 | C22⋊C4×#64408 | — | 2 | 2 | no | — |
| 14 | #17989523 | — | 2 | 1 | no | — |
| 15 | #17989459 | — | 2 | 2 | no | — |
| 16 | #17989458 | — | 2 | 1 | no | — |
| 17 | #124×D73 | C22 | 1 | 1 | yes | characteristic |
| 18 | #101×D73 | C22 | 1 | 2 | yes | — |
| 19 | #17988384 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #17988383 | C4 | 1 | 2 | yes | complemented |
| 21 | #355×C73 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C23×#64408 | — | 2 | 1 | no | — |
| 23 | D73⋊M16 | — | 2 | 1 | no | — |
| 24 | Q8×#64408 | — | 4 | 2 | no | — |
| 25 | #14556670 | — | 2 | 1 | no | — |
| 26 | #14556670 | — | 2 | 2 | no | — |
| 27 | C2×C4⋊#64408 | D4 | 1 | 2 | yes | — |
| 28 | C2×C22⋊#64408 | — | 2 | 2 | no | — |
| 29 | C2×C22⋊#64408 | — | 2 | 1 | no | — |
| 30 | C2×C4×#64408 | — | 2 | 2 | no | — |
| 31 | C2×C4×#64408 | D4 | 1 | 2 | yes | — |
| 32 | #14556660 | — | 4 | 2 | no | — |
| 33 | #14556659 | — | 4 | 2 | no | — |
| 34 | #14556653 | — | 4 | 2 | no | — |
| 35 | #14556652 | — | 2 | 2 | no | — |
| 36 | (C2×#64408)⋊C4 | — | 2 | 1 | no | — |
| 37 | (C2×#64408)⋊C4 | — | 2 | 1 | no | — |
| 38 | (C2×#64408)⋊C4 | — | 2 | 2 | no | — |
| 39 | (C2×#64408)⋊C4 | — | 2 | 2 | no | — |
| 40 | C2×Q8×D73 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×D4×D73 | D4 | 1 | 1 | yes | characteristic |
| 42 | M16×D73 | — | 2 | 2 | no | — |
| 43 | #14556580 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | #14556579 | Q8 | 1 | 1 | yes | characteristic |
| 45 | #14556578 | Q8 | 1 | 1 | yes | characteristic |
| 46 | #14556577 | — | 2 | 2 | no | — |
| 47 | C22⋊C4×D73 | — | 2 | 2 | no | — |
| 48 | C42×D73 | D4 | 1 | 1 | yes | characteristic |
| 49 | #14556510 | C2×C4 | 1 | 2 | yes | complemented |
| 50 | #14556509 | — | 2 | 2 | no | — |
| 51 | #124×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #101×C73 | C2×C4 | 1 | 2 | yes | complemented |
| 53 | C22×#64408 | — | 2 | 2 | no | — |
| 54 | C22×#64408 | — | 2 | 2 | no | — |
| 55 | C22×#64408 | — | 4 | 4 | no | — |
| 56 | C22×#64408 | — | 8 | 1 | no | — |
| 57 | #11924038 | — | 4 | 1 | no | — |
| 58 | #11924037 | — | 4 | 1 | no | — |
| 59 | #11924036 | — | 4 | 1 | no | — |
| 60 | C4⋊#64408 | — | 2 | 2 | no | — |
| 61 | C4⋊#64408 | — | 2 | 2 | no | — |
| 62 | C4⋊#64408 | — | 2 | 2 | no | — |
| 63 | C4⋊#64408 | — | 4 | 2 | no | — |
| 64 | C22⋊#64408 | — | 4 | 4 | no | — |
| 65 | C22⋊#64408 | — | 4 | 1 | no | — |
| 66 | C22⋊#64408 | — | 8 | 1 | no | — |
| 67 | C4×#64408 | — | 4 | 2 | no | — |
| 68 | C4×#64408 | — | 4 | 2 | no | — |
| 69 | C4×#64408 | — | 4 | 1 | no | — |
| 70 | C4×#64408 | — | 4 | 4 | no | — |
| 71 | C23×D73 | — | 2 | 1 | no | — |
| 72 | C2×Dic146 | C2×Q8 | 1 | 1 | yes | characteristic |
| 73 | C2×Dic146 | #41 | 1 | 2 | yes | — |
| 74 | Q8×D73 | — | 4 | 1 | no | — |
| 75 | Q8×D73 | — | 4 | 1 | no | — |
| 76 | C2×D292 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C22⋊D73 | — | 2 | 1 | no | — |
| 78 | D4×D73 | — | 4 | 1 | no | — |
| 79 | D4×D73 | — | 4 | 1 | no | — |
| 80 | C2×C4×D73 | — | 2 | 1 | no | — |
| 81 | C2×C4×D73 | C2×D4 | 1 | 2 | yes | — |
| 82 | C2×C4×D73 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | #11924012 | — | 2 | 2 | no | — |
| 84 | #11924011 | — | 4 | 2 | no | — |
| 85 | C8×D73 | — | 4 | 2 | no | — |
| 86 | C22⋊Dic73 | — | 2 | 2 | no | — |
| 87 | #11924005 | — | 2 | 2 | no | — |
| 88 | #11924005 | — | 2 | 2 | no | — |
| 89 | C4×Dic73 | — | 2 | 1 | no | — |
| 90 | C4×Dic73 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C2×Q8×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C2×D4×C73 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 93 | M16×C73 | — | 2 | 2 | no | — |
| 94 | C22⋊C4×C73 | — | 2 | 2 | no | — |
| 95 | C42×C73 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C23×C73 | — | 2 | 1 | no | — |
| 97 | C22×D73 | — | 2 | 1 | no | — |
| 98 | C22×D73 | — | 4 | 2 | no | — |
| 99 | C22×D73 | — | 4 | 2 | no | — |
| 100 | C22×D73 | — | 4 | 1 | no | — |
| 101 | C22×D73 | — | 4 | 1 | no | — |
| 102 | C22×D73 | #120 | 1 | 1 | yes | characteristic |
| 103 | C2×#64408 | — | 2 | 4 | no | — |
| 104 | C2×#64408 | — | 4 | 2 | no | — |
| 105 | C2×#64408 | — | 4 | 2 | no | — |
| 106 | C2×#64408 | — | 4 | 2 | no | — |
| 107 | C2×#64408 | — | 4 | 2 | no | — |
| 108 | C2×#64408 | — | 4 | 2 | no | — |
| 109 | C2×#64408 | — | 4 | 2 | no | — |
| 110 | C2×#64408 | — | 8 | 4 | no | — |
| 111 | Q8×C73 | — | 4 | 1 | no | — |
| 112 | D4×C73 | — | 4 | 1 | no | — |
| 113 | C2×C4×C73 | — | 2 | 1 | no | — |
| 114 | C2×C4×C73 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C73 | C4×D4 | 1 | 2 | yes | — |
| 116 | C22⋊D73 | — | 8 | 1 | no | — |
| 117 | C2×Dic73 | — | 2 | 1 | no | — |
| 118 | C2×Dic73 | C22⋊Q8 | 1 | 2 | yes | — |
| 119 | C2×Dic73 | #123 | 1 | 1 | yes | characteristic |
| 120 | D292 | — | 4 | 1 | no | — |
| 121 | C4×D73 | — | 2 | 2 | no | — |
| 122 | C4×D73 | — | 2 | 1 | no | — |
| 123 | C4×D73 | — | 2 | 2 | no | — |
| 124 | C4×D73 | — | 2 | 1 | no | — |
| 125 | C4×D73 | — | 4 | 2 | no | — |
| 126 | Dic146 | — | 4 | 1 | no | — |
| 127 | Dic146 | — | 4 | 2 | no | — |
| 128 | #10596833 | — | 8 | 1 | no | — |
| 129 | C8×C73 | — | 4 | 2 | no | — |
| 130 | #10596831 | — | 4 | 2 | no | — |
| 131 | C22×C73 | — | 4 | 2 | no | — |
| 132 | C22×C73 | — | 4 | 1 | no | — |
| 133 | C22×C73 | #385 | 1 | 1 | yes | characteristic |
| 134 | C2×D73 | — | 2 | 2 | no | — |
| 135 | C2×D73 | — | 4 | 2 | no | — |
| 136 | C2×D73 | — | 4 | 1 | no | — |
| 137 | C2×D73 | — | 8 | 1 | no | — |
| 138 | C2×D73 | — | 8 | 1 | no | — |
| 139 | C2×D73 | — | 8 | 2 | no | — |
| 140 | C2×D73 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | C2×D73 | #458 | 1 | 1 | yes | characteristic |
| 142 | #64408 | — | 4 | 4 | no | — |
| 143 | #64408 | — | 8 | 2 | no | — |
| 144 | #64408 | — | 8 | 2 | no | — |
| 145 | C4×C73 | — | 2 | 2 | no | — |
| 146 | C4×C73 | — | 2 | 1 | no | — |
| 147 | C4×C73 | — | 4 | 1 | no | — |
| 148 | Dic73 | — | 2 | 1 | no | — |
| 149 | Dic73 | — | 2 | 2 | no | — |
| 150 | Dic73 | — | 4 | 1 | no | — |
| 151 | #12972 | — | 73 | 1 | no | maximal, Sylow |
| 152 | C2×C73 | — | 2 | 1 | no | — |
| 153 | C2×C73 | — | 8 | 1 | no | — |
| 154 | C2×C73 | #1867 | 1 | 1 | yes | characteristic |
| 155 | D73 | — | 2 | 1 | no | — |
| 156 | D73 | — | 8 | 1 | no | — |
| 157 | D73 | #2204 | 1 | 2 | yes | — |
| 158 | #2372 | — | 73 | 1 | no | — |
| 159 | C2×#355 | — | 73 | 1 | no | — |
| 160 | #1888 | — | 73 | 1 | no | — |
| 161 | #1857 | — | 73 | 2 | no | — |
| 162 | #1394 | — | 73 | 2 | no | — |
| 163 | C73 | #12972 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C2×#124 | — | 73 | 1 | no | — |
| 165 | #519 | — | 73 | 1 | no | — |
| 166 | #428 | — | 146 | 2 | no | — |
| 167 | #418 | — | 146 | 1 | no | — |
| 168 | C2×#101 | — | 73 | 2 | no | — |
| 169 | C2×#101 | — | 73 | 1 | no | — |
| 170 | #391 | — | 73 | 2 | no | — |
| 171 | #388 | — | 146 | 2 | no | — |
| 172 | #386 | — | 146 | 1 | no | — |
| 173 | C4×C22⋊C4 | — | 146 | 2 | no | — |
| 174 | #376 | — | 146 | 1 | no | — |
| 175 | #355 | — | 73 | 1 | no | — |
| 176 | #355 | — | 73 | 2 | no | — |
| 177 | #355 | #64408 | 1 | 1 | yes | characteristic, complemented |
| 178 | #339 | — | 146 | 2 | no | — |
| 179 | #339 | — | 146 | 1 | no | — |
| 180 | C22×Q8 | — | 73 | 1 | no | — |
| 181 | C22×D4 | — | 73 | 1 | no | — |
| 182 | C23×C4 | — | 146 | 1 | no | — |
| 183 | C2×M16 | — | 146 | 2 | no | — |
| 184 | C2×M16 | — | 146 | 1 | no | — |
| 185 | #124 | — | 73 | 1 | no | — |
| 186 | #124 | — | 73 | 1 | no | — |
| 187 | #124 | — | 73 | 1 | no | — |
| 188 | #124 | — | 146 | 2 | no | — |
| 189 | #124 | C2×#64408 | 1 | 1 | yes | characteristic |
| 190 | C4×Q8 | — | 292 | 2 | no | — |
| 191 | #117 | — | 146 | 1 | no | — |
| 192 | #117 | — | 146 | 2 | no | — |
| 193 | C2×C4⋊C4 | — | 73 | 2 | no | — |
| 194 | C2×C22⋊C4 | — | 146 | 2 | no | — |
| 195 | C2×C22⋊C4 | — | 146 | 2 | no | — |
| 196 | C2×C22⋊C4 | — | 146 | 1 | no | — |
| 197 | C2×C42 | — | 73 | 2 | no | — |
| 198 | C2×C42 | — | 73 | 1 | no | — |
| 199 | C2×C42 | — | 146 | 2 | no | — |
| 200 | #107 | — | 292 | 2 | no | — |
| 201 | #106 | — | 292 | 2 | no | — |
| 202 | #102 | — | 292 | 2 | no | — |
| 203 | #101 | — | 73 | 2 | no | — |
| 204 | #101 | — | 146 | 2 | no | — |
| 205 | #101 | — | 146 | 2 | no | — |
| 206 | #101 | C2×#64408 | 1 | 2 | yes | complemented |
| 207 | #95 | — | 146 | 2 | no | — |
| 208 | #95 | — | 146 | 2 | no | — |
| 209 | #95 | — | 146 | 1 | no | — |
| 210 | #95 | — | 146 | 1 | no | — |
| 211 | C24 | — | 146 | 1 | no | — |
| 212 | C2×Q8 | — | 292 | 1 | no | — |
| 213 | C2×Q8 | — | 292 | 1 | no | — |
| 214 | C2×Q8 | — | 73 | 1 | no | — |
| 215 | C2×Q8 | — | 73 | 2 | no | — |
| 216 | C2×Q8 | C22×#64408 | 1 | 1 | yes | characteristic |
| 217 | C2×D4 | — | 292 | 1 | no | — |
| 218 | C2×D4 | — | 292 | 1 | no | — |
| 219 | C2×D4 | — | 73 | 1 | no | — |
| 220 | C2×D4 | — | 146 | 1 | no | — |
| 221 | C2×D4 | C4⋊#64408 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | — | 292 | 4 | no | — |
| 223 | C22×C4 | — | 584 | 1 | no | — |
| 224 | C22×C4 | — | 73 | 1 | no | — |
| 225 | C22×C4 | — | 73 | 2 | no | — |
| 226 | C22×C4 | — | 146 | 1 | no | — |
| 227 | C22×C4 | — | 146 | 2 | no | — |
| 228 | C22×C4 | — | 146 | 2 | no | — |
| 229 | M16 | — | 2 | 2 | no | — |
| 230 | M16 | — | 292 | 2 | no | — |
| 231 | M16 | — | 292 | 1 | no | — |
| 232 | M16 | — | 292 | 1 | no | — |
| 233 | M16 | — | 146 | 2 | no | — |
| 234 | C2×C8 | — | 292 | 2 | no | — |
| 235 | C2×C8 | — | 292 | 1 | no | — |
| 236 | C4⋊C4 | — | 292 | 2 | no | — |
| 237 | C4⋊C4 | — | 146 | 2 | no | — |
| 238 | C4⋊C4 | — | 146 | 2 | no | — |
| 239 | C4⋊C4 | — | 146 | 2 | no | — |
| 240 | C22⋊C4 | — | 2 | 2 | no | — |
| 241 | C22⋊C4 | — | 292 | 1 | no | — |
| 242 | C22⋊C4 | — | 292 | 4 | no | — |
| 243 | C22⋊C4 | — | 584 | 1 | no | — |
| 244 | C22⋊C4 | — | 146 | 2 | no | — |
| 245 | C22⋊C4 | — | 146 | 2 | no | — |
| 246 | C22⋊C4 | — | 146 | 2 | no | — |
| 247 | C42 | — | 292 | 4 | no | — |
| 248 | C42 | — | 292 | 2 | no | — |
| 249 | C42 | — | 292 | 1 | no | — |
| 250 | C42 | — | 292 | 2 | no | — |
| 251 | C42 | — | 73 | 1 | no | — |
| 252 | C42 | — | 146 | 1 | no | — |
| 253 | C42 | C4⋊#64408 | 1 | 1 | yes | characteristic |
| 254 | C23 | — | 2 | 1 | no | — |
| 255 | C23 | — | 292 | 1 | no | — |
| 256 | C23 | — | 292 | 1 | no | — |
| 257 | C23 | — | 292 | 2 | no | — |
| 258 | C23 | — | 292 | 2 | no | — |
| 259 | C23 | — | 73 | 1 | no | — |
| 260 | C23 | — | 146 | 1 | no | — |
| 261 | Q8 | — | 4 | 1 | no | — |
| 262 | Q8 | — | 292 | 2 | no | — |
| 263 | Q8 | — | 292 | 1 | no | — |
| 264 | D4 | — | 4 | 1 | no | — |
| 265 | D4 | — | 292 | 1 | no | — |
| 266 | D4 | — | 584 | 1 | no | — |
| 267 | C2×C4 | — | 2 | 1 | no | — |
| 268 | C2×C4 | — | 292 | 2 | no | — |
| 269 | C2×C4 | — | 292 | 2 | no | — |
| 270 | C2×C4 | — | 292 | 2 | no | — |
| 271 | C2×C4 | — | 292 | 2 | no | — |
| 272 | C2×C4 | — | 292 | 2 | no | — |
| 273 | C2×C4 | — | 292 | 2 | no | — |
| 274 | C2×C4 | — | 292 | 2 | no | — |
| 275 | C2×C4 | — | 584 | 4 | no | — |
| 276 | C2×C4 | — | 73 | 1 | no | — |
| 277 | C2×C4 | — | 73 | 2 | no | — |
| 278 | C2×C4 | — | 146 | 2 | no | — |
| 279 | C2×C4 | — | 146 | 1 | no | — |
| 280 | C2×C4 | — | 146 | 2 | no | — |
| 281 | C2×C4 | — | 146 | 1 | no | — |
| 282 | C2×C4 | — | 146 | 4 | no | — |
| 283 | C2×C4 | — | 146 | 1 | no | — |
| 284 | C2×C4 | C2×C4⋊#64408 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | D4×#64408 | 1 | 2 | yes | — |
| 286 | C8 | — | 4 | 2 | no | — |
| 287 | C8 | — | 292 | 2 | no | — |
| 288 | C8 | — | 584 | 1 | no | — |
| 289 | C22 | — | 4 | 2 | no | — |
| 290 | C22 | — | 4 | 1 | no | — |
| 291 | C22 | — | 292 | 2 | no | — |
| 292 | C22 | — | 292 | 1 | no | — |
| 293 | C22 | — | 584 | 2 | no | — |
| 294 | C22 | — | 584 | 1 | no | — |
| 295 | C22 | — | 584 | 1 | no | — |
| 296 | C22 | — | 73 | 1 | no | — |
| 297 | C22 | — | 73 | 1 | no | — |
| 298 | C22 | — | 146 | 2 | no | — |
| 299 | C22 | #17989534 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 2 | no | — |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 4 | 1 | no | — |
| 303 | C4 | — | 292 | 1 | no | — |
| 304 | C4 | — | 292 | 4 | no | — |
| 305 | C4 | — | 584 | 2 | no | — |
| 306 | C4 | — | 584 | 2 | no | — |
| 307 | C4 | — | 146 | 1 | no | — |
| 308 | C4 | — | 146 | 2 | no | — |
| 309 | C2 | — | 2 | 1 | no | — |
| 310 | C2 | — | 8 | 1 | no | — |
| 311 | C2 | — | 584 | 1 | no | — |
| 312 | C2 | — | 73 | 2 | no | — |
| 313 | C2 | — | 146 | 1 | no | — |
| 314 | C2 | #22625902 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22625887 | C2 | semidirect |
| 4. #22625027 | C2 | semidirect |
| 19. #17988384 | C4 | semidirect |
| 20. #17988383 | C4 | semidirect |
| 21. #355×C73 | C4 | semidirect |
| 49. #14556510 | C2×C4 | semidirect |
| 52. #101×C73 | C2×C4 | semidirect |
| 163. C73 | #12972 | semidirect |
| 177. #355 | #64408 | semidirect |
| 206. #101 | C2×#64408 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626467 | 2 |
| #22625939 | 2 |
| #22625887 | 2 |
| #22625027 | 2 |
| #355×D73 | 2 |
| #12972 | 73 |