Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
1208 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 220 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 | #22626542 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22626233 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #22626222 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #22623725 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #17989667 | — | 2 | 2 | no | — |
| 6 | #17989573 | C22 | 1 | 12 | yes | — |
| 7 | D73⋊#100 | — | 2 | 6 | no | — |
| 8 | #17989497 | C22 | 1 | 12 | yes | complemented |
| 9 | #17989495 | C22 | 1 | 4 | yes | complemented |
| 10 | C2×#14556637 | C22 | 1 | 1 | yes | characteristic |
| 11 | #17989248 | — | 2 | 2 | no | — |
| 12 | #17989225 | — | 2 | 6 | no | — |
| 13 | #17989161 | C22 | 1 | 3 | yes | — |
| 14 | #17989151 | C4 | 1 | 3 | yes | — |
| 15 | #17989133 | C22 | 1 | 3 | yes | — |
| 16 | C2×#14556580 | C4 | 1 | 3 | yes | — |
| 17 | #41×Dic73 | C4 | 1 | 1 | yes | characteristic |
| 18 | C2×2−1+4×C73 | C4 | 1 | 1 | yes | characteristic |
| 19 | #14556679 | — | 4 | 8 | no | — |
| 20 | D73⋊M16 | — | 2 | 12 | no | — |
| 21 | #14556652 | C23 | 1 | 8 | yes | — |
| 22 | #14556651 | — | 2 | 12 | no | — |
| 23 | #14556637 | — | 4 | 4 | no | — |
| 24 | C2×#11924029 | D4 | 1 | 2 | yes | — |
| 25 | C2×#11924028 | C23 | 1 | 3 | yes | — |
| 26 | C22×D292 | C2×C4 | 1 | 3 | yes | — |
| 27 | C2×D4×D73 | D4 | 1 | 6 | yes | — |
| 28 | #14556589 | C23 | 1 | 3 | yes | — |
| 29 | #14556587 | C2×C4 | 1 | 3 | yes | — |
| 30 | #14556580 | — | 2 | 6 | no | — |
| 31 | #14556580 | C2×C4 | 1 | 3 | yes | — |
| 32 | #14556580 | C2×C4 | 1 | 6 | yes | — |
| 33 | #14556578 | C23 | 1 | 1 | yes | characteristic |
| 34 | #14556574 | — | 2 | 6 | no | — |
| 35 | #14556567 | — | 2 | 6 | no | — |
| 36 | #14556565 | — | 2 | 6 | no | — |
| 37 | #14556563 | — | 2 | 6 | no | — |
| 38 | #14556559 | — | 2 | 6 | no | — |
| 39 | Q8×Dic73 | — | 2 | 1 | no | — |
| 40 | D4×Dic73 | — | 2 | 3 | no | — |
| 41 | C4×C22⋊D73 | — | 2 | 6 | no | — |
| 42 | #14556550 | C2×C4 | 1 | 3 | yes | — |
| 43 | C2×#11924005 | — | 2 | 6 | no | — |
| 44 | C2×C4×Dic73 | C2×C4 | 1 | 3 | yes | — |
| 45 | 2−1+4×C73 | — | 4 | 4 | no | — |
| 46 | C2×#41×C73 | — | 2 | 3 | no | — |
| 47 | C2×#41×C73 | C2×C4 | 1 | 3 | yes | — |
| 48 | C2×#41×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×Q8×C73 | — | 2 | 1 | no | — |
| 50 | C22×Q8×C73 | C2×C4 | 1 | 3 | yes | — |
| 51 | #11924038 | — | 4 | 12 | no | — |
| 52 | #11924037 | — | 2 | 8 | no | — |
| 53 | #11924036 | — | 4 | 12 | no | — |
| 54 | C23×D73 | C22⋊C4 | 1 | 6 | yes | — |
| 55 | #11924029 | — | 2 | 8 | no | — |
| 56 | #11924028 | — | 4 | 6 | no | — |
| 57 | C2×Dic146 | C24 | 1 | 1 | yes | characteristic |
| 58 | C2×D292 | — | 4 | 6 | no | — |
| 59 | C2×D292 | C22⋊C4 | 1 | 6 | yes | — |
| 60 | C2×D292 | C22×C4 | 1 | 3 | yes | — |
| 61 | C2×C22⋊D73 | — | 2 | 6 | no | — |
| 62 | C2×C22⋊D73 | C2×D4 | 1 | 6 | yes | — |
| 63 | D4×D73 | — | 4 | 12 | no | — |
| 64 | C22×Dic73 | — | 2 | 3 | no | — |
| 65 | C2×C4×D73 | C2×D4 | 1 | 6 | yes | — |
| 66 | C4⋊Dic73 | — | 2 | 3 | no | — |
| 67 | #11924007 | — | 2 | 6 | no | — |
| 68 | C22⋊Dic73 | — | 2 | 3 | no | — |
| 69 | C22⋊Dic73 | — | 2 | 2 | no | — |
| 70 | #11924005 | — | 2 | 6 | no | — |
| 71 | #11924005 | — | 2 | 12 | no | — |
| 72 | C4×Dic73 | — | 2 | 3 | no | — |
| 73 | C4×Dic73 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C4×Dic73 | C22×C4 | 1 | 3 | yes | — |
| 75 | #41×C73 | — | 2 | 4 | no | — |
| 76 | #41×C73 | — | 4 | 6 | no | — |
| 77 | #41×C73 | — | 4 | 12 | no | — |
| 78 | C2×Q8×C73 | — | 2 | 3 | no | — |
| 79 | C2×Q8×C73 | — | 4 | 6 | no | — |
| 80 | C2×Q8×C73 | — | 4 | 4 | no | — |
| 81 | C2×Q8×C73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C2×Q8×C73 | C22×C4 | 1 | 3 | yes | — |
| 83 | C2×D4×C73 | — | 2 | 3 | no | — |
| 84 | C2×D4×C73 | C22⋊C4 | 1 | 3 | yes | — |
| 85 | C2×D4×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C22×C4×C73 | — | 2 | 3 | no | — |
| 87 | C22×C4×C73 | — | 2 | 3 | no | — |
| 88 | C22×C4×C73 | C22×C4 | 1 | 3 | yes | — |
| 89 | C23×C73 | — | 2 | 1 | no | — |
| 90 | C23×C73 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 91 | C22×D73 | — | 4 | 6 | no | — |
| 92 | C22×D73 | — | 8 | 6 | no | — |
| 93 | C22×D73 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 94 | Q8×C73 | — | 4 | 3 | no | — |
| 95 | Q8×C73 | — | 4 | 6 | no | — |
| 96 | Q8×C73 | #99 | 1 | 4 | yes | — |
| 97 | D4×C73 | — | 2 | 6 | no | — |
| 98 | D4×C73 | — | 4 | 1 | no | — |
| 99 | D4×C73 | — | 4 | 6 | no | — |
| 100 | C2×C4×C73 | — | 2 | 6 | no | — |
| 101 | C2×C4×C73 | — | 2 | 3 | no | — |
| 102 | C2×C4×C73 | — | 4 | 6 | no | — |
| 103 | C2×C4×C73 | — | 4 | 6 | no | — |
| 104 | C2×C4×C73 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 105 | C2×C4×C73 | C23×C4 | 1 | 1 | yes | characteristic |
| 106 | C22⋊D73 | — | 4 | 6 | no | — |
| 107 | C22⋊D73 | — | 8 | 6 | no | — |
| 108 | C2×Dic73 | — | 2 | 1 | no | — |
| 109 | C2×Dic73 | — | 2 | 3 | no | — |
| 110 | C2×Dic73 | — | 8 | 3 | no | — |
| 111 | C2×Dic73 | C22×D4 | 1 | 2 | yes | — |
| 112 | D292 | — | 2 | 12 | no | — |
| 113 | D292 | — | 4 | 3 | no | — |
| 114 | C4×D73 | — | 2 | 12 | no | — |
| 115 | Dic146 | — | 4 | 1 | no | — |
| 116 | #10596833 | — | 4 | 8 | no | — |
| 117 | C22×C73 | — | 2 | 3 | no | — |
| 118 | C22×C73 | — | 4 | 3 | no | — |
| 119 | C22×C73 | — | 4 | 2 | no | — |
| 120 | C22×C73 | — | 4 | 1 | no | — |
| 121 | C22×C73 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C2×D73 | — | 2 | 6 | no | — |
| 123 | C2×D73 | — | 4 | 6 | no | — |
| 124 | C2×D73 | — | 16 | 6 | no | — |
| 125 | C4×C73 | — | 2 | 1 | no | — |
| 126 | C4×C73 | — | 4 | 3 | no | — |
| 127 | C4×C73 | C2×#99 | 1 | 6 | yes | — |
| 128 | Dic73 | — | 2 | 2 | no | — |
| 129 | Dic73 | — | 4 | 1 | no | — |
| 130 | Dic73 | — | 4 | 3 | no | — |
| 131 | #33371 | — | 73 | 1 | no | maximal, Sylow |
| 132 | C2×C73 | — | 2 | 1 | no | — |
| 133 | C2×C73 | — | 4 | 3 | no | — |
| 134 | C2×C73 | — | 8 | 1 | no | — |
| 135 | C2×C73 | C22×#99 | 1 | 1 | yes | characteristic |
| 136 | D73 | — | 4 | 6 | no | — |
| 137 | #3500 | — | 73 | 1 | no | — |
| 138 | #2888 | — | 73 | 2 | no | — |
| 139 | #2135 | — | 73 | 6 | no | — |
| 140 | #2132 | — | 73 | 6 | no | — |
| 141 | C73 | #33371 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 142 | C2×2−1+4 | #64408 | 1 | 1 | yes | characteristic |
| 143 | C2×H(5,𝔽2) | — | 73 | 1 | no | — |
| 144 | #568 | — | 146 | 2 | no | — |
| 145 | #548 | — | 146 | 2 | no | — |
| 146 | #546 | — | 146 | 6 | no | — |
| 147 | #536 | — | 73 | 3 | no | — |
| 148 | #535 | — | 73 | 3 | no | — |
| 149 | #532 | — | 73 | 3 | no | — |
| 150 | C2×#124 | — | 73 | 3 | no | — |
| 151 | C4×#41 | — | 73 | 1 | no | — |
| 152 | #413 | — | 73 | 12 | no | — |
| 153 | C2×#100 | — | 146 | 6 | no | — |
| 154 | #356 | — | 73 | 12 | no | — |
| 155 | #355 | — | 73 | 4 | no | — |
| 156 | 2−1+4 | — | 4 | 4 | no | — |
| 157 | H(5,𝔽2) | — | 292 | 4 | no | — |
| 158 | C2×#41 | — | 2 | 3 | no | — |
| 159 | C2×#41 | — | 73 | 2 | no | — |
| 160 | C2×#41 | — | 73 | 3 | no | — |
| 161 | C2×#41 | C2×#64408 | 1 | 3 | yes | — |
| 162 | C2×#41 | C2×#64408 | 1 | 1 | yes | characteristic |
| 163 | C22×Q8 | — | 2 | 1 | no | — |
| 164 | C22×Q8 | C2×#64408 | 1 | 3 | yes | — |
| 165 | C22×D4 | — | 73 | 6 | no | — |
| 166 | C22×D4 | — | 73 | 3 | no | — |
| 167 | #131 | — | 292 | 8 | no | — |
| 168 | C2×M16 | — | 146 | 12 | no | — |
| 169 | C4⋊Q8 | — | 73 | 3 | no | — |
| 170 | #127 | — | 73 | 3 | no | — |
| 171 | #124 | — | 73 | 3 | no | — |
| 172 | #124 | — | 73 | 6 | no | — |
| 173 | #124 | — | 73 | 1 | no | — |
| 174 | #124 | — | 146 | 6 | no | — |
| 175 | #123 | — | 146 | 6 | no | — |
| 176 | C22⋊Q8 | — | 146 | 6 | no | — |
| 177 | #121 | — | 146 | 6 | no | — |
| 178 | #121 | — | 146 | 6 | no | — |
| 179 | #120 | — | 146 | 6 | no | — |
| 180 | C4×Q8 | — | 146 | 1 | no | — |
| 181 | C4×D4 | — | 146 | 6 | no | — |
| 182 | C4×D4 | — | 146 | 3 | no | — |
| 183 | #117 | — | 73 | 3 | no | — |
| 184 | C2×C22⋊C4 | — | 146 | 6 | no | — |
| 185 | C2×C42 | — | 73 | 3 | no | — |
| 186 | #101 | — | 73 | 8 | no | — |
| 187 | #100 | — | 146 | 12 | no | — |
| 188 | C24 | — | 73 | 6 | no | — |
| 189 | #41 | — | 2 | 4 | no | — |
| 190 | #41 | — | 4 | 6 | no | — |
| 191 | #41 | — | 4 | 12 | no | — |
| 192 | #41 | — | 292 | 6 | no | — |
| 193 | #41 | — | 146 | 8 | no | — |
| 194 | C2×Q8 | — | 2 | 3 | no | — |
| 195 | C2×Q8 | — | 4 | 6 | no | — |
| 196 | C2×Q8 | — | 4 | 4 | no | — |
| 197 | C2×Q8 | — | 73 | 1 | no | — |
| 198 | C2×Q8 | C22⋊#64408 | 1 | 1 | yes | characteristic |
| 199 | C2×Q8 | C22×#64408 | 1 | 3 | yes | — |
| 200 | C2×D4 | — | 2 | 3 | no | — |
| 201 | C2×D4 | — | 292 | 6 | no | — |
| 202 | C2×D4 | — | 292 | 12 | no | — |
| 203 | C2×D4 | — | 73 | 6 | no | — |
| 204 | C2×D4 | — | 73 | 3 | no | — |
| 205 | C2×D4 | — | 73 | 6 | no | — |
| 206 | C2×D4 | — | 146 | 6 | no | — |
| 207 | C2×D4 | C22⋊#64408 | 1 | 3 | yes | — |
| 208 | C2×D4 | C22×#64408 | 1 | 1 | yes | characteristic |
| 209 | C22×C4 | — | 2 | 3 | no | — |
| 210 | C22×C4 | — | 2 | 3 | no | — |
| 211 | C22×C4 | — | 73 | 6 | no | — |
| 212 | C22×C4 | — | 146 | 3 | no | — |
| 213 | C22×C4 | C22×#64408 | 1 | 3 | yes | — |
| 214 | M16 | — | 292 | 12 | no | — |
| 215 | M16 | — | 146 | 8 | no | — |
| 216 | C2×C8 | — | 292 | 12 | no | — |
| 217 | C4⋊C4 | — | 146 | 6 | no | — |
| 218 | C4⋊C4 | — | 146 | 3 | no | — |
| 219 | C22⋊C4 | — | 146 | 6 | no | — |
| 220 | C22⋊C4 | — | 146 | 12 | no | — |
| 221 | C22⋊C4 | — | 146 | 2 | no | — |
| 222 | C22⋊C4 | — | 146 | 3 | no | — |
| 223 | C42 | — | 73 | 1 | no | — |
| 224 | C42 | — | 73 | 3 | no | — |
| 225 | C42 | — | 146 | 3 | no | — |
| 226 | C23 | — | 2 | 1 | no | — |
| 227 | C23 | — | 292 | 6 | no | — |
| 228 | C23 | — | 584 | 6 | no | — |
| 229 | C23 | — | 73 | 6 | no | — |
| 230 | C23 | C2×C22⋊#64408 | 1 | 3 | yes | — |
| 231 | Q8 | — | 4 | 3 | no | — |
| 232 | Q8 | — | 4 | 6 | no | — |
| 233 | Q8 | — | 292 | 1 | no | — |
| 234 | Q8 | #14556649 | 1 | 4 | yes | — |
| 235 | D4 | — | 2 | 6 | no | — |
| 236 | D4 | — | 4 | 1 | no | — |
| 237 | D4 | — | 4 | 6 | no | — |
| 238 | D4 | — | 292 | 3 | no | — |
| 239 | D4 | — | 292 | 6 | no | — |
| 240 | D4 | — | 584 | 6 | no | — |
| 241 | D4 | — | 146 | 12 | no | — |
| 242 | C2×C4 | — | 2 | 6 | no | — |
| 243 | C2×C4 | — | 2 | 3 | no | — |
| 244 | C2×C4 | — | 4 | 6 | no | — |
| 245 | C2×C4 | — | 4 | 6 | no | — |
| 246 | C2×C4 | — | 584 | 3 | no | — |
| 247 | C2×C4 | — | 73 | 2 | no | — |
| 248 | C2×C4 | — | 146 | 1 | no | — |
| 249 | C2×C4 | — | 146 | 3 | no | — |
| 250 | C2×C4 | — | 146 | 12 | no | — |
| 251 | C2×C4 | C2×C22⋊#64408 | 1 | 3 | yes | — |
| 252 | C2×C4 | C23×#64408 | 1 | 1 | yes | characteristic |
| 253 | C8 | — | 292 | 8 | no | — |
| 254 | C22 | — | 2 | 3 | no | — |
| 255 | C22 | — | 4 | 3 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | — | 292 | 6 | no | — |
| 259 | C22 | — | 1168 | 6 | no | — |
| 260 | C22 | — | 146 | 6 | no | — |
| 261 | C22 | C22×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 4 | 3 | no | — |
| 264 | C4 | — | 292 | 1 | no | — |
| 265 | C4 | — | 292 | 3 | no | — |
| 266 | C4 | — | 146 | 2 | no | — |
| 267 | C4 | C2×#14556649 | 1 | 6 | yes | — |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 4 | 3 | no | — |
| 270 | C2 | — | 8 | 1 | no | — |
| 271 | C2 | — | 292 | 6 | no | — |
| 272 | C2 | C22×#14556649 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22626542 | C2 | semidirect |
| 2. #22626233 | C2 | semidirect |
| 3. #22626222 | C2 | semidirect |
| 8. #17989497 | C22 | semidirect |
| 9. #17989495 | C22 | semidirect |
| 141. C73 | #33371 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626542 | 2 |
| #22626233 | 2 |
| #22626222 | 2 |
| #22623725 | 2 |
| #33371 | 73 |