Finite Group DatabasebetaFinite Group DatabasebetaOrder 21248.
362 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 57 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 | #24918925 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #24909391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #24909377 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #24908720 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #24905477 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #24905404 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C83×#1350 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D664 | C22 | 1 | 1 | yes | characteristic |
| 9 | #19128324 | — | 2 | 1 | no | — |
| 10 | C2×#14581735 | — | 2 | 1 | no | — |
| 11 | C2×D1328 | — | 2 | 1 | no | — |
| 12 | #19127994 | — | 2 | 1 | no | — |
| 13 | #19127980 | — | 2 | 1 | no | — |
| 14 | #19127949 | — | 2 | 1 | no | — |
| 15 | #19127937 | C22 | 1 | 1 | yes | characteristic |
| 16 | #19127925 | — | 2 | 1 | no | — |
| 17 | #19127919 | — | 2 | 1 | no | — |
| 18 | #19127727 | C22 | 1 | 1 | yes | characteristic |
| 19 | #19127707 | — | 2 | 1 | no | — |
| 20 | C2×#14581708 | — | 2 | 1 | no | — |
| 21 | #19127503 | — | 2 | 1 | no | — |
| 22 | #19127501 | — | 2 | 1 | no | — |
| 23 | C22⋊(C83⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 24 | #467×C83 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22⋊C16×C83 | C22 | 1 | 1 | yes | characteristic |
| 26 | #341×C83 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22×D332 | — | 2 | 1 | no | — |
| 28 | C2×#13050577 | — | 2 | 1 | no | — |
| 29 | C2×D664 | — | 2 | 1 | no | — |
| 30 | C2×D664 | — | 2 | 1 | no | — |
| 31 | C2×D664 | — | 4 | 2 | no | — |
| 32 | C2×D664 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#13050574 | D4 | 1 | 1 | yes | characteristic |
| 34 | #14581791 | — | 2 | 1 | no | — |
| 35 | #14581787 | — | 4 | 1 | no | — |
| 36 | #14581781 | — | 2 | 1 | no | — |
| 37 | #14581778 | — | 4 | 1 | no | — |
| 38 | #14581774 | — | 4 | 1 | no | — |
| 39 | #14581772 | — | 4 | 1 | no | — |
| 40 | #14581766 | — | 2 | 1 | no | — |
| 41 | Q8×Dic83 | — | 2 | 1 | no | — |
| 42 | #14581754 | — | 2 | 1 | no | — |
| 43 | #14581751 | D4 | 1 | 1 | yes | characteristic |
| 44 | #14581735 | — | 4 | 2 | no | — |
| 45 | D1328 | — | 4 | 2 | no | — |
| 46 | C2×C83⋊C16 | — | 2 | 1 | no | — |
| 47 | #14581726 | D4 | 1 | 1 | yes | characteristic |
| 48 | #14581725 | — | 2 | 1 | no | — |
| 49 | Dic83⋊C8 | — | 2 | 1 | no | — |
| 50 | Dic83⋊C8 | — | 2 | 1 | no | — |
| 51 | #14581716 | — | 2 | 1 | no | — |
| 52 | #14581714 | — | 2 | 1 | no | — |
| 53 | #14581714 | — | 4 | 1 | no | — |
| 54 | #14581714 | — | 4 | 1 | no | — |
| 55 | #14581712 | — | 2 | 1 | no | — |
| 56 | #14581708 | — | 4 | 2 | no | — |
| 57 | C2×Q16×C83 | D4 | 1 | 1 | yes | characteristic |
| 58 | C2×M16×C83 | — | 2 | 1 | no | — |
| 59 | C22×C8×C83 | C23 | 1 | 1 | yes | characteristic |
| 60 | C22⋊Q8×C83 | — | 2 | 1 | no | — |
| 61 | C2×C16×C83 | — | 2 | 1 | no | — |
| 62 | #107×C83 | D4 | 1 | 1 | yes | characteristic |
| 63 | #103×C83 | — | 2 | 1 | no | — |
| 64 | C23×D83 | — | 4 | 1 | no | — |
| 65 | C2×Dic166 | — | 4 | 1 | no | — |
| 66 | C2×D332 | — | 2 | 2 | no | — |
| 67 | C2×D332 | — | 2 | 1 | no | — |
| 68 | C2×D332 | — | 4 | 1 | no | — |
| 69 | C2×D332 | — | 4 | 1 | no | — |
| 70 | C2×D332 | — | 4 | 1 | no | — |
| 71 | C2×D332 | — | 4 | 1 | no | — |
| 72 | C2×D332 | — | 4 | 1 | no | — |
| 73 | C2×C22⋊D83 | — | 4 | 1 | no | — |
| 74 | C2×C4×D83 | — | 4 | 1 | no | — |
| 75 | #13050577 | — | 8 | 1 | no | — |
| 76 | D664 | — | 2 | 2 | no | — |
| 77 | D664 | — | 2 | 2 | no | — |
| 78 | D664 | — | 4 | 1 | no | — |
| 79 | D664 | — | 4 | 1 | no | — |
| 80 | D664 | — | 4 | 1 | no | — |
| 81 | #13050574 | — | 4 | 1 | no | — |
| 82 | C2×C83⋊C8 | — | 2 | 1 | no | — |
| 83 | C4⋊Dic83 | — | 2 | 1 | no | — |
| 84 | C4⋊Dic83 | — | 4 | 1 | no | — |
| 85 | C4⋊Dic83 | D8 | 1 | 1 | yes | characteristic |
| 86 | C4⋊Dic83 | D8 | 1 | 1 | yes | characteristic |
| 87 | Dic83⋊C4 | — | 4 | 1 | no | — |
| 88 | C22⋊Dic83 | — | 2 | 1 | no | — |
| 89 | (C2×D83)⋊C4 | — | 4 | 1 | no | — |
| 90 | (C2×D83)⋊C4 | — | 4 | 1 | no | — |
| 91 | (C2×D83)⋊C4 | — | 4 | 1 | no | — |
| 92 | (C2×D83)⋊C4 | — | 4 | 1 | no | — |
| 93 | C4×Dic83 | — | 2 | 1 | no | — |
| 94 | C4×Dic83 | — | 4 | 1 | no | — |
| 95 | C83⋊C16 | — | 4 | 1 | no | — |
| 96 | C2×Q8×C83 | — | 2 | 1 | no | — |
| 97 | C22×C4×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | Q16×C83 | — | 4 | 1 | no | — |
| 99 | M16×C83 | — | 4 | 1 | no | — |
| 100 | M16×C83 | — | 4 | 1 | no | — |
| 101 | C2×C8×C83 | — | 4 | 1 | no | — |
| 102 | C2×C8×C83 | — | 4 | 1 | no | — |
| 103 | C2×C8×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C2×C8×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C4⋊C4×C83 | — | 2 | 1 | no | — |
| 106 | C4⋊C4×C83 | — | 4 | 1 | no | — |
| 107 | C22⋊C4×C83 | — | 4 | 1 | no | — |
| 108 | C16×C83 | — | 4 | 1 | no | — |
| 109 | C23×C83 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 110 | C22×D83 | — | 4 | 1 | no | — |
| 111 | C22×D83 | — | 4 | 2 | no | — |
| 112 | C22×D83 | — | 8 | 1 | no | — |
| 113 | C22×D83 | — | 8 | 1 | no | — |
| 114 | C22×D83 | — | 16 | 1 | no | — |
| 115 | C22×D83 | — | 16 | 1 | no | — |
| 116 | Q8×C83 | — | 4 | 1 | no | — |
| 117 | Q8×C83 | — | 4 | 1 | no | — |
| 118 | C2×C4×C83 | — | 2 | 1 | no | — |
| 119 | C2×C4×C83 | — | 2 | 1 | no | — |
| 120 | C2×C4×C83 | — | 4 | 1 | no | — |
| 121 | C2×C4×C83 | — | 4 | 1 | no | — |
| 122 | C2×C4×C83 | #120 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C83 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | C22⋊D83 | — | 16 | 1 | no | — |
| 125 | C2×Dic83 | — | 2 | 1 | no | — |
| 126 | C2×Dic83 | — | 2 | 1 | no | — |
| 127 | C2×Dic83 | — | 4 | 1 | no | — |
| 128 | D332 | — | 4 | 2 | no | — |
| 129 | D332 | — | 4 | 1 | no | — |
| 130 | D332 | — | 4 | 2 | no | — |
| 131 | D332 | — | 4 | 1 | no | — |
| 132 | D332 | — | 8 | 1 | no | — |
| 133 | D332 | — | 8 | 1 | no | — |
| 134 | C4×D83 | — | 16 | 1 | no | — |
| 135 | Dic166 | — | 16 | 1 | no | — |
| 136 | C8×C83 | — | 2 | 1 | no | — |
| 137 | C8×C83 | — | 2 | 1 | no | — |
| 138 | C8×C83 | — | 8 | 1 | no | — |
| 139 | C83⋊C8 | — | 4 | 1 | no | — |
| 140 | C22×C83 | — | 2 | 1 | no | — |
| 141 | C22×C83 | — | 2 | 1 | no | — |
| 142 | C22×C83 | — | 2 | 1 | no | — |
| 143 | C22×C83 | #447 | 1 | 1 | yes | characteristic |
| 144 | C2×D83 | — | 8 | 2 | no | — |
| 145 | C2×D83 | — | 8 | 1 | no | — |
| 146 | C2×D83 | — | 8 | 1 | no | — |
| 147 | C2×D83 | — | 8 | 1 | no | — |
| 148 | C2×D83 | — | 16 | 2 | no | — |
| 149 | C2×D83 | — | 16 | 1 | no | — |
| 150 | C2×D83 | — | 16 | 1 | no | — |
| 151 | C2×D83 | — | 16 | 2 | no | — |
| 152 | C4×C83 | — | 2 | 1 | no | — |
| 153 | C4×C83 | — | 8 | 1 | no | — |
| 154 | C4×C83 | — | 8 | 1 | no | — |
| 155 | C4×C83 | #454 | 1 | 1 | yes | characteristic |
| 156 | C4×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 157 | Dic83 | — | 4 | 1 | no | — |
| 158 | Dic83 | — | 4 | 1 | no | — |
| 159 | Dic83 | — | 8 | 1 | no | — |
| 160 | #12166 | — | 83 | 1 | no | maximal, Sylow |
| 161 | C2×C83 | — | 4 | 1 | no | — |
| 162 | C2×C83 | #1598 | 1 | 1 | yes | characteristic |
| 163 | C2×C83 | #2192 | 1 | 1 | yes | characteristic |
| 164 | C2×C83 | #2193 | 1 | 1 | yes | characteristic |
| 165 | D83 | — | 16 | 1 | no | — |
| 166 | D83 | — | 16 | 2 | no | — |
| 167 | #3262 | — | 83 | 1 | no | — |
| 168 | #2187 | — | 83 | 1 | no | — |
| 169 | #2184 | — | 83 | 1 | no | — |
| 170 | #2045 | — | 83 | 1 | no | — |
| 171 | #1350 | D83 | 1 | 1 | yes | characteristic, complemented |
| 172 | #1349 | — | 83 | 1 | no | — |
| 173 | #1323 | — | 83 | 1 | no | — |
| 174 | C83 | #12166 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 175 | C22×D8 | — | 83 | 1 | no | — |
| 176 | #542 | — | 166 | 1 | no | — |
| 177 | C2×SD32 | — | 166 | 1 | no | — |
| 178 | C2×D16 | — | 166 | 1 | no | — |
| 179 | #477 | — | 166 | 1 | no | — |
| 180 | #474 | — | 166 | 1 | no | — |
| 181 | #468 | — | 166 | 1 | no | — |
| 182 | #467 | C2×D83 | 1 | 1 | yes | characteristic |
| 183 | #466 | — | 83 | 1 | no | — |
| 184 | #464 | — | 166 | 1 | no | — |
| 185 | #463 | — | 166 | 1 | no | — |
| 186 | #424 | — | 83 | 1 | no | — |
| 187 | #418 | — | 166 | 1 | no | — |
| 188 | C2×#100 | — | 166 | 1 | no | — |
| 189 | #357 | — | 166 | 1 | no | — |
| 190 | #357 | — | 166 | 1 | no | — |
| 191 | C22⋊C16 | — | 83 | 1 | no | — |
| 192 | C22⋊C16 | C2×D83 | 1 | 1 | yes | characteristic |
| 193 | #341 | C2×D83 | 1 | 1 | yes | characteristic |
| 194 | C22×D4 | — | 166 | 1 | no | — |
| 195 | C2×Q16 | D332 | 1 | 1 | yes | characteristic |
| 196 | C2×SD16 | — | 166 | 1 | no | — |
| 197 | C2×D8 | — | 332 | 2 | no | — |
| 198 | C2×D8 | — | 83 | 1 | no | — |
| 199 | C2×D8 | — | 166 | 1 | no | — |
| 200 | C2×D8 | — | 166 | 1 | no | — |
| 201 | C2×M16 | — | 2 | 1 | no | — |
| 202 | C2×M16 | — | 83 | 1 | no | — |
| 203 | C22×C8 | C22×D83 | 1 | 1 | yes | characteristic |
| 204 | C4⋊Q8 | — | 166 | 1 | no | — |
| 205 | #126 | — | 332 | 1 | no | — |
| 206 | #124 | — | 332 | 1 | no | — |
| 207 | #124 | — | 166 | 1 | no | — |
| 208 | #123 | — | 332 | 1 | no | — |
| 209 | C22⋊Q8 | — | 2 | 1 | no | — |
| 210 | C22⋊Q8 | — | 332 | 1 | no | — |
| 211 | #121 | — | 166 | 1 | no | — |
| 212 | C4×Q8 | — | 166 | 1 | no | — |
| 213 | C4×D4 | — | 166 | 1 | no | — |
| 214 | #117 | — | 83 | 1 | no | — |
| 215 | SD32 | — | 332 | 2 | no | — |
| 216 | D16 | — | 332 | 2 | no | — |
| 217 | C2×C16 | — | 2 | 1 | no | — |
| 218 | C2×C16 | — | 166 | 1 | no | — |
| 219 | #107 | — | 83 | 1 | no | — |
| 220 | #107 | — | 166 | 1 | no | — |
| 221 | #107 | D332 | 1 | 1 | yes | characteristic |
| 222 | C4⋊C8 | — | 166 | 1 | no | — |
| 223 | C4⋊C8 | — | 166 | 1 | no | — |
| 224 | #103 | — | 2 | 1 | no | — |
| 225 | #103 | — | 166 | 1 | no | — |
| 226 | #102 | — | 332 | 1 | no | — |
| 227 | #102 | — | 332 | 1 | no | — |
| 228 | #102 | — | 166 | 1 | no | — |
| 229 | #102 | — | 166 | 1 | no | — |
| 230 | #100 | — | 332 | 2 | no | — |
| 231 | C24 | — | 332 | 1 | no | — |
| 232 | C2×Q8 | — | 2 | 1 | no | — |
| 233 | C2×Q8 | — | 332 | 1 | no | — |
| 234 | C2×D4 | — | 332 | 1 | no | — |
| 235 | C2×D4 | — | 332 | 1 | no | — |
| 236 | C2×D4 | — | 332 | 1 | no | — |
| 237 | C2×D4 | — | 332 | 1 | no | — |
| 238 | C2×D4 | — | 332 | 1 | no | — |
| 239 | C2×D4 | — | 332 | 1 | no | — |
| 240 | C2×D4 | — | 166 | 2 | no | — |
| 241 | C2×D4 | — | 166 | 1 | no | — |
| 242 | C22×C4 | — | 332 | 1 | no | — |
| 243 | C22×C4 | D4×D83 | 1 | 1 | yes | characteristic |
| 244 | Q16 | — | 4 | 1 | no | — |
| 245 | SD16 | — | 664 | 1 | no | — |
| 246 | D8 | — | 332 | 1 | no | — |
| 247 | D8 | — | 332 | 1 | no | — |
| 248 | D8 | — | 332 | 1 | no | — |
| 249 | D8 | — | 166 | 2 | no | — |
| 250 | D8 | — | 166 | 2 | no | — |
| 251 | M16 | — | 4 | 1 | no | — |
| 252 | M16 | — | 4 | 1 | no | — |
| 253 | M16 | — | 332 | 1 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 4 | 1 | no | — |
| 256 | C2×C8 | — | 166 | 1 | no | — |
| 257 | C2×C8 | D4×D83 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | C2×D332 | 1 | 1 | yes | characteristic |
| 259 | C4⋊C4 | — | 2 | 1 | no | — |
| 260 | C4⋊C4 | — | 4 | 1 | no | — |
| 261 | C4⋊C4 | — | 332 | 1 | no | — |
| 262 | C4⋊C4 | — | 332 | 1 | no | — |
| 263 | C4⋊C4 | — | 83 | 1 | no | — |
| 264 | C4⋊C4 | — | 83 | 1 | no | — |
| 265 | C4⋊C4 | — | 166 | 1 | no | — |
| 266 | C22⋊C4 | — | 4 | 1 | no | — |
| 267 | C22⋊C4 | — | 332 | 1 | no | — |
| 268 | C22⋊C4 | — | 332 | 1 | no | — |
| 269 | C22⋊C4 | — | 332 | 1 | no | — |
| 270 | C22⋊C4 | — | 332 | 1 | no | — |
| 271 | C22⋊C4 | — | 166 | 1 | no | — |
| 272 | C42 | — | 332 | 1 | no | — |
| 273 | C42 | — | 166 | 1 | no | — |
| 274 | C16 | — | 4 | 1 | no | — |
| 275 | C16 | — | 332 | 1 | no | — |
| 276 | C23 | — | 1328 | 1 | no | — |
| 277 | C23 | — | 1328 | 1 | no | — |
| 278 | C23 | — | 332 | 2 | no | — |
| 279 | C23 | — | 332 | 1 | no | — |
| 280 | C23 | — | 664 | 1 | no | — |
| 281 | C23 | — | 664 | 1 | no | — |
| 282 | C23 | #14581819 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 4 | 1 | no | — |
| 284 | Q8 | — | 4 | 1 | no | — |
| 285 | Q8 | — | 1328 | 1 | no | — |
| 286 | D4 | — | 1328 | 1 | no | — |
| 287 | D4 | — | 332 | 2 | no | — |
| 288 | D4 | — | 332 | 1 | no | — |
| 289 | D4 | — | 332 | 2 | no | — |
| 290 | D4 | — | 332 | 1 | no | — |
| 291 | D4 | — | 664 | 1 | no | — |
| 292 | D4 | — | 664 | 1 | no | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 2 | 1 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 4 | 1 | no | — |
| 297 | C2×C4 | — | 1328 | 1 | no | — |
| 298 | C2×C4 | — | 332 | 1 | no | — |
| 299 | C2×C4 | — | 166 | 1 | no | — |
| 300 | C2×C4 | — | 166 | 1 | no | — |
| 301 | C2×C4 | #14581762 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | D8×D83 | 1 | 1 | yes | characteristic |
| 303 | C8 | — | 2 | 1 | no | — |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 8 | 1 | no | — |
| 306 | C8 | — | 332 | 1 | no | — |
| 307 | C22 | — | 2 | 1 | no | — |
| 308 | C22 | — | 2 | 1 | no | — |
| 309 | C22 | — | 2 | 1 | no | — |
| 310 | C22 | — | 1328 | 2 | no | — |
| 311 | C22 | — | 1328 | 2 | no | — |
| 312 | C22 | — | 1328 | 1 | no | — |
| 313 | C22 | — | 1328 | 1 | no | — |
| 314 | C22 | — | 664 | 1 | no | — |
| 315 | C22 | — | 664 | 1 | no | — |
| 316 | C22 | — | 664 | 2 | no | — |
| 317 | C22 | — | 664 | 1 | no | — |
| 318 | C22 | #19127836 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 2 | 1 | no | — |
| 320 | C4 | — | 8 | 1 | no | — |
| 321 | C4 | — | 8 | 1 | no | — |
| 322 | C4 | — | 332 | 1 | no | — |
| 323 | C4 | — | 332 | 1 | no | — |
| 324 | C4 | — | 664 | 1 | no | — |
| 325 | C4 | #19127875 | 1 | 1 | yes | characteristic |
| 326 | C4 | #19127889 | 1 | 1 | yes | characteristic |
| 327 | C2 | — | 4 | 1 | no | — |
| 328 | C2 | — | 1328 | 2 | no | — |
| 329 | C2 | — | 1328 | 1 | no | — |
| 330 | C2 | #24906399 | 1 | 1 | yes | characteristic |
| 331 | C2 | #24909421 | 1 | 1 | yes | characteristic |
| 332 | C2 | #24909430 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #24918925 | C2 | semidirect |
| 2. #24909391 | C2 | semidirect |
| 4. #24908720 | C2 | semidirect |
| 5. #24905477 | C2 | semidirect |
| 6. #24905404 | C2 | semidirect |
| 7. C83×#1350 | C2 | semidirect |
| 171. #1350 | D83 | semidirect |
| 174. C83 | #12166 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24918925 | 2 |
| #24909391 | 2 |
| #24909377 | 2 |
| #24908720 | 2 |
| #24905477 | 2 |
| #24905404 | 2 |
| C83×#1350 | 2 |
| #12166 | 83 |