Finite Group DatabasebetaFinite Group DatabasebetaOrder 21248.
540 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 73 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 | #24915451 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #24907424 | C2 | 1 | 2 | yes | maximal |
| 3 | #24907421 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #24905589 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C83×#1394 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14581840 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#14581720 | — | 2 | 1 | no | — |
| 8 | C2×#14581718 | — | 2 | 2 | no | — |
| 9 | #19127703 | — | 2 | 2 | no | — |
| 10 | #19127700 | — | 2 | 1 | no | — |
| 11 | C2×#14581711 | C22 | 1 | 1 | yes | characteristic |
| 12 | #19127640 | — | 2 | 1 | no | — |
| 13 | #19127633 | — | 2 | 2 | no | — |
| 14 | #19127608 | C22 | 1 | 2 | yes | — |
| 15 | #19127599 | — | 2 | 1 | no | — |
| 16 | #19127588 | — | 2 | 2 | no | — |
| 17 | C2×#101×C83 | C22 | 1 | 2 | yes | — |
| 18 | #391×C83 | C22 | 1 | 1 | yes | characteristic |
| 19 | #14581840 | — | 2 | 1 | no | — |
| 20 | #14581840 | — | 2 | 2 | no | — |
| 21 | #14581840 | — | 4 | 2 | no | — |
| 22 | #14581840 | Q8 | 1 | 2 | yes | — |
| 23 | C2×#13050591 | — | 4 | 1 | no | — |
| 24 | C2×#13050590 | — | 2 | 1 | no | — |
| 25 | C2×#13050590 | — | 2 | 2 | no | — |
| 26 | C2×Q8×D83 | — | 4 | 1 | no | — |
| 27 | C2×#13050574 | — | 2 | 1 | no | — |
| 28 | C2×C4⋊Dic83 | — | 2 | 1 | no | — |
| 29 | C2×C4⋊Dic83 | — | 2 | 1 | no | — |
| 30 | C2×C4⋊Dic83 | — | 2 | 1 | no | — |
| 31 | C2×C4⋊Dic83 | D4 | 1 | 2 | yes | — |
| 32 | C2×Dic83⋊C4 | — | 2 | 2 | no | — |
| 33 | C2×C4×Dic83 | D4 | 1 | 2 | yes | — |
| 34 | #14581720 | — | 4 | 2 | no | — |
| 35 | #14581718 | — | 4 | 4 | no | — |
| 36 | #14581717 | — | 4 | 2 | no | — |
| 37 | #14581715 | — | 4 | 1 | no | — |
| 38 | #14581714 | — | 4 | 2 | no | — |
| 39 | #14581712 | — | 4 | 1 | no | — |
| 40 | #14581711 | — | 2 | 2 | no | — |
| 41 | C83⋊#95 | — | 2 | 1 | no | — |
| 42 | C83⋊#95 | — | 2 | 2 | no | — |
| 43 | C83⋊#95 | — | 2 | 2 | no | — |
| 44 | C83⋊#95 | — | 2 | 2 | no | — |
| 45 | C22×Q8×C83 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×M16×C83 | — | 2 | 2 | no | — |
| 47 | C2×C4⋊C4×C83 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C42×C83 | D4 | 1 | 1 | yes | characteristic |
| 49 | #101×C83 | — | 2 | 2 | no | — |
| 50 | #101×C83 | — | 2 | 2 | no | — |
| 51 | #95×C83 | — | 2 | 2 | no | — |
| 52 | #13050591 | — | 4 | 2 | no | — |
| 53 | #13050591 | — | 4 | 2 | no | — |
| 54 | #13050591 | — | 4 | 4 | no | — |
| 55 | #13050590 | — | 2 | 4 | no | — |
| 56 | #13050590 | — | 2 | 2 | no | — |
| 57 | #13050590 | — | 2 | 4 | no | — |
| 58 | #13050590 | — | 2 | 2 | no | — |
| 59 | #13050590 | — | 4 | 1 | no | — |
| 60 | #13050590 | — | 4 | 2 | no | — |
| 61 | #13050590 | — | 4 | 2 | no | — |
| 62 | #13050590 | — | 4 | 1 | no | — |
| 63 | C2×Dic166 | — | 2 | 1 | no | — |
| 64 | C2×Dic166 | — | 2 | 2 | no | — |
| 65 | Q8×D83 | — | 4 | 2 | no | — |
| 66 | Q8×D83 | — | 4 | 2 | no | — |
| 67 | Q8×D83 | — | 4 | 4 | no | — |
| 68 | C2×D332 | — | 2 | 1 | no | — |
| 69 | C2×D332 | — | 2 | 2 | no | — |
| 70 | C2×C22⋊D83 | — | 4 | 1 | no | — |
| 71 | C22×Dic83 | — | 2 | 2 | no | — |
| 72 | C22×Dic83 | — | 2 | 2 | no | — |
| 73 | C2×C4×D83 | — | 4 | 1 | no | — |
| 74 | C2×C4×D83 | — | 4 | 2 | no | — |
| 75 | #13050574 | — | 4 | 1 | no | — |
| 76 | #13050574 | — | 4 | 1 | no | — |
| 77 | C2×C83⋊C8 | — | 4 | 1 | no | — |
| 78 | C4⋊Dic83 | — | 4 | 2 | no | — |
| 79 | C4⋊Dic83 | — | 4 | 2 | no | — |
| 80 | C4⋊Dic83 | — | 8 | 1 | no | — |
| 81 | C4⋊Dic83 | — | 8 | 1 | no | — |
| 82 | Dic83⋊C4 | — | 4 | 4 | no | — |
| 83 | C4×Dic83 | — | 4 | 2 | no | — |
| 84 | C2×Q8×C83 | — | 4 | 1 | no | — |
| 85 | C2×Q8×C83 | — | 4 | 1 | no | — |
| 86 | C2×Q8×C83 | C2×Q8 | 1 | 1 | yes | characteristic |
| 87 | C2×Q8×C83 | #41 | 1 | 1 | yes | characteristic |
| 88 | C2×Q8×C83 | #41 | 1 | 2 | yes | — |
| 89 | C22×C4×C83 | — | 2 | 1 | no | — |
| 90 | C22×C4×C83 | — | 2 | 1 | no | — |
| 91 | C22×C4×C83 | C2×D4 | 1 | 2 | yes | — |
| 92 | C22×C4×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | M16×C83 | — | 4 | 2 | no | — |
| 94 | M16×C83 | — | 4 | 2 | no | — |
| 95 | C2×C8×C83 | — | 4 | 2 | no | — |
| 96 | C4⋊C4×C83 | — | 4 | 1 | no | — |
| 97 | C42×C83 | — | 4 | 1 | no | — |
| 98 | C23×C83 | #120 | 1 | 1 | yes | characteristic |
| 99 | C22×D83 | — | 4 | 1 | no | — |
| 100 | Q8×C83 | — | 4 | 1 | no | — |
| 101 | Q8×C83 | — | 4 | 1 | no | — |
| 102 | Q8×C83 | — | 4 | 2 | no | — |
| 103 | C2×C4×C83 | — | 2 | 1 | no | — |
| 104 | C2×C4×C83 | — | 2 | 1 | no | — |
| 105 | C2×C4×C83 | — | 2 | 2 | no | — |
| 106 | C2×C4×C83 | — | 2 | 2 | no | — |
| 107 | C2×C4×C83 | — | 2 | 2 | no | — |
| 108 | C2×C4×C83 | — | 4 | 1 | no | — |
| 109 | C2×C4×C83 | — | 4 | 1 | no | — |
| 110 | C2×C4×C83 | — | 4 | 2 | no | — |
| 111 | C2×C4×C83 | — | 4 | 1 | no | — |
| 112 | C2×C4×C83 | C22⋊Q8 | 1 | 2 | yes | — |
| 113 | C2×C4×C83 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C83 | #124 | 1 | 2 | yes | — |
| 115 | C2×C4×C83 | #124 | 1 | 1 | yes | characteristic |
| 116 | C22⋊D83 | — | 4 | 2 | no | — |
| 117 | C22⋊D83 | — | 4 | 2 | no | — |
| 118 | C2×Dic83 | — | 4 | 1 | no | — |
| 119 | C2×Dic83 | — | 4 | 2 | no | — |
| 120 | C2×Dic83 | — | 4 | 2 | no | — |
| 121 | C2×Dic83 | — | 4 | 2 | no | — |
| 122 | C2×Dic83 | — | 4 | 2 | no | — |
| 123 | C2×Dic83 | — | 4 | 2 | no | — |
| 124 | C2×Dic83 | — | 4 | 2 | no | — |
| 125 | D332 | — | 2 | 2 | no | — |
| 126 | D332 | — | 2 | 4 | no | — |
| 127 | D332 | — | 4 | 1 | no | — |
| 128 | D332 | — | 4 | 2 | no | — |
| 129 | C4×D83 | — | 4 | 2 | no | — |
| 130 | C4×D83 | — | 4 | 4 | no | — |
| 131 | C4×D83 | — | 4 | 4 | no | — |
| 132 | C4×D83 | — | 4 | 2 | no | — |
| 133 | Dic166 | — | 2 | 2 | no | — |
| 134 | Dic166 | — | 2 | 4 | no | — |
| 135 | Dic166 | — | 4 | 1 | no | — |
| 136 | Dic166 | — | 4 | 2 | no | — |
| 137 | C8×C83 | — | 8 | 2 | no | — |
| 138 | C83⋊C8 | — | 8 | 1 | no | — |
| 139 | C22×C83 | — | 2 | 1 | no | — |
| 140 | C22×C83 | — | 2 | 1 | no | — |
| 141 | C22×C83 | #393 | 1 | 1 | yes | characteristic |
| 142 | C22×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 143 | C22×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 144 | C2×D83 | — | 4 | 2 | no | — |
| 145 | C2×D83 | — | 8 | 1 | no | — |
| 146 | C2×D83 | — | 8 | 1 | no | — |
| 147 | C4×C83 | — | 2 | 1 | no | — |
| 148 | C4×C83 | — | 2 | 2 | no | — |
| 149 | C4×C83 | — | 2 | 2 | no | — |
| 150 | C4×C83 | — | 2 | 1 | no | — |
| 151 | C4×C83 | — | 4 | 2 | no | — |
| 152 | C4×C83 | — | 8 | 1 | no | — |
| 153 | Dic83 | — | 4 | 2 | no | — |
| 154 | Dic83 | — | 8 | 2 | no | — |
| 155 | Dic83 | — | 8 | 2 | no | — |
| 156 | #13379 | — | 83 | 1 | no | maximal, Sylow |
| 157 | C2×C83 | — | 2 | 1 | no | — |
| 158 | C2×C83 | — | 2 | 1 | no | — |
| 159 | C2×C83 | #2029 | 1 | 1 | yes | characteristic |
| 160 | C2×C83 | #2203 | 1 | 1 | yes | characteristic |
| 161 | C2×C83 | #2204 | 1 | 1 | yes | characteristic |
| 162 | D83 | — | 8 | 2 | no | — |
| 163 | #2809 | — | 83 | 1 | no | — |
| 164 | #1793 | — | 83 | 2 | no | — |
| 165 | #1793 | — | 83 | 1 | no | — |
| 166 | #1394 | — | 83 | 2 | no | — |
| 167 | #1394 | D83 | 1 | 1 | yes | characteristic, complemented |
| 168 | C83 | #13379 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 169 | C2×2−1+4 | — | 83 | 1 | no | — |
| 170 | C2×#104 | — | 166 | 2 | no | — |
| 171 | C2×#104 | — | 166 | 1 | no | — |
| 172 | #417 | — | 166 | 2 | no | — |
| 173 | #417 | — | 166 | 1 | no | — |
| 174 | C2×#101 | — | 83 | 1 | no | — |
| 175 | C2×#101 | C2×D83 | 1 | 2 | yes | — |
| 176 | #400 | — | 166 | 1 | no | — |
| 177 | #398 | — | 166 | 2 | no | — |
| 178 | #391 | — | 83 | 2 | no | — |
| 179 | #391 | C2×D83 | 1 | 1 | yes | characteristic |
| 180 | #389 | — | 166 | 1 | no | — |
| 181 | #387 | — | 166 | 2 | no | — |
| 182 | 2−1+4 | — | 332 | 2 | no | — |
| 183 | 2−1+4 | — | 83 | 2 | no | — |
| 184 | 2−1+4 | — | 166 | 1 | no | — |
| 185 | 2−1+4 | — | 166 | 2 | no | — |
| 186 | C2×#41 | — | 332 | 1 | no | — |
| 187 | C2×#41 | — | 166 | 2 | no | — |
| 188 | C2×#41 | — | 166 | 1 | no | — |
| 189 | C22×Q8 | — | 332 | 1 | no | — |
| 190 | C22×Q8 | C22×D83 | 1 | 1 | yes | characteristic |
| 191 | C2×M16 | — | 2 | 2 | no | — |
| 192 | C2×M16 | — | 166 | 1 | no | — |
| 193 | C2×C4⋊C4 | — | 83 | 2 | no | — |
| 194 | C2×C4⋊C4 | — | 166 | 2 | no | — |
| 195 | C2×C4⋊C4 | — | 166 | 1 | no | — |
| 196 | C2×C4⋊C4 | — | 166 | 1 | no | — |
| 197 | C2×C4⋊C4 | — | 166 | 1 | no | — |
| 198 | C2×C4⋊C4 | D332 | 1 | 1 | yes | characteristic |
| 199 | C2×C42 | — | 83 | 2 | no | — |
| 200 | C2×C42 | D332 | 1 | 1 | yes | characteristic |
| 201 | #104 | — | 332 | 2 | no | — |
| 202 | #104 | — | 332 | 4 | no | — |
| 203 | #103 | — | 332 | 1 | no | — |
| 204 | #103 | — | 332 | 2 | no | — |
| 205 | #102 | — | 332 | 1 | no | — |
| 206 | #102 | — | 332 | 2 | no | — |
| 207 | #101 | — | 2 | 2 | no | — |
| 208 | #101 | — | 2 | 2 | no | — |
| 209 | #101 | — | 166 | 2 | no | — |
| 210 | #95 | — | 2 | 2 | no | — |
| 211 | #95 | — | 166 | 1 | no | — |
| 212 | #95 | — | 166 | 2 | no | — |
| 213 | #95 | — | 166 | 2 | no | — |
| 214 | #95 | — | 166 | 2 | no | — |
| 215 | #41 | — | 332 | 1 | no | — |
| 216 | #41 | — | 332 | 2 | no | — |
| 217 | #41 | — | 332 | 2 | no | — |
| 218 | #41 | — | 332 | 4 | no | — |
| 219 | #41 | — | 332 | 2 | no | — |
| 220 | #41 | — | 332 | 2 | no | — |
| 221 | #41 | — | 332 | 1 | no | — |
| 222 | #41 | — | 166 | 2 | no | — |
| 223 | #41 | — | 166 | 4 | no | — |
| 224 | #41 | — | 166 | 4 | no | — |
| 225 | #41 | — | 166 | 2 | no | — |
| 226 | C2×Q8 | — | 4 | 1 | no | — |
| 227 | C2×Q8 | — | 4 | 1 | no | — |
| 228 | C2×Q8 | — | 332 | 2 | no | — |
| 229 | C2×Q8 | — | 332 | 4 | no | — |
| 230 | C2×Q8 | — | 332 | 2 | no | — |
| 231 | C2×Q8 | — | 166 | 1 | no | — |
| 232 | C2×Q8 | — | 166 | 2 | no | — |
| 233 | C2×Q8 | Q8×D83 | 1 | 1 | yes | characteristic |
| 234 | C2×Q8 | #13050589 | 1 | 1 | yes | characteristic |
| 235 | C2×Q8 | #13050590 | 1 | 2 | yes | — |
| 236 | C2×D4 | — | 332 | 1 | no | — |
| 237 | C2×D4 | — | 166 | 1 | no | — |
| 238 | C2×D4 | — | 166 | 2 | no | — |
| 239 | C22×C4 | — | 2 | 1 | no | — |
| 240 | C22×C4 | — | 2 | 1 | no | — |
| 241 | C22×C4 | — | 332 | 1 | no | — |
| 242 | C22×C4 | — | 332 | 2 | no | — |
| 243 | C22×C4 | — | 166 | 2 | no | — |
| 244 | C22×C4 | — | 166 | 2 | no | — |
| 245 | C22×C4 | D4×D83 | 1 | 2 | yes | — |
| 246 | C22×C4 | C2×D332 | 1 | 1 | yes | characteristic |
| 247 | M16 | — | 4 | 2 | no | — |
| 248 | M16 | — | 4 | 2 | no | — |
| 249 | M16 | — | 332 | 1 | no | — |
| 250 | M16 | — | 332 | 1 | no | — |
| 251 | C2×C8 | — | 4 | 2 | no | — |
| 252 | C2×C8 | — | 332 | 1 | no | — |
| 253 | C4⋊C4 | — | 4 | 1 | no | — |
| 254 | C4⋊C4 | — | 332 | 2 | no | — |
| 255 | C4⋊C4 | — | 332 | 4 | no | — |
| 256 | C4⋊C4 | — | 332 | 2 | no | — |
| 257 | C4⋊C4 | — | 664 | 1 | no | — |
| 258 | C4⋊C4 | — | 664 | 1 | no | — |
| 259 | C42 | — | 4 | 1 | no | — |
| 260 | C42 | — | 332 | 2 | no | — |
| 261 | C23 | — | 332 | 1 | no | — |
| 262 | C23 | #14581762 | 1 | 1 | yes | characteristic |
| 263 | Q8 | — | 4 | 1 | no | — |
| 264 | Q8 | — | 4 | 1 | no | — |
| 265 | Q8 | — | 4 | 2 | no | — |
| 266 | Q8 | — | 332 | 2 | no | — |
| 267 | Q8 | — | 332 | 1 | no | — |
| 268 | Q8 | — | 166 | 4 | no | — |
| 269 | Q8 | — | 166 | 2 | no | — |
| 270 | D4 | — | 332 | 2 | no | — |
| 271 | D4 | — | 332 | 1 | no | — |
| 272 | D4 | — | 332 | 2 | no | — |
| 273 | D4 | — | 332 | 2 | no | — |
| 274 | D4 | — | 166 | 4 | no | — |
| 275 | D4 | — | 166 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 1 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 1 | no | — |
| 282 | C2×C4 | — | 4 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 332 | 4 | no | — |
| 286 | C2×C4 | — | 332 | 2 | no | — |
| 287 | C2×C4 | — | 332 | 4 | no | — |
| 288 | C2×C4 | — | 332 | 2 | no | — |
| 289 | C2×C4 | — | 332 | 1 | no | — |
| 290 | C2×C4 | — | 332 | 2 | no | — |
| 291 | C2×C4 | — | 332 | 2 | no | — |
| 292 | C2×C4 | — | 332 | 2 | no | — |
| 293 | C2×C4 | — | 332 | 2 | no | — |
| 294 | C2×C4 | — | 332 | 2 | no | — |
| 295 | C2×C4 | — | 332 | 2 | no | — |
| 296 | C2×C4 | #14581770 | 1 | 2 | yes | — |
| 297 | C2×C4 | #14581772 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #14581778 | 1 | 2 | yes | — |
| 299 | C2×C4 | #14581780 | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 8 | 2 | no | — |
| 301 | C8 | — | 664 | 1 | no | — |
| 302 | C22 | — | 2 | 1 | no | — |
| 303 | C22 | — | 2 | 1 | no | — |
| 304 | C22 | — | 332 | 2 | no | — |
| 305 | C22 | — | 664 | 1 | no | — |
| 306 | C22 | — | 664 | 1 | no | — |
| 307 | C22 | #19127614 | 1 | 1 | yes | characteristic |
| 308 | C22 | #19127889 | 1 | 1 | yes | characteristic |
| 309 | C22 | #19127889 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 2 | 2 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 2 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 4 | 2 | no | — |
| 315 | C4 | — | 8 | 1 | no | — |
| 316 | C4 | — | 332 | 2 | no | — |
| 317 | C4 | — | 664 | 2 | no | — |
| 318 | C4 | — | 664 | 2 | no | — |
| 319 | C2 | — | 2 | 1 | no | — |
| 320 | C2 | — | 2 | 1 | no | — |
| 321 | C2 | — | 664 | 2 | no | — |
| 322 | C2 | #24908628 | 1 | 1 | yes | characteristic |
| 323 | C2 | #24909486 | 1 | 1 | yes | characteristic |
| 324 | C2 | #24909491 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #24915451 | C2 | semidirect |
| 4. #24905589 | C2 | semidirect |
| 5. C83×#1394 | C2 | semidirect |
| 167. #1394 | D83 | semidirect |
| 168. C83 | #13379 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24915451 | 2 |
| #24907424 | 2 |
| #24907421 | 2 |
| #24905589 | 2 |
| C83×#1394 | 2 |
| #13379 | 83 |