Finite Group DatabasebetaFinite Group DatabasebetaOrder 21248.
1184 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #24916811 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #24909493 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #24909489 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #24909148 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C83×#3483 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19128537 | C22 | 1 | 1 | yes | characteristic |
| 7 | #19128499 | — | 2 | 3 | no | — |
| 8 | #19127883 | C22 | 1 | 6 | yes | — |
| 9 | #19127878 | — | 2 | 3 | no | — |
| 10 | #19127873 | C22 | 1 | 6 | yes | — |
| 11 | #19127868 | — | 2 | 3 | no | — |
| 12 | C2×#14581720 | — | 2 | 3 | no | — |
| 13 | #19127685 | C22 | 1 | 3 | yes | — |
| 14 | #19127500 | C22 | 1 | 6 | yes | complemented |
| 15 | #19127498 | C22 | 1 | 6 | yes | complemented |
| 16 | Q82×C83 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #555×C83 | — | 2 | 3 | no | — |
| 18 | #555×C83 | C22 | 1 | 3 | yes | complemented |
| 19 | #550×C83 | — | 2 | 3 | no | — |
| 20 | #532×C83 | — | 2 | 3 | no | — |
| 21 | #532×C83 | C22 | 1 | 3 | yes | — |
| 22 | C4×#41×C83 | — | 2 | 3 | no | — |
| 23 | #14581840 | C23 | 1 | 2 | yes | — |
| 24 | #14581839 | — | 4 | 1 | no | — |
| 25 | #14581838 | — | 4 | 1 | no | — |
| 26 | #14581837 | — | 2 | 3 | no | — |
| 27 | C2×#13050590 | — | 2 | 3 | no | — |
| 28 | #41×D83 | — | 4 | 2 | no | — |
| 29 | #14581823 | — | 2 | 6 | no | — |
| 30 | #14581822 | — | 4 | 3 | no | — |
| 31 | #14581818 | — | 2 | 6 | no | — |
| 32 | #14581817 | — | 4 | 3 | no | — |
| 33 | #14581812 | — | 4 | 6 | no | — |
| 34 | C2×#13050574 | — | 2 | 3 | no | — |
| 35 | #14581720 | — | 2 | 6 | no | — |
| 36 | #14581720 | — | 2 | 6 | no | — |
| 37 | #14581711 | C23 | 1 | 6 | yes | — |
| 38 | #14581709 | — | 2 | 3 | no | — |
| 39 | C2×#41×C83 | — | 2 | 3 | no | — |
| 40 | C2×#41×C83 | — | 2 | 1 | no | — |
| 41 | C2×#41×C83 | D4 | 1 | 3 | yes | — |
| 42 | C2×#41×C83 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4⋊Q8×C83 | — | 2 | 3 | no | — |
| 44 | C4⋊Q8×C83 | C23 | 1 | 3 | yes | — |
| 45 | #127×C83 | — | 2 | 3 | no | — |
| 46 | #127×C83 | D4 | 1 | 3 | yes | — |
| 47 | #124×C83 | — | 2 | 3 | no | — |
| 48 | #124×C83 | — | 2 | 3 | no | — |
| 49 | #124×C83 | D4 | 1 | 3 | yes | — |
| 50 | #124×C83 | C23 | 1 | 3 | yes | — |
| 51 | #121×C83 | — | 2 | 6 | no | — |
| 52 | #121×C83 | — | 2 | 6 | no | — |
| 53 | #121×C83 | — | 2 | 6 | no | — |
| 54 | C4×Q8×C83 | — | 2 | 1 | no | — |
| 55 | C4×Q8×C83 | — | 2 | 3 | no | — |
| 56 | C4×D4×C83 | — | 2 | 3 | no | — |
| 57 | C4×D4×C83 | — | 2 | 6 | no | — |
| 58 | C4×D4×C83 | — | 2 | 6 | no | — |
| 59 | C4×D4×C83 | — | 2 | 3 | no | — |
| 60 | #117×C83 | — | 2 | 3 | no | — |
| 61 | #117×C83 | — | 2 | 6 | no | — |
| 62 | C2×C42×C83 | — | 2 | 3 | no | — |
| 63 | C2×C42×C83 | D4 | 1 | 3 | yes | — |
| 64 | #13050591 | — | 4 | 2 | no | — |
| 65 | #13050590 | — | 2 | 6 | no | — |
| 66 | #13050590 | — | 2 | 6 | no | — |
| 67 | #13050590 | — | 4 | 3 | no | — |
| 68 | #13050590 | — | 4 | 1 | no | — |
| 69 | #13050589 | — | 4 | 6 | no | — |
| 70 | C2×Dic166 | — | 2 | 3 | no | — |
| 71 | Q8×D83 | — | 4 | 2 | no | — |
| 72 | C2×D332 | — | 2 | 3 | no | — |
| 73 | C2×C22⋊D83 | — | 4 | 3 | no | — |
| 74 | D4×D83 | — | 4 | 6 | no | — |
| 75 | C2×C4×D83 | — | 4 | 3 | no | — |
| 76 | #13050580 | — | 4 | 6 | no | — |
| 77 | #13050578 | — | 4 | 6 | no | — |
| 78 | #13050577 | — | 4 | 6 | no | — |
| 79 | #13050575 | — | 4 | 6 | no | — |
| 80 | #13050574 | — | 2 | 6 | no | — |
| 81 | #13050574 | — | 4 | 3 | no | — |
| 82 | C2×C83⋊C8 | — | 4 | 3 | no | — |
| 83 | #41×C83 | — | 4 | 1 | no | — |
| 84 | #41×C83 | — | 4 | 1 | no | — |
| 85 | #41×C83 | — | 4 | 3 | no | — |
| 86 | #41×C83 | — | 4 | 3 | no | — |
| 87 | #41×C83 | — | 4 | 12 | no | — |
| 88 | #41×C83 | — | 8 | 1 | no | — |
| 89 | #41×C83 | — | 8 | 1 | no | — |
| 90 | C2×Q8×C83 | — | 2 | 1 | no | — |
| 91 | C2×Q8×C83 | — | 2 | 1 | no | — |
| 92 | C2×Q8×C83 | C2×D4 | 1 | 3 | yes | — |
| 93 | C2×Q8×C83 | C24 | 1 | 1 | yes | characteristic |
| 94 | C2×D4×C83 | — | 2 | 3 | no | — |
| 95 | C2×D4×C83 | — | 2 | 6 | no | — |
| 96 | C2×D4×C83 | — | 2 | 3 | no | — |
| 97 | C2×D4×C83 | — | 2 | 3 | no | — |
| 98 | C2×D4×C83 | C2×D4 | 1 | 3 | yes | — |
| 99 | C2×D4×C83 | C2×D4 | 1 | 3 | yes | — |
| 100 | C22×C4×C83 | — | 2 | 6 | no | — |
| 101 | C22×C4×C83 | — | 2 | 3 | no | — |
| 102 | C22×C4×C83 | — | 2 | 3 | no | — |
| 103 | C22×C4×C83 | C2×D4 | 1 | 3 | yes | — |
| 104 | C4⋊C4×C83 | — | 2 | 3 | no | — |
| 105 | C4⋊C4×C83 | — | 2 | 3 | no | — |
| 106 | C4⋊C4×C83 | — | 2 | 6 | no | — |
| 107 | C22⋊C4×C83 | — | 2 | 6 | no | — |
| 108 | C22⋊C4×C83 | — | 2 | 6 | no | — |
| 109 | C22⋊C4×C83 | — | 2 | 6 | no | — |
| 110 | C42×C83 | — | 2 | 3 | no | — |
| 111 | C42×C83 | — | 2 | 3 | no | — |
| 112 | C42×C83 | — | 2 | 3 | no | — |
| 113 | C42×C83 | C2×D4 | 1 | 3 | yes | — |
| 114 | C42×C83 | C2×D4 | 1 | 3 | yes | — |
| 115 | C23×C83 | — | 2 | 3 | no | — |
| 116 | C23×C83 | #120 | 1 | 3 | yes | — |
| 117 | C22×D83 | — | 4 | 3 | no | — |
| 118 | Q8×C83 | — | 2 | 4 | no | — |
| 119 | Q8×C83 | — | 4 | 1 | no | — |
| 120 | Q8×C83 | — | 4 | 3 | no | — |
| 121 | Q8×C83 | — | 8 | 1 | no | — |
| 122 | D4×C83 | — | 4 | 3 | no | — |
| 123 | D4×C83 | — | 4 | 3 | no | — |
| 124 | D4×C83 | — | 4 | 12 | no | — |
| 125 | D4×C83 | — | 4 | 6 | no | — |
| 126 | D4×C83 | — | 8 | 3 | no | — |
| 127 | D4×C83 | — | 8 | 3 | no | — |
| 128 | C2×C4×C83 | — | 2 | 3 | no | — |
| 129 | C2×C4×C83 | — | 2 | 3 | no | — |
| 130 | C2×C4×C83 | — | 2 | 3 | no | — |
| 131 | C2×C4×C83 | — | 2 | 3 | no | — |
| 132 | C2×C4×C83 | — | 2 | 3 | no | — |
| 133 | C2×C4×C83 | — | 4 | 6 | no | — |
| 134 | C2×C4×C83 | — | 4 | 12 | no | — |
| 135 | C2×C4×C83 | — | 4 | 3 | no | — |
| 136 | C2×C4×C83 | — | 4 | 3 | no | — |
| 137 | C2×C4×C83 | #120 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C83 | C22×D4 | 1 | 3 | yes | — |
| 139 | C22⋊D83 | — | 4 | 6 | no | — |
| 140 | C22⋊D83 | — | 4 | 2 | no | — |
| 141 | C2×Dic83 | — | 4 | 3 | no | — |
| 142 | D332 | — | 2 | 6 | no | — |
| 143 | D332 | — | 4 | 1 | no | — |
| 144 | C4×D83 | — | 4 | 6 | no | — |
| 145 | C4×D83 | — | 4 | 2 | no | — |
| 146 | Dic166 | — | 2 | 6 | no | — |
| 147 | Dic166 | — | 4 | 1 | no | — |
| 148 | C83⋊C8 | — | 4 | 6 | no | — |
| 149 | C22×C83 | — | 2 | 3 | no | — |
| 150 | C22×C83 | — | 4 | 3 | no | — |
| 151 | C22×C83 | — | 4 | 6 | no | — |
| 152 | C22×C83 | — | 4 | 3 | no | — |
| 153 | C22×C83 | C2×#120 | 1 | 1 | yes | characteristic |
| 154 | C2×D83 | — | 4 | 2 | no | — |
| 155 | C2×D83 | — | 16 | 3 | no | — |
| 156 | C4×C83 | — | 2 | 6 | no | — |
| 157 | C4×C83 | — | 2 | 3 | no | — |
| 158 | C4×C83 | — | 4 | 3 | no | — |
| 159 | C4×C83 | — | 4 | 3 | no | — |
| 160 | C4×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 161 | C4×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 162 | Dic83 | — | 4 | 2 | no | — |
| 163 | #33567 | — | 83 | 1 | no | maximal, Sylow |
| 164 | C2×C83 | — | 2 | 1 | no | — |
| 165 | C2×C83 | — | 4 | 3 | no | — |
| 166 | C2×C83 | — | 8 | 3 | no | — |
| 167 | C2×C83 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 168 | D83 | — | 8 | 2 | no | — |
| 169 | #3483 | D83 | 1 | 1 | yes | characteristic, complemented |
| 170 | #3018 | — | 83 | 3 | no | — |
| 171 | #2205 | — | 83 | 2 | no | — |
| 172 | #2204 | — | 83 | 6 | no | — |
| 173 | #2132 | — | 83 | 3 | no | — |
| 174 | C83 | #33567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 175 | #585 | — | 83 | 1 | no | — |
| 176 | #575 | — | 166 | 3 | no | — |
| 177 | Q82 | C2×D83 | 1 | 1 | yes | characteristic, complemented |
| 178 | #555 | — | 2 | 3 | no | — |
| 179 | #555 | C2×D83 | 1 | 3 | yes | complemented |
| 180 | #550 | — | 2 | 3 | no | — |
| 181 | #532 | — | 2 | 3 | no | — |
| 182 | #532 | C2×D83 | 1 | 3 | yes | — |
| 183 | C4×#41 | — | 2 | 3 | no | — |
| 184 | #456 | — | 83 | 6 | no | — |
| 185 | #455 | — | 166 | 3 | no | — |
| 186 | #454 | — | 83 | 6 | no | — |
| 187 | #453 | — | 166 | 3 | no | — |
| 188 | C2×#104 | — | 166 | 3 | no | — |
| 189 | #413 | — | 83 | 3 | no | — |
| 190 | #356 | — | 83 | 6 | no | — |
| 191 | #355 | — | 83 | 6 | no | — |
| 192 | 2−1+4 | — | 332 | 1 | no | — |
| 193 | 2−1+4 | — | 83 | 2 | no | — |
| 194 | H(5,𝔽2) | — | 332 | 1 | no | — |
| 195 | H(5,𝔽2) | — | 166 | 3 | no | — |
| 196 | C2×#41 | — | 2 | 3 | no | — |
| 197 | C2×#41 | — | 2 | 1 | no | — |
| 198 | C2×#41 | — | 332 | 2 | no | — |
| 199 | C2×#41 | — | 166 | 3 | no | — |
| 200 | C2×#41 | C22⋊D83 | 1 | 3 | yes | — |
| 201 | C2×#41 | C22×D83 | 1 | 1 | yes | characteristic |
| 202 | #137 | — | 332 | 3 | no | — |
| 203 | #137 | — | 166 | 6 | no | — |
| 204 | AGL(1,ℤ/8) | — | 332 | 3 | no | — |
| 205 | AGL(1,ℤ/8) | — | 166 | 6 | no | — |
| 206 | #135 | — | 332 | 6 | no | — |
| 207 | C2×M16 | — | 166 | 3 | no | — |
| 208 | C4⋊Q8 | — | 2 | 3 | no | — |
| 209 | C4⋊Q8 | C22×D83 | 1 | 3 | yes | — |
| 210 | #127 | — | 2 | 3 | no | — |
| 211 | #127 | C22⋊D83 | 1 | 3 | yes | — |
| 212 | #124 | — | 2 | 3 | no | — |
| 213 | #124 | — | 2 | 3 | no | — |
| 214 | #124 | C22⋊D83 | 1 | 3 | yes | — |
| 215 | #124 | C22×D83 | 1 | 3 | yes | — |
| 216 | #121 | — | 2 | 6 | no | — |
| 217 | #121 | — | 2 | 6 | no | — |
| 218 | #121 | — | 2 | 6 | no | — |
| 219 | C4×Q8 | — | 2 | 1 | no | — |
| 220 | C4×Q8 | — | 2 | 3 | no | — |
| 221 | C4×D4 | — | 2 | 3 | no | — |
| 222 | C4×D4 | — | 2 | 6 | no | — |
| 223 | C4×D4 | — | 2 | 6 | no | — |
| 224 | C4×D4 | — | 2 | 3 | no | — |
| 225 | #117 | — | 2 | 3 | no | — |
| 226 | #117 | — | 2 | 6 | no | — |
| 227 | C2×C42 | — | 2 | 3 | no | — |
| 228 | C2×C42 | C22⋊D83 | 1 | 3 | yes | — |
| 229 | #104 | — | 166 | 6 | no | — |
| 230 | #104 | — | 166 | 6 | no | — |
| 231 | #101 | — | 83 | 6 | no | — |
| 232 | #100 | — | 166 | 3 | no | — |
| 233 | #41 | — | 4 | 1 | no | — |
| 234 | #41 | — | 4 | 1 | no | — |
| 235 | #41 | — | 4 | 3 | no | — |
| 236 | #41 | — | 4 | 3 | no | — |
| 237 | #41 | — | 4 | 12 | no | — |
| 238 | #41 | — | 8 | 1 | no | — |
| 239 | #41 | — | 8 | 1 | no | — |
| 240 | #41 | — | 332 | 3 | no | — |
| 241 | #41 | — | 332 | 6 | no | — |
| 242 | #41 | — | 332 | 2 | no | — |
| 243 | #41 | — | 332 | 1 | no | — |
| 244 | #41 | — | 166 | 6 | no | — |
| 245 | #41 | — | 166 | 6 | no | — |
| 246 | C2×Q8 | — | 2 | 1 | no | — |
| 247 | C2×Q8 | — | 2 | 1 | no | — |
| 248 | C2×Q8 | — | 332 | 2 | no | — |
| 249 | C2×Q8 | — | 166 | 3 | no | — |
| 250 | C2×Q8 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 251 | C2×Q8 | C23×D83 | 1 | 1 | yes | characteristic |
| 252 | C2×D4 | — | 2 | 3 | no | — |
| 253 | C2×D4 | — | 2 | 6 | no | — |
| 254 | C2×D4 | — | 2 | 3 | no | — |
| 255 | C2×D4 | — | 2 | 3 | no | — |
| 256 | C2×D4 | — | 332 | 6 | no | — |
| 257 | C2×D4 | — | 332 | 3 | no | — |
| 258 | C2×D4 | — | 166 | 3 | no | — |
| 259 | C2×D4 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 260 | C2×D4 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 261 | C22×C4 | — | 2 | 6 | no | — |
| 262 | C22×C4 | — | 2 | 3 | no | — |
| 263 | C22×C4 | — | 2 | 3 | no | — |
| 264 | C22×C4 | — | 332 | 3 | no | — |
| 265 | C22×C4 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 266 | Q16 | — | 332 | 6 | no | — |
| 267 | SD16 | — | 332 | 6 | no | — |
| 268 | SD16 | — | 332 | 6 | no | — |
| 269 | D8 | — | 332 | 6 | no | — |
| 270 | M16 | — | 332 | 3 | no | — |
| 271 | M16 | — | 166 | 6 | no | — |
| 272 | C2×C8 | — | 332 | 3 | no | — |
| 273 | C4⋊C4 | — | 2 | 3 | no | — |
| 274 | C4⋊C4 | — | 2 | 3 | no | — |
| 275 | C4⋊C4 | — | 2 | 6 | no | — |
| 276 | C22⋊C4 | — | 2 | 6 | no | — |
| 277 | C22⋊C4 | — | 2 | 6 | no | — |
| 278 | C22⋊C4 | — | 2 | 6 | no | — |
| 279 | C42 | — | 2 | 3 | no | — |
| 280 | C42 | — | 2 | 3 | no | — |
| 281 | C42 | — | 2 | 3 | no | — |
| 282 | C42 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 283 | C42 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 284 | C23 | — | 2 | 3 | no | — |
| 285 | C23 | — | 332 | 3 | no | — |
| 286 | C23 | #14581761 | 1 | 3 | yes | — |
| 287 | Q8 | — | 2 | 4 | no | — |
| 288 | Q8 | — | 4 | 1 | no | — |
| 289 | Q8 | — | 4 | 3 | no | — |
| 290 | Q8 | — | 8 | 1 | no | — |
| 291 | Q8 | — | 332 | 1 | no | — |
| 292 | Q8 | — | 166 | 6 | no | — |
| 293 | D4 | — | 4 | 3 | no | — |
| 294 | D4 | — | 4 | 3 | no | — |
| 295 | D4 | — | 4 | 12 | no | — |
| 296 | D4 | — | 4 | 6 | no | — |
| 297 | D4 | — | 8 | 3 | no | — |
| 298 | D4 | — | 8 | 3 | no | — |
| 299 | D4 | — | 332 | 6 | no | — |
| 300 | D4 | — | 332 | 2 | no | — |
| 301 | D4 | — | 332 | 1 | no | — |
| 302 | D4 | — | 166 | 6 | no | — |
| 303 | C2×C4 | — | 2 | 3 | no | — |
| 304 | C2×C4 | — | 2 | 3 | no | — |
| 305 | C2×C4 | — | 2 | 3 | no | — |
| 306 | C2×C4 | — | 2 | 3 | no | — |
| 307 | C2×C4 | — | 2 | 3 | no | — |
| 308 | C2×C4 | — | 4 | 6 | no | — |
| 309 | C2×C4 | — | 4 | 12 | no | — |
| 310 | C2×C4 | — | 4 | 3 | no | — |
| 311 | C2×C4 | — | 4 | 3 | no | — |
| 312 | C2×C4 | — | 332 | 6 | no | — |
| 313 | C2×C4 | — | 332 | 3 | no | — |
| 314 | C2×C4 | — | 332 | 2 | no | — |
| 315 | C2×C4 | #14581761 | 1 | 1 | yes | characteristic |
| 316 | C2×C4 | C22×C22⋊D83 | 1 | 3 | yes | — |
| 317 | C8 | — | 332 | 6 | no | — |
| 318 | C22 | — | 2 | 3 | no | — |
| 319 | C22 | — | 4 | 3 | no | — |
| 320 | C22 | — | 4 | 6 | no | — |
| 321 | C22 | — | 4 | 3 | no | — |
| 322 | C22 | — | 1328 | 3 | no | — |
| 323 | C22 | — | 332 | 2 | no | — |
| 324 | C22 | C2×#14581761 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 2 | 6 | no | — |
| 326 | C4 | — | 2 | 3 | no | — |
| 327 | C4 | — | 4 | 3 | no | — |
| 328 | C4 | — | 4 | 3 | no | — |
| 329 | C4 | — | 332 | 2 | no | — |
| 330 | C4 | #19127888 | 1 | 1 | yes | characteristic |
| 331 | C4 | #19127888 | 1 | 1 | yes | characteristic |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 4 | 3 | no | — |
| 334 | C2 | — | 8 | 3 | no | — |
| 335 | C2 | — | 664 | 2 | no | — |
| 336 | C2 | C2×#19127888 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #24916811 | C2 | semidirect |
| 2. #24909493 | C2 | semidirect |
| 3. #24909489 | C2 | semidirect |
| 4. #24909148 | C2 | semidirect |
| 5. C83×#3483 | C2 | semidirect |
| 14. #19127500 | C22 | semidirect |
| 15. #19127498 | C22 | semidirect |
| 16. Q82×C83 | C22 | semidirect |
| 18. #555×C83 | C22 | semidirect |
| 169. #3483 | D83 | semidirect |
| 174. C83 | #33567 | semidirect |
| 177. Q82 | C2×D83 | semidirect |
| 179. #555 | C2×D83 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24916811 | 2 |
| #24909493 | 2 |
| #24909489 | 2 |
| #24909148 | 2 |
| C83×#3483 | 2 |
| #33567 | 83 |