Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
588 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #16811622 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #16811615 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #16811609 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16811602 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#13133263 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | D53⋊M16 | C22 | 1 | 2 | yes | complemented |
| 7 | #13133296 | C22 | 1 | 2 | yes | complemented |
| 8 | #13133282 | — | 2 | 4 | no | — |
| 9 | #13133279 | — | 2 | 4 | no | — |
| 10 | #13133278 | — | 2 | 1 | no | — |
| 11 | #13133277 | — | 2 | 1 | no | — |
| 12 | #13133276 | — | 2 | 1 | no | — |
| 13 | #13133275 | — | 2 | 1 | no | — |
| 14 | #13133263 | — | 2 | 4 | no | — |
| 15 | #13133263 | — | 2 | 4 | no | — |
| 16 | C2×#11719949 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#11719949 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#11719945 | — | 2 | 1 | no | — |
| 19 | C2×#11719945 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×D212 | — | 2 | 1 | no | — |
| 21 | C22×D212 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×D4×D53 | — | 2 | 1 | no | — |
| 23 | C2×D4×D53 | — | 2 | 1 | no | — |
| 24 | C2×D4×D53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 25 | C2×D4×D53 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#41×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 27 | C23×D53 | — | 2 | 1 | no | — |
| 28 | C23×D53 | — | 2 | 1 | no | — |
| 29 | C23×D53 | — | 2 | 1 | no | — |
| 30 | #41×C53 | — | 2 | 2 | no | — |
| 31 | #41×C53 | — | 2 | 2 | no | — |
| 32 | C2×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×D4×C53 | — | 2 | 1 | no | — |
| 34 | C2×D4×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C22×C4×C53 | — | 2 | 1 | no | — |
| 36 | C22×C4×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 38 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 39 | C2×C22⋊D53 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×C22⋊D53 | D4 | 1 | 1 | yes | characteristic |
| 41 | #11719949 | — | 2 | 2 | no | — |
| 42 | #11719949 | — | 2 | 2 | no | — |
| 43 | #11719949 | — | 2 | 2 | no | — |
| 44 | #11719949 | D4 | 1 | 4 | yes | — |
| 45 | D4×D53 | — | 2 | 8 | no | — |
| 46 | D4×D53 | — | 2 | 8 | no | — |
| 47 | D4×D53 | — | 2 | 2 | no | — |
| 48 | D4×D53 | — | 2 | 2 | no | — |
| 49 | D4×D53 | — | 2 | 2 | no | — |
| 50 | D4×D53 | D4 | 1 | 4 | yes | — |
| 51 | #11719945 | — | 2 | 8 | no | — |
| 52 | #11719945 | — | 2 | 2 | no | — |
| 53 | #11719945 | — | 2 | 2 | no | — |
| 54 | C2×D212 | — | 2 | 8 | no | — |
| 55 | C2×D212 | — | 2 | 1 | no | — |
| 56 | C2×D212 | — | 2 | 1 | no | — |
| 57 | C2×D212 | — | 2 | 1 | no | — |
| 58 | C2×D212 | — | 2 | 2 | no | — |
| 59 | C2×D212 | — | 2 | 2 | no | — |
| 60 | C2×D212 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×D212 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×D212 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×D53 | — | 2 | 1 | no | — |
| 64 | C2×C4×D53 | — | 2 | 1 | no | — |
| 65 | C2×C4×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×D53 | C23 | 1 | 1 | yes | characteristic |
| 67 | C2×Dic106 | D4 | 1 | 1 | yes | characteristic |
| 68 | C22⋊#6157 | — | 2 | 2 | no | — |
| 69 | #11719940 | — | 2 | 2 | no | — |
| 70 | C4⋊#6157 | — | 2 | 2 | no | — |
| 71 | C4×#6157 | — | 2 | 2 | no | — |
| 72 | #11719936 | — | 2 | 2 | no | — |
| 73 | D53⋊C8 | — | 2 | 2 | no | — |
| 74 | C23×C53 | — | 2 | 1 | no | — |
| 75 | C23×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C22×D53 | — | 2 | 1 | no | — |
| 77 | C22×D53 | — | 2 | 1 | no | — |
| 78 | C22×D53 | — | 2 | 4 | no | — |
| 79 | C22×D53 | — | 2 | 4 | no | — |
| 80 | C22×D53 | — | 2 | 4 | no | — |
| 81 | C22×D53 | — | 8 | 2 | no | — |
| 82 | C22×D53 | — | 8 | 2 | no | — |
| 83 | C22×D53 | — | 8 | 2 | no | — |
| 84 | C22×D53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C22×D53 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×#6157 | — | 4 | 2 | no | — |
| 87 | Q8×C53 | — | 2 | 1 | no | — |
| 88 | Q8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 89 | D4×C53 | — | 2 | 4 | no | — |
| 90 | D4×C53 | — | 2 | 1 | no | — |
| 91 | D4×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 92 | C2×C4×C53 | — | 2 | 4 | no | — |
| 93 | C2×C4×C53 | — | 2 | 1 | no | — |
| 94 | C2×C4×C53 | — | 2 | 1 | no | — |
| 95 | C2×C4×C53 | — | 2 | 1 | no | — |
| 96 | C2×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C22⋊D53 | — | 2 | 4 | no | — |
| 99 | C22⋊D53 | — | 2 | 2 | no | — |
| 100 | C22⋊D53 | — | 2 | 2 | no | — |
| 101 | C22⋊D53 | — | 2 | 4 | no | — |
| 102 | C2×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 103 | C2×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | D212 | — | 2 | 4 | no | — |
| 105 | D212 | — | 2 | 4 | no | — |
| 106 | D212 | — | 2 | 2 | no | — |
| 107 | D212 | — | 2 | 4 | no | — |
| 108 | D212 | — | 2 | 1 | no | — |
| 109 | D212 | — | 2 | 1 | no | — |
| 110 | D212 | C22⋊C4 | 1 | 2 | yes | — |
| 111 | D212 | C22⋊C4 | 1 | 2 | yes | — |
| 112 | C4×D53 | — | 2 | 4 | no | — |
| 113 | C4×D53 | — | 2 | 1 | no | — |
| 114 | C4×D53 | — | 2 | 1 | no | — |
| 115 | C4×D53 | — | 2 | 4 | no | — |
| 116 | C4×D53 | C2×D4 | 1 | 2 | yes | — |
| 117 | C4×D53 | C2×D4 | 1 | 2 | yes | — |
| 118 | Dic106 | — | 2 | 2 | no | — |
| 119 | #88324 | — | 4 | 2 | no | — |
| 120 | C22×C53 | — | 2 | 1 | no | — |
| 121 | C22×C53 | — | 2 | 2 | no | — |
| 122 | C22×C53 | — | 4 | 1 | no | — |
| 123 | C22×C53 | — | 4 | 2 | no | — |
| 124 | C22×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C2×D53 | — | 2 | 1 | no | — |
| 126 | C2×D53 | — | 2 | 2 | no | — |
| 127 | C2×D53 | — | 2 | 2 | no | — |
| 128 | C2×D53 | — | 4 | 1 | no | — |
| 129 | C2×D53 | — | 4 | 2 | no | — |
| 130 | C2×D53 | — | 4 | 2 | no | — |
| 131 | C2×D53 | — | 4 | 1 | no | — |
| 132 | C2×D53 | — | 8 | 4 | no | — |
| 133 | C2×D53 | — | 8 | 4 | no | — |
| 134 | C2×D53 | — | 8 | 4 | no | — |
| 135 | C2×D53 | #120 | 1 | 2 | yes | — |
| 136 | #6157 | — | 8 | 2 | no | — |
| 137 | C4×C53 | — | 2 | 1 | no | — |
| 138 | C4×C53 | — | 2 | 2 | no | — |
| 139 | C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 140 | C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 141 | Dic53 | — | 2 | 1 | no | — |
| 142 | Dic53 | #120 | 1 | 2 | yes | — |
| 143 | #1795 | — | 53 | 1 | no | maximal, Sylow |
| 144 | C2×C53 | — | 2 | 1 | no | — |
| 145 | C2×C53 | — | 4 | 1 | no | — |
| 146 | C2×C53 | — | 4 | 2 | no | — |
| 147 | C2×C53 | #379 | 1 | 1 | yes | characteristic |
| 148 | D53 | — | 2 | 2 | no | — |
| 149 | D53 | — | 4 | 2 | no | — |
| 150 | D53 | — | 4 | 2 | no | — |
| 151 | D53 | — | 4 | 1 | no | — |
| 152 | C2×H(5,𝔽2) | — | 53 | 1 | no | — |
| 153 | #421 | — | 53 | 2 | no | — |
| 154 | #418 | — | 53 | 2 | no | — |
| 155 | #413 | — | 53 | 1 | no | — |
| 156 | #410 | — | 53 | 1 | no | — |
| 157 | C53 | #1795 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 158 | H(5,𝔽2) | — | 106 | 4 | no | — |
| 159 | H(5,𝔽2) | — | 106 | 4 | no | — |
| 160 | C2×#41 | — | 53 | 1 | no | — |
| 161 | C2×#41 | — | 53 | 1 | no | — |
| 162 | C2×#41 | — | 53 | 1 | no | — |
| 163 | C2×#41 | — | 106 | 1 | no | — |
| 164 | C2×#41 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 165 | C22×D4 | — | 53 | 1 | no | — |
| 166 | C22×D4 | — | 53 | 1 | no | — |
| 167 | C22×D4 | — | 53 | 1 | no | — |
| 168 | C22×D4 | — | 106 | 1 | no | — |
| 169 | C22×D4 | — | 106 | 1 | no | — |
| 170 | C22×D4 | — | 106 | 1 | no | — |
| 171 | C2×M16 | — | 53 | 2 | no | — |
| 172 | #117 | — | 53 | 2 | no | — |
| 173 | #104 | — | 106 | 4 | no | — |
| 174 | #102 | — | 106 | 4 | no | — |
| 175 | #101 | — | 106 | 1 | no | — |
| 176 | #100 | — | 106 | 1 | no | — |
| 177 | #99 | — | 106 | 1 | no | — |
| 178 | #99 | — | 106 | 1 | no | — |
| 179 | C24 | — | 106 | 1 | no | — |
| 180 | C24 | — | 106 | 1 | no | — |
| 181 | C24 | — | 106 | 1 | no | — |
| 182 | #41 | — | 2 | 2 | no | — |
| 183 | #41 | — | 2 | 2 | no | — |
| 184 | #41 | — | 53 | 4 | no | — |
| 185 | #41 | — | 106 | 2 | no | — |
| 186 | #41 | — | 106 | 2 | no | — |
| 187 | #41 | — | 106 | 8 | no | — |
| 188 | #41 | — | 106 | 2 | no | — |
| 189 | #41 | — | 106 | 2 | no | — |
| 190 | #41 | — | 106 | 2 | no | — |
| 191 | C2×Q8 | — | 53 | 1 | no | — |
| 192 | C2×Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 193 | C2×D4 | — | 2 | 1 | no | — |
| 194 | C2×D4 | — | 53 | 4 | no | — |
| 195 | C2×D4 | — | 53 | 1 | no | — |
| 196 | C2×D4 | — | 53 | 1 | no | — |
| 197 | C2×D4 | — | 53 | 1 | no | — |
| 198 | C2×D4 | — | 53 | 1 | no | — |
| 199 | C2×D4 | — | 53 | 1 | no | — |
| 200 | C2×D4 | — | 106 | 2 | no | — |
| 201 | C2×D4 | — | 106 | 2 | no | — |
| 202 | C2×D4 | — | 106 | 1 | no | — |
| 203 | C2×D4 | — | 106 | 1 | no | — |
| 204 | C2×D4 | — | 106 | 8 | no | — |
| 205 | C2×D4 | — | 106 | 2 | no | — |
| 206 | C2×D4 | — | 106 | 1 | no | — |
| 207 | C2×D4 | — | 106 | 1 | no | — |
| 208 | C2×D4 | — | 106 | 2 | no | — |
| 209 | C2×D4 | — | 106 | 8 | no | — |
| 210 | C2×D4 | — | 106 | 8 | no | — |
| 211 | C2×D4 | — | 106 | 2 | no | — |
| 212 | C2×D4 | — | 106 | 1 | no | — |
| 213 | C2×D4 | C2×#6157 | 1 | 1 | yes | characteristic |
| 214 | C22×C4 | — | 2 | 1 | no | — |
| 215 | C22×C4 | — | 53 | 1 | no | — |
| 216 | C22×C4 | — | 53 | 1 | no | — |
| 217 | C22×C4 | — | 106 | 1 | no | — |
| 218 | C22×C4 | — | 106 | 1 | no | — |
| 219 | C22×C4 | C2×#6157 | 1 | 1 | yes | characteristic |
| 220 | M16 | — | 106 | 2 | no | — |
| 221 | M16 | — | 106 | 2 | no | — |
| 222 | C2×C8 | — | 106 | 2 | no | — |
| 223 | C4⋊C4 | — | 106 | 2 | no | — |
| 224 | C22⋊C4 | — | 106 | 2 | no | — |
| 225 | C42 | — | 106 | 2 | no | — |
| 226 | C23 | — | 2 | 1 | no | — |
| 227 | C23 | — | 53 | 1 | no | — |
| 228 | C23 | — | 53 | 1 | no | — |
| 229 | C23 | — | 106 | 4 | no | — |
| 230 | C23 | — | 106 | 4 | no | — |
| 231 | C23 | — | 106 | 4 | no | — |
| 232 | C23 | — | 106 | 1 | no | — |
| 233 | C23 | — | 106 | 1 | no | — |
| 234 | C23 | — | 424 | 2 | no | — |
| 235 | C23 | — | 424 | 2 | no | — |
| 236 | C23 | — | 424 | 2 | no | — |
| 237 | C23 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 238 | Q8 | — | 2 | 1 | no | — |
| 239 | Q8 | — | 106 | 2 | no | — |
| 240 | Q8 | C22⋊#6157 | 1 | 2 | yes | — |
| 241 | D4 | — | 2 | 4 | no | — |
| 242 | D4 | — | 2 | 1 | no | — |
| 243 | D4 | — | 53 | 2 | no | — |
| 244 | D4 | — | 53 | 2 | no | — |
| 245 | D4 | — | 106 | 2 | no | — |
| 246 | D4 | — | 106 | 4 | no | — |
| 247 | D4 | — | 106 | 4 | no | — |
| 248 | D4 | — | 106 | 4 | no | — |
| 249 | D4 | — | 106 | 2 | no | — |
| 250 | D4 | — | 106 | 4 | no | — |
| 251 | D4 | — | 106 | 2 | no | — |
| 252 | D4 | — | 106 | 4 | no | — |
| 253 | D4 | — | 106 | 1 | no | — |
| 254 | D4 | — | 106 | 1 | no | — |
| 255 | D4 | C22⋊#6157 | 1 | 2 | yes | — |
| 256 | C2×C4 | — | 2 | 4 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 1 | no | — |
| 260 | C2×C4 | — | 53 | 1 | no | — |
| 261 | C2×C4 | — | 53 | 2 | no | — |
| 262 | C2×C4 | — | 53 | 2 | no | — |
| 263 | C2×C4 | — | 53 | 1 | no | — |
| 264 | C2×C4 | — | 106 | 4 | no | — |
| 265 | C2×C4 | — | 106 | 4 | no | — |
| 266 | C2×C4 | — | 106 | 1 | no | — |
| 267 | C2×C4 | — | 106 | 1 | no | — |
| 268 | C2×C4 | — | 212 | 2 | no | — |
| 269 | C2×C4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 271 | C8 | — | 212 | 2 | no | — |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 2 | 2 | no | — |
| 274 | C22 | — | 4 | 1 | no | — |
| 275 | C22 | — | 4 | 2 | no | — |
| 276 | C22 | — | 53 | 2 | no | — |
| 277 | C22 | — | 106 | 2 | no | — |
| 278 | C22 | — | 106 | 2 | no | — |
| 279 | C22 | — | 106 | 1 | no | — |
| 280 | C22 | — | 424 | 4 | no | — |
| 281 | C22 | — | 424 | 4 | no | — |
| 282 | C22 | — | 424 | 4 | no | — |
| 283 | C22 | — | 212 | 1 | no | — |
| 284 | C22 | — | 212 | 1 | no | — |
| 285 | C22 | — | 212 | 2 | no | — |
| 286 | C22 | — | 212 | 2 | no | — |
| 287 | C22 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 288 | C4 | — | 2 | 1 | no | — |
| 289 | C4 | — | 2 | 2 | no | — |
| 290 | C4 | — | 53 | 2 | no | — |
| 291 | C4 | — | 106 | 1 | no | — |
| 292 | C4 | — | 424 | 2 | no | — |
| 293 | C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 294 | C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 295 | C2 | — | 2 | 1 | no | — |
| 296 | C2 | — | 4 | 2 | no | — |
| 297 | C2 | — | 4 | 1 | no | — |
| 298 | C2 | — | 106 | 2 | no | — |
| 299 | C2 | — | 212 | 2 | no | — |
| 300 | C2 | — | 212 | 2 | no | — |
| 301 | C2 | — | 212 | 1 | no | — |
| 302 | C2 | #16811564 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811622 | C2 | semidirect |
| 2. #16811615 | C2 | semidirect |
| 3. #16811609 | C2 | semidirect |
| 4. #16811602 | C2 | semidirect |
| 6. D53⋊M16 | C22 | semidirect |
| 7. #13133296 | C22 | semidirect |
| 16. C2×#11719949 | C4 | semidirect |
| 21. C22×D212 | C4 | semidirect |
| 24. C2×D4×D53 | C4 | semidirect |
| 26. C2×#41×C53 | C4 | semidirect |
| 157. C53 | #1795 | semidirect |
| 164. C2×#41 | #6157 | semidirect |