Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1176 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 191 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 | C3×#3459 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #82373 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #71894 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #71849 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #71619 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #29685 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#543 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×#539 | — | 2 | 3 | no | — |
| 9 | C3×#538 | — | 2 | 3 | no | — |
| 10 | C3×#534 | C22 | 1 | 3 | yes | — |
| 11 | C3×#529 | — | 2 | 3 | no | — |
| 12 | C3×#525 | — | 2 | 3 | no | — |
| 13 | C2×C3×C4×D4 | C22 | 1 | 3 | yes | — |
| 14 | C2×#935 | C22 | 1 | 6 | yes | — |
| 15 | C22×C4⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#834 | C22 | 1 | 6 | yes | — |
| 17 | #5212 | — | 2 | 3 | no | — |
| 18 | #5212 | C22 | 1 | 3 | yes | — |
| 19 | #4981 | — | 2 | 3 | no | — |
| 20 | #4953 | — | 2 | 3 | no | — |
| 21 | #4943 | C22 | 1 | 6 | yes | — |
| 22 | #4928 | — | 2 | 3 | no | — |
| 23 | #4526 | C22 | 1 | 6 | yes | — |
| 24 | #3459 | D3 | 1 | 1 | yes | characteristic |
| 25 | #2486 | — | 3 | 3 | no | — |
| 26 | #2030 | — | 3 | 2 | no | — |
| 27 | #2022 | — | 3 | 6 | no | — |
| 28 | #1799 | — | 3 | 3 | no | — |
| 29 | C2×C3×#41 | — | 2 | 1 | no | — |
| 30 | C22×C3×D4 | C23 | 1 | 3 | yes | — |
| 31 | C23×C3×C4 | D4 | 1 | 3 | yes | — |
| 32 | C23×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 33 | C23×Dic3 | C23 | 1 | 2 | yes | — |
| 34 | C3×#126 | — | 2 | 6 | no | — |
| 35 | C3×#125 | — | 2 | 3 | no | — |
| 36 | C3×#124 | — | 2 | 3 | no | — |
| 37 | C3×#123 | — | 2 | 6 | no | — |
| 38 | C3×#123 | — | 2 | 6 | no | — |
| 39 | C3×C22⋊Q8 | — | 2 | 3 | no | — |
| 40 | C3×C22⋊Q8 | — | 2 | 3 | no | — |
| 41 | C3×C22⋊Q8 | — | 2 | 3 | no | — |
| 42 | C3×#121 | — | 2 | 3 | no | — |
| 43 | C3×#121 | — | 2 | 3 | no | — |
| 44 | C3×#121 | — | 2 | 3 | no | — |
| 45 | C3×#120 | — | 2 | 6 | no | — |
| 46 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 47 | C3×C4×D4 | — | 2 | 3 | no | — |
| 48 | C3×C4×D4 | — | 2 | 6 | no | — |
| 49 | C3×C4×D4 | — | 2 | 6 | no | — |
| 50 | C3×#117 | — | 2 | 3 | no | — |
| 51 | C2×C3×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 52 | C2×C3×C22⋊C4 | D4 | 1 | 3 | yes | — |
| 53 | C2×C3×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 54 | C2×C3×C42 | D4 | 1 | 3 | yes | — |
| 55 | C2×#217 | — | 2 | 6 | no | — |
| 56 | C2×#217 | C23 | 1 | 6 | yes | — |
| 57 | #935 | — | 4 | 12 | no | — |
| 58 | C2×C4⋊Dic3 | — | 2 | 3 | no | — |
| 59 | C2×C4⋊Dic3 | — | 2 | 3 | no | — |
| 60 | C2×C4⋊Dic3 | — | 2 | 3 | no | — |
| 61 | C2×Dic3⋊C4 | — | 2 | 6 | no | — |
| 62 | #834 | — | 4 | 6 | no | — |
| 63 | C3⋊#95 | — | 2 | 6 | no | — |
| 64 | C3⋊#95 | — | 2 | 6 | no | — |
| 65 | C3⋊#95 | — | 2 | 3 | no | — |
| 66 | #543 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | #539 | — | 2 | 3 | no | — |
| 68 | #538 | — | 2 | 3 | no | — |
| 69 | #534 | C2×D3 | 1 | 3 | yes | — |
| 70 | #529 | — | 2 | 3 | no | — |
| 71 | #525 | — | 2 | 3 | no | — |
| 72 | C2×#123 | — | 3 | 6 | no | — |
| 73 | C2×C4×D4 | C2×D3 | 1 | 3 | yes | — |
| 74 | C22×C4⋊C4 | — | 3 | 1 | no | — |
| 75 | C2×#99 | — | 3 | 6 | no | — |
| 76 | #398 | — | 6 | 3 | no | — |
| 77 | #397 | — | 6 | 3 | no | — |
| 78 | #389 | — | 6 | 3 | no | — |
| 79 | #381 | — | 3 | 6 | no | — |
| 80 | #380 | — | 3 | 3 | no | — |
| 81 | #380 | — | 6 | 3 | no | — |
| 82 | #342 | — | 3 | 6 | no | — |
| 83 | C24×C3 | C2×D4 | 1 | 3 | yes | — |
| 84 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 85 | C3×#41 | — | 8 | 2 | no | — |
| 86 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 87 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 88 | C2×C3×D4 | — | 2 | 6 | no | — |
| 89 | C2×C3×D4 | — | 2 | 3 | no | — |
| 90 | C2×C3×D4 | — | 2 | 6 | no | — |
| 91 | C2×C3×D4 | — | 2 | 3 | no | — |
| 92 | C2×C3×D4 | — | 2 | 3 | no | — |
| 93 | C2×C3×D4 | — | 2 | 3 | no | — |
| 94 | C22×C3×C4 | — | 2 | 3 | no | — |
| 95 | C22×C3×C4 | — | 2 | 3 | no | — |
| 96 | C22×C3×C4 | — | 4 | 2 | no | — |
| 97 | C22×C3×C4 | — | 4 | 6 | no | — |
| 98 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 99 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 100 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 101 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 102 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 103 | C22×Dic3 | — | 2 | 6 | no | — |
| 104 | C22×Dic3 | — | 2 | 6 | no | — |
| 105 | C22×Dic3 | — | 2 | 6 | no | — |
| 106 | C22×Dic3 | — | 2 | 2 | no | — |
| 107 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 108 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 109 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 110 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 111 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 112 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 113 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 114 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 115 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 116 | C3×C42 | — | 2 | 3 | no | — |
| 117 | C3×C42 | — | 2 | 3 | no | — |
| 118 | #217 | — | 2 | 12 | no | — |
| 119 | #217 | — | 4 | 12 | no | — |
| 120 | C4⋊Dic3 | — | 4 | 6 | no | — |
| 121 | C4⋊Dic3 | — | 4 | 3 | no | — |
| 122 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 123 | Dic3⋊C4 | — | 4 | 12 | no | — |
| 124 | C2×#41 | — | 2 | 1 | no | — |
| 125 | C22×D4 | C22×D3 | 1 | 3 | yes | — |
| 126 | C23×C4 | — | 3 | 2 | no | — |
| 127 | C23×C4 | #67 | 1 | 3 | yes | — |
| 128 | C23×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 129 | #126 | — | 2 | 6 | no | — |
| 130 | #125 | — | 2 | 3 | no | — |
| 131 | #124 | — | 2 | 3 | no | — |
| 132 | #123 | — | 2 | 6 | no | — |
| 133 | #123 | — | 2 | 6 | no | — |
| 134 | #123 | — | 12 | 12 | no | — |
| 135 | C22⋊Q8 | — | 2 | 3 | no | — |
| 136 | C22⋊Q8 | — | 2 | 3 | no | — |
| 137 | C22⋊Q8 | — | 2 | 3 | no | — |
| 138 | #121 | — | 2 | 3 | no | — |
| 139 | #121 | — | 2 | 3 | no | — |
| 140 | #121 | — | 2 | 3 | no | — |
| 141 | #120 | — | 2 | 6 | no | — |
| 142 | C4×Q8 | — | 2 | 1 | no | — |
| 143 | C4×D4 | — | 2 | 3 | no | — |
| 144 | C4×D4 | — | 2 | 6 | no | — |
| 145 | C4×D4 | — | 2 | 6 | no | — |
| 146 | #117 | — | 2 | 3 | no | — |
| 147 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 148 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 149 | C2×C4⋊C4 | — | 6 | 6 | no | — |
| 150 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 151 | C2×C4⋊C4 | #67 | 1 | 3 | yes | — |
| 152 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 153 | C2×C22⋊C4 | — | 6 | 6 | no | — |
| 154 | C2×C22⋊C4 | #67 | 1 | 3 | yes | — |
| 155 | C2×C22⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 156 | C2×C42 | #67 | 1 | 3 | yes | — |
| 157 | #99 | — | 12 | 6 | no | — |
| 158 | #95 | — | 6 | 3 | no | — |
| 159 | #95 | — | 6 | 6 | no | — |
| 160 | #95 | — | 6 | 6 | no | — |
| 161 | C23×C3 | — | 2 | 3 | no | — |
| 162 | C23×C3 | — | 2 | 3 | no | — |
| 163 | C23×C3 | — | 2 | 3 | no | — |
| 164 | C23×C3 | — | 4 | 1 | no | — |
| 165 | C23×C3 | — | 4 | 1 | no | — |
| 166 | C23×C3 | — | 4 | 3 | no | — |
| 167 | C23×C3 | — | 4 | 3 | no | — |
| 168 | C23×C3 | C22×D4 | 1 | 3 | yes | — |
| 169 | C23×C3 | H(5,𝔽2) | 1 | 3 | yes | — |
| 170 | C23×C3 | 2−1+4 | 1 | 1 | yes | characteristic |
| 171 | C3×Q8 | — | 8 | 1 | no | — |
| 172 | C3×Q8 | — | 8 | 1 | no | — |
| 173 | C3×D4 | — | 4 | 6 | no | — |
| 174 | C3×D4 | — | 4 | 6 | no | — |
| 175 | C3×D4 | — | 8 | 3 | no | — |
| 176 | C3×D4 | — | 8 | 3 | no | — |
| 177 | C2×C3×C4 | — | 2 | 6 | no | — |
| 178 | C2×C3×C4 | — | 2 | 3 | no | — |
| 179 | C2×C3×C4 | — | 2 | 3 | no | — |
| 180 | C2×C3×C4 | — | 2 | 3 | no | — |
| 181 | C2×C3×C4 | — | 2 | 3 | no | — |
| 182 | C2×C3×C4 | — | 2 | 3 | no | — |
| 183 | C2×C3×C4 | — | 4 | 3 | no | — |
| 184 | C2×C3×C4 | — | 4 | 6 | no | — |
| 185 | C2×C3×C4 | — | 4 | 3 | no | — |
| 186 | C2×C3×C4 | — | 4 | 3 | no | — |
| 187 | C2×C3×C4 | — | 8 | 3 | no | — |
| 188 | C2×C3×C4 | #120 | 1 | 3 | yes | — |
| 189 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 190 | C2×Dic3 | — | 4 | 6 | no | — |
| 191 | C2×Dic3 | — | 4 | 6 | no | — |
| 192 | C2×Dic3 | — | 4 | 2 | no | — |
| 193 | C2×Dic3 | — | 4 | 6 | no | — |
| 194 | C2×Dic3 | — | 4 | 6 | no | — |
| 195 | C2×Dic3 | — | 4 | 6 | no | — |
| 196 | C24 | C2×#67 | 1 | 3 | yes | — |
| 197 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 198 | #41 | — | 8 | 2 | no | — |
| 199 | C2×Q8 | — | 2 | 1 | no | — |
| 200 | C2×Q8 | — | 2 | 1 | no | — |
| 201 | C2×D4 | — | 2 | 3 | no | — |
| 202 | C2×D4 | — | 2 | 6 | no | — |
| 203 | C2×D4 | — | 2 | 3 | no | — |
| 204 | C2×D4 | — | 2 | 6 | no | — |
| 205 | C2×D4 | — | 2 | 3 | no | — |
| 206 | C2×D4 | — | 2 | 3 | no | — |
| 207 | C22×C4 | — | 2 | 3 | no | — |
| 208 | C22×C4 | — | 2 | 3 | no | — |
| 209 | C22×C4 | — | 4 | 2 | no | — |
| 210 | C22×C4 | — | 4 | 6 | no | — |
| 211 | C22×C4 | — | 6 | 6 | no | — |
| 212 | C22×C4 | — | 6 | 2 | no | — |
| 213 | C22×C4 | — | 6 | 6 | no | — |
| 214 | C22×C4 | — | 6 | 6 | no | — |
| 215 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 216 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 217 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 218 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 219 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 220 | C4⋊C4 | — | 2 | 3 | no | — |
| 221 | C4⋊C4 | — | 2 | 3 | no | — |
| 222 | C4⋊C4 | — | 2 | 3 | no | — |
| 223 | C4⋊C4 | — | 2 | 3 | no | — |
| 224 | C4⋊C4 | — | 2 | 6 | no | — |
| 225 | C4⋊C4 | — | 12 | 12 | no | — |
| 226 | C4⋊C4 | — | 12 | 6 | no | — |
| 227 | C4⋊C4 | — | 12 | 3 | no | — |
| 228 | C4⋊C4 | — | 12 | 1 | no | — |
| 229 | C22⋊C4 | — | 2 | 6 | no | — |
| 230 | C22⋊C4 | — | 2 | 6 | no | — |
| 231 | C22⋊C4 | — | 2 | 6 | no | — |
| 232 | C22⋊C4 | — | 2 | 6 | no | — |
| 233 | C22⋊C4 | — | 6 | 12 | no | — |
| 234 | C22⋊C4 | — | 12 | 12 | no | — |
| 235 | C42 | — | 2 | 3 | no | — |
| 236 | C42 | — | 2 | 3 | no | — |
| 237 | C22×C3 | — | 2 | 3 | no | — |
| 238 | C22×C3 | — | 2 | 3 | no | — |
| 239 | C22×C3 | — | 4 | 3 | no | — |
| 240 | C22×C3 | — | 4 | 1 | no | — |
| 241 | C22×C3 | — | 4 | 6 | no | — |
| 242 | C22×C3 | — | 4 | 3 | no | — |
| 243 | C22×C3 | — | 4 | 3 | no | — |
| 244 | C22×C3 | — | 4 | 3 | no | — |
| 245 | C22×C3 | — | 8 | 3 | no | — |
| 246 | C22×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 247 | C22×C3 | #535 | 1 | 3 | yes | — |
| 248 | C22×C3 | #537 | 1 | 3 | yes | — |
| 249 | C3×C4 | — | 2 | 3 | no | — |
| 250 | C3×C4 | — | 4 | 3 | no | — |
| 251 | C3×C4 | — | 4 | 3 | no | — |
| 252 | C3×C4 | — | 4 | 3 | no | — |
| 253 | C3×C4 | #458 | 1 | 2 | yes | — |
| 254 | Dic3 | — | 8 | 6 | no | — |
| 255 | Dic3 | — | 8 | 2 | no | — |
| 256 | C23 | — | 2 | 3 | no | — |
| 257 | C23 | — | 2 | 3 | no | — |
| 258 | C23 | — | 2 | 3 | no | — |
| 259 | C23 | — | 4 | 1 | no | — |
| 260 | C23 | — | 4 | 1 | no | — |
| 261 | C23 | — | 4 | 3 | no | — |
| 262 | C23 | — | 4 | 3 | no | — |
| 263 | C23 | #1009 | 1 | 3 | yes | — |
| 264 | C23 | #1010 | 1 | 1 | yes | characteristic |
| 265 | C23 | C22×#67 | 1 | 3 | yes | — |
| 266 | Q8 | — | 8 | 1 | no | — |
| 267 | Q8 | — | 8 | 1 | no | — |
| 268 | D4 | — | 4 | 6 | no | — |
| 269 | D4 | — | 4 | 6 | no | — |
| 270 | D4 | — | 8 | 3 | no | — |
| 271 | D4 | — | 8 | 3 | no | — |
| 272 | C2×C4 | — | 2 | 6 | no | — |
| 273 | C2×C4 | — | 2 | 3 | no | — |
| 274 | C2×C4 | — | 2 | 3 | no | — |
| 275 | C2×C4 | — | 2 | 3 | no | — |
| 276 | C2×C4 | — | 2 | 3 | no | — |
| 277 | C2×C4 | — | 2 | 3 | no | — |
| 278 | C2×C4 | — | 4 | 3 | no | — |
| 279 | C2×C4 | — | 4 | 6 | no | — |
| 280 | C2×C4 | — | 4 | 3 | no | — |
| 281 | C2×C4 | — | 4 | 3 | no | — |
| 282 | C2×C4 | — | 8 | 3 | no | — |
| 283 | C2×C4 | — | 12 | 6 | no | — |
| 284 | C2×C4 | — | 12 | 6 | no | — |
| 285 | C2×C4 | — | 12 | 6 | no | — |
| 286 | C2×C4 | — | 12 | 6 | no | — |
| 287 | C2×C4 | — | 12 | 6 | no | — |
| 288 | C2×C4 | — | 12 | 2 | no | — |
| 289 | C2×C4 | #953 | 1 | 3 | yes | — |
| 290 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 291 | C2×C3 | — | 2 | 3 | no | — |
| 292 | C2×C3 | — | 2 | 3 | no | — |
| 293 | C2×C3 | — | 8 | 3 | no | — |
| 294 | C2×C3 | #2615 | 1 | 1 | yes | characteristic |
| 295 | C2×C3 | C2×#458 | 1 | 1 | yes | characteristic |
| 296 | C2×C3 | #3028 | 1 | 1 | yes | characteristic |
| 297 | C22 | — | 2 | 3 | no | — |
| 298 | C22 | — | 2 | 3 | no | — |
| 299 | C22 | — | 4 | 3 | no | — |
| 300 | C22 | — | 4 | 1 | no | — |
| 301 | C22 | — | 4 | 6 | no | — |
| 302 | C22 | — | 4 | 3 | no | — |
| 303 | C22 | — | 4 | 3 | no | — |
| 304 | C22 | — | 4 | 3 | no | — |
| 305 | C22 | — | 8 | 3 | no | — |
| 306 | C22 | #5830 | 1 | 3 | yes | — |
| 307 | C22 | #5833 | 1 | 3 | yes | — |
| 308 | C22 | C2×#953 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 3 | no | — |
| 310 | C4 | — | 4 | 3 | no | — |
| 311 | C4 | — | 4 | 3 | no | — |
| 312 | C4 | — | 4 | 3 | no | — |
| 313 | C4 | — | 24 | 6 | no | — |
| 314 | C4 | — | 24 | 2 | no | — |
| 315 | C4 | #5243 | 1 | 2 | yes | — |
| 316 | C3 | #29685 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 317 | C2 | — | 2 | 3 | no | — |
| 318 | C2 | — | 2 | 3 | no | — |
| 319 | C2 | — | 8 | 3 | no | — |
| 320 | C2 | #84098 | 1 | 1 | yes | characteristic |
| 321 | C2 | C2×#5243 | 1 | 1 | yes | characteristic |
| 322 | C2 | #84124 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #82373 | C2 | semidirect |
| 3. #71894 | C2 | semidirect |
| 4. #71849 | C2 | semidirect |
| 5. #71619 | C2 | semidirect |
| 316. C3 | #29685 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3459 | 2 |
| #82373 | 2 |
| #71894 | 2 |
| #71849 | 2 |
| #71619 | 2 |
| #29685 | 3 |