Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
893 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 893 conjugacy classes of subgroups — too many to draw. Only its 119 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 | #3504⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85423 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #85414 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #85365 | C2 | 1 | 2 | yes | maximal |
| 5 | #85361 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×#985 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #85339 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #61376 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×#985 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5688 | — | 4 | 1 | no | maximal |
| 11 | #557⋊C3 | C22 | 1 | 4 | yes | — |
| 12 | #520⋊C3 | C22 | 1 | 2 | yes | — |
| 13 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 14 | #5426 | C22 | 1 | 2 | yes | — |
| 15 | #5424 | C22 | 1 | 1 | yes | characteristic |
| 16 | #5394 | C22 | 1 | 4 | yes | — |
| 17 | C4×GL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 18 | #5389 | C22 | 1 | 4 | yes | — |
| 19 | #5389 | C22 | 1 | 4 | yes | — |
| 20 | C4×#226 | C22 | 1 | 2 | yes | — |
| 21 | C4×#226 | C22 | 1 | 4 | yes | — |
| 22 | #5386 | C22 | 1 | 4 | yes | — |
| 23 | #3504 | — | 3 | 1 | no | — |
| 24 | #3504 | D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | #3388 | — | 3 | 4 | no | — |
| 26 | #3388 | — | 3 | 4 | no | — |
| 27 | #3383 | — | 3 | 4 | no | — |
| 28 | #3380 | — | 3 | 4 | no | — |
| 29 | #3164 | — | 3 | 2 | no | — |
| 30 | C2×#499 | — | 3 | 1 | no | — |
| 31 | #3084 | — | 3 | 2 | no | — |
| 32 | #3084 | — | 3 | 2 | no | — |
| 33 | #3080 | — | 3 | 1 | no | — |
| 34 | #3080 | — | 3 | 1 | no | — |
| 35 | #2960 | — | 3 | 1 | no | — |
| 36 | #2951 | — | 3 | 2 | no | — |
| 37 | C4×#135 | — | 3 | 1 | no | — |
| 38 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 39 | #985 | Q8 | 1 | 4 | yes | complemented |
| 40 | C2×GL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×#226 | C23 | 1 | 1 | yes | characteristic |
| 42 | C3×#125 | — | 4 | 1 | no | — |
| 43 | #896 | — | 4 | 4 | no | — |
| 44 | #892 | — | 4 | 2 | no | — |
| 45 | #888 | — | 4 | 2 | no | — |
| 46 | #887 | — | 4 | 4 | no | — |
| 47 | C42×D3 | — | 4 | 1 | no | — |
| 48 | #870 | — | 4 | 1 | no | — |
| 49 | C4×SL(2,𝔽3) | C23 | 1 | 4 | yes | — |
| 50 | C4×SL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 51 | #859 | C23 | 1 | 2 | yes | — |
| 52 | #859 | C23 | 1 | 4 | yes | — |
| 53 | C2×#135 | — | 3 | 1 | no | — |
| 54 | #557 | — | 3 | 4 | no | — |
| 55 | #557 | C2×D3 | 1 | 4 | yes | — |
| 56 | #554 | — | 3 | 4 | no | — |
| 57 | #554 | — | 3 | 4 | no | — |
| 58 | #554 | — | 6 | 4 | no | — |
| 59 | #544 | — | 3 | 2 | no | — |
| 60 | #544 | — | 3 | 2 | no | — |
| 61 | #544 | — | 6 | 2 | no | — |
| 62 | #533 | — | 3 | 1 | no | — |
| 63 | #533 | — | 6 | 1 | no | — |
| 64 | C2×#125 | — | 3 | 1 | no | — |
| 65 | C2×#125 | — | 3 | 1 | no | — |
| 66 | C2×#125 | — | 6 | 1 | no | — |
| 67 | #520 | — | 3 | 2 | no | — |
| 68 | #520 | C2×D3 | 1 | 2 | yes | — |
| 69 | C4×#41 | — | 3 | 1 | no | — |
| 70 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 71 | #501 | — | 3 | 4 | no | — |
| 72 | #501 | — | 3 | 4 | no | — |
| 73 | #500 | — | 3 | 4 | no | — |
| 74 | #499 | — | 3 | 4 | no | — |
| 75 | #499 | — | 3 | 4 | no | — |
| 76 | #499 | — | 3 | 4 | no | — |
| 77 | #479 | — | 3 | 4 | no | — |
| 78 | #479 | — | 3 | 4 | no | — |
| 79 | #479 | — | 3 | 4 | no | — |
| 80 | #479 | — | 3 | 4 | no | — |
| 81 | #478 | — | 3 | 4 | no | — |
| 82 | #478 | — | 3 | 4 | no | — |
| 83 | #478 | — | 3 | 4 | no | — |
| 84 | #478 | — | 3 | 4 | no | — |
| 85 | #476 | — | 3 | 4 | no | — |
| 86 | #476 | — | 3 | 4 | no | — |
| 87 | #475 | — | 3 | 4 | no | — |
| 88 | #475 | — | 3 | 4 | no | — |
| 89 | #446 | — | 3 | 4 | no | — |
| 90 | C8×Q8 | — | 3 | 4 | no | — |
| 91 | #440 | — | 3 | 4 | no | — |
| 92 | #440 | — | 3 | 4 | no | — |
| 93 | C4×Q16 | — | 3 | 2 | no | — |
| 94 | C4×Q16 | — | 3 | 4 | no | — |
| 95 | C4×SD16 | — | 3 | 2 | no | — |
| 96 | C4×SD16 | — | 3 | 2 | no | — |
| 97 | C4×D8 | — | 3 | 2 | no | — |
| 98 | C4×D8 | — | 3 | 4 | no | — |
| 99 | #428 | — | 3 | 2 | no | — |
| 100 | #428 | — | 3 | 2 | no | — |
| 101 | #427 | — | 3 | 1 | no | — |
| 102 | C2×#107 | — | 3 | 2 | no | — |
| 103 | C2×#106 | — | 3 | 2 | no | — |
| 104 | #424 | — | 3 | 2 | no | — |
| 105 | #423 | — | 3 | 1 | no | — |
| 106 | #417 | — | 3 | 2 | no | — |
| 107 | #417 | — | 3 | 2 | no | — |
| 108 | #416 | — | 3 | 1 | no | — |
| 109 | #416 | — | 3 | 1 | no | — |
| 110 | #405 | — | 3 | 2 | no | — |
| 111 | C2×C4×C8 | — | 3 | 1 | no | — |
| 112 | C2×C4×D3 | — | 4 | 1 | no | — |
| 113 | C2×C4×D3 | — | 4 | 2 | no | — |
| 114 | C2×Dic6 | — | 4 | 4 | no | — |
| 115 | #41⋊C3 | C2×Q8 | 1 | 2 | yes | — |
| 116 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 117 | GL(2,𝔽3) | C2×Q8 | 1 | 2 | yes | — |
| 118 | #226 | C2×Q8 | 1 | 2 | yes | — |
| 119 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 120 | C3×C4⋊C4 | — | 4 | 4 | no | — |
| 121 | C3×C42 | — | 4 | 1 | no | — |
| 122 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 123 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 124 | Dic3⋊C4 | — | 4 | 4 | no | — |
| 125 | Dic3⋊C4 | — | 4 | 4 | no | — |
| 126 | C4×Dic3 | — | 4 | 2 | no | — |
| 127 | C4×Dic3 | — | 4 | 4 | no | — |
| 128 | C4×Dic3 | — | 4 | 1 | no | — |
| 129 | C2×#41 | — | 3 | 1 | no | — |
| 130 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 131 | #135 | — | 3 | 4 | no | — |
| 132 | #135 | — | 3 | 4 | no | — |
| 133 | C2×Q16 | — | 3 | 1 | no | — |
| 134 | C2×SD16 | — | 3 | 1 | no | — |
| 135 | C2×SD16 | — | 3 | 1 | no | — |
| 136 | C2×D8 | — | 3 | 1 | no | — |
| 137 | C2×M16 | — | 3 | 2 | no | — |
| 138 | C22×C8 | — | 3 | 1 | no | — |
| 139 | C22×C8 | — | 3 | 1 | no | — |
| 140 | C4⋊Q8 | — | 3 | 2 | no | — |
| 141 | C4⋊Q8 | — | 3 | 4 | no | — |
| 142 | C4⋊Q8 | — | 3 | 4 | no | — |
| 143 | C4⋊Q8 | — | 6 | 4 | no | — |
| 144 | C4⋊Q8 | — | 6 | 2 | no | — |
| 145 | #125 | — | 3 | 4 | no | — |
| 146 | #125 | — | 3 | 4 | no | — |
| 147 | #125 | — | 3 | 4 | no | — |
| 148 | #125 | — | 3 | 4 | no | — |
| 149 | #125 | — | 3 | 1 | no | — |
| 150 | #125 | — | 6 | 1 | no | — |
| 151 | #125 | — | 6 | 1 | no | — |
| 152 | #125 | — | 6 | 2 | no | — |
| 153 | #125 | — | 6 | 4 | no | — |
| 154 | #125 | — | 6 | 4 | no | — |
| 155 | #125 | — | 6 | 1 | no | — |
| 156 | #125 | S4 | 1 | 1 | yes | characteristic |
| 157 | C22⋊Q8 | — | 3 | 4 | no | — |
| 158 | C22⋊Q8 | — | 6 | 4 | no | — |
| 159 | C22⋊Q8 | — | 6 | 4 | no | — |
| 160 | C22⋊Q8 | — | 6 | 4 | no | — |
| 161 | C22⋊Q8 | — | 6 | 4 | no | — |
| 162 | C22⋊Q8 | — | 6 | 4 | no | — |
| 163 | C4×Q8 | — | 3 | 4 | no | — |
| 164 | C4×Q8 | — | 3 | 2 | no | — |
| 165 | C4×Q8 | — | 3 | 4 | no | — |
| 166 | C4×Q8 | — | 6 | 4 | no | — |
| 167 | C4×Q8 | C22×D3 | 1 | 2 | yes | — |
| 168 | C4×Q8 | C22×D3 | 1 | 4 | yes | — |
| 169 | C4×D4 | — | 3 | 4 | no | — |
| 170 | C4×D4 | — | 3 | 2 | no | — |
| 171 | C4×D4 | — | 3 | 2 | no | — |
| 172 | C4×D4 | — | 3 | 4 | no | — |
| 173 | C4×D4 | — | 6 | 4 | no | — |
| 174 | C4×D4 | — | 6 | 2 | no | — |
| 175 | #117 | — | 3 | 2 | no | — |
| 176 | #117 | — | 3 | 1 | no | — |
| 177 | #117 | — | 3 | 1 | no | — |
| 178 | #117 | — | 6 | 1 | no | — |
| 179 | #117 | — | 6 | 2 | no | — |
| 180 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 181 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 182 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 183 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 184 | C2×C4⋊C4 | — | 6 | 4 | no | — |
| 185 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 186 | C2×C42 | — | 3 | 1 | no | — |
| 187 | C2×C42 | — | 6 | 1 | no | — |
| 188 | #107 | — | 3 | 4 | no | — |
| 189 | #107 | — | 3 | 4 | no | — |
| 190 | #107 | — | 3 | 2 | no | — |
| 191 | #107 | — | 3 | 4 | no | — |
| 192 | #107 | — | 3 | 4 | no | — |
| 193 | #106 | — | 3 | 4 | no | — |
| 194 | #106 | — | 3 | 4 | no | — |
| 195 | #106 | — | 3 | 2 | no | — |
| 196 | #106 | — | 3 | 4 | no | — |
| 197 | #106 | — | 3 | 4 | no | — |
| 198 | C4⋊C8 | — | 3 | 4 | no | — |
| 199 | C4⋊C8 | — | 3 | 4 | no | — |
| 200 | C4⋊C8 | — | 3 | 4 | no | — |
| 201 | #103 | — | 3 | 2 | no | — |
| 202 | #103 | — | 3 | 4 | no | — |
| 203 | #103 | — | 3 | 2 | no | — |
| 204 | #103 | — | 3 | 4 | no | — |
| 205 | #102 | — | 3 | 2 | no | — |
| 206 | #102 | — | 3 | 2 | no | — |
| 207 | #102 | — | 3 | 4 | no | — |
| 208 | #102 | — | 3 | 4 | no | — |
| 209 | #97 | — | 3 | 4 | no | — |
| 210 | C4×C8 | — | 3 | 2 | no | — |
| 211 | C4×C8 | — | 3 | 2 | no | — |
| 212 | C4×C8 | — | 3 | 4 | no | — |
| 213 | C22×D3 | — | 4 | 1 | no | — |
| 214 | C2×C3×C4 | — | 4 | 1 | no | — |
| 215 | C2×C3×C4 | — | 4 | 2 | no | — |
| 216 | C2×C3×C4 | — | 4 | 4 | no | — |
| 217 | C2×Dic3 | — | 4 | 2 | no | — |
| 218 | C2×Dic3 | — | 4 | 1 | no | — |
| 219 | C2×Dic3 | — | 4 | 4 | no | — |
| 220 | C4×D3 | — | 4 | 4 | no | — |
| 221 | C4×D3 | — | 8 | 4 | no | — |
| 222 | Dic6 | — | 8 | 8 | no | — |
| 223 | SL(2,𝔽3) | C22×Q8 | 1 | 1 | yes | characteristic |
| 224 | #41 | — | 3 | 2 | no | — |
| 225 | #41 | — | 3 | 4 | no | — |
| 226 | #41 | — | 6 | 2 | no | — |
| 227 | #41 | D3×Q8 | 1 | 2 | yes | — |
| 228 | C2×Q8 | — | 3 | 4 | no | — |
| 229 | C2×Q8 | — | 3 | 1 | no | — |
| 230 | C2×Q8 | — | 6 | 4 | no | — |
| 231 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 232 | C2×D4 | — | 3 | 1 | no | — |
| 233 | C2×D4 | — | 3 | 1 | no | — |
| 234 | C2×D4 | — | 6 | 1 | no | — |
| 235 | C22×C4 | — | 3 | 1 | no | — |
| 236 | C22×C4 | — | 3 | 2 | no | — |
| 237 | C22×C4 | — | 6 | 2 | no | — |
| 238 | C22×C4 | — | 6 | 4 | no | — |
| 239 | C22×C4 | — | 6 | 1 | no | — |
| 240 | C22×C4 | — | 6 | 2 | no | — |
| 241 | C22×C4 | — | 6 | 2 | no | — |
| 242 | Q16 | — | 3 | 2 | no | — |
| 243 | Q16 | — | 3 | 2 | no | — |
| 244 | SD16 | — | 3 | 4 | no | — |
| 245 | SD16 | — | 3 | 2 | no | — |
| 246 | SD16 | — | 3 | 2 | no | — |
| 247 | D8 | — | 3 | 4 | no | — |
| 248 | M16 | — | 6 | 4 | no | — |
| 249 | C2×C8 | — | 3 | 1 | no | — |
| 250 | C2×C8 | — | 3 | 1 | no | — |
| 251 | C2×C8 | — | 3 | 4 | no | — |
| 252 | C2×C8 | — | 3 | 2 | no | — |
| 253 | C2×C8 | — | 3 | 4 | no | — |
| 254 | C2×C8 | — | 6 | 2 | no | — |
| 255 | C4⋊C4 | — | 3 | 4 | no | — |
| 256 | C4⋊C4 | — | 3 | 2 | no | — |
| 257 | C4⋊C4 | — | 3 | 2 | no | — |
| 258 | C4⋊C4 | — | 3 | 4 | no | — |
| 259 | C4⋊C4 | — | 3 | 4 | no | — |
| 260 | C4⋊C4 | — | 3 | 4 | no | — |
| 261 | C4⋊C4 | — | 3 | 2 | no | — |
| 262 | C4⋊C4 | — | 6 | 4 | no | — |
| 263 | C4⋊C4 | — | 6 | 4 | no | — |
| 264 | C4⋊C4 | — | 6 | 4 | no | — |
| 265 | C4⋊C4 | — | 6 | 4 | no | — |
| 266 | C4⋊C4 | — | 6 | 4 | no | — |
| 267 | C4⋊C4 | — | 6 | 2 | no | — |
| 268 | C4⋊C4 | — | 6 | 2 | no | — |
| 269 | C4⋊C4 | — | 6 | 4 | no | — |
| 270 | C4⋊C4 | — | 6 | 4 | no | — |
| 271 | C4⋊C4 | C2×S4 | 1 | 2 | yes | — |
| 272 | C4⋊C4 | C2×S4 | 1 | 4 | yes | — |
| 273 | C22⋊C4 | — | 3 | 4 | no | — |
| 274 | C22⋊C4 | — | 6 | 1 | no | — |
| 275 | C22⋊C4 | — | 6 | 2 | no | — |
| 276 | C22⋊C4 | — | 6 | 1 | no | — |
| 277 | C22⋊C4 | — | 6 | 4 | no | — |
| 278 | C42 | — | 3 | 2 | no | — |
| 279 | C42 | — | 3 | 4 | no | — |
| 280 | C42 | — | 3 | 1 | no | — |
| 281 | C42 | — | 6 | 1 | no | — |
| 282 | C42 | — | 6 | 2 | no | — |
| 283 | C42 | — | 6 | 4 | no | — |
| 284 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 285 | C22×C3 | — | 4 | 1 | no | — |
| 286 | C2×D3 | — | 4 | 2 | no | — |
| 287 | C2×D3 | — | 8 | 2 | no | — |
| 288 | C3×C4 | — | 4 | 2 | no | — |
| 289 | C3×C4 | — | 8 | 4 | no | — |
| 290 | C3×C4 | — | 8 | 2 | no | — |
| 291 | Dic3 | — | 4 | 2 | no | — |
| 292 | Dic3 | — | 4 | 4 | no | — |
| 293 | Dic3 | — | 8 | 4 | no | — |
| 294 | C23 | — | 3 | 1 | no | — |
| 295 | C23 | — | 6 | 1 | no | — |
| 296 | Q8 | — | 3 | 1 | no | — |
| 297 | Q8 | — | 3 | 2 | no | — |
| 298 | Q8 | — | 6 | 1 | no | — |
| 299 | Q8 | — | 12 | 4 | no | — |
| 300 | Q8 | — | 24 | 4 | no | — |
| 301 | Q8 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 302 | D4 | — | 3 | 2 | no | — |
| 303 | D4 | — | 3 | 2 | no | — |
| 304 | D4 | — | 6 | 1 | no | — |
| 305 | D4 | — | 6 | 1 | no | — |
| 306 | D4 | — | 6 | 4 | no | — |
| 307 | C2×C4 | — | 3 | 1 | no | — |
| 308 | C2×C4 | — | 3 | 2 | no | — |
| 309 | C2×C4 | — | 3 | 2 | no | — |
| 310 | C2×C4 | — | 3 | 2 | no | — |
| 311 | C2×C4 | — | 3 | 4 | no | — |
| 312 | C2×C4 | — | 6 | 1 | no | — |
| 313 | C2×C4 | — | 6 | 4 | no | — |
| 314 | C2×C4 | — | 6 | 2 | no | — |
| 315 | C2×C4 | — | 6 | 4 | no | — |
| 316 | C2×C4 | — | 12 | 2 | no | — |
| 317 | C2×C4 | — | 24 | 2 | no | — |
| 318 | C2×C4 | — | 24 | 2 | no | — |
| 319 | C2×C4 | — | 24 | 4 | no | — |
| 320 | C2×C4 | — | 24 | 2 | no | — |
| 321 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 322 | C2×C4 | C22×S4 | 1 | 4 | yes | — |
| 323 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 324 | C8 | — | 3 | 2 | no | — |
| 325 | C8 | — | 3 | 2 | no | — |
| 326 | C8 | — | 6 | 2 | no | — |
| 327 | C8 | — | 6 | 4 | no | — |
| 328 | C2×C3 | — | 4 | 1 | no | — |
| 329 | C2×C3 | — | 4 | 2 | no | — |
| 330 | D3 | — | 8 | 2 | no | — |
| 331 | C22 | — | 3 | 2 | no | — |
| 332 | C22 | — | 6 | 2 | no | — |
| 333 | C22 | — | 6 | 2 | no | — |
| 334 | C22 | — | 12 | 2 | no | — |
| 335 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 336 | C4 | — | 2 | 2 | no | — |
| 337 | C4 | — | 2 | 4 | no | — |
| 338 | C4 | — | 3 | 1 | no | — |
| 339 | C4 | — | 3 | 1 | no | — |
| 340 | C4 | — | 6 | 2 | no | — |
| 341 | C4 | — | 6 | 2 | no | — |
| 342 | C4 | — | 6 | 4 | no | — |
| 343 | C4 | — | 12 | 4 | no | — |
| 344 | C4 | — | 12 | 2 | no | — |
| 345 | C4 | Q8×S4 | 1 | 2 | yes | — |
| 346 | C3 | — | 4 | 1 | no | Sylow |
| 347 | C2 | — | 6 | 2 | no | — |
| 348 | C2 | — | 12 | 2 | no | — |
| 349 | C2 | C2×Q8×S4 | 1 | 1 | yes | characteristic |
| 350 | C2 | #87497 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3504⋊C3 | C2 | semidirect |
| 2. #85423 | C2 | semidirect |
| 3. #85414 | C2 | semidirect |
| 5. #85361 | C2 | semidirect |
| 7. #85339 | C2 | semidirect |
| 24. #3504 | D3 | semidirect |
| 39. #985 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3504⋊C3 | 2 |
| #85423 | 2 |
| #85414 | 2 |
| #85365 | 2 |
| #85361 | 2 |
| C4×#985 | 2 |
| #85339 | 2 |
| #61376 | 3 |
| #5688 | 4 |