Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
6588 conjugacy classes of subgroups.
The subgroup lattice has 380 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#3531 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #87139 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #87134 | C2 | 1 | 20 | yes | maximal, complemented |
| 4 | #87099 | C2 | 1 | 10 | yes | maximal, complemented |
| 5 | #86288 | C2 | 1 | 5 | yes | maximal, complemented |
| 6 | #86282 | C2 | 1 | 5 | yes | maximal, complemented |
| 7 | #86140 | C2 | 1 | 10 | yes | maximal, complemented |
| 8 | #86126 | C2 | 1 | 10 | yes | maximal, complemented |
| 9 | #63070 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C3×#564 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#563 | C22 | 1 | 10 | yes | — |
| 12 | C3×#562 | C22 | 1 | 5 | yes | — |
| 13 | C3×#533 | C22 | 1 | 5 | yes | — |
| 14 | C3×#529 | C22 | 1 | 10 | yes | — |
| 15 | #5729 | C22 | 1 | 20 | yes | — |
| 16 | #5726 | C22 | 1 | 20 | yes | — |
| 17 | #5709 | C22 | 1 | 40 | yes | — |
| 18 | #5707 | C22 | 1 | 40 | yes | — |
| 19 | #5700 | C22 | 1 | 10 | yes | — |
| 20 | #5700 | C22 | 1 | 10 | yes | — |
| 21 | #5699 | C22 | 1 | 20 | yes | — |
| 22 | #5691 | C22 | 1 | 20 | yes | — |
| 23 | #5688 | C22 | 1 | 10 | yes | — |
| 24 | D3×#125 | C22 | 1 | 10 | yes | — |
| 25 | #5684 | C22 | 1 | 20 | yes | — |
| 26 | #5668 | C22 | 1 | 20 | yes | — |
| 27 | #5663 | C22 | 1 | 40 | yes | — |
| 28 | #5655 | C22 | 1 | 40 | yes | — |
| 29 | #5645 | C22 | 1 | 20 | yes | — |
| 30 | #5636 | C22 | 1 | 40 | yes | — |
| 31 | #5631 | C22 | 1 | 20 | yes | — |
| 32 | #5629 | C22 | 1 | 40 | yes | — |
| 33 | #5599 | C22 | 1 | 20 | yes | — |
| 34 | #5597 | C22 | 1 | 20 | yes | — |
| 35 | #5579 | C22 | 1 | 20 | yes | — |
| 36 | #5577 | C22 | 1 | 10 | yes | — |
| 37 | #5567 | C22 | 1 | 10 | yes | — |
| 38 | #5565 | C22 | 1 | 20 | yes | — |
| 39 | #5555 | C22 | 1 | 10 | yes | — |
| 40 | #5553 | C22 | 1 | 10 | yes | — |
| 41 | #5535 | C22 | 1 | 20 | yes | — |
| 42 | #5523 | C22 | 1 | 10 | yes | — |
| 43 | #5519 | C22 | 1 | 10 | yes | — |
| 44 | #5477 | C22 | 1 | 10 | yes | — |
| 45 | C4×#235 | C22 | 1 | 5 | yes | — |
| 46 | C2×#870 | C22 | 1 | 5 | yes | — |
| 47 | #3559 | — | 3 | 20 | no | — |
| 48 | #3559 | — | 3 | 10 | no | — |
| 49 | #3548 | — | 3 | 10 | no | — |
| 50 | #3548 | — | 3 | 5 | no | — |
| 51 | #3531 | — | 3 | 2 | no | — |
| 52 | #3531 | D3 | 1 | 1 | yes | characteristic, complemented |
| 53 | #3517 | — | 3 | 10 | no | — |
| 54 | #3504 | — | 3 | 5 | no | — |
| 55 | C2×#239 | C23 | 1 | 20 | yes | — |
| 56 | C2×#237 | C23 | 1 | 20 | yes | — |
| 57 | C2×#235 | C23 | 1 | 5 | yes | — |
| 58 | C2×#235 | C23 | 1 | 20 | yes | — |
| 59 | C3×C4⋊Q8 | C23 | 1 | 10 | yes | — |
| 60 | C3×#126 | C23 | 1 | 20 | yes | — |
| 61 | C3×#125 | C23 | 1 | 5 | yes | — |
| 62 | C3×#125 | C23 | 1 | 5 | yes | — |
| 63 | C3×#125 | C23 | 1 | 10 | yes | — |
| 64 | C3×#124 | C23 | 1 | 10 | yes | — |
| 65 | C3×#123 | C23 | 1 | 20 | yes | — |
| 66 | C3×C22⋊Q8 | C23 | 1 | 20 | yes | — |
| 67 | C3×C22⋊Q8 | C23 | 1 | 10 | yes | — |
| 68 | C3×#121 | C23 | 1 | 10 | yes | — |
| 69 | C3×C4×Q8 | C23 | 1 | 10 | yes | — |
| 70 | C3×C4×D4 | C23 | 1 | 10 | yes | — |
| 71 | C3×#117 | C23 | 1 | 10 | yes | — |
| 72 | C2×C3×C42 | C23 | 1 | 5 | yes | — |
| 73 | #946 | C23 | 1 | 40 | yes | — |
| 74 | #944 | C23 | 1 | 20 | yes | — |
| 75 | #940 | C23 | 1 | 20 | yes | — |
| 76 | #939 | C23 | 1 | 40 | yes | — |
| 77 | #930 | C23 | 1 | 20 | yes | — |
| 78 | #930 | C23 | 1 | 20 | yes | — |
| 79 | C4×#67 | C23 | 1 | 20 | yes | — |
| 80 | #926 | C23 | 1 | 5 | yes | — |
| 81 | C2×C4⋊Dic3 | C23 | 1 | 10 | yes | — |
| 82 | C2×Dic3⋊C4 | C23 | 1 | 20 | yes | — |
| 83 | #898 | C23 | 1 | 40 | yes | — |
| 84 | #897 | C23 | 1 | 20 | yes | — |
| 85 | #896 | C23 | 1 | 40 | yes | — |
| 86 | #896 | C23 | 1 | 40 | yes | — |
| 87 | #895 | C23 | 1 | 40 | yes | — |
| 88 | #895 | C23 | 1 | 20 | yes | — |
| 89 | #894 | C23 | 1 | 40 | yes | — |
| 90 | #894 | C23 | 1 | 40 | yes | — |
| 91 | #893 | C23 | 1 | 20 | yes | — |
| 92 | #892 | C23 | 1 | 20 | yes | — |
| 93 | D3×C4⋊C4 | C23 | 1 | 40 | yes | — |
| 94 | D3×C4⋊C4 | C23 | 1 | 20 | yes | — |
| 95 | #890 | C23 | 1 | 20 | yes | — |
| 96 | #889 | C23 | 1 | 40 | yes | — |
| 97 | #888 | C23 | 1 | 20 | yes | — |
| 98 | #887 | C23 | 1 | 20 | yes | — |
| 99 | #886 | C23 | 1 | 40 | yes | — |
| 100 | #884 | C23 | 1 | 40 | yes | — |
| 101 | #884 | C23 | 1 | 40 | yes | — |
| 102 | #884 | C23 | 1 | 40 | yes | — |
| 103 | #883 | C23 | 1 | 40 | yes | — |
| 104 | #67⋊C4 | C23 | 1 | 40 | yes | — |
| 105 | #67⋊C4 | C23 | 1 | 40 | yes | — |
| 106 | #878 | C23 | 1 | 40 | yes | — |
| 107 | #875 | C23 | 1 | 20 | yes | — |
| 108 | #874 | C23 | 1 | 10 | yes | — |
| 109 | C4×D12 | C23 | 1 | 20 | yes | — |
| 110 | C4×D12 | C23 | 1 | 10 | yes | — |
| 111 | C4×D12 | C23 | 1 | 20 | yes | — |
| 112 | #872 | C23 | 1 | 10 | yes | — |
| 113 | C42×D3 | C23 | 1 | 10 | yes | — |
| 114 | #870 | C23 | 1 | 10 | yes | — |
| 115 | #870 | C23 | 1 | 10 | yes | — |
| 116 | #870 | C23 | 1 | 20 | yes | — |
| 117 | #869 | C23 | 1 | 10 | yes | — |
| 118 | C4×Dic6 | C23 | 1 | 10 | yes | — |
| 119 | #564 | C2×D3 | 1 | 1 | yes | characteristic |
| 120 | #563 | — | 3 | 20 | no | — |
| 121 | #563 | C2×D3 | 1 | 10 | yes | — |
| 122 | #562 | — | 3 | 10 | no | — |
| 123 | #562 | — | 3 | 10 | no | — |
| 124 | #562 | — | 3 | 20 | no | — |
| 125 | #562 | C2×D3 | 1 | 5 | yes | — |
| 126 | #557 | — | 3 | 10 | no | — |
| 127 | #554 | — | 3 | 40 | no | — |
| 128 | #554 | — | 3 | 20 | no | — |
| 129 | #553 | — | 3 | 40 | no | — |
| 130 | #553 | — | 3 | 20 | no | — |
| 131 | #550 | — | 3 | 20 | no | — |
| 132 | #550 | — | 3 | 10 | no | — |
| 133 | #550 | — | 3 | 10 | no | — |
| 134 | #547 | — | 3 | 20 | no | — |
| 135 | #547 | — | 3 | 40 | no | — |
| 136 | #547 | — | 3 | 20 | no | — |
| 137 | #547 | — | 3 | 10 | no | — |
| 138 | #544 | — | 3 | 20 | no | — |
| 139 | #544 | — | 3 | 10 | no | — |
| 140 | #540 | — | 3 | 40 | no | — |
| 141 | #540 | — | 3 | 20 | no | — |
| 142 | #539 | — | 3 | 40 | no | — |
| 143 | #539 | — | 3 | 20 | no | — |
| 144 | #537 | — | 3 | 40 | no | — |
| 145 | #537 | — | 3 | 20 | no | — |
| 146 | #533 | — | 3 | 10 | no | — |
| 147 | #533 | C2×D3 | 1 | 5 | yes | — |
| 148 | #532 | — | 3 | 20 | no | — |
| 149 | #532 | — | 3 | 10 | no | — |
| 150 | #529 | — | 3 | 20 | no | — |
| 151 | #529 | C2×D3 | 1 | 10 | yes | — |
| 152 | C2×#125 | — | 3 | 10 | no | — |
| 153 | C2×#125 | — | 3 | 5 | no | — |
| 154 | #520 | — | 3 | 10 | no | — |
| 155 | C4×#41 | — | 3 | 5 | no | — |
| 156 | C2×C3×Q8 | C24 | 1 | 10 | yes | — |
| 157 | C2×C3×D4 | C24 | 1 | 10 | yes | — |
| 158 | C22×C3×C4 | C24 | 1 | 5 | yes | — |
| 159 | C22×C3×C4 | C24 | 1 | 10 | yes | — |
| 160 | C2×#67 | C24 | 1 | 10 | yes | — |
| 161 | C2×#67 | C24 | 1 | 40 | yes | — |
| 162 | C22×Dic3 | C24 | 1 | 20 | yes | — |
| 163 | #239 | — | 8 | 20 | no | — |
| 164 | #237 | — | 8 | 20 | no | — |
| 165 | #235 | — | 2 | 20 | no | — |
| 166 | #235 | — | 8 | 20 | no | — |
| 167 | C2×D12 | C24 | 1 | 5 | yes | — |
| 168 | C2×D12 | C24 | 1 | 20 | yes | — |
| 169 | C2×D12 | C24 | 1 | 20 | yes | — |
| 170 | C2×C4×D3 | C24 | 1 | 10 | yes | — |
| 171 | C2×C4×D3 | C24 | 1 | 20 | yes | — |
| 172 | C2×C4×D3 | C24 | 1 | 40 | yes | — |
| 173 | C2×Dic6 | C24 | 1 | 20 | yes | — |
| 174 | C2×Dic6 | C24 | 1 | 1 | yes | characteristic |
| 175 | C3×C4⋊C4 | C24 | 1 | 10 | yes | — |
| 176 | C3×C4⋊C4 | C24 | 1 | 10 | yes | — |
| 177 | C3×C4⋊C4 | C24 | 1 | 20 | yes | — |
| 178 | C3×C4⋊C4 | C24 | 1 | 10 | yes | — |
| 179 | C3×C4⋊C4 | C24 | 1 | 10 | yes | — |
| 180 | C3×C22⋊C4 | C24 | 1 | 20 | yes | — |
| 181 | C3×C22⋊C4 | C24 | 1 | 20 | yes | — |
| 182 | C3×C42 | C24 | 1 | 10 | yes | — |
| 183 | C3×C42 | C24 | 1 | 5 | yes | — |
| 184 | C3×C42 | C24 | 1 | 5 | yes | — |
| 185 | #217 | C24 | 1 | 10 | yes | — |
| 186 | (C2×D3)⋊C4 | C24 | 1 | 20 | yes | — |
| 187 | (C2×D3)⋊C4 | C24 | 1 | 40 | yes | — |
| 188 | (C2×D3)⋊C4 | C24 | 1 | 40 | yes | — |
| 189 | C4⋊Dic3 | C24 | 1 | 10 | yes | — |
| 190 | C4⋊Dic3 | C24 | 1 | 20 | yes | — |
| 191 | C4⋊Dic3 | C24 | 1 | 20 | yes | — |
| 192 | Dic3⋊C4 | C24 | 1 | 20 | yes | — |
| 193 | Dic3⋊C4 | C24 | 1 | 40 | yes | — |
| 194 | Dic3⋊C4 | C24 | 1 | 40 | yes | — |
| 195 | C4×Dic3 | C24 | 1 | 20 | yes | — |
| 196 | C4×Dic3 | C24 | 1 | 10 | yes | — |
| 197 | C2×#41 | — | 3 | 20 | no | — |
| 198 | C2×#41 | — | 3 | 20 | no | — |
| 199 | C2×#41 | — | 3 | 5 | no | — |
| 200 | C2×#41 | — | 3 | 20 | no | — |
| 201 | C4⋊Q8 | — | 3 | 20 | no | — |
| 202 | C4⋊Q8 | — | 3 | 20 | no | — |
| 203 | C4⋊Q8 | — | 3 | 10 | no | — |
| 204 | C4⋊Q8 | C22×D3 | 1 | 10 | yes | — |
| 205 | #127 | — | 3 | 20 | no | — |
| 206 | #127 | — | 3 | 10 | no | — |
| 207 | #126 | — | 3 | 40 | no | — |
| 208 | #126 | C22×D3 | 1 | 20 | yes | — |
| 209 | #125 | — | 3 | 20 | no | — |
| 210 | #125 | — | 3 | 40 | no | — |
| 211 | #125 | — | 3 | 10 | no | — |
| 212 | #125 | — | 3 | 10 | no | — |
| 213 | #125 | — | 3 | 20 | no | — |
| 214 | #125 | C22×D3 | 1 | 5 | yes | — |
| 215 | #125 | C22×D3 | 1 | 10 | yes | — |
| 216 | #125 | C22×D3 | 1 | 5 | yes | — |
| 217 | #124 | — | 3 | 20 | no | — |
| 218 | #124 | C22×D3 | 1 | 10 | yes | — |
| 219 | #123 | — | 3 | 40 | no | — |
| 220 | #123 | — | 3 | 40 | no | — |
| 221 | #123 | — | 3 | 40 | no | — |
| 222 | #123 | — | 3 | 40 | no | — |
| 223 | #123 | C22×D3 | 1 | 20 | yes | — |
| 224 | C22⋊Q8 | — | 3 | 40 | no | — |
| 225 | C22⋊Q8 | — | 3 | 20 | no | — |
| 226 | C22⋊Q8 | — | 3 | 40 | no | — |
| 227 | C22⋊Q8 | — | 3 | 40 | no | — |
| 228 | C22⋊Q8 | — | 3 | 40 | no | — |
| 229 | C22⋊Q8 | C22×D3 | 1 | 10 | yes | — |
| 230 | C22⋊Q8 | C22×D3 | 1 | 20 | yes | — |
| 231 | #121 | — | 3 | 40 | no | — |
| 232 | #121 | — | 3 | 20 | no | — |
| 233 | #121 | — | 3 | 20 | no | — |
| 234 | #121 | — | 3 | 20 | no | — |
| 235 | #121 | — | 3 | 40 | no | — |
| 236 | #121 | — | 3 | 40 | no | — |
| 237 | #121 | — | 3 | 40 | no | — |
| 238 | #121 | — | 3 | 40 | no | — |
| 239 | #121 | C22×D3 | 1 | 10 | yes | — |
| 240 | C4×Q8 | — | 3 | 20 | no | — |
| 241 | C4×Q8 | — | 3 | 10 | no | — |
| 242 | C4×Q8 | C22×D3 | 1 | 10 | yes | — |
| 243 | C4×D4 | — | 3 | 20 | no | — |
| 244 | C4×D4 | — | 3 | 20 | no | — |
| 245 | C4×D4 | — | 3 | 40 | no | — |
| 246 | C4×D4 | — | 3 | 40 | no | — |
| 247 | C4×D4 | — | 3 | 10 | no | — |
| 248 | C4×D4 | — | 3 | 20 | no | — |
| 249 | C4×D4 | — | 3 | 20 | no | — |
| 250 | C4×D4 | C22×D3 | 1 | 10 | yes | — |
| 251 | #117 | — | 3 | 5 | no | — |
| 252 | #117 | — | 3 | 20 | no | — |
| 253 | #117 | — | 3 | 10 | no | — |
| 254 | #117 | C22×D3 | 1 | 10 | yes | — |
| 255 | C2×C4⋊C4 | — | 3 | 10 | no | — |
| 256 | C2×C4⋊C4 | — | 3 | 20 | no | — |
| 257 | C2×C4⋊C4 | — | 3 | 20 | no | — |
| 258 | C2×C4⋊C4 | — | 3 | 40 | no | — |
| 259 | C2×C42 | — | 3 | 10 | no | — |
| 260 | C2×C42 | C22×D3 | 1 | 5 | yes | — |
| 261 | C23×C3 | C25 | 1 | 5 | yes | — |
| 262 | C22×D3 | C25 | 1 | 10 | yes | — |
| 263 | C3×Q8 | — | 4 | 10 | no | — |
| 264 | C3×D4 | — | 4 | 10 | no | — |
| 265 | C2×C3×C4 | — | 2 | 10 | no | — |
| 266 | C2×C3×C4 | — | 4 | 10 | no | — |
| 267 | C2×C3×C4 | C25 | 1 | 10 | yes | — |
| 268 | C2×C3×C4 | C25 | 1 | 5 | yes | — |
| 269 | C2×C3×C4 | C25 | 1 | 10 | yes | — |
| 270 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 271 | #67 | — | 2 | 20 | no | — |
| 272 | #67 | — | 4 | 40 | no | — |
| 273 | C2×Dic3 | — | 4 | 20 | no | — |
| 274 | C2×Dic3 | C25 | 1 | 20 | yes | — |
| 275 | C2×Dic3 | C25 | 1 | 2 | yes | — |
| 276 | D12 | — | 2 | 10 | no | — |
| 277 | D12 | — | 4 | 20 | no | — |
| 278 | D12 | — | 4 | 20 | no | — |
| 279 | C4×D3 | — | 2 | 20 | no | — |
| 280 | C4×D3 | — | 4 | 40 | no | — |
| 281 | C4×D3 | — | 4 | 20 | no | — |
| 282 | Dic6 | — | 4 | 20 | no | — |
| 283 | Dic6 | 2−1+4 | 1 | 4 | yes | — |
| 284 | #41 | — | 6 | 20 | no | — |
| 285 | #41 | — | 24 | 20 | no | — |
| 286 | #41 | — | 24 | 20 | no | — |
| 287 | #41 | — | 24 | 20 | no | — |
| 288 | C2×Q8 | — | 3 | 1 | no | — |
| 289 | C2×Q8 | — | 3 | 20 | no | — |
| 290 | C2×Q8 | C23×D3 | 1 | 10 | yes | — |
| 291 | C2×D4 | — | 3 | 10 | no | — |
| 292 | C2×D4 | — | 3 | 40 | no | — |
| 293 | C2×D4 | — | 3 | 5 | no | — |
| 294 | C2×D4 | — | 3 | 20 | no | — |
| 295 | C2×D4 | — | 3 | 20 | no | — |
| 296 | C2×D4 | C23×D3 | 1 | 10 | yes | — |
| 297 | C22×C4 | — | 3 | 20 | no | — |
| 298 | C22×C4 | — | 3 | 10 | no | — |
| 299 | C22×C4 | — | 3 | 20 | no | — |
| 300 | C22×C4 | — | 3 | 40 | no | — |
| 301 | C22×C4 | C23×D3 | 1 | 5 | yes | — |
| 302 | C22×C4 | C23×D3 | 1 | 10 | yes | — |
| 303 | C4⋊C4 | — | 3 | 20 | no | — |
| 304 | C4⋊C4 | — | 3 | 10 | no | — |
| 305 | C4⋊C4 | — | 3 | 20 | no | — |
| 306 | C4⋊C4 | — | 3 | 40 | no | — |
| 307 | C4⋊C4 | — | 3 | 20 | no | — |
| 308 | C4⋊C4 | — | 3 | 40 | no | — |
| 309 | C4⋊C4 | C23×D3 | 1 | 10 | yes | — |
| 310 | C4⋊C4 | C23×D3 | 1 | 20 | yes | — |
| 311 | C4⋊C4 | C23×D3 | 1 | 10 | yes | — |
| 312 | C4⋊C4 | C23×D3 | 1 | 10 | yes | — |
| 313 | C4⋊C4 | C23×D3 | 1 | 10 | yes | — |
| 314 | C22⋊C4 | — | 3 | 10 | no | — |
| 315 | C22⋊C4 | — | 3 | 40 | no | — |
| 316 | C22⋊C4 | — | 3 | 20 | no | — |
| 317 | C22⋊C4 | — | 3 | 40 | no | — |
| 318 | C22⋊C4 | C23×D3 | 1 | 20 | yes | — |
| 319 | C22⋊C4 | C23×D3 | 1 | 20 | yes | — |
| 320 | C42 | — | 3 | 20 | no | — |
| 321 | C42 | — | 3 | 10 | no | — |
| 322 | C42 | C23×D3 | 1 | 5 | yes | — |
| 323 | C42 | C23×D3 | 1 | 5 | yes | — |
| 324 | C42 | C23×D3 | 1 | 10 | yes | — |
| 325 | C22×C3 | — | 2 | 5 | no | — |
| 326 | C22×C3 | — | 2 | 10 | no | — |
| 327 | C22×C3 | C26 | 1 | 1 | yes | characteristic |
| 328 | C2×D3 | — | 2 | 10 | no | — |
| 329 | C2×D3 | — | 2 | 20 | no | — |
| 330 | C3×C4 | — | 2 | 5 | no | — |
| 331 | C3×C4 | — | 2 | 10 | no | — |
| 332 | C3×C4 | — | 2 | 10 | no | — |
| 333 | C3×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 334 | Dic3 | — | 2 | 20 | no | — |
| 335 | Dic3 | C2×2−1+4 | 1 | 4 | yes | — |
| 336 | C23 | — | 3 | 10 | no | — |
| 337 | C23 | C24×D3 | 1 | 5 | yes | — |
| 338 | Q8 | — | 3 | 4 | no | — |
| 339 | Q8 | — | 4 | 10 | no | — |
| 340 | Q8 | — | 12 | 20 | no | — |
| 341 | D4 | — | 4 | 10 | no | — |
| 342 | D4 | — | 6 | 20 | no | — |
| 343 | D4 | — | 6 | 10 | no | — |
| 344 | D4 | — | 12 | 20 | no | — |
| 345 | D4 | — | 12 | 20 | no | — |
| 346 | D4 | — | 12 | 40 | no | — |
| 347 | C2×C4 | — | 2 | 10 | no | — |
| 348 | C2×C4 | — | 3 | 2 | no | — |
| 349 | C2×C4 | — | 3 | 20 | no | — |
| 350 | C2×C4 | — | 4 | 10 | no | — |
| 351 | C2×C4 | — | 6 | 20 | no | — |
| 352 | C2×C4 | — | 12 | 40 | no | — |
| 353 | C2×C4 | — | 12 | 20 | no | — |
| 354 | C2×C4 | — | 12 | 20 | no | — |
| 355 | C2×C4 | C24×D3 | 1 | 5 | yes | — |
| 356 | C2×C4 | C24×D3 | 1 | 10 | yes | — |
| 357 | C2×C4 | C24×D3 | 1 | 10 | yes | — |
| 358 | C2×C4 | C24×D3 | 1 | 1 | yes | characteristic |
| 359 | C2×C3 | — | 4 | 5 | no | — |
| 360 | C2×C3 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 361 | C2×C3 | H(7,𝔽2) | 1 | 2 | yes | — |
| 362 | D3 | — | 4 | 10 | no | — |
| 363 | C22 | — | 2 | 5 | no | — |
| 364 | C22 | — | 2 | 10 | no | — |
| 365 | C22 | — | 6 | 10 | no | — |
| 366 | C22 | — | 6 | 20 | no | — |
| 367 | C22 | C25×D3 | 1 | 1 | yes | characteristic |
| 368 | C4 | — | 2 | 5 | no | — |
| 369 | C4 | — | 2 | 10 | no | — |
| 370 | C4 | — | 2 | 10 | no | — |
| 371 | C4 | — | 3 | 4 | no | — |
| 372 | C4 | — | 6 | 20 | no | — |
| 373 | C4 | D3×2−1+4 | 1 | 2 | yes | — |
| 374 | C3 | #63070 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 375 | C2 | — | 4 | 5 | no | — |
| 376 | C2 | — | 12 | 10 | no | — |
| 377 | C2 | C2×D3×2−1+4 | 1 | 1 | yes | characteristic |
| 378 | C2 | #87574 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C3×#3531 | C2 |
| hypercenter | #3531 | D3 |
| hypocenter | C3 | #63070 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #3531 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3531 | C2 | semidirect |
| 2. #87139 | C2 | semidirect |
| 3. #87134 | C2 | semidirect |
| 4. #87099 | C2 | semidirect |
| 5. #86288 | C2 | semidirect |
| 6. #86282 | C2 | semidirect |
| 7. #86140 | C2 | semidirect |
| 8. #86126 | C2 | semidirect |
| 52. #3531 | D3 | semidirect |
| 374. C3 | #63070 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3531 | 2 |
| #87139 | 2 |
| #87134 | 2 |
| #87099 | 2 |
| #86288 | 2 |
| #86282 | 2 |
| #86140 | 2 |
| #86126 | 2 |
| #63070 | 3 |