Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5070 conjugacy classes of subgroups.
The subgroup lattice has 644 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×D42 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D4×#67 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C2×D3×#127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #84049 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #83173 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #82649 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #82640 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #82604 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #82594 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | C4×#938 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #82378 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #59126 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C23×C3×D4 | C22 | 1 | 4 | yes | complemented |
| 14 | C23×#67 | C22 | 1 | 4 | yes | complemented |
| 15 | C22×D3×D4 | C22 | 1 | 4 | yes | complemented |
| 16 | C22×D3×D4 | C22 | 1 | 2 | yes | complemented |
| 17 | C3×D42 | — | 2 | 8 | no | — |
| 18 | C2×C3×#127 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×#121 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×C3×#120 | C22 | 1 | 4 | yes | complemented |
| 21 | C2×C3×C4×D4 | C22 | 1 | 2 | yes | complemented |
| 22 | C2×#953 | C22 | 1 | 8 | yes | complemented |
| 23 | C2×#953 | C22 | 1 | 4 | yes | complemented |
| 24 | C2×#940 | C22 | 1 | 4 | yes | complemented |
| 25 | C2×#940 | C22 | 1 | 4 | yes | complemented |
| 26 | C2×#940 | C22 | 1 | 2 | yes | complemented |
| 27 | C2×#939 | C22 | 1 | 8 | yes | complemented |
| 28 | D4×#67 | — | 2 | 32 | no | — |
| 29 | C2×#938 | C22 | 1 | 2 | yes | — |
| 30 | C2×#937 | C22 | 1 | 8 | yes | complemented |
| 31 | C2×D4×Dic3 | C22 | 1 | 4 | yes | complemented |
| 32 | C2×#930 | C22 | 1 | 4 | yes | complemented |
| 33 | C2×#930 | C22 | 1 | 4 | yes | complemented |
| 34 | C2×C4×#67 | C22 | 1 | 4 | yes | complemented |
| 35 | D3×#127 | — | 2 | 8 | no | — |
| 36 | C2×#874 | C22 | 1 | 1 | yes | characteristic, complemented |
| 37 | C2×C42×D3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 38 | #5223 | C22 | 1 | 8 | yes | complemented |
| 39 | #5223 | C22 | 1 | 8 | yes | complemented |
| 40 | #5221 | C22 | 1 | 2 | yes | complemented |
| 41 | #5221 | C22 | 1 | 4 | yes | complemented |
| 42 | #5220 | C22 | 1 | 4 | yes | complemented |
| 43 | C4×#217 | C22 | 1 | 4 | yes | complemented |
| 44 | #4988 | C22 | 1 | 4 | yes | complemented |
| 45 | #4987 | C22 | 1 | 2 | yes | complemented |
| 46 | #4972 | C22 | 1 | 2 | yes | complemented |
| 47 | #4960 | C22 | 1 | 8 | yes | complemented |
| 48 | #4959 | C22 | 1 | 8 | yes | complemented |
| 49 | #4958 | C22 | 1 | 4 | yes | complemented |
| 50 | #4956 | C22 | 1 | 4 | yes | complemented |
| 51 | #4955 | C22 | 1 | 4 | yes | complemented |
| 52 | #4944 | C22 | 1 | 4 | yes | complemented |
| 53 | C4×C4⋊Dic3 | C22 | 1 | 2 | yes | complemented |
| 54 | C2×D42 | — | 3 | 4 | no | — |
| 55 | C2×D42 | D3 | 1 | 1 | yes | characteristic, complemented |
| 56 | C22×#127 | — | 3 | 1 | no | — |
| 57 | #2669 | — | 3 | 4 | no | — |
| 58 | #2541 | — | 3 | 2 | no | — |
| 59 | #2433 | — | 3 | 4 | no | — |
| 60 | #2428 | — | 3 | 4 | no | — |
| 61 | #2408 | — | 3 | 4 | no | — |
| 62 | #2403 | — | 3 | 4 | no | — |
| 63 | #2366 | — | 3 | 1 | no | — |
| 64 | C4×#121 | — | 3 | 2 | no | — |
| 65 | C25×C3 | C23 | 1 | 4 | yes | — |
| 66 | C24×D3 | C23 | 1 | 4 | yes | — |
| 67 | C22×C3×D4 | — | 2 | 16 | no | — |
| 68 | C22×C3×D4 | — | 2 | 16 | no | — |
| 69 | C22×C3×D4 | — | 4 | 8 | no | — |
| 70 | C22×C3×D4 | C23 | 1 | 8 | yes | — |
| 71 | C22×C3×D4 | C23 | 1 | 4 | yes | — |
| 72 | C22×C3×D4 | C23 | 1 | 2 | yes | — |
| 73 | C22×C3×D4 | C23 | 1 | 2 | yes | — |
| 74 | C23×C3×C4 | C23 | 1 | 4 | yes | — |
| 75 | C22×#67 | — | 2 | 32 | no | — |
| 76 | C22×#67 | — | 4 | 8 | no | — |
| 77 | C22×#67 | C23 | 1 | 4 | yes | — |
| 78 | C22×#67 | C23 | 1 | 8 | yes | — |
| 79 | C22×#67 | C23 | 1 | 8 | yes | — |
| 80 | C22×#67 | C23 | 1 | 4 | yes | — |
| 81 | C22×#67 | C23 | 1 | 8 | yes | — |
| 82 | C23×Dic3 | C23 | 1 | 4 | yes | — |
| 83 | C2×D3×D4 | — | 2 | 16 | no | — |
| 84 | C2×D3×D4 | — | 2 | 8 | no | — |
| 85 | C2×D3×D4 | — | 2 | 16 | no | — |
| 86 | C2×D3×D4 | — | 4 | 8 | no | — |
| 87 | C2×D3×D4 | — | 4 | 4 | no | — |
| 88 | C2×D3×D4 | — | 4 | 4 | no | — |
| 89 | C22×D12 | C23 | 1 | 4 | yes | — |
| 90 | C22×D12 | C23 | 1 | 2 | yes | — |
| 91 | C22×C4×D3 | C23 | 1 | 1 | yes | characteristic |
| 92 | C22×C4×D3 | C23 | 1 | 2 | yes | — |
| 93 | C3×#127 | D4 | 1 | 8 | yes | complemented |
| 94 | C3×#121 | — | 2 | 16 | no | — |
| 95 | C3×#120 | — | 2 | 16 | no | — |
| 96 | C3×C4×D4 | — | 2 | 8 | no | — |
| 97 | C2×C3×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 98 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 99 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 100 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 101 | #953 | — | 2 | 32 | no | — |
| 102 | #953 | — | 4 | 8 | no | — |
| 103 | C2×#217 | C23 | 1 | 4 | yes | — |
| 104 | C2×#217 | C23 | 1 | 4 | yes | — |
| 105 | C2×#217 | C23 | 1 | 8 | yes | — |
| 106 | C2×#217 | C23 | 1 | 4 | yes | — |
| 107 | C2×#217 | C23 | 1 | 4 | yes | — |
| 108 | #940 | — | 2 | 16 | no | — |
| 109 | #940 | — | 2 | 8 | no | — |
| 110 | #940 | — | 2 | 16 | no | — |
| 111 | #939 | — | 2 | 32 | no | — |
| 112 | #938 | D4 | 1 | 16 | yes | complemented |
| 113 | #937 | — | 2 | 32 | no | — |
| 114 | D4×Dic3 | — | 2 | 16 | no | — |
| 115 | #930 | — | 2 | 16 | no | — |
| 116 | #930 | — | 4 | 8 | no | — |
| 117 | C4×#67 | — | 2 | 16 | no | — |
| 118 | C2×(C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 119 | C2×(C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 120 | C2×(C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 121 | C2×C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 122 | C2×C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 123 | C2×Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 124 | C2×C4×Dic3 | C23 | 1 | 4 | yes | — |
| 125 | C2×C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×Dic3 | C23 | 1 | 2 | yes | — |
| 127 | #874 | — | 2 | 4 | no | — |
| 128 | C42×D3 | — | 2 | 4 | no | — |
| 129 | C3⋊#95 | C23 | 1 | 4 | yes | — |
| 130 | C3⋊#95 | C23 | 1 | 4 | yes | — |
| 131 | C3⋊#95 | C23 | 1 | 4 | yes | — |
| 132 | C23×D4 | — | 3 | 4 | no | — |
| 133 | C23×D4 | — | 3 | 4 | no | — |
| 134 | C23×D4 | — | 3 | 2 | no | — |
| 135 | C23×D4 | C2×D3 | 1 | 4 | yes | complemented |
| 136 | D42 | — | 2 | 8 | no | — |
| 137 | D42 | — | 6 | 32 | no | — |
| 138 | C2×#127 | — | 3 | 2 | no | — |
| 139 | C2×#127 | — | 3 | 4 | no | — |
| 140 | C2×#127 | — | 3 | 4 | no | — |
| 141 | C2×#127 | — | 3 | 1 | no | — |
| 142 | C2×#127 | — | 6 | 8 | no | — |
| 143 | C2×#127 | C2×D3 | 1 | 1 | yes | characteristic |
| 144 | C2×#121 | — | 3 | 2 | no | — |
| 145 | C2×#121 | — | 3 | 8 | no | — |
| 146 | C2×#121 | — | 3 | 4 | no | — |
| 147 | C2×#121 | — | 3 | 4 | no | — |
| 148 | C2×#121 | C2×D3 | 1 | 4 | yes | complemented |
| 149 | C2×#120 | — | 3 | 8 | no | — |
| 150 | C2×#120 | — | 3 | 8 | no | — |
| 151 | C2×#120 | — | 3 | 4 | no | — |
| 152 | C2×#120 | C2×D3 | 1 | 4 | yes | complemented |
| 153 | C2×C4×D4 | — | 3 | 4 | no | — |
| 154 | C2×C4×D4 | — | 3 | 4 | no | — |
| 155 | C2×C4×D4 | C2×D3 | 1 | 2 | yes | complemented |
| 156 | C22×C42 | — | 3 | 1 | no | — |
| 157 | #394 | — | 3 | 8 | no | — |
| 158 | #392 | — | 3 | 8 | no | — |
| 159 | #392 | — | 3 | 8 | no | — |
| 160 | #392 | — | 3 | 8 | no | — |
| 161 | #390 | — | 3 | 4 | no | — |
| 162 | #390 | — | 3 | 2 | no | — |
| 163 | #390 | — | 3 | 4 | no | — |
| 164 | #389 | — | 3 | 2 | no | — |
| 165 | #388 | — | 3 | 4 | no | — |
| 166 | #386 | — | 3 | 4 | no | — |
| 167 | #386 | — | 3 | 4 | no | — |
| 168 | #386 | — | 3 | 4 | no | — |
| 169 | #380 | — | 3 | 2 | no | — |
| 170 | #379 | — | 3 | 4 | no | — |
| 171 | C4×C4⋊C4 | — | 3 | 2 | no | — |
| 172 | C4×C22⋊C4 | — | 3 | 4 | no | — |
| 173 | C24×C3 | — | 2 | 16 | no | — |
| 174 | C24×C3 | — | 2 | 8 | no | — |
| 175 | C24×C3 | — | 4 | 4 | no | — |
| 176 | C24×C3 | — | 4 | 8 | no | — |
| 177 | C24×C3 | — | 4 | 4 | no | — |
| 178 | C24×C3 | C24 | 1 | 4 | yes | — |
| 179 | C24×C3 | C24 | 1 | 4 | yes | — |
| 180 | C23×D3 | — | 2 | 8 | no | — |
| 181 | C23×D3 | — | 2 | 16 | no | — |
| 182 | C23×D3 | — | 2 | 8 | no | — |
| 183 | C23×D3 | — | 4 | 8 | no | — |
| 184 | C23×D3 | — | 4 | 4 | no | — |
| 185 | C23×D3 | C24 | 1 | 4 | yes | — |
| 186 | C23×D3 | C24 | 1 | 1 | yes | characteristic |
| 187 | C2×C3×D4 | — | 2 | 16 | no | — |
| 188 | C2×C3×D4 | — | 2 | 16 | no | — |
| 189 | C2×C3×D4 | — | 2 | 8 | no | — |
| 190 | C2×C3×D4 | — | 2 | 4 | no | — |
| 191 | C2×C3×D4 | — | 2 | 8 | no | — |
| 192 | C2×C3×D4 | — | 2 | 4 | no | — |
| 193 | C2×C3×D4 | — | 4 | 2 | no | — |
| 194 | C2×C3×D4 | — | 4 | 2 | no | — |
| 195 | C2×C3×D4 | — | 4 | 8 | no | — |
| 196 | C2×C3×D4 | — | 4 | 64 | no | — |
| 197 | C2×C3×D4 | C2×D4 | 1 | 8 | yes | complemented |
| 198 | C2×C3×D4 | C2×D4 | 1 | 16 | yes | complemented |
| 199 | C2×C3×D4 | C2×D4 | 1 | 8 | yes | complemented |
| 200 | C22×C3×C4 | — | 2 | 8 | no | — |
| 201 | C22×C3×C4 | — | 2 | 8 | no | — |
| 202 | C22×C3×C4 | — | 2 | 8 | no | — |
| 203 | C22×C3×C4 | C24 | 1 | 4 | yes | — |
| 204 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 205 | C22×C3×C4 | C24 | 1 | 2 | yes | — |
| 206 | C2×#67 | — | 2 | 8 | no | — |
| 207 | C2×#67 | — | 2 | 8 | no | — |
| 208 | C2×#67 | — | 2 | 16 | no | — |
| 209 | C2×#67 | — | 2 | 8 | no | — |
| 210 | C2×#67 | — | 2 | 16 | no | — |
| 211 | C2×#67 | — | 2 | 16 | no | — |
| 212 | C2×#67 | — | 2 | 16 | no | — |
| 213 | C2×#67 | — | 2 | 16 | no | — |
| 214 | C2×#67 | — | 2 | 8 | no | — |
| 215 | C2×#67 | — | 4 | 8 | no | — |
| 216 | C2×#67 | — | 4 | 4 | no | — |
| 217 | C2×#67 | — | 4 | 8 | no | — |
| 218 | C2×#67 | — | 4 | 8 | no | — |
| 219 | C2×#67 | — | 4 | 8 | no | — |
| 220 | C2×#67 | — | 4 | 64 | no | — |
| 221 | C2×#67 | C2×D4 | 1 | 16 | yes | complemented |
| 222 | C22×Dic3 | — | 2 | 16 | no | — |
| 223 | C22×Dic3 | — | 4 | 4 | no | — |
| 224 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 225 | C22×Dic3 | C24 | 1 | 2 | yes | — |
| 226 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 227 | C22×Dic3 | C24 | 1 | 1 | yes | characteristic |
| 228 | D3×D4 | — | 4 | 64 | no | — |
| 229 | D3×D4 | — | 4 | 32 | no | — |
| 230 | C2×D12 | — | 2 | 4 | no | — |
| 231 | C2×D12 | — | 2 | 8 | no | — |
| 232 | C2×D12 | — | 2 | 8 | no | — |
| 233 | C2×D12 | — | 2 | 8 | no | — |
| 234 | C2×D12 | — | 4 | 2 | no | — |
| 235 | C2×D12 | — | 4 | 2 | no | — |
| 236 | C2×C4×D3 | — | 2 | 2 | no | — |
| 237 | C2×C4×D3 | — | 2 | 2 | no | — |
| 238 | C2×C4×D3 | — | 2 | 4 | no | — |
| 239 | C2×C4×D3 | — | 2 | 4 | no | — |
| 240 | C2×C4×D3 | — | 2 | 2 | no | — |
| 241 | C2×C4×D3 | C2×D4 | 1 | 8 | yes | complemented |
| 242 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 243 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 244 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 245 | C3×C42 | C2×D4 | 1 | 4 | yes | complemented |
| 246 | #217 | — | 2 | 8 | no | — |
| 247 | #217 | — | 2 | 16 | no | — |
| 248 | #217 | — | 2 | 8 | no | — |
| 249 | #217 | — | 4 | 4 | no | — |
| 250 | #217 | C2×D4 | 1 | 16 | yes | complemented |
| 251 | (C2×D3)⋊C4 | — | 2 | 8 | no | — |
| 252 | (C2×D3)⋊C4 | — | 2 | 16 | no | — |
| 253 | (C2×D3)⋊C4 | — | 4 | 8 | no | — |
| 254 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 255 | C4⋊Dic3 | C2×D4 | 1 | 8 | yes | complemented |
| 256 | Dic3⋊C4 | — | 2 | 8 | no | — |
| 257 | C4×Dic3 | — | 2 | 2 | no | — |
| 258 | C4×Dic3 | — | 2 | 4 | no | — |
| 259 | C4×Dic3 | — | 2 | 8 | no | — |
| 260 | C25 | — | 3 | 4 | no | — |
| 261 | C25 | C22×D3 | 1 | 4 | yes | — |
| 262 | C22×D4 | — | 2 | 16 | no | — |
| 263 | C22×D4 | — | 2 | 16 | no | — |
| 264 | C22×D4 | — | 3 | 4 | no | — |
| 265 | C22×D4 | — | 3 | 8 | no | — |
| 266 | C22×D4 | — | 3 | 8 | no | — |
| 267 | C22×D4 | — | 3 | 4 | no | — |
| 268 | C22×D4 | — | 3 | 8 | no | — |
| 269 | C22×D4 | — | 3 | 4 | no | — |
| 270 | C22×D4 | — | 3 | 2 | no | — |
| 271 | C22×D4 | — | 4 | 8 | no | — |
| 272 | C22×D4 | — | 6 | 16 | no | — |
| 273 | C22×D4 | — | 6 | 32 | no | — |
| 274 | C22×D4 | — | 6 | 16 | no | — |
| 275 | C22×D4 | — | 6 | 8 | no | — |
| 276 | C22×D4 | — | 12 | 8 | no | — |
| 277 | C22×D4 | — | 12 | 4 | no | — |
| 278 | C22×D4 | — | 12 | 4 | no | — |
| 279 | C22×D4 | — | 12 | 8 | no | — |
| 280 | C22×D4 | C22×D3 | 1 | 8 | yes | — |
| 281 | C22×D4 | C22×D3 | 1 | 2 | yes | — |
| 282 | C22×D4 | C22×D3 | 1 | 4 | yes | — |
| 283 | C22×D4 | C22×D3 | 1 | 2 | yes | — |
| 284 | C23×C4 | — | 3 | 2 | no | — |
| 285 | C23×C4 | — | 3 | 4 | no | — |
| 286 | C23×C4 | — | 3 | 1 | no | — |
| 287 | C23×C4 | C22×D3 | 1 | 4 | yes | — |
| 288 | #127 | — | 6 | 16 | no | — |
| 289 | #127 | — | 6 | 16 | no | — |
| 290 | #127 | — | 6 | 8 | no | — |
| 291 | #127 | — | 6 | 4 | no | — |
| 292 | #127 | #67 | 1 | 8 | yes | complemented |
| 293 | #121 | — | 2 | 16 | no | — |
| 294 | #121 | — | 3 | 16 | no | — |
| 295 | #121 | — | 6 | 32 | no | — |
| 296 | #121 | — | 6 | 16 | no | — |
| 297 | #121 | — | 12 | 8 | no | — |
| 298 | #120 | — | 2 | 16 | no | — |
| 299 | #120 | — | 6 | 32 | no | — |
| 300 | #120 | — | 6 | 32 | no | — |
| 301 | #120 | — | 12 | 8 | no | — |
| 302 | C4×D4 | — | 2 | 8 | no | — |
| 303 | C4×D4 | — | 6 | 16 | no | — |
| 304 | C4×D4 | — | 6 | 16 | no | — |
| 305 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 306 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 307 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 308 | C2×C4⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 309 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 310 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 311 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 312 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 313 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 314 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 315 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 316 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 317 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 318 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 319 | C2×C42 | — | 3 | 4 | no | — |
| 320 | C2×C42 | — | 3 | 2 | no | — |
| 321 | C2×C42 | — | 3 | 1 | no | — |
| 322 | C2×C42 | — | 6 | 4 | no | — |
| 323 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 324 | #95 | — | 3 | 4 | no | — |
| 325 | #95 | — | 3 | 4 | no | — |
| 326 | #95 | — | 3 | 4 | no | — |
| 327 | C23×C3 | — | 2 | 8 | no | — |
| 328 | C23×C3 | — | 2 | 4 | no | — |
| 329 | C23×C3 | — | 2 | 4 | no | — |
| 330 | C23×C3 | — | 2 | 4 | no | — |
| 331 | C23×C3 | — | 2 | 4 | no | — |
| 332 | C23×C3 | — | 2 | 4 | no | — |
| 333 | C23×C3 | — | 2 | 4 | no | — |
| 334 | C23×C3 | — | 2 | 4 | no | — |
| 335 | C23×C3 | — | 2 | 8 | no | — |
| 336 | C23×C3 | — | 2 | 4 | no | — |
| 337 | C23×C3 | — | 4 | 16 | no | — |
| 338 | C23×C3 | — | 4 | 32 | no | — |
| 339 | C23×C3 | — | 8 | 8 | no | — |
| 340 | C23×C3 | — | 8 | 16 | no | — |
| 341 | C23×C3 | — | 8 | 8 | no | — |
| 342 | C23×C3 | C22×D4 | 1 | 8 | yes | — |
| 343 | C23×C3 | C22×D4 | 1 | 8 | yes | — |
| 344 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 345 | C22×D3 | — | 2 | 1 | no | — |
| 346 | C22×D3 | — | 2 | 2 | no | — |
| 347 | C22×D3 | — | 2 | 1 | no | — |
| 348 | C22×D3 | — | 2 | 8 | no | — |
| 349 | C22×D3 | — | 2 | 4 | no | — |
| 350 | C22×D3 | — | 2 | 8 | no | — |
| 351 | C22×D3 | — | 2 | 4 | no | — |
| 352 | C22×D3 | — | 2 | 4 | no | — |
| 353 | C22×D3 | — | 4 | 32 | no | — |
| 354 | C22×D3 | — | 4 | 16 | no | — |
| 355 | C22×D3 | — | 4 | 16 | no | — |
| 356 | C22×D3 | — | 8 | 16 | no | — |
| 357 | C22×D3 | — | 8 | 8 | no | — |
| 358 | C22×D3 | C22×D4 | 1 | 4 | yes | — |
| 359 | C22×D3 | H(5,𝔽2) | 1 | 2 | yes | — |
| 360 | C3×D4 | — | 2 | 32 | no | — |
| 361 | C3×D4 | — | 2 | 16 | no | — |
| 362 | C3×D4 | — | 4 | 8 | no | — |
| 363 | C3×D4 | — | 4 | 32 | no | — |
| 364 | C2×C3×C4 | — | 2 | 1 | no | — |
| 365 | C2×C3×C4 | — | 2 | 1 | no | — |
| 366 | C2×C3×C4 | — | 2 | 4 | no | — |
| 367 | C2×C3×C4 | — | 2 | 4 | no | — |
| 368 | C2×C3×C4 | — | 2 | 4 | no | — |
| 369 | C2×C3×C4 | — | 4 | 16 | no | — |
| 370 | C2×C3×C4 | C22×D4 | 1 | 4 | yes | — |
| 371 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 372 | C2×C3×C4 | C22×D4 | 1 | 4 | yes | — |
| 373 | C2×C3×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 374 | #67 | — | 2 | 32 | no | — |
| 375 | #67 | — | 4 | 32 | no | — |
| 376 | #67 | — | 4 | 16 | no | — |
| 377 | #67 | — | 4 | 32 | no | — |
| 378 | #67 | — | 8 | 16 | no | — |
| 379 | C2×Dic3 | — | 2 | 2 | no | — |
| 380 | C2×Dic3 | — | 2 | 1 | no | — |
| 381 | C2×Dic3 | — | 2 | 4 | no | — |
| 382 | C2×Dic3 | — | 2 | 2 | no | — |
| 383 | C2×Dic3 | — | 2 | 1 | no | — |
| 384 | C2×Dic3 | — | 2 | 4 | no | — |
| 385 | C2×Dic3 | — | 2 | 4 | no | — |
| 386 | C2×Dic3 | — | 2 | 4 | no | — |
| 387 | C2×Dic3 | — | 2 | 4 | no | — |
| 388 | C2×Dic3 | — | 4 | 16 | no | — |
| 389 | C2×Dic3 | C22×D4 | 1 | 4 | yes | — |
| 390 | C2×Dic3 | C22×D4 | 1 | 8 | yes | — |
| 391 | C2×Dic3 | H(5,𝔽2) | 1 | 2 | yes | — |
| 392 | D12 | — | 4 | 8 | no | — |
| 393 | D12 | — | 4 | 16 | no | — |
| 394 | C4×D3 | — | 2 | 8 | no | — |
| 395 | C4×D3 | — | 2 | 16 | no | — |
| 396 | C24 | — | 2 | 16 | no | — |
| 397 | C24 | — | 2 | 8 | no | — |
| 398 | C24 | — | 3 | 1 | no | — |
| 399 | C24 | — | 3 | 4 | no | — |
| 400 | C24 | — | 4 | 8 | no | — |
| 401 | C24 | — | 4 | 4 | no | — |
| 402 | C24 | — | 4 | 4 | no | — |
| 403 | C24 | — | 6 | 16 | no | — |
| 404 | C24 | — | 6 | 8 | no | — |
| 405 | C24 | — | 6 | 8 | no | — |
| 406 | C24 | — | 12 | 8 | no | — |
| 407 | C24 | — | 12 | 4 | no | — |
| 408 | C24 | C23×D3 | 1 | 4 | yes | — |
| 409 | C24 | C23×D3 | 1 | 4 | yes | — |
| 410 | C2×D4 | — | 2 | 16 | no | — |
| 411 | C2×D4 | — | 2 | 16 | no | — |
| 412 | C2×D4 | — | 2 | 8 | no | — |
| 413 | C2×D4 | — | 2 | 4 | no | — |
| 414 | C2×D4 | — | 2 | 8 | no | — |
| 415 | C2×D4 | — | 2 | 4 | no | — |
| 416 | C2×D4 | — | 3 | 16 | no | — |
| 417 | C2×D4 | — | 4 | 8 | no | — |
| 418 | C2×D4 | — | 4 | 2 | no | — |
| 419 | C2×D4 | — | 4 | 2 | no | — |
| 420 | C2×D4 | — | 4 | 64 | no | — |
| 421 | C2×D4 | — | 6 | 16 | no | — |
| 422 | C2×D4 | — | 6 | 16 | no | — |
| 423 | C2×D4 | — | 6 | 8 | no | — |
| 424 | C2×D4 | — | 6 | 8 | no | — |
| 425 | C2×D4 | — | 6 | 8 | no | — |
| 426 | C2×D4 | — | 6 | 16 | no | — |
| 427 | C2×D4 | — | 6 | 16 | no | — |
| 428 | C2×D4 | — | 6 | 8 | no | — |
| 429 | C2×D4 | — | 6 | 16 | no | — |
| 430 | C2×D4 | — | 6 | 8 | no | — |
| 431 | C2×D4 | — | 6 | 8 | no | — |
| 432 | C2×D4 | — | 6 | 4 | no | — |
| 433 | C2×D4 | — | 6 | 8 | no | — |
| 434 | C2×D4 | — | 12 | 2 | no | — |
| 435 | C2×D4 | — | 12 | 2 | no | — |
| 436 | C2×D4 | — | 12 | 64 | no | — |
| 437 | C2×D4 | — | 12 | 32 | no | — |
| 438 | C2×D4 | — | 12 | 8 | no | — |
| 439 | C2×D4 | — | 12 | 4 | no | — |
| 440 | C2×D4 | — | 12 | 8 | no | — |
| 441 | C2×D4 | — | 12 | 8 | no | — |
| 442 | C2×D4 | — | 12 | 8 | no | — |
| 443 | C2×D4 | — | 12 | 64 | no | — |
| 444 | C2×D4 | D3×D4 | 1 | 8 | yes | complemented |
| 445 | C2×D4 | C2×#67 | 1 | 16 | yes | complemented |
| 446 | C2×D4 | C2×#67 | 1 | 8 | yes | complemented |
| 447 | C22×C4 | — | 2 | 8 | no | — |
| 448 | C22×C4 | — | 2 | 8 | no | — |
| 449 | C22×C4 | — | 2 | 8 | no | — |
| 450 | C22×C4 | — | 3 | 1 | no | — |
| 451 | C22×C4 | — | 3 | 2 | no | — |
| 452 | C22×C4 | — | 3 | 4 | no | — |
| 453 | C22×C4 | — | 3 | 8 | no | — |
| 454 | C22×C4 | — | 3 | 4 | no | — |
| 455 | C22×C4 | — | 6 | 2 | no | — |
| 456 | C22×C4 | — | 6 | 2 | no | — |
| 457 | C22×C4 | — | 6 | 16 | no | — |
| 458 | C22×C4 | — | 6 | 4 | no | — |
| 459 | C22×C4 | — | 6 | 2 | no | — |
| 460 | C22×C4 | — | 6 | 4 | no | — |
| 461 | C22×C4 | — | 12 | 4 | no | — |
| 462 | C22×C4 | C23×D3 | 1 | 4 | yes | — |
| 463 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 464 | C22×C4 | C23×D3 | 1 | 2 | yes | — |
| 465 | C4⋊C4 | — | 2 | 4 | no | — |
| 466 | C4⋊C4 | — | 3 | 8 | no | — |
| 467 | C4⋊C4 | — | 6 | 8 | no | — |
| 468 | C4⋊C4 | — | 6 | 4 | no | — |
| 469 | C22⋊C4 | — | 2 | 8 | no | — |
| 470 | C22⋊C4 | — | 2 | 8 | no | — |
| 471 | C22⋊C4 | — | 3 | 16 | no | — |
| 472 | C22⋊C4 | — | 6 | 16 | no | — |
| 473 | C22⋊C4 | — | 6 | 8 | no | — |
| 474 | C22⋊C4 | — | 6 | 8 | no | — |
| 475 | C22⋊C4 | — | 6 | 16 | no | — |
| 476 | C22⋊C4 | — | 6 | 8 | no | — |
| 477 | C22⋊C4 | — | 12 | 8 | no | — |
| 478 | C22⋊C4 | — | 12 | 4 | no | — |
| 479 | C42 | — | 6 | 8 | no | — |
| 480 | C42 | — | 6 | 2 | no | — |
| 481 | C42 | — | 6 | 4 | no | — |
| 482 | C42 | C2×#67 | 1 | 4 | yes | complemented |
| 483 | C22×C3 | — | 2 | 8 | no | — |
| 484 | C22×C3 | — | 2 | 16 | no | — |
| 485 | C22×C3 | — | 2 | 8 | no | — |
| 486 | C22×C3 | — | 2 | 8 | no | — |
| 487 | C22×C3 | — | 2 | 8 | no | — |
| 488 | C22×C3 | — | 4 | 4 | no | — |
| 489 | C22×C3 | — | 4 | 4 | no | — |
| 490 | C22×C3 | — | 4 | 4 | no | — |
| 491 | C22×C3 | — | 4 | 8 | no | — |
| 492 | C22×C3 | — | 4 | 4 | no | — |
| 493 | C22×C3 | — | 4 | 4 | no | — |
| 494 | C22×C3 | — | 4 | 4 | no | — |
| 495 | C22×C3 | — | 8 | 32 | no | — |
| 496 | C22×C3 | C23×D4 | 1 | 2 | yes | — |
| 497 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 498 | C22×C3 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 499 | C22×C3 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 500 | C22×C3 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 501 | C2×D3 | — | 2 | 4 | no | — |
| 502 | C2×D3 | — | 2 | 2 | no | — |
| 503 | C2×D3 | — | 2 | 4 | no | — |
| 504 | C2×D3 | — | 2 | 2 | no | — |
| 505 | C2×D3 | — | 4 | 8 | no | — |
| 506 | C2×D3 | — | 4 | 4 | no | — |
| 507 | C2×D3 | — | 4 | 4 | no | — |
| 508 | C2×D3 | — | 4 | 8 | no | — |
| 509 | C2×D3 | — | 4 | 4 | no | — |
| 510 | C2×D3 | — | 8 | 32 | no | — |
| 511 | C2×D3 | D42 | 1 | 4 | yes | complemented |
| 512 | C3×C4 | — | 2 | 2 | no | — |
| 513 | C3×C4 | — | 4 | 4 | no | — |
| 514 | C3×C4 | D42 | 1 | 8 | yes | complemented |
| 515 | Dic3 | — | 2 | 4 | no | — |
| 516 | Dic3 | — | 2 | 8 | no | — |
| 517 | Dic3 | — | 2 | 2 | no | — |
| 518 | Dic3 | — | 4 | 4 | no | — |
| 519 | C23 | — | 2 | 4 | no | — |
| 520 | C23 | — | 2 | 8 | no | — |
| 521 | C23 | — | 2 | 4 | no | — |
| 522 | C23 | — | 2 | 4 | no | — |
| 523 | C23 | — | 2 | 8 | no | — |
| 524 | C23 | — | 2 | 4 | no | — |
| 525 | C23 | — | 2 | 4 | no | — |
| 526 | C23 | — | 2 | 4 | no | — |
| 527 | C23 | — | 2 | 4 | no | — |
| 528 | C23 | — | 2 | 4 | no | — |
| 529 | C23 | — | 3 | 4 | no | — |
| 530 | C23 | — | 3 | 2 | no | — |
| 531 | C23 | — | 4 | 32 | no | — |
| 532 | C23 | — | 4 | 16 | no | — |
| 533 | C23 | — | 6 | 2 | no | — |
| 534 | C23 | — | 6 | 1 | no | — |
| 535 | C23 | — | 6 | 1 | no | — |
| 536 | C23 | — | 6 | 8 | no | — |
| 537 | C23 | — | 6 | 8 | no | — |
| 538 | C23 | — | 6 | 4 | no | — |
| 539 | C23 | — | 6 | 4 | no | — |
| 540 | C23 | — | 6 | 4 | no | — |
| 541 | C23 | — | 8 | 16 | no | — |
| 542 | C23 | — | 8 | 8 | no | — |
| 543 | C23 | — | 8 | 8 | no | — |
| 544 | C23 | — | 12 | 32 | no | — |
| 545 | C23 | — | 12 | 16 | no | — |
| 546 | C23 | — | 12 | 16 | no | — |
| 547 | C23 | — | 24 | 16 | no | — |
| 548 | C23 | — | 24 | 8 | no | — |
| 549 | C23 | C2×D3×D4 | 1 | 8 | yes | — |
| 550 | C23 | C22×#67 | 1 | 8 | yes | — |
| 551 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 552 | D4 | — | 2 | 16 | no | — |
| 553 | D4 | — | 2 | 32 | no | — |
| 554 | D4 | — | 4 | 8 | no | — |
| 555 | D4 | — | 4 | 32 | no | — |
| 556 | D4 | — | 6 | 32 | no | — |
| 557 | D4 | — | 12 | 8 | no | — |
| 558 | D4 | — | 12 | 32 | no | — |
| 559 | D4 | — | 12 | 32 | no | — |
| 560 | D4 | — | 12 | 16 | no | — |
| 561 | D4 | — | 12 | 16 | no | — |
| 562 | D4 | — | 24 | 16 | no | — |
| 563 | C2×C4 | — | 2 | 1 | no | — |
| 564 | C2×C4 | — | 2 | 1 | no | — |
| 565 | C2×C4 | — | 2 | 4 | no | — |
| 566 | C2×C4 | — | 2 | 4 | no | — |
| 567 | C2×C4 | — | 2 | 4 | no | — |
| 568 | C2×C4 | — | 3 | 4 | no | — |
| 569 | C2×C4 | — | 3 | 8 | no | — |
| 570 | C2×C4 | — | 3 | 2 | no | — |
| 571 | C2×C4 | — | 4 | 16 | no | — |
| 572 | C2×C4 | — | 6 | 8 | no | — |
| 573 | C2×C4 | — | 6 | 4 | no | — |
| 574 | C2×C4 | — | 6 | 2 | no | — |
| 575 | C2×C4 | — | 6 | 1 | no | — |
| 576 | C2×C4 | — | 6 | 4 | no | — |
| 577 | C2×C4 | — | 6 | 2 | no | — |
| 578 | C2×C4 | — | 6 | 1 | no | — |
| 579 | C2×C4 | — | 6 | 16 | no | — |
| 580 | C2×C4 | — | 6 | 4 | no | — |
| 581 | C2×C4 | — | 6 | 4 | no | — |
| 582 | C2×C4 | — | 6 | 4 | no | — |
| 583 | C2×C4 | — | 12 | 16 | no | — |
| 584 | C2×C4 | C2×D3×D4 | 1 | 4 | yes | — |
| 585 | C2×C4 | #1004 | 1 | 4 | yes | — |
| 586 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 587 | C2×C4 | C22×#67 | 1 | 4 | yes | — |
| 588 | C2×C3 | — | 4 | 8 | no | — |
| 589 | C2×C3 | — | 4 | 8 | no | — |
| 590 | C2×C3 | C2×D42 | 1 | 1 | yes | characteristic |
| 591 | C2×C3 | C2×D42 | 1 | 2 | yes | — |
| 592 | C2×C3 | #3484 | 1 | 1 | yes | characteristic |
| 593 | C2×C3 | #3498 | 1 | 2 | yes | — |
| 594 | C2×C3 | #3543 | 1 | 1 | yes | characteristic |
| 595 | D3 | — | 2 | 4 | no | — |
| 596 | D3 | — | 8 | 4 | no | — |
| 597 | C22 | — | 2 | 8 | no | — |
| 598 | C22 | — | 2 | 8 | no | — |
| 599 | C22 | — | 2 | 8 | no | — |
| 600 | C22 | — | 2 | 16 | no | — |
| 601 | C22 | — | 2 | 8 | no | — |
| 602 | C22 | — | 3 | 4 | no | — |
| 603 | C22 | — | 4 | 4 | no | — |
| 604 | C22 | — | 4 | 8 | no | — |
| 605 | C22 | — | 4 | 4 | no | — |
| 606 | C22 | — | 4 | 4 | no | — |
| 607 | C22 | — | 4 | 4 | no | — |
| 608 | C22 | — | 4 | 4 | no | — |
| 609 | C22 | — | 4 | 4 | no | — |
| 610 | C22 | — | 6 | 4 | no | — |
| 611 | C22 | — | 6 | 2 | no | — |
| 612 | C22 | — | 6 | 4 | no | — |
| 613 | C22 | — | 6 | 2 | no | — |
| 614 | C22 | — | 8 | 32 | no | — |
| 615 | C22 | — | 12 | 8 | no | — |
| 616 | C22 | — | 12 | 4 | no | — |
| 617 | C22 | — | 12 | 4 | no | — |
| 618 | C22 | — | 12 | 8 | no | — |
| 619 | C22 | — | 12 | 4 | no | — |
| 620 | C22 | — | 24 | 32 | no | — |
| 621 | C22 | C22×D3×D4 | 1 | 2 | yes | — |
| 622 | C22 | C2×#1004 | 1 | 2 | yes | — |
| 623 | C22 | D3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 624 | C22 | D3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 625 | C22 | C23×#67 | 1 | 1 | yes | characteristic |
| 626 | C4 | — | 2 | 2 | no | — |
| 627 | C4 | — | 4 | 4 | no | — |
| 628 | C4 | — | 6 | 8 | no | — |
| 629 | C4 | — | 6 | 4 | no | — |
| 630 | C4 | — | 6 | 2 | no | — |
| 631 | C4 | — | 12 | 4 | no | — |
| 632 | C4 | D4×#67 | 1 | 8 | yes | complemented |
| 633 | C3 | #59126 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 634 | C2 | — | 4 | 8 | no | — |
| 635 | C2 | — | 4 | 8 | no | — |
| 636 | C2 | — | 6 | 4 | no | — |
| 637 | C2 | — | 24 | 4 | no | — |
| 638 | C2 | #86608 | 1 | 2 | yes | — |
| 639 | C2 | D3×D42 | 1 | 1 | yes | characteristic |
| 640 | C2 | #86624 | 1 | 1 | yes | characteristic |
| 641 | C2 | C2×D4×#67 | 1 | 2 | yes | — |
| 642 | C2 | #87284 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C2×C3×D42 | C2 |
| hypercenter | C2×D42 | D3 |
| hypocenter | C3 | #59126 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×D42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D42 | C2 | semidirect |
| 2. C2×D4×#67 | C2 | semidirect |
| 3. C2×D3×#127 | C2 | semidirect |
| 4. #84049 | C2 | semidirect |
| 5. #83173 | C2 | semidirect |
| 6. #82649 | C2 | semidirect |
| 7. #82640 | C2 | semidirect |
| 8. #82604 | C2 | semidirect |
| 9. #82594 | C2 | semidirect |
| 10. C4×#938 | C2 | semidirect |
| 11. #82378 | C2 | semidirect |
| 13. C23×C3×D4 | C22 | semidirect |
| 14. C23×#67 | C22 | semidirect |
| 15. C22×D3×D4 | C22 | semidirect |
| 16. C22×D3×D4 | C22 | semidirect |
| 19. C2×C3×#121 | C22 | semidirect |
| 20. C2×C3×#120 | C22 | semidirect |
| 21. C2×C3×C4×D4 | C22 | semidirect |
| 22. C2×#953 | C22 | semidirect |
| 23. C2×#953 | C22 | semidirect |
| 24. C2×#940 | C22 | semidirect |
| 25. C2×#940 | C22 | semidirect |
| 26. C2×#940 | C22 | semidirect |
| 27. C2×#939 | C22 | semidirect |
| 30. C2×#937 | C22 | semidirect |
| 31. C2×D4×Dic3 | C22 | semidirect |
| 32. C2×#930 | C22 | semidirect |
| 33. C2×#930 | C22 | semidirect |
| 34. C2×C4×#67 | C22 | semidirect |
| 36. C2×#874 | C22 | semidirect |
| 37. C2×C42×D3 | C22 | semidirect |
| 38. #5223 | C22 | semidirect |
| 39. #5223 | C22 | semidirect |
| 40. #5221 | C22 | semidirect |
| 41. #5221 | C22 | semidirect |
| 42. #5220 | C22 | semidirect |
| 43. C4×#217 | C22 | semidirect |
| 44. #4988 | C22 | semidirect |
| 45. #4987 | C22 | semidirect |
| 46. #4972 | C22 | semidirect |
| 47. #4960 | C22 | semidirect |
| 48. #4959 | C22 | semidirect |
| 49. #4958 | C22 | semidirect |
| 50. #4956 | C22 | semidirect |
| 51. #4955 | C22 | semidirect |
| 52. #4944 | C22 | semidirect |
| 53. C4×C4⋊Dic3 | C22 | semidirect |
| 55. C2×D42 | D3 | semidirect |
| 93. C3×#127 | D4 | semidirect |
| 112. #938 | D4 | semidirect |
| 135. C23×D4 | C2×D3 | semidirect |
| 148. C2×#121 | C2×D3 | semidirect |
| 152. C2×#120 | C2×D3 | semidirect |
| 155. C2×C4×D4 | C2×D3 | semidirect |
| 197. C2×C3×D4 | C2×D4 | semidirect |
| 198. C2×C3×D4 | C2×D4 | semidirect |
| 199. C2×C3×D4 | C2×D4 | semidirect |
| 221. C2×#67 | C2×D4 | semidirect |
| 241. C2×C4×D3 | C2×D4 | semidirect |
| 245. C3×C42 | C2×D4 | semidirect |
| 250. #217 | C2×D4 | semidirect |
| 255. C4⋊Dic3 | C2×D4 | semidirect |
| 292. #127 | #67 | semidirect |
| 444. C2×D4 | D3×D4 | semidirect |
| 445. C2×D4 | C2×#67 | semidirect |
| 446. C2×D4 | C2×#67 | semidirect |
| 482. C42 | C2×#67 | semidirect |
| 511. C2×D3 | D42 | semidirect |
| 514. C3×C4 | D42 | semidirect |
| 632. C4 | D4×#67 | semidirect |
| 633. C3 | #59126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D42 | 2 |
| C2×D4×#67 | 2 |
| C2×D3×#127 | 2 |
| #84049 | 2 |
| #83173 | 2 |
| #82649 | 2 |
| #82640 | 2 |
| #82604 | 2 |
| #82594 | 2 |
| C4×#938 | 2 |
| #82378 | 2 |
| #59126 | 3 |