Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5126 conjugacy classes of subgroups.
The subgroup lattice has 544 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×C3×C22⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#953 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#5493 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #84032 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #82130 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #82127 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #82121 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #82108 | C2 | 1 | 8 | yes | maximal, complemented |
| 9 | #82094 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #34981 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C26×C3 | C22 | 1 | 1 | yes | characteristic |
| 12 | C24×C3×C4 | C22 | 1 | 2 | yes | complemented |
| 13 | C23×#67 | C22 | 1 | 2 | yes | — |
| 14 | C23×#67 | C22 | 1 | 4 | yes | complemented |
| 15 | C22×C3×C22⋊C4 | — | 2 | 8 | no | — |
| 16 | C22×C3×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×C3×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×C3×C22⋊C4 | C22 | 1 | 8 | yes | complemented |
| 19 | C22×C3×C22⋊C4 | C22 | 1 | 2 | yes | complemented |
| 20 | C2×#953 | — | 2 | 16 | no | — |
| 21 | C2×#953 | C22 | 1 | 8 | yes | — |
| 22 | C2×#953 | C22 | 1 | 4 | yes | complemented |
| 23 | C2×#953 | C22 | 1 | 4 | yes | complemented |
| 24 | C22×#217 | C22 | 1 | 2 | yes | — |
| 25 | C22×#217 | C22 | 1 | 4 | yes | complemented |
| 26 | C2×#930 | C22 | 1 | 4 | yes | complemented |
| 27 | C2×#930 | C22 | 1 | 8 | yes | complemented |
| 28 | #5493 | — | 2 | 8 | no | — |
| 29 | C2×#886 | C22 | 1 | 4 | yes | complemented |
| 30 | C2×#882 | C22 | 1 | 4 | yes | complemented |
| 31 | #5213 | C22 | 1 | 16 | yes | complemented |
| 32 | #5213 | C22 | 1 | 8 | yes | complemented |
| 33 | #5212 | C22 | 1 | 4 | yes | complemented |
| 34 | #5212 | C22 | 1 | 8 | yes | complemented |
| 35 | #4961 | C22 | 1 | 16 | yes | complemented |
| 36 | #4960 | C22 | 1 | 8 | yes | complemented |
| 37 | #4958 | C22 | 1 | 8 | yes | complemented |
| 38 | #4955 | C22 | 1 | 4 | yes | complemented |
| 39 | #4953 | C22 | 1 | 8 | yes | complemented |
| 40 | #4950 | C22 | 1 | 8 | yes | complemented |
| 41 | #4943 | C22 | 1 | 4 | yes | complemented |
| 42 | C2×#534 | — | 3 | 1 | no | — |
| 43 | C22×#120 | — | 3 | 2 | no | — |
| 44 | C23×C22⋊C4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 45 | #2847 | — | 3 | 4 | no | — |
| 46 | #2617 | — | 3 | 4 | no | — |
| 47 | #2412 | — | 3 | 8 | no | — |
| 48 | #2410 | — | 3 | 8 | no | — |
| 49 | #2315 | — | 3 | 1 | no | — |
| 50 | #2308 | — | 3 | 2 | no | — |
| 51 | C25×C3 | — | 2 | 4 | no | — |
| 52 | C25×C3 | — | 2 | 4 | no | — |
| 53 | C25×C3 | — | 4 | 8 | no | — |
| 54 | C25×C3 | D4 | 1 | 8 | yes | complemented |
| 55 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 56 | C25×C3 | C23 | 1 | 4 | yes | — |
| 57 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 58 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 59 | C24×D3 | C23 | 1 | 2 | yes | — |
| 60 | C23×C3×C4 | — | 2 | 8 | no | — |
| 61 | C23×C3×C4 | — | 4 | 8 | no | — |
| 62 | C23×C3×C4 | C23 | 1 | 8 | yes | — |
| 63 | C23×C3×C4 | C23 | 1 | 2 | yes | — |
| 64 | C23×C3×C4 | C23 | 1 | 2 | yes | — |
| 65 | C22×#67 | — | 2 | 8 | no | — |
| 66 | C22×#67 | — | 2 | 8 | no | — |
| 67 | C22×#67 | — | 2 | 16 | no | — |
| 68 | C22×#67 | — | 2 | 8 | no | — |
| 69 | C22×#67 | — | 4 | 16 | no | — |
| 70 | C22×#67 | C23 | 1 | 4 | yes | — |
| 71 | C22×#67 | C23 | 1 | 4 | yes | — |
| 72 | C22×#67 | C23 | 1 | 8 | yes | — |
| 73 | C22×#67 | C23 | 1 | 8 | yes | — |
| 74 | C23×Dic3 | C23 | 1 | 2 | yes | — |
| 75 | C23×Dic3 | C23 | 1 | 4 | yes | — |
| 76 | C22×D12 | C23 | 1 | 4 | yes | complemented |
| 77 | C2×C3×C22⋊C4 | — | 2 | 4 | no | — |
| 78 | C2×C3×C22⋊C4 | — | 2 | 8 | no | — |
| 79 | C2×C3×C22⋊C4 | — | 2 | 32 | no | — |
| 80 | C2×C3×C22⋊C4 | D4 | 1 | 8 | yes | complemented |
| 81 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 82 | C2×C3×C22⋊C4 | C23 | 1 | 8 | yes | — |
| 83 | C2×C3×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | #953 | — | 2 | 32 | no | — |
| 85 | #953 | — | 2 | 16 | no | — |
| 86 | #953 | — | 2 | 16 | no | — |
| 87 | C2×#217 | — | 2 | 8 | no | — |
| 88 | C2×#217 | — | 2 | 16 | no | — |
| 89 | C2×#217 | C23 | 1 | 4 | yes | — |
| 90 | C2×#217 | C23 | 1 | 8 | yes | — |
| 91 | C2×#217 | C23 | 1 | 4 | yes | — |
| 92 | C2×#217 | C23 | 1 | 8 | yes | — |
| 93 | #930 | — | 2 | 16 | no | — |
| 94 | #930 | — | 4 | 16 | no | — |
| 95 | C2×(C2×D3)⋊C4 | C23 | 1 | 8 | yes | complemented |
| 96 | C2×(C2×D3)⋊C4 | C23 | 1 | 16 | yes | complemented |
| 97 | C2×C4⋊Dic3 | C23 | 1 | 4 | yes | complemented |
| 98 | C2×C4⋊Dic3 | C23 | 1 | 8 | yes | complemented |
| 99 | #886 | — | 2 | 16 | no | — |
| 100 | #882 | — | 2 | 16 | no | — |
| 101 | C3⋊#95 | C23 | 1 | 16 | yes | complemented |
| 102 | C3⋊#95 | C23 | 1 | 8 | yes | complemented |
| 103 | C26 | C2×D3 | 1 | 1 | yes | characteristic |
| 104 | C23×D4 | — | 3 | 2 | no | — |
| 105 | C23×D4 | — | 3 | 4 | no | — |
| 106 | C24×C4 | C2×D3 | 1 | 2 | yes | complemented |
| 107 | #534 | — | 6 | 8 | no | — |
| 108 | C2×#123 | — | 3 | 4 | no | — |
| 109 | C2×#121 | — | 3 | 4 | no | — |
| 110 | C2×#121 | — | 3 | 8 | no | — |
| 111 | C2×#120 | — | 3 | 4 | no | — |
| 112 | C2×#120 | — | 3 | 4 | no | — |
| 113 | C2×#120 | — | 3 | 8 | no | — |
| 114 | C2×#120 | — | 3 | 4 | no | — |
| 115 | C2×#120 | — | 6 | 16 | no | — |
| 116 | C22×C22⋊C4 | — | 2 | 8 | no | — |
| 117 | C22×C22⋊C4 | — | 3 | 2 | no | — |
| 118 | C22×C22⋊C4 | — | 3 | 4 | no | — |
| 119 | C22×C22⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 120 | C22×C22⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 121 | C22×C22⋊C4 | C2×D3 | 1 | 2 | yes | complemented |
| 122 | C22×C22⋊C4 | C2×D3 | 1 | 8 | yes | complemented |
| 123 | #397 | — | 3 | 8 | no | — |
| 124 | #394 | — | 3 | 16 | no | — |
| 125 | #392 | — | 3 | 8 | no | — |
| 126 | #386 | — | 3 | 8 | no | — |
| 127 | #386 | — | 3 | 16 | no | — |
| 128 | #386 | — | 3 | 8 | no | — |
| 129 | #385 | — | 3 | 8 | no | — |
| 130 | #381 | — | 3 | 4 | no | — |
| 131 | #380 | — | 3 | 4 | no | — |
| 132 | #380 | — | 3 | 8 | no | — |
| 133 | #379 | — | 3 | 4 | no | — |
| 134 | C24×C3 | — | 2 | 2 | no | — |
| 135 | C24×C3 | — | 2 | 2 | no | — |
| 136 | C24×C3 | — | 2 | 8 | no | — |
| 137 | C24×C3 | — | 2 | 32 | no | — |
| 138 | C24×C3 | — | 2 | 8 | no | — |
| 139 | C24×C3 | — | 2 | 2 | no | — |
| 140 | C24×C3 | — | 2 | 8 | no | — |
| 141 | C24×C3 | — | 2 | 2 | no | — |
| 142 | C24×C3 | — | 2 | 16 | no | — |
| 143 | C24×C3 | — | 4 | 32 | no | — |
| 144 | C24×C3 | — | 4 | 4 | no | — |
| 145 | C24×C3 | — | 4 | 8 | no | — |
| 146 | C24×C3 | — | 4 | 4 | no | — |
| 147 | C24×C3 | — | 4 | 32 | no | — |
| 148 | C24×C3 | — | 4 | 32 | no | — |
| 149 | C24×C3 | C2×D4 | 1 | 4 | yes | — |
| 150 | C24×C3 | C2×D4 | 1 | 16 | yes | — |
| 151 | C24×C3 | C2×D4 | 1 | 4 | yes | — |
| 152 | C24×C3 | C2×D4 | 1 | 4 | yes | — |
| 153 | C24×C3 | C2×D4 | 1 | 4 | yes | — |
| 154 | C24×C3 | #41 | 1 | 4 | yes | — |
| 155 | C24×C3 | C24 | 1 | 2 | yes | — |
| 156 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 157 | C24×C3 | C24 | 1 | 2 | yes | — |
| 158 | C24×C3 | C24 | 1 | 2 | yes | — |
| 159 | C23×D3 | — | 2 | 4 | no | — |
| 160 | C23×D3 | — | 2 | 4 | no | — |
| 161 | C23×D3 | — | 2 | 4 | no | — |
| 162 | C23×D3 | — | 4 | 8 | no | — |
| 163 | C23×D3 | C24 | 1 | 4 | yes | — |
| 164 | C22×C3×C4 | — | 2 | 4 | no | — |
| 165 | C22×C3×C4 | — | 2 | 16 | no | — |
| 166 | C22×C3×C4 | — | 2 | 16 | no | — |
| 167 | C22×C3×C4 | — | 4 | 4 | no | — |
| 168 | C22×C3×C4 | — | 4 | 8 | no | — |
| 169 | C22×C3×C4 | — | 4 | 4 | no | — |
| 170 | C22×C3×C4 | — | 4 | 32 | no | — |
| 171 | C22×C3×C4 | C2×D4 | 1 | 8 | yes | complemented |
| 172 | C22×C3×C4 | C24 | 1 | 4 | yes | — |
| 173 | C22×C3×C4 | C24 | 1 | 4 | yes | — |
| 174 | C2×#67 | — | 2 | 8 | no | — |
| 175 | C2×#67 | — | 2 | 8 | no | — |
| 176 | C2×#67 | — | 2 | 16 | no | — |
| 177 | C2×#67 | — | 2 | 8 | no | — |
| 178 | C2×#67 | — | 2 | 8 | no | — |
| 179 | C2×#67 | — | 2 | 16 | no | — |
| 180 | C2×#67 | — | 2 | 8 | no | — |
| 181 | C2×#67 | — | 2 | 8 | no | — |
| 182 | C2×#67 | — | 2 | 32 | no | — |
| 183 | C2×#67 | — | 4 | 16 | no | — |
| 184 | C2×#67 | — | 4 | 64 | no | — |
| 185 | C2×#67 | — | 4 | 32 | no | — |
| 186 | C22×Dic3 | — | 2 | 4 | no | — |
| 187 | C22×Dic3 | — | 2 | 8 | no | — |
| 188 | C22×Dic3 | — | 2 | 4 | no | — |
| 189 | C22×Dic3 | — | 2 | 4 | no | — |
| 190 | C22×Dic3 | — | 4 | 8 | no | — |
| 191 | C22×Dic3 | C24 | 1 | 4 | yes | — |
| 192 | C22×Dic3 | C24 | 1 | 8 | yes | — |
| 193 | C2×D12 | — | 2 | 8 | no | — |
| 194 | C2×D12 | — | 4 | 8 | no | — |
| 195 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 196 | C3×C22⋊C4 | — | 2 | 16 | no | — |
| 197 | C3×C22⋊C4 | C2×D4 | 1 | 16 | yes | complemented |
| 198 | #217 | — | 2 | 16 | no | — |
| 199 | #217 | — | 2 | 8 | no | — |
| 200 | #217 | — | 2 | 8 | no | — |
| 201 | #217 | — | 2 | 16 | no | — |
| 202 | (C2×D3)⋊C4 | — | 2 | 16 | no | — |
| 203 | (C2×D3)⋊C4 | — | 4 | 16 | no | — |
| 204 | C4⋊Dic3 | — | 2 | 8 | no | — |
| 205 | C4⋊Dic3 | — | 4 | 8 | no | — |
| 206 | C25 | — | 2 | 4 | no | — |
| 207 | C25 | — | 2 | 4 | no | — |
| 208 | C25 | — | 3 | 2 | no | — |
| 209 | C25 | — | 4 | 8 | no | — |
| 210 | C25 | D12 | 1 | 8 | yes | complemented |
| 211 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 212 | C25 | C22×D3 | 1 | 4 | yes | — |
| 213 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 214 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 215 | C22×D4 | — | 3 | 4 | no | — |
| 216 | C22×D4 | — | 3 | 4 | no | — |
| 217 | C22×D4 | — | 3 | 8 | no | — |
| 218 | C22×D4 | — | 3 | 8 | no | — |
| 219 | C22×D4 | — | 3 | 4 | no | — |
| 220 | C22×D4 | — | 6 | 16 | no | — |
| 221 | C22×D4 | — | 6 | 8 | no | — |
| 222 | C22×D4 | — | 6 | 8 | no | — |
| 223 | C22×D4 | — | 6 | 8 | no | — |
| 224 | C22×D4 | — | 12 | 16 | no | — |
| 225 | C23×C4 | — | 2 | 8 | no | — |
| 226 | C23×C4 | — | 3 | 2 | no | — |
| 227 | C23×C4 | — | 3 | 4 | no | — |
| 228 | C23×C4 | — | 4 | 8 | no | — |
| 229 | C23×C4 | C22×D3 | 1 | 8 | yes | — |
| 230 | C23×C4 | C22×D3 | 1 | 2 | yes | — |
| 231 | C23×C4 | C22×D3 | 1 | 2 | yes | — |
| 232 | #123 | — | 6 | 16 | no | — |
| 233 | #121 | — | 6 | 16 | no | — |
| 234 | #121 | — | 12 | 16 | no | — |
| 235 | #120 | — | 6 | 16 | no | — |
| 236 | #120 | — | 6 | 32 | no | — |
| 237 | #120 | — | 6 | 16 | no | — |
| 238 | #120 | — | 6 | 16 | no | — |
| 239 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 240 | C2×C4⋊C4 | — | 3 | 8 | no | — |
| 241 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 242 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 243 | C2×C22⋊C4 | — | 2 | 32 | no | — |
| 244 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 245 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 246 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 247 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 248 | C2×C22⋊C4 | — | 3 | 8 | no | — |
| 249 | C2×C22⋊C4 | — | 3 | 16 | no | — |
| 250 | C2×C22⋊C4 | — | 6 | 16 | no | — |
| 251 | C2×C22⋊C4 | — | 6 | 8 | no | — |
| 252 | C2×C22⋊C4 | #67 | 1 | 8 | yes | complemented |
| 253 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 254 | C2×C22⋊C4 | C22×D3 | 1 | 8 | yes | — |
| 255 | C2×C22⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 256 | #95 | — | 3 | 16 | no | — |
| 257 | #95 | — | 3 | 8 | no | — |
| 258 | C23×C3 | — | 2 | 16 | no | — |
| 259 | C23×C3 | — | 2 | 16 | no | — |
| 260 | C23×C3 | — | 2 | 8 | no | — |
| 261 | C23×C3 | — | 2 | 16 | no | — |
| 262 | C23×C3 | — | 2 | 2 | no | — |
| 263 | C23×C3 | — | 2 | 2 | no | — |
| 264 | C23×C3 | — | 2 | 8 | no | — |
| 265 | C23×C3 | — | 2 | 4 | no | — |
| 266 | C23×C3 | — | 2 | 4 | no | — |
| 267 | C23×C3 | — | 2 | 32 | no | — |
| 268 | C23×C3 | — | 2 | 8 | no | — |
| 269 | C23×C3 | — | 2 | 32 | no | — |
| 270 | C23×C3 | — | 2 | 2 | no | — |
| 271 | C23×C3 | — | 2 | 2 | no | — |
| 272 | C23×C3 | — | 2 | 8 | no | — |
| 273 | C23×C3 | — | 4 | 8 | no | — |
| 274 | C23×C3 | — | 4 | 16 | no | — |
| 275 | C23×C3 | — | 4 | 8 | no | — |
| 276 | C23×C3 | — | 4 | 8 | no | — |
| 277 | C23×C3 | — | 4 | 8 | no | — |
| 278 | C23×C3 | — | 4 | 32 | no | — |
| 279 | C23×C3 | — | 4 | 32 | no | — |
| 280 | C23×C3 | — | 4 | 128 | no | — |
| 281 | C23×C3 | — | 4 | 8 | no | — |
| 282 | C23×C3 | — | 4 | 8 | no | — |
| 283 | C23×C3 | #121 | 1 | 16 | yes | complemented |
| 284 | C23×C3 | C22×D4 | 1 | 4 | yes | — |
| 285 | C23×C3 | C22×D4 | 1 | 2 | yes | — |
| 286 | C23×C3 | C22×D4 | 1 | 4 | yes | — |
| 287 | C23×C3 | C22×D4 | 1 | 4 | yes | — |
| 288 | C23×C3 | C22×D4 | 1 | 4 | yes | — |
| 289 | C23×C3 | C22×D4 | 1 | 2 | yes | — |
| 290 | C23×C3 | C2×#41 | 1 | 4 | yes | — |
| 291 | C23×C3 | C2×#41 | 1 | 2 | yes | — |
| 292 | C23×C3 | H(5,𝔽2) | 1 | 8 | yes | — |
| 293 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 294 | C22×D3 | — | 2 | 4 | no | — |
| 295 | C22×D3 | — | 2 | 4 | no | — |
| 296 | C22×D3 | — | 2 | 4 | no | — |
| 297 | C22×D3 | — | 2 | 16 | no | — |
| 298 | C22×D3 | — | 4 | 16 | no | — |
| 299 | C22×D3 | — | 4 | 8 | no | — |
| 300 | C22×D3 | — | 8 | 8 | no | — |
| 301 | C22×D3 | — | 8 | 8 | no | — |
| 302 | C2×C3×C4 | — | 2 | 8 | no | — |
| 303 | C2×C3×C4 | — | 2 | 4 | no | — |
| 304 | C2×C3×C4 | — | 2 | 8 | no | — |
| 305 | C2×C3×C4 | — | 4 | 8 | no | — |
| 306 | C2×C3×C4 | — | 4 | 32 | no | — |
| 307 | C2×C3×C4 | — | 4 | 8 | no | — |
| 308 | C2×C3×C4 | C22×D4 | 1 | 8 | yes | — |
| 309 | #67 | — | 4 | 32 | no | — |
| 310 | #67 | — | 4 | 16 | no | — |
| 311 | #67 | — | 4 | 16 | no | — |
| 312 | #67 | — | 4 | 32 | no | — |
| 313 | C2×Dic3 | — | 2 | 4 | no | — |
| 314 | C2×Dic3 | — | 2 | 4 | no | — |
| 315 | C2×Dic3 | — | 2 | 8 | no | — |
| 316 | C2×Dic3 | — | 2 | 4 | no | — |
| 317 | C2×Dic3 | — | 2 | 16 | no | — |
| 318 | C2×Dic3 | — | 4 | 16 | no | — |
| 319 | C2×Dic3 | — | 4 | 8 | no | — |
| 320 | D12 | — | 8 | 8 | no | — |
| 321 | C24 | — | 2 | 2 | no | — |
| 322 | C24 | — | 2 | 8 | no | — |
| 323 | C24 | — | 2 | 2 | no | — |
| 324 | C24 | — | 2 | 32 | no | — |
| 325 | C24 | — | 2 | 8 | no | — |
| 326 | C24 | — | 2 | 2 | no | — |
| 327 | C24 | — | 2 | 16 | no | — |
| 328 | C24 | — | 2 | 8 | no | — |
| 329 | C24 | — | 2 | 2 | no | — |
| 330 | C24 | — | 3 | 4 | no | — |
| 331 | C24 | — | 4 | 32 | no | — |
| 332 | C24 | — | 4 | 4 | no | — |
| 333 | C24 | — | 4 | 8 | no | — |
| 334 | C24 | — | 4 | 4 | no | — |
| 335 | C24 | — | 4 | 32 | no | — |
| 336 | C24 | — | 4 | 32 | no | — |
| 337 | C24 | — | 6 | 4 | no | — |
| 338 | C24 | — | 6 | 4 | no | — |
| 339 | C24 | — | 6 | 4 | no | — |
| 340 | C24 | — | 12 | 8 | no | — |
| 341 | C24 | C2×D12 | 1 | 4 | yes | — |
| 342 | C24 | C2×D12 | 1 | 16 | yes | — |
| 343 | C24 | C2×D12 | 1 | 4 | yes | — |
| 344 | C24 | C2×D12 | 1 | 4 | yes | — |
| 345 | C24 | #235 | 1 | 4 | yes | — |
| 346 | C24 | C2×#67 | 1 | 4 | yes | — |
| 347 | C24 | C23×D3 | 1 | 2 | yes | — |
| 348 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 349 | C24 | C23×D3 | 1 | 2 | yes | — |
| 350 | C24 | C23×D3 | 1 | 2 | yes | — |
| 351 | C2×D4 | — | 6 | 16 | no | — |
| 352 | C2×D4 | — | 6 | 16 | no | — |
| 353 | C2×D4 | — | 6 | 8 | no | — |
| 354 | C2×D4 | — | 6 | 8 | no | — |
| 355 | C2×D4 | — | 6 | 8 | no | — |
| 356 | C2×D4 | — | 6 | 8 | no | — |
| 357 | C2×D4 | — | 6 | 32 | no | — |
| 358 | C2×D4 | — | 6 | 8 | no | — |
| 359 | C2×D4 | — | 6 | 8 | no | — |
| 360 | C2×D4 | — | 6 | 8 | no | — |
| 361 | C2×D4 | — | 12 | 8 | no | — |
| 362 | C2×D4 | — | 12 | 16 | no | — |
| 363 | C2×D4 | — | 12 | 64 | no | — |
| 364 | C2×D4 | — | 12 | 32 | no | — |
| 365 | C22×C4 | — | 2 | 16 | no | — |
| 366 | C22×C4 | — | 2 | 4 | no | — |
| 367 | C22×C4 | — | 2 | 16 | no | — |
| 368 | C22×C4 | — | 3 | 4 | no | — |
| 369 | C22×C4 | — | 3 | 8 | no | — |
| 370 | C22×C4 | — | 4 | 4 | no | — |
| 371 | C22×C4 | — | 4 | 8 | no | — |
| 372 | C22×C4 | — | 4 | 4 | no | — |
| 373 | C22×C4 | — | 4 | 32 | no | — |
| 374 | C22×C4 | — | 6 | 8 | no | — |
| 375 | C22×C4 | — | 6 | 4 | no | — |
| 376 | C22×C4 | — | 6 | 4 | no | — |
| 377 | C22×C4 | — | 6 | 4 | no | — |
| 378 | C22×C4 | — | 12 | 8 | no | — |
| 379 | C22×C4 | C2×#67 | 1 | 8 | yes | complemented |
| 380 | C22×C4 | C23×D3 | 1 | 4 | yes | — |
| 381 | C22×C4 | C23×D3 | 1 | 4 | yes | — |
| 382 | C4⋊C4 | — | 6 | 8 | no | — |
| 383 | C4⋊C4 | — | 12 | 8 | no | — |
| 384 | C22⋊C4 | — | 2 | 8 | no | — |
| 385 | C22⋊C4 | — | 2 | 16 | no | — |
| 386 | C22⋊C4 | — | 6 | 16 | no | — |
| 387 | C22⋊C4 | — | 6 | 16 | no | — |
| 388 | C22⋊C4 | — | 6 | 8 | no | — |
| 389 | C22⋊C4 | — | 6 | 8 | no | — |
| 390 | C22⋊C4 | — | 6 | 16 | no | — |
| 391 | C22⋊C4 | — | 12 | 16 | no | — |
| 392 | C22⋊C4 | C2×#67 | 1 | 16 | yes | complemented |
| 393 | C22×C3 | — | 2 | 32 | no | — |
| 394 | C22×C3 | — | 2 | 4 | no | — |
| 395 | C22×C3 | — | 2 | 8 | no | — |
| 396 | C22×C3 | — | 2 | 4 | no | — |
| 397 | C22×C3 | — | 2 | 4 | no | — |
| 398 | C22×C3 | — | 2 | 4 | no | — |
| 399 | C22×C3 | — | 2 | 8 | no | — |
| 400 | C22×C3 | — | 2 | 8 | no | — |
| 401 | C22×C3 | — | 2 | 8 | no | — |
| 402 | C22×C3 | — | 2 | 4 | no | — |
| 403 | C22×C3 | — | 2 | 8 | no | — |
| 404 | C22×C3 | — | 2 | 4 | no | — |
| 405 | C22×C3 | — | 2 | 8 | no | — |
| 406 | C22×C3 | — | 4 | 16 | no | — |
| 407 | C22×C3 | — | 4 | 64 | no | — |
| 408 | C22×C3 | — | 4 | 16 | no | — |
| 409 | C22×C3 | — | 4 | 4 | no | — |
| 410 | C22×C3 | — | 4 | 4 | no | — |
| 411 | C22×C3 | C2×#121 | 1 | 8 | yes | — |
| 412 | C22×C3 | C2×#121 | 1 | 4 | yes | — |
| 413 | C22×C3 | #534 | 1 | 8 | yes | complemented |
| 414 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 415 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 416 | C22×C3 | C22×#41 | 1 | 1 | yes | characteristic |
| 417 | C22×C3 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 418 | C2×D3 | — | 4 | 4 | no | — |
| 419 | C2×D3 | — | 4 | 8 | no | — |
| 420 | C2×D3 | — | 4 | 8 | no | — |
| 421 | C2×D3 | — | 4 | 8 | no | — |
| 422 | C2×D3 | — | 8 | 16 | no | — |
| 423 | C3×C4 | — | 2 | 8 | no | — |
| 424 | C3×C4 | — | 4 | 4 | no | — |
| 425 | Dic3 | — | 4 | 4 | no | — |
| 426 | Dic3 | — | 4 | 8 | no | — |
| 427 | C23 | — | 2 | 16 | no | — |
| 428 | C23 | — | 2 | 16 | no | — |
| 429 | C23 | — | 2 | 2 | no | — |
| 430 | C23 | — | 2 | 4 | no | — |
| 431 | C23 | — | 2 | 4 | no | — |
| 432 | C23 | — | 2 | 8 | no | — |
| 433 | C23 | — | 2 | 2 | no | — |
| 434 | C23 | — | 2 | 16 | no | — |
| 435 | C23 | — | 2 | 8 | no | — |
| 436 | C23 | — | 2 | 2 | no | — |
| 437 | C23 | — | 2 | 8 | no | — |
| 438 | C23 | — | 2 | 2 | no | — |
| 439 | C23 | — | 2 | 32 | no | — |
| 440 | C23 | — | 2 | 32 | no | — |
| 441 | C23 | — | 2 | 8 | no | — |
| 442 | C23 | — | 4 | 8 | no | — |
| 443 | C23 | — | 4 | 16 | no | — |
| 444 | C23 | — | 4 | 8 | no | — |
| 445 | C23 | — | 4 | 8 | no | — |
| 446 | C23 | — | 4 | 8 | no | — |
| 447 | C23 | — | 4 | 32 | no | — |
| 448 | C23 | — | 4 | 32 | no | — |
| 449 | C23 | — | 4 | 128 | no | — |
| 450 | C23 | — | 4 | 8 | no | — |
| 451 | C23 | — | 4 | 8 | no | — |
| 452 | C23 | — | 6 | 4 | no | — |
| 453 | C23 | — | 6 | 4 | no | — |
| 454 | C23 | — | 6 | 16 | no | — |
| 455 | C23 | — | 6 | 4 | no | — |
| 456 | C23 | — | 12 | 16 | no | — |
| 457 | C23 | — | 12 | 8 | no | — |
| 458 | C23 | — | 24 | 8 | no | — |
| 459 | C23 | — | 24 | 8 | no | — |
| 460 | C23 | #930 | 1 | 16 | yes | complemented |
| 461 | C23 | C22×D12 | 1 | 4 | yes | — |
| 462 | C23 | C22×D12 | 1 | 4 | yes | — |
| 463 | C23 | C22×D12 | 1 | 2 | yes | — |
| 464 | C23 | C22×D12 | 1 | 4 | yes | — |
| 465 | C23 | C2×#235 | 1 | 4 | yes | — |
| 466 | C23 | C2×#235 | 1 | 2 | yes | — |
| 467 | C23 | #1004 | 1 | 8 | yes | — |
| 468 | C23 | C22×#67 | 1 | 4 | yes | — |
| 469 | C23 | C22×#67 | 1 | 2 | yes | — |
| 470 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 471 | D4 | — | 12 | 32 | no | — |
| 472 | D4 | — | 12 | 32 | no | — |
| 473 | D4 | — | 12 | 16 | no | — |
| 474 | D4 | — | 12 | 16 | no | — |
| 475 | D4 | — | 24 | 8 | no | — |
| 476 | C2×C4 | — | 2 | 8 | no | — |
| 477 | C2×C4 | — | 2 | 8 | no | — |
| 478 | C2×C4 | — | 2 | 4 | no | — |
| 479 | C2×C4 | — | 4 | 8 | no | — |
| 480 | C2×C4 | — | 4 | 32 | no | — |
| 481 | C2×C4 | — | 4 | 8 | no | — |
| 482 | C2×C4 | — | 6 | 4 | no | — |
| 483 | C2×C4 | — | 6 | 8 | no | — |
| 484 | C2×C4 | — | 6 | 4 | no | — |
| 485 | C2×C4 | — | 6 | 16 | no | — |
| 486 | C2×C4 | — | 6 | 4 | no | — |
| 487 | C2×C4 | — | 12 | 16 | no | — |
| 488 | C2×C4 | — | 12 | 8 | no | — |
| 489 | C2×C4 | C22×#67 | 1 | 8 | yes | — |
| 490 | C2×C3 | — | 2 | 8 | no | — |
| 491 | C2×C3 | — | 2 | 4 | no | — |
| 492 | C2×C3 | — | 2 | 8 | no | — |
| 493 | C2×C3 | — | 4 | 4 | no | — |
| 494 | C2×C3 | C22×#121 | 1 | 1 | yes | characteristic |
| 495 | C2×C3 | C2×#534 | 1 | 2 | yes | — |
| 496 | C2×C3 | #3484 | 1 | 2 | yes | — |
| 497 | C2×C3 | #3490 | 1 | 2 | yes | — |
| 498 | D3 | — | 8 | 4 | no | — |
| 499 | C22 | — | 2 | 32 | no | — |
| 500 | C22 | — | 2 | 4 | no | — |
| 501 | C22 | — | 2 | 8 | no | — |
| 502 | C22 | — | 2 | 4 | no | — |
| 503 | C22 | — | 2 | 4 | no | — |
| 504 | C22 | — | 2 | 4 | no | — |
| 505 | C22 | — | 2 | 8 | no | — |
| 506 | C22 | — | 2 | 8 | no | — |
| 507 | C22 | — | 2 | 8 | no | — |
| 508 | C22 | — | 2 | 4 | no | — |
| 509 | C22 | — | 2 | 8 | no | — |
| 510 | C22 | — | 2 | 4 | no | — |
| 511 | C22 | — | 2 | 8 | no | — |
| 512 | C22 | — | 4 | 16 | no | — |
| 513 | C22 | — | 4 | 64 | no | — |
| 514 | C22 | — | 4 | 16 | no | — |
| 515 | C22 | — | 4 | 4 | no | — |
| 516 | C22 | — | 4 | 4 | no | — |
| 517 | C22 | — | 12 | 4 | no | — |
| 518 | C22 | — | 12 | 8 | no | — |
| 519 | C22 | — | 12 | 8 | no | — |
| 520 | C22 | — | 12 | 8 | no | — |
| 521 | C22 | — | 24 | 16 | no | — |
| 522 | C22 | #5493 | 1 | 8 | yes | complemented |
| 523 | C22 | C2×#930 | 1 | 8 | yes | — |
| 524 | C22 | C2×#930 | 1 | 4 | yes | — |
| 525 | C22 | C23×D12 | 1 | 1 | yes | characteristic |
| 526 | C22 | C22×#235 | 1 | 1 | yes | characteristic |
| 527 | C22 | C2×#1004 | 1 | 4 | yes | — |
| 528 | C22 | C23×#67 | 1 | 1 | yes | characteristic |
| 529 | C4 | — | 2 | 8 | no | — |
| 530 | C4 | — | 4 | 4 | no | — |
| 531 | C4 | — | 12 | 4 | no | — |
| 532 | C4 | — | 12 | 8 | no | — |
| 533 | C3 | #34981 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 534 | C2 | — | 2 | 8 | no | — |
| 535 | C2 | — | 2 | 4 | no | — |
| 536 | C2 | — | 2 | 8 | no | — |
| 537 | C2 | — | 4 | 4 | no | — |
| 538 | C2 | — | 24 | 4 | no | — |
| 539 | C2 | C2×#5493 | 1 | 2 | yes | — |
| 540 | C2 | #85697 | 1 | 2 | yes | — |
| 541 | C2 | C22×#930 | 1 | 1 | yes | characteristic |
| 542 | C2 | #87239 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C23×C3×C22⋊C4 | C2 |
| hypercenter | C23×C22⋊C4 | D3 |
| hypocenter | C3 | #34981 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×C22⋊C4 | C2 | semidirect |
| 2. C22×#953 | C2 | semidirect |
| 3. C2×#5493 | C2 | semidirect |
| 4. #84032 | C2 | semidirect |
| 5. #82130 | C2 | semidirect |
| 6. #82127 | C2 | semidirect |
| 7. #82121 | C2 | semidirect |
| 8. #82108 | C2 | semidirect |
| 9. #82094 | C2 | semidirect |
| 12. C24×C3×C4 | C22 | semidirect |
| 14. C23×#67 | C22 | semidirect |
| 18. C22×C3×C22⋊C4 | C22 | semidirect |
| 19. C22×C3×C22⋊C4 | C22 | semidirect |
| 22. C2×#953 | C22 | semidirect |
| 23. C2×#953 | C22 | semidirect |
| 25. C22×#217 | C22 | semidirect |
| 26. C2×#930 | C22 | semidirect |
| 27. C2×#930 | C22 | semidirect |
| 29. C2×#886 | C22 | semidirect |
| 30. C2×#882 | C22 | semidirect |
| 31. #5213 | C22 | semidirect |
| 32. #5213 | C22 | semidirect |
| 33. #5212 | C22 | semidirect |
| 34. #5212 | C22 | semidirect |
| 35. #4961 | C22 | semidirect |
| 36. #4960 | C22 | semidirect |
| 37. #4958 | C22 | semidirect |
| 38. #4955 | C22 | semidirect |
| 39. #4953 | C22 | semidirect |
| 40. #4950 | C22 | semidirect |
| 41. #4943 | C22 | semidirect |
| 44. C23×C22⋊C4 | D3 | semidirect |
| 54. C25×C3 | D4 | semidirect |
| 76. C22×D12 | C23 | semidirect |
| 80. C2×C3×C22⋊C4 | D4 | semidirect |
| 95. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 96. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 97. C2×C4⋊Dic3 | C23 | semidirect |
| 98. C2×C4⋊Dic3 | C23 | semidirect |
| 101. C3⋊#95 | C23 | semidirect |
| 102. C3⋊#95 | C23 | semidirect |
| 106. C24×C4 | C2×D3 | semidirect |
| 121. C22×C22⋊C4 | C2×D3 | semidirect |
| 122. C22×C22⋊C4 | C2×D3 | semidirect |
| 171. C22×C3×C4 | C2×D4 | semidirect |
| 197. C3×C22⋊C4 | C2×D4 | semidirect |
| 210. C25 | D12 | semidirect |
| 252. C2×C22⋊C4 | #67 | semidirect |
| 283. C23×C3 | #121 | semidirect |
| 379. C22×C4 | C2×#67 | semidirect |
| 392. C22⋊C4 | C2×#67 | semidirect |
| 413. C22×C3 | #534 | semidirect |
| 460. C23 | #930 | semidirect |
| 522. C22 | #5493 | semidirect |
| 533. C3 | #34981 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C22⋊C4 | 2 |
| C22×#953 | 2 |
| C2×#5493 | 2 |
| #84032 | 2 |
| #82130 | 2 |
| #82127 | 2 |
| #82121 | 2 |
| #82108 | 2 |
| #82094 | 2 |
| #34981 | 3 |