Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
4156 conjugacy classes of subgroups.
The subgroup lattice has 608 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C3×#120 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5594 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #84128 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #84052 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #84050 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #82611 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #82598 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #82597 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #82595 | C2 | 1 | 6 | yes | maximal, complemented |
| 10 | #82580 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #82111 | C2 | 1 | 6 | yes | maximal, complemented |
| 12 | #45439 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C26×C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C23×C3×D4 | C22 | 1 | 3 | yes | — |
| 15 | C23×#67 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×C3×#120 | — | 2 | 8 | no | — |
| 17 | C2×C3×#120 | C22 | 1 | 3 | yes | — |
| 18 | C2×C3×#120 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×#120 | C22 | 1 | 3 | yes | complemented |
| 20 | C2×C3×#120 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | C22×C3×C22⋊C4 | C22 | 1 | 3 | yes | complemented |
| 22 | C2×#953 | C22 | 1 | 6 | yes | — |
| 23 | C2×#953 | C22 | 1 | 2 | yes | complemented |
| 24 | C2×#953 | C22 | 1 | 6 | yes | complemented |
| 25 | C2×#937 | C22 | 1 | 3 | yes | complemented |
| 26 | C2×#937 | C22 | 1 | 3 | yes | complemented |
| 27 | C2×#936 | C22 | 1 | 3 | yes | complemented |
| 28 | C2×#936 | C22 | 1 | 3 | yes | complemented |
| 29 | C2×#924 | C22 | 1 | 3 | yes | complemented |
| 30 | #5594 | — | 2 | 8 | no | — |
| 31 | C2×#885 | C22 | 1 | 6 | yes | complemented |
| 32 | #5247 | C22 | 1 | 6 | yes | complemented |
| 33 | #5247 | C22 | 1 | 2 | yes | complemented |
| 34 | #5247 | C22 | 1 | 6 | yes | complemented |
| 35 | #5247 | C22 | 1 | 3 | yes | complemented |
| 36 | #5223 | C22 | 1 | 6 | yes | complemented |
| 37 | #5223 | C22 | 1 | 6 | yes | complemented |
| 38 | #5222 | C22 | 1 | 6 | yes | complemented |
| 39 | #5222 | C22 | 1 | 6 | yes | complemented |
| 40 | #5221 | C22 | 1 | 3 | yes | complemented |
| 41 | #5221 | C22 | 1 | 3 | yes | complemented |
| 42 | #5220 | C22 | 1 | 3 | yes | complemented |
| 43 | #5220 | C22 | 1 | 3 | yes | complemented |
| 44 | #5213 | C22 | 1 | 6 | yes | complemented |
| 45 | C4×#217 | C22 | 1 | 3 | yes | complemented |
| 46 | #4961 | C22 | 1 | 6 | yes | complemented |
| 47 | #4959 | C22 | 1 | 6 | yes | complemented |
| 48 | #4957 | C22 | 1 | 6 | yes | complemented |
| 49 | #4952 | C22 | 1 | 6 | yes | complemented |
| 50 | #4951 | C22 | 1 | 6 | yes | complemented |
| 51 | #4947 | C22 | 1 | 6 | yes | complemented |
| 52 | #4944 | C22 | 1 | 6 | yes | complemented |
| 53 | C2×H(3,𝔽4) | — | 3 | 1 | no | — |
| 54 | C22×#120 | D3 | 1 | 1 | yes | characteristic, complemented |
| 55 | #2846 | — | 3 | 2 | no | — |
| 56 | #2812 | — | 3 | 2 | no | — |
| 57 | #2684 | — | 3 | 3 | no | — |
| 58 | #2679 | — | 3 | 3 | no | — |
| 59 | #2670 | — | 3 | 6 | no | — |
| 60 | #2472 | — | 3 | 6 | no | — |
| 61 | #2435 | — | 3 | 3 | no | — |
| 62 | #2403 | — | 3 | 3 | no | — |
| 63 | #2375 | — | 3 | 1 | no | — |
| 64 | C25×C3 | — | 2 | 4 | no | — |
| 65 | C25×C3 | — | 4 | 6 | no | — |
| 66 | C25×C3 | — | 4 | 6 | no | — |
| 67 | C25×C3 | C23 | 1 | 3 | yes | — |
| 68 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 69 | C25×C3 | C23 | 1 | 3 | yes | — |
| 70 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 71 | C24×D3 | C23 | 1 | 1 | yes | characteristic, complemented |
| 72 | C22×C3×D4 | — | 2 | 12 | no | — |
| 73 | C22×C3×D4 | — | 4 | 6 | no | — |
| 74 | C22×C3×D4 | — | 4 | 6 | no | — |
| 75 | C22×C3×D4 | C23 | 1 | 3 | yes | — |
| 76 | C22×C3×D4 | C23 | 1 | 3 | yes | — |
| 77 | C22×C3×D4 | C23 | 1 | 3 | yes | — |
| 78 | C22×C3×D4 | C23 | 1 | 3 | yes | — |
| 79 | C23×C3×C4 | C23 | 1 | 3 | yes | — |
| 80 | C22×#67 | — | 4 | 6 | no | — |
| 81 | C22×#67 | C23 | 1 | 2 | yes | — |
| 82 | C22×#67 | C23 | 1 | 6 | yes | — |
| 83 | C22×#67 | C23 | 1 | 6 | yes | — |
| 84 | C22×#67 | C23 | 1 | 2 | yes | complemented |
| 85 | C23×Dic3 | C23 | 1 | 1 | yes | characteristic, complemented |
| 86 | C22×Dic6 | C23 | 1 | 3 | yes | complemented |
| 87 | C3×#120 | — | 2 | 4 | no | — |
| 88 | C3×#120 | — | 2 | 12 | no | — |
| 89 | C3×#120 | — | 2 | 12 | no | — |
| 90 | C3×#120 | D4 | 1 | 8 | yes | complemented |
| 91 | C2×C3×C22⋊C4 | — | 2 | 12 | no | — |
| 92 | C2×C3×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 93 | C2×C3×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 94 | C2×C3×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 95 | #953 | — | 2 | 8 | no | — |
| 96 | #953 | — | 4 | 12 | no | — |
| 97 | #953 | — | 4 | 12 | no | — |
| 98 | C2×#217 | C23 | 1 | 6 | yes | — |
| 99 | C2×#217 | C23 | 1 | 6 | yes | — |
| 100 | C2×#217 | C23 | 1 | 6 | yes | — |
| 101 | C2×#217 | C23 | 1 | 6 | yes | — |
| 102 | C2×#217 | C23 | 1 | 6 | yes | — |
| 103 | C2×#217 | C23 | 1 | 6 | yes | — |
| 104 | C2×#217 | C23 | 1 | 6 | yes | — |
| 105 | C2×#217 | C23 | 1 | 3 | yes | complemented |
| 106 | C2×#217 | C23 | 1 | 3 | yes | complemented |
| 107 | C2×#217 | C23 | 1 | 6 | yes | complemented |
| 108 | #937 | — | 2 | 12 | no | — |
| 109 | #937 | — | 4 | 6 | no | — |
| 110 | #936 | — | 2 | 12 | no | — |
| 111 | #936 | — | 4 | 6 | no | — |
| 112 | C2×(C2×D3)⋊C4 | C23 | 1 | 6 | yes | complemented |
| 113 | C2×(C2×D3)⋊C4 | C23 | 1 | 6 | yes | complemented |
| 114 | C2×C4⋊Dic3 | C23 | 1 | 3 | yes | complemented |
| 115 | #924 | — | 4 | 6 | no | — |
| 116 | C2×Dic3⋊C4 | C23 | 1 | 6 | yes | complemented |
| 117 | C2×C4×Dic3 | C23 | 1 | 3 | yes | complemented |
| 118 | C2×C4×Dic3 | C23 | 1 | 3 | yes | complemented |
| 119 | #885 | — | 2 | 24 | no | — |
| 120 | C3⋊#95 | C23 | 1 | 6 | yes | complemented |
| 121 | C3⋊#95 | C23 | 1 | 6 | yes | complemented |
| 122 | C3⋊#95 | C23 | 1 | 6 | yes | complemented |
| 123 | C26 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 124 | C23×D4 | — | 3 | 1 | no | — |
| 125 | C23×D4 | C2×D3 | 1 | 3 | yes | — |
| 126 | H(3,𝔽4) | — | 6 | 8 | no | — |
| 127 | C2×#124 | — | 3 | 3 | no | — |
| 128 | C2×#124 | — | 3 | 3 | no | — |
| 129 | C2×#124 | — | 3 | 6 | no | — |
| 130 | C2×C22⋊Q8 | — | 3 | 3 | no | — |
| 131 | C2×#120 | — | 2 | 8 | no | — |
| 132 | C2×#120 | — | 3 | 2 | no | — |
| 133 | C2×#120 | — | 3 | 3 | no | — |
| 134 | C2×#120 | — | 3 | 6 | no | — |
| 135 | C2×#120 | — | 3 | 6 | no | — |
| 136 | C2×#120 | — | 3 | 3 | no | — |
| 137 | C2×#120 | C2×D3 | 1 | 3 | yes | — |
| 138 | C2×#120 | C2×D3 | 1 | 1 | yes | characteristic |
| 139 | C2×#120 | C2×D3 | 1 | 3 | yes | complemented |
| 140 | C2×#120 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 141 | C22×C22⋊C4 | C2×D3 | 1 | 3 | yes | complemented |
| 142 | #397 | — | 3 | 6 | no | — |
| 143 | #394 | — | 3 | 6 | no | — |
| 144 | #394 | — | 3 | 6 | no | — |
| 145 | #394 | — | 3 | 6 | no | — |
| 146 | #394 | — | 3 | 6 | no | — |
| 147 | #393 | — | 3 | 6 | no | — |
| 148 | #392 | — | 3 | 6 | no | — |
| 149 | #392 | — | 3 | 6 | no | — |
| 150 | #390 | — | 3 | 3 | no | — |
| 151 | #390 | — | 3 | 3 | no | — |
| 152 | #390 | — | 3 | 6 | no | — |
| 153 | #388 | — | 3 | 6 | no | — |
| 154 | #388 | — | 3 | 6 | no | — |
| 155 | #386 | — | 3 | 3 | no | — |
| 156 | #386 | — | 3 | 3 | no | — |
| 157 | #386 | — | 3 | 6 | no | — |
| 158 | #379 | — | 3 | 2 | no | — |
| 159 | #379 | — | 3 | 6 | no | — |
| 160 | #379 | — | 3 | 6 | no | — |
| 161 | #379 | — | 3 | 3 | no | — |
| 162 | C4×C22⋊C4 | — | 3 | 3 | no | — |
| 163 | C24×C3 | — | 2 | 6 | no | — |
| 164 | C24×C3 | — | 2 | 6 | no | — |
| 165 | C24×C3 | — | 2 | 6 | no | — |
| 166 | C24×C3 | — | 2 | 6 | no | — |
| 167 | C24×C3 | — | 2 | 2 | no | — |
| 168 | C24×C3 | — | 2 | 12 | no | — |
| 169 | C24×C3 | — | 2 | 12 | no | — |
| 170 | C24×C3 | — | 2 | 6 | no | — |
| 171 | C24×C3 | — | 4 | 6 | no | — |
| 172 | C24×C3 | — | 4 | 3 | no | — |
| 173 | C24×C3 | — | 4 | 3 | no | — |
| 174 | C24×C3 | — | 4 | 6 | no | — |
| 175 | C24×C3 | — | 4 | 3 | no | — |
| 176 | C24×C3 | — | 4 | 3 | no | — |
| 177 | C24×C3 | — | 4 | 3 | no | — |
| 178 | C24×C3 | — | 4 | 48 | no | — |
| 179 | C24×C3 | — | 4 | 48 | no | — |
| 180 | C24×C3 | — | 4 | 16 | no | — |
| 181 | C24×C3 | C2×D4 | 1 | 4 | yes | complemented |
| 182 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 183 | C24×C3 | C24 | 1 | 3 | yes | — |
| 184 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 185 | C24×C3 | C24 | 1 | 3 | yes | — |
| 186 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 187 | C23×D3 | — | 2 | 2 | no | — |
| 188 | C23×D3 | — | 4 | 3 | no | — |
| 189 | C23×D3 | — | 4 | 3 | no | — |
| 190 | C23×D3 | C24 | 1 | 2 | yes | — |
| 191 | C2×C3×D4 | — | 2 | 6 | no | — |
| 192 | C2×C3×D4 | — | 2 | 6 | no | — |
| 193 | C2×C3×D4 | — | 2 | 6 | no | — |
| 194 | C2×C3×D4 | — | 2 | 6 | no | — |
| 195 | C2×C3×D4 | — | 2 | 6 | no | — |
| 196 | C2×C3×D4 | — | 4 | 3 | no | — |
| 197 | C2×C3×D4 | — | 4 | 3 | no | — |
| 198 | C2×C3×D4 | — | 4 | 3 | no | — |
| 199 | C2×C3×D4 | — | 4 | 3 | no | — |
| 200 | C2×C3×D4 | — | 4 | 3 | no | — |
| 201 | C2×C3×D4 | — | 4 | 3 | no | — |
| 202 | C2×C3×D4 | — | 4 | 48 | no | — |
| 203 | C2×C3×D4 | C2×D4 | 1 | 12 | yes | — |
| 204 | C22×C3×C4 | — | 2 | 6 | no | — |
| 205 | C22×C3×C4 | — | 2 | 6 | no | — |
| 206 | C22×C3×C4 | — | 4 | 3 | no | — |
| 207 | C22×C3×C4 | C24 | 1 | 3 | yes | — |
| 208 | C22×C3×C4 | C24 | 1 | 3 | yes | — |
| 209 | C2×#67 | — | 2 | 12 | no | — |
| 210 | C2×#67 | — | 2 | 12 | no | — |
| 211 | C2×#67 | — | 2 | 12 | no | — |
| 212 | C2×#67 | — | 4 | 6 | no | — |
| 213 | C2×#67 | — | 4 | 6 | no | — |
| 214 | C2×#67 | — | 4 | 6 | no | — |
| 215 | C2×#67 | — | 4 | 6 | no | — |
| 216 | C2×#67 | — | 4 | 6 | no | — |
| 217 | C2×#67 | — | 4 | 16 | no | — |
| 218 | C22×Dic3 | — | 4 | 3 | no | — |
| 219 | C22×Dic3 | C24 | 1 | 2 | yes | — |
| 220 | C22×Dic3 | C24 | 1 | 6 | yes | — |
| 221 | C22×Dic3 | C24 | 1 | 6 | yes | — |
| 222 | C2×Dic6 | — | 2 | 6 | no | — |
| 223 | C2×Dic6 | — | 4 | 3 | no | — |
| 224 | C2×Dic6 | — | 4 | 3 | no | — |
| 225 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 226 | C3×C22⋊C4 | — | 2 | 12 | no | — |
| 227 | C3×C22⋊C4 | C2×D4 | 1 | 12 | yes | complemented |
| 228 | #217 | — | 2 | 12 | no | — |
| 229 | #217 | — | 2 | 6 | no | — |
| 230 | #217 | — | 2 | 12 | no | — |
| 231 | #217 | — | 2 | 12 | no | — |
| 232 | #217 | — | 4 | 6 | no | — |
| 233 | #217 | — | 4 | 3 | no | — |
| 234 | #217 | — | 4 | 6 | no | — |
| 235 | #217 | — | 4 | 6 | no | — |
| 236 | #217 | — | 4 | 6 | no | — |
| 237 | #217 | — | 4 | 6 | no | — |
| 238 | (C2×D3)⋊C4 | — | 2 | 12 | no | — |
| 239 | (C2×D3)⋊C4 | — | 4 | 6 | no | — |
| 240 | C4⋊Dic3 | — | 4 | 3 | no | — |
| 241 | Dic3⋊C4 | — | 4 | 6 | no | — |
| 242 | C4×Dic3 | — | 2 | 6 | no | — |
| 243 | C4×Dic3 | — | 4 | 3 | no | — |
| 244 | C25 | — | 2 | 4 | no | — |
| 245 | C25 | — | 3 | 1 | no | — |
| 246 | C25 | — | 4 | 6 | no | — |
| 247 | C25 | — | 4 | 6 | no | — |
| 248 | C25 | C22×D3 | 1 | 3 | yes | — |
| 249 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 250 | C25 | C22×D3 | 1 | 3 | yes | — |
| 251 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 252 | C22×Q8 | — | 3 | 3 | no | — |
| 253 | C22×D4 | — | 2 | 12 | no | — |
| 254 | C22×D4 | — | 3 | 2 | no | — |
| 255 | C22×D4 | — | 3 | 6 | no | — |
| 256 | C22×D4 | — | 3 | 6 | no | — |
| 257 | C22×D4 | — | 3 | 2 | no | — |
| 258 | C22×D4 | — | 4 | 6 | no | — |
| 259 | C22×D4 | — | 4 | 6 | no | — |
| 260 | C22×D4 | — | 12 | 6 | no | — |
| 261 | C22×D4 | C22×D3 | 1 | 3 | yes | — |
| 262 | C22×D4 | C22×D3 | 1 | 3 | yes | — |
| 263 | C22×D4 | C22×D3 | 1 | 3 | yes | — |
| 264 | C22×D4 | C22×D3 | 1 | 3 | yes | — |
| 265 | C23×C4 | — | 3 | 1 | no | — |
| 266 | C23×C4 | C22×D3 | 1 | 3 | yes | — |
| 267 | #124 | — | 6 | 12 | no | — |
| 268 | #124 | — | 6 | 24 | no | — |
| 269 | #124 | — | 12 | 6 | no | — |
| 270 | C22⋊Q8 | — | 12 | 6 | no | — |
| 271 | #120 | — | 2 | 4 | no | — |
| 272 | #120 | — | 2 | 12 | no | — |
| 273 | #120 | — | 2 | 12 | no | — |
| 274 | #120 | — | 6 | 8 | no | — |
| 275 | #120 | — | 6 | 12 | no | — |
| 276 | #120 | — | 12 | 12 | no | — |
| 277 | #120 | — | 12 | 12 | no | — |
| 278 | #120 | — | 12 | 6 | no | — |
| 279 | #120 | #67 | 1 | 8 | yes | complemented |
| 280 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 281 | C2×C4⋊C4 | — | 3 | 6 | no | — |
| 282 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 283 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 284 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 285 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 286 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 287 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 288 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 289 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 290 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 291 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 292 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 293 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 294 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 295 | C2×C22⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 296 | C2×C22⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 297 | C2×C22⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 298 | C2×C42 | — | 3 | 3 | no | — |
| 299 | C2×C42 | — | 3 | 3 | no | — |
| 300 | #95 | — | 3 | 6 | no | — |
| 301 | #95 | — | 3 | 6 | no | — |
| 302 | #95 | — | 3 | 6 | no | — |
| 303 | C23×C3 | — | 2 | 1 | no | — |
| 304 | C23×C3 | — | 2 | 3 | no | — |
| 305 | C23×C3 | — | 2 | 24 | no | — |
| 306 | C23×C3 | — | 2 | 3 | no | — |
| 307 | C23×C3 | — | 2 | 6 | no | — |
| 308 | C23×C3 | — | 2 | 6 | no | — |
| 309 | C23×C3 | — | 2 | 6 | no | — |
| 310 | C23×C3 | — | 2 | 6 | no | — |
| 311 | C23×C3 | — | 2 | 3 | no | — |
| 312 | C23×C3 | — | 2 | 3 | no | — |
| 313 | C23×C3 | — | 2 | 3 | no | — |
| 314 | C23×C3 | — | 2 | 3 | no | — |
| 315 | C23×C3 | — | 2 | 3 | no | — |
| 316 | C23×C3 | — | 2 | 1 | no | — |
| 317 | C23×C3 | — | 2 | 3 | no | — |
| 318 | C23×C3 | — | 2 | 3 | no | — |
| 319 | C23×C3 | — | 4 | 12 | no | — |
| 320 | C23×C3 | — | 4 | 24 | no | — |
| 321 | C23×C3 | — | 4 | 24 | no | — |
| 322 | C23×C3 | — | 4 | 24 | no | — |
| 323 | C23×C3 | — | 4 | 12 | no | — |
| 324 | C23×C3 | — | 4 | 12 | no | — |
| 325 | C23×C3 | — | 4 | 12 | no | — |
| 326 | C23×C3 | — | 4 | 12 | no | — |
| 327 | C23×C3 | — | 4 | 24 | no | — |
| 328 | C23×C3 | — | 4 | 128 | no | — |
| 329 | C23×C3 | — | 4 | 4 | no | — |
| 330 | C23×C3 | — | 4 | 12 | no | — |
| 331 | C23×C3 | — | 4 | 12 | no | — |
| 332 | C23×C3 | — | 4 | 12 | no | — |
| 333 | C23×C3 | — | 4 | 4 | no | — |
| 334 | C23×C3 | — | 8 | 6 | no | — |
| 335 | C23×C3 | C22×D4 | 1 | 2 | yes | — |
| 336 | C23×C3 | C22×D4 | 1 | 6 | yes | — |
| 337 | C23×C3 | H(5,𝔽2) | 1 | 6 | yes | — |
| 338 | C23×C3 | H(5,𝔽2) | 1 | 6 | yes | — |
| 339 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 340 | C22×D3 | — | 2 | 2 | no | — |
| 341 | C22×D3 | — | 2 | 6 | no | — |
| 342 | C22×D3 | — | 2 | 6 | no | — |
| 343 | C22×D3 | — | 4 | 4 | no | — |
| 344 | C22×D3 | — | 8 | 6 | no | — |
| 345 | C22×D3 | — | 8 | 6 | no | — |
| 346 | C3×D4 | — | 4 | 12 | no | — |
| 347 | C3×D4 | — | 4 | 12 | no | — |
| 348 | C3×D4 | — | 4 | 12 | no | — |
| 349 | C3×D4 | — | 4 | 12 | no | — |
| 350 | C2×C3×C4 | — | 2 | 3 | no | — |
| 351 | C2×C3×C4 | — | 2 | 3 | no | — |
| 352 | C2×C3×C4 | — | 2 | 3 | no | — |
| 353 | C2×C3×C4 | — | 2 | 3 | no | — |
| 354 | C2×C3×C4 | — | 2 | 3 | no | — |
| 355 | C2×C3×C4 | — | 4 | 12 | no | — |
| 356 | C2×C3×C4 | C22×D4 | 1 | 6 | yes | — |
| 357 | #67 | — | 4 | 8 | no | — |
| 358 | #67 | — | 4 | 8 | no | — |
| 359 | #67 | — | 8 | 12 | no | — |
| 360 | #67 | — | 8 | 12 | no | — |
| 361 | C2×Dic3 | — | 2 | 6 | no | — |
| 362 | C2×Dic3 | — | 2 | 6 | no | — |
| 363 | C2×Dic3 | — | 2 | 6 | no | — |
| 364 | C2×Dic3 | — | 2 | 6 | no | — |
| 365 | C2×Dic3 | — | 2 | 6 | no | — |
| 366 | C2×Dic3 | — | 2 | 6 | no | — |
| 367 | C2×Dic3 | — | 2 | 6 | no | — |
| 368 | C2×Dic3 | — | 4 | 4 | no | — |
| 369 | Dic6 | — | 8 | 6 | no | — |
| 370 | C24 | — | 2 | 6 | no | — |
| 371 | C24 | — | 2 | 6 | no | — |
| 372 | C24 | — | 2 | 6 | no | — |
| 373 | C24 | — | 2 | 12 | no | — |
| 374 | C24 | — | 2 | 12 | no | — |
| 375 | C24 | — | 2 | 6 | no | — |
| 376 | C24 | — | 2 | 2 | no | — |
| 377 | C24 | — | 2 | 6 | no | — |
| 378 | C24 | — | 3 | 2 | no | — |
| 379 | C24 | — | 4 | 6 | no | — |
| 380 | C24 | — | 4 | 3 | no | — |
| 381 | C24 | — | 4 | 3 | no | — |
| 382 | C24 | — | 4 | 3 | no | — |
| 383 | C24 | — | 4 | 6 | no | — |
| 384 | C24 | — | 4 | 3 | no | — |
| 385 | C24 | — | 4 | 3 | no | — |
| 386 | C24 | — | 4 | 48 | no | — |
| 387 | C24 | — | 4 | 48 | no | — |
| 388 | C24 | — | 4 | 16 | no | — |
| 389 | C24 | — | 6 | 2 | no | — |
| 390 | C24 | — | 12 | 3 | no | — |
| 391 | C24 | — | 12 | 3 | no | — |
| 392 | C24 | C2×#67 | 1 | 4 | yes | complemented |
| 393 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 394 | C24 | C23×D3 | 1 | 3 | yes | — |
| 395 | C24 | C23×D3 | 1 | 3 | yes | — |
| 396 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 397 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 398 | C2×Q8 | — | 6 | 6 | no | — |
| 399 | C2×Q8 | — | 12 | 3 | no | — |
| 400 | C2×Q8 | — | 12 | 3 | no | — |
| 401 | C2×D4 | — | 2 | 6 | no | — |
| 402 | C2×D4 | — | 2 | 6 | no | — |
| 403 | C2×D4 | — | 2 | 6 | no | — |
| 404 | C2×D4 | — | 2 | 6 | no | — |
| 405 | C2×D4 | — | 2 | 6 | no | — |
| 406 | C2×D4 | — | 4 | 3 | no | — |
| 407 | C2×D4 | — | 4 | 3 | no | — |
| 408 | C2×D4 | — | 4 | 3 | no | — |
| 409 | C2×D4 | — | 4 | 3 | no | — |
| 410 | C2×D4 | — | 4 | 3 | no | — |
| 411 | C2×D4 | — | 4 | 3 | no | — |
| 412 | C2×D4 | — | 4 | 48 | no | — |
| 413 | C2×D4 | — | 6 | 12 | no | — |
| 414 | C2×D4 | — | 6 | 12 | no | — |
| 415 | C2×D4 | — | 6 | 12 | no | — |
| 416 | C2×D4 | — | 12 | 6 | no | — |
| 417 | C2×D4 | — | 12 | 6 | no | — |
| 418 | C2×D4 | — | 12 | 6 | no | — |
| 419 | C2×D4 | — | 12 | 6 | no | — |
| 420 | C2×D4 | — | 12 | 6 | no | — |
| 421 | C2×D4 | — | 12 | 16 | no | — |
| 422 | C2×D4 | C2×#67 | 1 | 12 | yes | — |
| 423 | C22×C4 | — | 2 | 6 | no | — |
| 424 | C22×C4 | — | 2 | 6 | no | — |
| 425 | C22×C4 | — | 3 | 2 | no | — |
| 426 | C22×C4 | — | 3 | 6 | no | — |
| 427 | C22×C4 | — | 3 | 6 | no | — |
| 428 | C22×C4 | — | 4 | 3 | no | — |
| 429 | C22×C4 | — | 12 | 3 | no | — |
| 430 | C22×C4 | C23×D3 | 1 | 3 | yes | — |
| 431 | C22×C4 | C23×D3 | 1 | 3 | yes | — |
| 432 | C4⋊C4 | — | 12 | 6 | no | — |
| 433 | C4⋊C4 | — | 12 | 3 | no | — |
| 434 | C22⋊C4 | — | 2 | 6 | no | — |
| 435 | C22⋊C4 | — | 2 | 12 | no | — |
| 436 | C22⋊C4 | — | 6 | 12 | no | — |
| 437 | C22⋊C4 | — | 6 | 12 | no | — |
| 438 | C22⋊C4 | — | 6 | 12 | no | — |
| 439 | C22⋊C4 | — | 6 | 6 | no | — |
| 440 | C22⋊C4 | — | 6 | 12 | no | — |
| 441 | C22⋊C4 | — | 12 | 6 | no | — |
| 442 | C22⋊C4 | — | 12 | 6 | no | — |
| 443 | C22⋊C4 | — | 12 | 6 | no | — |
| 444 | C22⋊C4 | — | 12 | 6 | no | — |
| 445 | C22⋊C4 | — | 12 | 6 | no | — |
| 446 | C22⋊C4 | — | 12 | 3 | no | — |
| 447 | C22⋊C4 | — | 12 | 6 | no | — |
| 448 | C22⋊C4 | C2×#67 | 1 | 12 | yes | complemented |
| 449 | C42 | — | 6 | 6 | no | — |
| 450 | C42 | — | 12 | 3 | no | — |
| 451 | C22×C3 | — | 2 | 6 | no | — |
| 452 | C22×C3 | — | 2 | 6 | no | — |
| 453 | C22×C3 | — | 2 | 12 | no | — |
| 454 | C22×C3 | — | 2 | 6 | no | — |
| 455 | C22×C3 | — | 2 | 6 | no | — |
| 456 | C22×C3 | — | 2 | 6 | no | — |
| 457 | C22×C3 | — | 2 | 12 | no | — |
| 458 | C22×C3 | — | 4 | 48 | no | — |
| 459 | C22×C3 | — | 4 | 16 | no | — |
| 460 | C22×C3 | — | 4 | 48 | no | — |
| 461 | C22×C3 | — | 4 | 6 | no | — |
| 462 | C22×C3 | — | 4 | 1 | no | — |
| 463 | C22×C3 | — | 4 | 3 | no | — |
| 464 | C22×C3 | — | 4 | 3 | no | — |
| 465 | C22×C3 | — | 4 | 3 | no | — |
| 466 | C22×C3 | — | 4 | 6 | no | — |
| 467 | C22×C3 | — | 4 | 3 | no | — |
| 468 | C22×C3 | — | 4 | 3 | no | — |
| 469 | C22×C3 | — | 4 | 3 | no | — |
| 470 | C22×C3 | — | 4 | 1 | no | — |
| 471 | C22×C3 | — | 4 | 3 | no | — |
| 472 | C22×C3 | — | 8 | 8 | no | — |
| 473 | C22×C3 | H(3,𝔽4) | 1 | 4 | yes | complemented |
| 474 | C22×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 475 | C22×C3 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 476 | C22×C3 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 477 | C2×D3 | — | 4 | 6 | no | — |
| 478 | C2×D3 | — | 4 | 2 | no | — |
| 479 | C2×D3 | — | 4 | 6 | no | — |
| 480 | C2×D3 | — | 8 | 8 | no | — |
| 481 | C3×C4 | — | 2 | 6 | no | — |
| 482 | C3×C4 | — | 4 | 3 | no | — |
| 483 | Dic3 | — | 4 | 6 | no | — |
| 484 | Dic3 | — | 4 | 2 | no | — |
| 485 | Dic3 | — | 4 | 6 | no | — |
| 486 | C23 | — | 2 | 3 | no | — |
| 487 | C23 | — | 2 | 1 | no | — |
| 488 | C23 | — | 2 | 24 | no | — |
| 489 | C23 | — | 2 | 3 | no | — |
| 490 | C23 | — | 2 | 6 | no | — |
| 491 | C23 | — | 2 | 6 | no | — |
| 492 | C23 | — | 2 | 6 | no | — |
| 493 | C23 | — | 2 | 6 | no | — |
| 494 | C23 | — | 2 | 3 | no | — |
| 495 | C23 | — | 2 | 3 | no | — |
| 496 | C23 | — | 2 | 3 | no | — |
| 497 | C23 | — | 2 | 3 | no | — |
| 498 | C23 | — | 2 | 3 | no | — |
| 499 | C23 | — | 2 | 3 | no | — |
| 500 | C23 | — | 2 | 3 | no | — |
| 501 | C23 | — | 2 | 1 | no | — |
| 502 | C23 | — | 4 | 12 | no | — |
| 503 | C23 | — | 4 | 24 | no | — |
| 504 | C23 | — | 4 | 24 | no | — |
| 505 | C23 | — | 4 | 12 | no | — |
| 506 | C23 | — | 4 | 24 | no | — |
| 507 | C23 | — | 4 | 12 | no | — |
| 508 | C23 | — | 4 | 12 | no | — |
| 509 | C23 | — | 4 | 24 | no | — |
| 510 | C23 | — | 4 | 12 | no | — |
| 511 | C23 | — | 4 | 128 | no | — |
| 512 | C23 | — | 4 | 4 | no | — |
| 513 | C23 | — | 4 | 12 | no | — |
| 514 | C23 | — | 4 | 12 | no | — |
| 515 | C23 | — | 4 | 12 | no | — |
| 516 | C23 | — | 4 | 4 | no | — |
| 517 | C23 | — | 6 | 6 | no | — |
| 518 | C23 | — | 6 | 6 | no | — |
| 519 | C23 | — | 6 | 2 | no | — |
| 520 | C23 | — | 8 | 6 | no | — |
| 521 | C23 | — | 12 | 4 | no | — |
| 522 | C23 | — | 24 | 6 | no | — |
| 523 | C23 | — | 24 | 6 | no | — |
| 524 | C23 | #1004 | 1 | 6 | yes | — |
| 525 | C23 | #1004 | 1 | 6 | yes | — |
| 526 | C23 | C22×#67 | 1 | 6 | yes | — |
| 527 | C23 | C22×#67 | 1 | 2 | yes | — |
| 528 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 529 | Q8 | — | 24 | 6 | no | — |
| 530 | D4 | — | 4 | 12 | no | — |
| 531 | D4 | — | 4 | 12 | no | — |
| 532 | D4 | — | 4 | 12 | no | — |
| 533 | D4 | — | 4 | 12 | no | — |
| 534 | D4 | — | 12 | 8 | no | — |
| 535 | D4 | — | 12 | 8 | no | — |
| 536 | D4 | — | 24 | 12 | no | — |
| 537 | D4 | — | 24 | 12 | no | — |
| 538 | C2×C4 | — | 2 | 3 | no | — |
| 539 | C2×C4 | — | 2 | 3 | no | — |
| 540 | C2×C4 | — | 2 | 3 | no | — |
| 541 | C2×C4 | — | 2 | 3 | no | — |
| 542 | C2×C4 | — | 2 | 3 | no | — |
| 543 | C2×C4 | — | 4 | 12 | no | — |
| 544 | C2×C4 | — | 6 | 6 | no | — |
| 545 | C2×C4 | — | 6 | 6 | no | — |
| 546 | C2×C4 | — | 6 | 6 | no | — |
| 547 | C2×C4 | — | 6 | 6 | no | — |
| 548 | C2×C4 | — | 6 | 6 | no | — |
| 549 | C2×C4 | — | 6 | 6 | no | — |
| 550 | C2×C4 | — | 6 | 6 | no | — |
| 551 | C2×C4 | — | 12 | 4 | no | — |
| 552 | C2×C4 | C22×#67 | 1 | 6 | yes | — |
| 553 | C2×C3 | — | 2 | 4 | no | — |
| 554 | C2×C3 | — | 4 | 2 | no | — |
| 555 | C2×C3 | — | 4 | 6 | no | — |
| 556 | C2×C3 | — | 4 | 6 | no | — |
| 557 | C2×C3 | — | 8 | 1 | no | — |
| 558 | C2×C3 | #3484 | 1 | 3 | yes | — |
| 559 | C2×C3 | C2×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 560 | C2×C3 | #3532 | 1 | 3 | yes | — |
| 561 | D3 | — | 8 | 2 | no | — |
| 562 | C22 | — | 2 | 6 | no | — |
| 563 | C22 | — | 2 | 6 | no | — |
| 564 | C22 | — | 2 | 12 | no | — |
| 565 | C22 | — | 2 | 6 | no | — |
| 566 | C22 | — | 2 | 6 | no | — |
| 567 | C22 | — | 2 | 6 | no | — |
| 568 | C22 | — | 2 | 12 | no | — |
| 569 | C22 | — | 4 | 48 | no | — |
| 570 | C22 | — | 4 | 16 | no | — |
| 571 | C22 | — | 4 | 48 | no | — |
| 572 | C22 | — | 4 | 1 | no | — |
| 573 | C22 | — | 4 | 3 | no | — |
| 574 | C22 | — | 4 | 3 | no | — |
| 575 | C22 | — | 4 | 6 | no | — |
| 576 | C22 | — | 4 | 3 | no | — |
| 577 | C22 | — | 4 | 3 | no | — |
| 578 | C22 | — | 4 | 3 | no | — |
| 579 | C22 | — | 4 | 6 | no | — |
| 580 | C22 | — | 4 | 1 | no | — |
| 581 | C22 | — | 4 | 3 | no | — |
| 582 | C22 | — | 4 | 3 | no | — |
| 583 | C22 | — | 8 | 8 | no | — |
| 584 | C22 | — | 12 | 2 | no | — |
| 585 | C22 | — | 12 | 6 | no | — |
| 586 | C22 | — | 12 | 6 | no | — |
| 587 | C22 | — | 24 | 8 | no | — |
| 588 | C22 | #5594 | 1 | 4 | yes | complemented |
| 589 | C22 | C2×#1004 | 1 | 3 | yes | — |
| 590 | C22 | C2×#1004 | 1 | 3 | yes | — |
| 591 | C22 | C23×#67 | 1 | 1 | yes | characteristic |
| 592 | C4 | — | 2 | 6 | no | — |
| 593 | C4 | — | 4 | 3 | no | — |
| 594 | C4 | — | 12 | 2 | no | — |
| 595 | C4 | — | 12 | 6 | no | — |
| 596 | C4 | — | 12 | 6 | no | — |
| 597 | C3 | #45439 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 598 | C2 | — | 2 | 4 | no | — |
| 599 | C2 | — | 4 | 6 | no | — |
| 600 | C2 | — | 4 | 6 | no | — |
| 601 | C2 | — | 4 | 2 | no | — |
| 602 | C2 | — | 8 | 1 | no | — |
| 603 | C2 | — | 24 | 2 | no | — |
| 604 | C2 | C2×#5594 | 1 | 1 | yes | characteristic |
| 605 | C2 | #85921 | 1 | 3 | yes | — |
| 606 | C2 | #87239 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C22×C3×#120 | C2 |
| hypercenter | C22×#120 | D3 |
| hypocenter | C3 | #45439 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#120 | C2 | semidirect |
| 2. C2×#5594 | C2 | semidirect |
| 3. #84128 | C2 | semidirect |
| 4. #84052 | C2 | semidirect |
| 5. #84050 | C2 | semidirect |
| 6. #82611 | C2 | semidirect |
| 7. #82598 | C2 | semidirect |
| 8. #82597 | C2 | semidirect |
| 9. #82595 | C2 | semidirect |
| 10. #82580 | C2 | semidirect |
| 11. #82111 | C2 | semidirect |
| 13. C26×C3 | C22 | semidirect |
| 15. C23×#67 | C22 | semidirect |
| 19. C2×C3×#120 | C22 | semidirect |
| 20. C2×C3×#120 | C22 | semidirect |
| 21. C22×C3×C22⋊C4 | C22 | semidirect |
| 23. C2×#953 | C22 | semidirect |
| 24. C2×#953 | C22 | semidirect |
| 25. C2×#937 | C22 | semidirect |
| 26. C2×#937 | C22 | semidirect |
| 27. C2×#936 | C22 | semidirect |
| 28. C2×#936 | C22 | semidirect |
| 29. C2×#924 | C22 | semidirect |
| 31. C2×#885 | C22 | semidirect |
| 32. #5247 | C22 | semidirect |
| 33. #5247 | C22 | semidirect |
| 34. #5247 | C22 | semidirect |
| 35. #5247 | C22 | semidirect |
| 36. #5223 | C22 | semidirect |
| 37. #5223 | C22 | semidirect |
| 38. #5222 | C22 | semidirect |
| 39. #5222 | C22 | semidirect |
| 40. #5221 | C22 | semidirect |
| 41. #5221 | C22 | semidirect |
| 42. #5220 | C22 | semidirect |
| 43. #5220 | C22 | semidirect |
| 44. #5213 | C22 | semidirect |
| 45. C4×#217 | C22 | semidirect |
| 46. #4961 | C22 | semidirect |
| 47. #4959 | C22 | semidirect |
| 48. #4957 | C22 | semidirect |
| 49. #4952 | C22 | semidirect |
| 50. #4951 | C22 | semidirect |
| 51. #4947 | C22 | semidirect |
| 52. #4944 | C22 | semidirect |
| 54. C22×#120 | D3 | semidirect |
| 71. C24×D3 | C23 | semidirect |
| 84. C22×#67 | C23 | semidirect |
| 85. C23×Dic3 | C23 | semidirect |
| 86. C22×Dic6 | C23 | semidirect |
| 90. C3×#120 | D4 | semidirect |
| 105. C2×#217 | C23 | semidirect |
| 106. C2×#217 | C23 | semidirect |
| 107. C2×#217 | C23 | semidirect |
| 112. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 113. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 114. C2×C4⋊Dic3 | C23 | semidirect |
| 116. C2×Dic3⋊C4 | C23 | semidirect |
| 117. C2×C4×Dic3 | C23 | semidirect |
| 118. C2×C4×Dic3 | C23 | semidirect |
| 120. C3⋊#95 | C23 | semidirect |
| 121. C3⋊#95 | C23 | semidirect |
| 122. C3⋊#95 | C23 | semidirect |
| 123. C26 | C2×D3 | semidirect |
| 139. C2×#120 | C2×D3 | semidirect |
| 140. C2×#120 | C2×D3 | semidirect |
| 141. C22×C22⋊C4 | C2×D3 | semidirect |
| 181. C24×C3 | C2×D4 | semidirect |
| 227. C3×C22⋊C4 | C2×D4 | semidirect |
| 279. #120 | #67 | semidirect |
| 392. C24 | C2×#67 | semidirect |
| 448. C22⋊C4 | C2×#67 | semidirect |
| 473. C22×C3 | H(3,𝔽4) | semidirect |
| 588. C22 | #5594 | semidirect |
| 597. C3 | #45439 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#120 | 2 |
| C2×#5594 | 2 |
| #84128 | 2 |
| #84052 | 2 |
| #84050 | 2 |
| #82611 | 2 |
| #82598 | 2 |
| #82597 | 2 |
| #82595 | 2 |
| #82580 | 2 |
| #82111 | 2 |
| #45439 | 3 |