Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1290 conjugacy classes of subgroups.
The subgroup lattice has 504 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#3584 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #84121 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #83928 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #83787 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72145 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #72121 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #72117 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #72085 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #72014 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #24288 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C3×#585 | C22 | 1 | 2 | yes | — |
| 12 | C3×#578 | — | 2 | 2 | no | — |
| 13 | C3×#577 | — | 2 | 4 | no | — |
| 14 | C3×#576 | — | 2 | 2 | no | — |
| 15 | C3×#575 | — | 2 | 2 | no | — |
| 16 | C3×#575 | C22 | 1 | 2 | yes | — |
| 17 | C2×C3×#135 | — | 2 | 2 | no | — |
| 18 | C2×C3×#135 | C22 | 1 | 2 | yes | — |
| 19 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 20 | #5832 | C22 | 1 | 1 | yes | characteristic |
| 21 | #5822 | C22 | 1 | 4 | yes | complemented |
| 22 | C2×#901 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #5246 | — | 2 | 2 | no | — |
| 24 | #5245 | — | 2 | 2 | no | — |
| 25 | #5242 | — | 2 | 2 | no | — |
| 26 | #5241 | — | 2 | 2 | no | — |
| 27 | #5237 | C22 | 1 | 2 | yes | — |
| 28 | #5237 | C22 | 1 | 4 | yes | complemented |
| 29 | C2×#837 | C22 | 1 | 4 | yes | complemented |
| 30 | #5234 | C22 | 1 | 1 | yes | characteristic |
| 31 | #5234 | C22 | 1 | 4 | yes | complemented |
| 32 | #5192 | — | 2 | 2 | no | — |
| 33 | #5177 | — | 2 | 4 | no | — |
| 34 | #5160 | — | 2 | 2 | no | — |
| 35 | #5129 | C22 | 1 | 1 | yes | characteristic, complemented |
| 36 | #5127 | — | 2 | 2 | no | — |
| 37 | #5118 | C22 | 1 | 1 | yes | characteristic, complemented |
| 38 | #5111 | — | 2 | 2 | no | — |
| 39 | C2×#819 | C22 | 1 | 2 | yes | complemented |
| 40 | C2×#815 | C22 | 1 | 1 | yes | characteristic, complemented |
| 41 | C3⋊#338 | C22 | 1 | 2 | yes | complemented |
| 42 | #3584 | D3 | 1 | 1 | yes | characteristic, complemented |
| 43 | #3067 | — | 3 | 2 | no | — |
| 44 | #3018 | — | 3 | 2 | no | — |
| 45 | #2932 | — | 3 | 1 | no | — |
| 46 | #2050 | — | 3 | 2 | no | — |
| 47 | #2017 | — | 3 | 2 | no | — |
| 48 | #1975 | — | 3 | 2 | no | — |
| 49 | #1813 | — | 3 | 1 | no | — |
| 50 | #1791 | — | 3 | 2 | no | — |
| 51 | C3×2−1+4 | — | 2 | 4 | no | — |
| 52 | C3×2−1+4 | — | 2 | 2 | no | — |
| 53 | C3×H(5,𝔽2) | — | 2 | 4 | no | — |
| 54 | C3×H(5,𝔽2) | — | 2 | 4 | no | — |
| 55 | C3×H(5,𝔽2) | — | 2 | 2 | no | — |
| 56 | C2×C3×#41 | — | 2 | 4 | no | — |
| 57 | C2×C3×#41 | — | 4 | 4 | no | — |
| 58 | C2×C3×#41 | C23 | 1 | 4 | yes | — |
| 59 | C2×C3×#41 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×#235 | C23 | 1 | 1 | yes | characteristic |
| 61 | C3×#137 | — | 2 | 8 | no | — |
| 62 | C3×#137 | — | 2 | 4 | no | — |
| 63 | C3×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 64 | C3×AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 65 | C3×#135 | — | 2 | 4 | no | — |
| 66 | C3×#135 | — | 2 | 4 | no | — |
| 67 | C3×#135 | — | 2 | 2 | no | — |
| 68 | C3×#135 | — | 2 | 4 | no | — |
| 69 | C3×#135 | — | 2 | 2 | no | — |
| 70 | C3×#135 | — | 2 | 2 | no | — |
| 71 | C3×#135 | — | 2 | 2 | no | — |
| 72 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 73 | C2×C3×Q16 | D4 | 1 | 2 | yes | complemented |
| 74 | C2×C3×SD16 | — | 2 | 4 | no | — |
| 75 | C2×C3×SD16 | D4 | 1 | 4 | yes | complemented |
| 76 | C2×C3×D8 | — | 2 | 2 | no | — |
| 77 | C2×C3×D8 | D4 | 1 | 2 | yes | complemented |
| 78 | C3×#131 | — | 2 | 2 | no | — |
| 79 | C3×#131 | — | 2 | 2 | no | — |
| 80 | C2×C3×M16 | D4 | 1 | 2 | yes | — |
| 81 | C2×C3×M16 | C23 | 1 | 1 | yes | characteristic |
| 82 | C22×C3×C8 | D4 | 1 | 2 | yes | — |
| 83 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 84 | #951 | — | 2 | 4 | no | — |
| 85 | #950 | — | 4 | 4 | no | — |
| 86 | #949 | — | 2 | 4 | no | — |
| 87 | #947 | D4 | 1 | 1 | yes | characteristic |
| 88 | #944 | D4 | 1 | 1 | yes | characteristic |
| 89 | #942 | — | 2 | 4 | no | — |
| 90 | #940 | D4 | 1 | 1 | yes | characteristic |
| 91 | #939 | — | 2 | 2 | no | — |
| 92 | #936 | D4 | 1 | 1 | yes | characteristic |
| 93 | #932 | — | 2 | 4 | no | — |
| 94 | C4×#67 | — | 2 | 2 | no | — |
| 95 | C2×C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 96 | C2×#208 | C23 | 1 | 1 | yes | characteristic |
| 97 | C2×#208 | C23 | 1 | 4 | yes | — |
| 98 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 99 | #901 | — | 2 | 2 | no | — |
| 100 | #901 | — | 2 | 2 | no | — |
| 101 | C2×ΓL(1,𝔽25) | — | 2 | 1 | no | — |
| 102 | C2×D3×C8 | — | 2 | 1 | no | — |
| 103 | #837 | — | 2 | 4 | no | — |
| 104 | #837 | — | 2 | 4 | no | — |
| 105 | #836 | — | 2 | 4 | no | — |
| 106 | #836 | — | 2 | 2 | no | — |
| 107 | #833 | — | 2 | 4 | no | — |
| 108 | #833 | — | 2 | 2 | no | — |
| 109 | #830 | — | 2 | 2 | no | — |
| 110 | #830 | — | 4 | 4 | no | — |
| 111 | (C2×D3)⋊C8 | — | 2 | 2 | no | — |
| 112 | #819 | — | 2 | 2 | no | — |
| 113 | #819 | — | 2 | 2 | no | — |
| 114 | #815 | — | 2 | 1 | no | — |
| 115 | #815 | — | 2 | 1 | no | — |
| 116 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 117 | #585 | C2×D3 | 1 | 2 | yes | — |
| 118 | #578 | — | 2 | 2 | no | — |
| 119 | #577 | — | 2 | 4 | no | — |
| 120 | #576 | — | 2 | 2 | no | — |
| 121 | #575 | — | 2 | 2 | no | — |
| 122 | #575 | — | 3 | 4 | no | — |
| 123 | #575 | C2×D3 | 1 | 2 | yes | — |
| 124 | C2×#135 | — | 2 | 2 | no | — |
| 125 | C2×#135 | C2×D3 | 1 | 2 | yes | — |
| 126 | C2×#131 | — | 3 | 1 | no | — |
| 127 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 128 | #532 | — | 3 | 1 | no | — |
| 129 | #473 | — | 6 | 2 | no | — |
| 130 | #472 | — | 6 | 2 | no | — |
| 131 | #471 | — | 6 | 4 | no | — |
| 132 | #456 | — | 6 | 2 | no | — |
| 133 | #455 | — | 6 | 2 | no | — |
| 134 | #454 | — | 6 | 2 | no | — |
| 135 | #453 | — | 6 | 2 | no | — |
| 136 | #435 | — | 6 | 2 | no | — |
| 137 | #435 | — | 6 | 2 | no | — |
| 138 | C2×#108 | — | 3 | 2 | no | — |
| 139 | #423 | — | 3 | 1 | no | — |
| 140 | C2×#104 | — | 3 | 4 | no | — |
| 141 | #413 | — | 3 | 4 | no | — |
| 142 | #413 | — | 3 | 2 | no | — |
| 143 | #408 | — | 3 | 1 | no | — |
| 144 | #408 | — | 3 | 1 | no | — |
| 145 | #408 | — | 3 | 4 | no | — |
| 146 | C2×#97 | — | 3 | 1 | no | — |
| 147 | #338 | — | 3 | 2 | no | — |
| 148 | C3×#41 | — | 2 | 2 | no | — |
| 149 | C3×#41 | — | 2 | 2 | no | — |
| 150 | C3×#41 | — | 2 | 8 | no | — |
| 151 | C3×#41 | — | 2 | 4 | no | — |
| 152 | C3×#41 | — | 2 | 4 | no | — |
| 153 | C3×#41 | — | 2 | 4 | no | — |
| 154 | C3×#41 | — | 2 | 4 | no | — |
| 155 | C3×#41 | — | 4 | 8 | no | — |
| 156 | C3×#41 | — | 4 | 4 | no | — |
| 157 | C3×#41 | — | 4 | 4 | no | — |
| 158 | C3×#41 | — | 4 | 4 | no | — |
| 159 | C3×#41 | — | 4 | 4 | no | — |
| 160 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 161 | C2×C3×Q8 | — | 4 | 4 | no | — |
| 162 | C2×C3×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 163 | C2×C3×Q8 | C2×D4 | 1 | 4 | yes | — |
| 164 | C2×C3×D4 | — | 2 | 4 | no | — |
| 165 | C2×C3×D4 | — | 2 | 4 | no | — |
| 166 | C2×C3×D4 | — | 4 | 4 | no | — |
| 167 | C2×C3×D4 | — | 4 | 8 | no | — |
| 168 | C2×C3×D4 | — | 4 | 4 | no | — |
| 169 | C2×C3×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C2×C3×D4 | C2×D4 | 1 | 4 | yes | — |
| 171 | C2×C3×D4 | #41 | 1 | 2 | yes | — |
| 172 | C22×C3×C4 | — | 2 | 4 | no | — |
| 173 | C22×C3×C4 | — | 4 | 4 | no | — |
| 174 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 175 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 176 | C2×#67 | — | 2 | 1 | no | — |
| 177 | C2×#67 | — | 2 | 2 | no | — |
| 178 | C22×Dic3 | — | 2 | 1 | no | — |
| 179 | #235 | — | 2 | 2 | no | — |
| 180 | #235 | — | 2 | 2 | no | — |
| 181 | C2×D12 | C2×D4 | 1 | 1 | yes | characteristic |
| 182 | C2×C4×D3 | — | 2 | 1 | no | — |
| 183 | C2×Dic6 | C2×D4 | 1 | 1 | yes | characteristic |
| 184 | C3×Q16 | — | 2 | 4 | no | — |
| 185 | C3×Q16 | — | 2 | 2 | no | — |
| 186 | C3×Q16 | — | 2 | 2 | no | — |
| 187 | C3×SD16 | — | 2 | 8 | no | — |
| 188 | C3×SD16 | — | 2 | 4 | no | — |
| 189 | C3×SD16 | — | 2 | 4 | no | — |
| 190 | C3×D8 | — | 2 | 4 | no | — |
| 191 | C3×D8 | — | 2 | 2 | no | — |
| 192 | C3×D8 | — | 2 | 2 | no | — |
| 193 | C3×M16 | — | 2 | 2 | no | — |
| 194 | C3×M16 | — | 2 | 4 | no | — |
| 195 | C2×C3×C8 | — | 2 | 2 | no | — |
| 196 | C2×C3×C8 | — | 2 | 2 | no | — |
| 197 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 198 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 199 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 200 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 201 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 202 | #217 | — | 2 | 2 | no | — |
| 203 | #216 | — | 4 | 4 | no | — |
| 204 | #215 | — | 4 | 4 | no | — |
| 205 | #214 | — | 4 | 4 | no | — |
| 206 | #213 | — | 4 | 4 | no | — |
| 207 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 208 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 209 | C4×Dic3 | — | 2 | 1 | no | — |
| 210 | C4×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 211 | C4×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 212 | #208 | — | 2 | 4 | no | — |
| 213 | #208 | — | 2 | 2 | no | — |
| 214 | #208 | — | 2 | 4 | no | — |
| 215 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 216 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 217 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 218 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 219 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 220 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 221 | D3×C8 | — | 4 | 2 | no | — |
| 222 | 2−1+4 | — | 2 | 4 | no | — |
| 223 | 2−1+4 | — | 2 | 2 | no | — |
| 224 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 225 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 226 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 227 | C2×#41 | — | 2 | 4 | no | — |
| 228 | C2×#41 | — | 3 | 1 | no | — |
| 229 | C2×#41 | — | 4 | 4 | no | — |
| 230 | C2×#41 | C22×D3 | 1 | 4 | yes | — |
| 231 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 232 | #137 | — | 2 | 8 | no | — |
| 233 | #137 | — | 2 | 4 | no | — |
| 234 | #137 | — | 6 | 4 | no | — |
| 235 | #137 | — | 6 | 4 | no | — |
| 236 | AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 237 | AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 238 | AGL(1,ℤ/8) | — | 6 | 4 | no | — |
| 239 | AGL(1,ℤ/8) | — | 6 | 4 | no | — |
| 240 | #135 | — | 2 | 4 | no | — |
| 241 | #135 | — | 2 | 4 | no | — |
| 242 | #135 | — | 2 | 2 | no | — |
| 243 | #135 | — | 2 | 4 | no | — |
| 244 | #135 | — | 2 | 2 | no | — |
| 245 | #135 | — | 2 | 2 | no | — |
| 246 | #135 | — | 2 | 2 | no | — |
| 247 | #135 | — | 12 | 4 | no | — |
| 248 | C2×Q16 | — | 2 | 2 | no | — |
| 249 | C2×Q16 | #67 | 1 | 2 | yes | complemented |
| 250 | C2×SD16 | — | 2 | 4 | no | — |
| 251 | C2×SD16 | #67 | 1 | 4 | yes | complemented |
| 252 | C2×D8 | — | 2 | 2 | no | — |
| 253 | C2×D8 | #67 | 1 | 2 | yes | complemented |
| 254 | #131 | — | 2 | 2 | no | — |
| 255 | #131 | — | 2 | 2 | no | — |
| 256 | #131 | — | 6 | 2 | no | — |
| 257 | #131 | — | 6 | 2 | no | — |
| 258 | C2×M16 | — | 3 | 1 | no | — |
| 259 | C2×M16 | — | 3 | 4 | no | — |
| 260 | C2×M16 | — | 6 | 1 | no | — |
| 261 | C2×M16 | #67 | 1 | 2 | yes | — |
| 262 | C2×M16 | C22×D3 | 1 | 1 | yes | characteristic |
| 263 | C22×C8 | — | 3 | 1 | no | — |
| 264 | C22×C8 | — | 6 | 1 | no | — |
| 265 | C22×C8 | #67 | 1 | 2 | yes | — |
| 266 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 267 | C4⋊Q8 | — | 3 | 1 | no | — |
| 268 | #127 | — | 3 | 1 | no | — |
| 269 | #124 | — | 3 | 1 | no | — |
| 270 | #124 | — | 3 | 1 | no | — |
| 271 | #121 | — | 6 | 2 | no | — |
| 272 | C4×D4 | — | 6 | 2 | no | — |
| 273 | C2×C42 | — | 3 | 1 | no | — |
| 274 | #108 | — | 6 | 2 | no | — |
| 275 | #108 | — | 6 | 2 | no | — |
| 276 | C4⋊C8 | — | 6 | 2 | no | — |
| 277 | #104 | — | 6 | 4 | no | — |
| 278 | #104 | — | 6 | 4 | no | — |
| 279 | #101 | — | 6 | 2 | no | — |
| 280 | #101 | — | 6 | 4 | no | — |
| 281 | #100 | — | 6 | 2 | no | — |
| 282 | #100 | — | 6 | 4 | no | — |
| 283 | C22⋊C8 | — | 6 | 2 | no | — |
| 284 | C22⋊C8 | — | 6 | 2 | no | — |
| 285 | C22⋊C8 | — | 12 | 4 | no | — |
| 286 | #97 | — | 6 | 1 | no | — |
| 287 | #97 | — | 6 | 1 | no | — |
| 288 | C23×C3 | — | 2 | 4 | no | — |
| 289 | C23×C3 | — | 4 | 4 | no | — |
| 290 | C23×C3 | #121 | 1 | 2 | yes | — |
| 291 | C23×C3 | C2×#41 | 1 | 1 | yes | characteristic |
| 292 | C22×D3 | — | 2 | 1 | no | — |
| 293 | C3×Q8 | — | 2 | 2 | no | — |
| 294 | C3×Q8 | — | 2 | 4 | no | — |
| 295 | C3×Q8 | — | 2 | 4 | no | — |
| 296 | C3×Q8 | — | 4 | 4 | no | — |
| 297 | C3×D4 | — | 2 | 4 | no | — |
| 298 | C3×D4 | — | 2 | 4 | no | — |
| 299 | C3×D4 | — | 2 | 2 | no | — |
| 300 | C3×D4 | — | 2 | 4 | no | — |
| 301 | C3×D4 | — | 4 | 4 | no | — |
| 302 | C3×D4 | — | 4 | 8 | no | — |
| 303 | C3×D4 | — | 4 | 4 | no | — |
| 304 | C3×D4 | — | 4 | 4 | no | — |
| 305 | C2×C3×C4 | — | 2 | 2 | no | — |
| 306 | C2×C3×C4 | — | 2 | 2 | no | — |
| 307 | C2×C3×C4 | — | 2 | 4 | no | — |
| 308 | C2×C3×C4 | — | 4 | 8 | no | — |
| 309 | C2×C3×C4 | — | 4 | 4 | no | — |
| 310 | C2×C3×C4 | — | 4 | 4 | no | — |
| 311 | C2×C3×C4 | — | 4 | 4 | no | — |
| 312 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 313 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 314 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 315 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 316 | C2×C3×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 317 | #67 | — | 4 | 2 | no | — |
| 318 | #67 | — | 8 | 2 | no | — |
| 319 | C2×Dic3 | — | 2 | 1 | no | — |
| 320 | C2×Dic3 | — | 2 | 1 | no | — |
| 321 | C2×Dic3 | — | 2 | 1 | no | — |
| 322 | C2×Dic3 | — | 8 | 1 | no | — |
| 323 | D12 | — | 2 | 2 | no | — |
| 324 | C4×D3 | — | 4 | 1 | no | — |
| 325 | C4×D3 | — | 4 | 1 | no | — |
| 326 | Dic6 | — | 2 | 2 | no | — |
| 327 | C3×C8 | — | 2 | 2 | no | — |
| 328 | C3×C8 | #120 | 1 | 2 | yes | — |
| 329 | C3×C8 | #120 | 1 | 1 | yes | characteristic |
| 330 | C3×C8 | #120 | 1 | 1 | yes | characteristic |
| 331 | C3⋊C8 | — | 4 | 1 | no | — |
| 332 | C3⋊C8 | — | 4 | 1 | no | — |
| 333 | C3⋊C8 | — | 4 | 4 | no | — |
| 334 | #41 | — | 2 | 2 | no | — |
| 335 | #41 | — | 2 | 2 | no | — |
| 336 | #41 | — | 2 | 8 | no | — |
| 337 | #41 | — | 2 | 4 | no | — |
| 338 | #41 | — | 2 | 4 | no | — |
| 339 | #41 | — | 2 | 4 | no | — |
| 340 | #41 | — | 2 | 4 | no | — |
| 341 | #41 | — | 4 | 8 | no | — |
| 342 | #41 | — | 4 | 4 | no | — |
| 343 | #41 | — | 4 | 4 | no | — |
| 344 | #41 | — | 4 | 4 | no | — |
| 345 | #41 | — | 4 | 4 | no | — |
| 346 | #41 | — | 6 | 2 | no | — |
| 347 | #41 | — | 6 | 2 | no | — |
| 348 | C2×Q8 | — | 2 | 4 | no | — |
| 349 | C2×Q8 | — | 3 | 1 | no | — |
| 350 | C2×Q8 | — | 4 | 4 | no | — |
| 351 | C2×Q8 | D3×D4 | 1 | 1 | yes | characteristic |
| 352 | C2×Q8 | C2×#67 | 1 | 4 | yes | — |
| 353 | C2×D4 | — | 2 | 4 | no | — |
| 354 | C2×D4 | — | 2 | 4 | no | — |
| 355 | C2×D4 | — | 3 | 1 | no | — |
| 356 | C2×D4 | — | 4 | 4 | no | — |
| 357 | C2×D4 | — | 4 | 8 | no | — |
| 358 | C2×D4 | — | 4 | 4 | no | — |
| 359 | C2×D4 | — | 6 | 1 | no | — |
| 360 | C2×D4 | — | 6 | 2 | no | — |
| 361 | C2×D4 | D3×D4 | 1 | 1 | yes | characteristic |
| 362 | C2×D4 | #237 | 1 | 2 | yes | — |
| 363 | C2×D4 | C2×#67 | 1 | 4 | yes | — |
| 364 | C22×C4 | — | 2 | 4 | no | — |
| 365 | C22×C4 | — | 4 | 4 | no | — |
| 366 | C22×C4 | — | 6 | 1 | no | — |
| 367 | C22×C4 | — | 6 | 1 | no | — |
| 368 | C22×C4 | C2×#67 | 1 | 2 | yes | — |
| 369 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 370 | Q16 | — | 2 | 4 | no | — |
| 371 | Q16 | — | 2 | 2 | no | — |
| 372 | Q16 | — | 2 | 2 | no | — |
| 373 | Q16 | — | 12 | 4 | no | — |
| 374 | SD16 | — | 2 | 8 | no | — |
| 375 | SD16 | — | 2 | 4 | no | — |
| 376 | SD16 | — | 2 | 4 | no | — |
| 377 | SD16 | — | 12 | 4 | no | — |
| 378 | SD16 | — | 12 | 4 | no | — |
| 379 | D8 | — | 2 | 4 | no | — |
| 380 | D8 | — | 2 | 2 | no | — |
| 381 | D8 | — | 2 | 2 | no | — |
| 382 | D8 | — | 12 | 4 | no | — |
| 383 | M16 | — | 2 | 2 | no | — |
| 384 | M16 | — | 2 | 4 | no | — |
| 385 | M16 | — | 6 | 4 | no | — |
| 386 | M16 | — | 6 | 2 | no | — |
| 387 | M16 | — | 6 | 4 | no | — |
| 388 | M16 | — | 12 | 1 | no | — |
| 389 | M16 | — | 12 | 1 | no | — |
| 390 | C2×C8 | — | 2 | 2 | no | — |
| 391 | C2×C8 | — | 2 | 2 | no | — |
| 392 | C2×C8 | — | 6 | 2 | no | — |
| 393 | C2×C8 | — | 6 | 4 | no | — |
| 394 | C2×C8 | — | 6 | 1 | no | — |
| 395 | C2×C8 | — | 6 | 1 | no | — |
| 396 | C2×C8 | — | 12 | 2 | no | — |
| 397 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 398 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 399 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 400 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 401 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 402 | C4⋊C4 | — | 6 | 2 | no | — |
| 403 | C22⋊C4 | — | 6 | 2 | no | — |
| 404 | C22⋊C4 | — | 6 | 2 | no | — |
| 405 | C42 | — | 3 | 1 | no | — |
| 406 | C42 | — | 3 | 1 | no | — |
| 407 | C42 | — | 6 | 1 | no | — |
| 408 | C22×C3 | — | 2 | 2 | no | — |
| 409 | C22×C3 | — | 4 | 1 | no | — |
| 410 | C22×C3 | — | 4 | 2 | no | — |
| 411 | C22×C3 | — | 4 | 4 | no | — |
| 412 | C22×C3 | — | 8 | 4 | no | — |
| 413 | C22×C3 | — | 16 | 4 | no | — |
| 414 | C22×C3 | C2×#121 | 1 | 1 | yes | characteristic |
| 415 | C22×C3 | #546 | 1 | 2 | yes | — |
| 416 | C2×D3 | — | 4 | 2 | no | — |
| 417 | C2×D3 | — | 4 | 1 | no | — |
| 418 | C3×C4 | — | 2 | 2 | no | — |
| 419 | C3×C4 | — | 4 | 4 | no | — |
| 420 | C3×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 421 | C3×C4 | #535 | 1 | 1 | yes | characteristic |
| 422 | C3×C4 | D42 | 1 | 2 | yes | — |
| 423 | Dic3 | — | 4 | 1 | no | — |
| 424 | Dic3 | — | 4 | 1 | no | — |
| 425 | Dic3 | — | 4 | 1 | no | — |
| 426 | C23 | — | 2 | 4 | no | — |
| 427 | C23 | — | 4 | 4 | no | — |
| 428 | C23 | — | 6 | 1 | no | — |
| 429 | C23 | #938 | 1 | 2 | yes | — |
| 430 | C23 | C2×#237 | 1 | 1 | yes | characteristic |
| 431 | Q8 | — | 2 | 2 | no | — |
| 432 | Q8 | — | 2 | 4 | no | — |
| 433 | Q8 | — | 2 | 4 | no | — |
| 434 | Q8 | — | 4 | 4 | no | — |
| 435 | Q8 | — | 6 | 2 | no | — |
| 436 | D4 | — | 2 | 4 | no | — |
| 437 | D4 | — | 2 | 2 | no | — |
| 438 | D4 | — | 2 | 4 | no | — |
| 439 | D4 | — | 2 | 4 | no | — |
| 440 | D4 | — | 4 | 4 | no | — |
| 441 | D4 | — | 4 | 8 | no | — |
| 442 | D4 | — | 4 | 4 | no | — |
| 443 | D4 | — | 4 | 4 | no | — |
| 444 | D4 | — | 6 | 2 | no | — |
| 445 | D4 | — | 12 | 2 | no | — |
| 446 | D4 | — | 24 | 2 | no | — |
| 447 | C2×C4 | — | 2 | 2 | no | — |
| 448 | C2×C4 | — | 2 | 2 | no | — |
| 449 | C2×C4 | — | 2 | 4 | no | — |
| 450 | C2×C4 | — | 4 | 8 | no | — |
| 451 | C2×C4 | — | 4 | 4 | no | — |
| 452 | C2×C4 | — | 4 | 4 | no | — |
| 453 | C2×C4 | — | 4 | 4 | no | — |
| 454 | C2×C4 | — | 6 | 1 | no | — |
| 455 | C2×C4 | — | 6 | 1 | no | — |
| 456 | C2×C4 | — | 6 | 1 | no | — |
| 457 | C2×C4 | — | 12 | 1 | no | — |
| 458 | C2×C4 | — | 12 | 1 | no | — |
| 459 | C2×C4 | — | 24 | 1 | no | — |
| 460 | C2×C4 | #938 | 1 | 2 | yes | — |
| 461 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 462 | C2×C4 | #1004 | 1 | 2 | yes | — |
| 463 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 464 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 465 | C8 | — | 2 | 2 | no | — |
| 466 | C8 | — | 12 | 4 | no | — |
| 467 | C8 | — | 12 | 1 | no | — |
| 468 | C8 | — | 12 | 1 | no | — |
| 469 | C8 | #953 | 1 | 1 | yes | characteristic |
| 470 | C8 | #953 | 1 | 1 | yes | characteristic |
| 471 | C8 | #953 | 1 | 2 | yes | — |
| 472 | C2×C3 | — | 2 | 1 | no | — |
| 473 | C2×C3 | — | 2 | 2 | no | — |
| 474 | C2×C3 | — | 4 | 2 | no | — |
| 475 | C2×C3 | — | 8 | 4 | no | — |
| 476 | C2×C3 | #2408 | 1 | 1 | yes | characteristic |
| 477 | D3 | — | 8 | 1 | no | — |
| 478 | C22 | — | 2 | 2 | no | — |
| 479 | C22 | — | 4 | 1 | no | — |
| 480 | C22 | — | 4 | 2 | no | — |
| 481 | C22 | — | 4 | 4 | no | — |
| 482 | C22 | — | 8 | 4 | no | — |
| 483 | C22 | — | 12 | 2 | no | — |
| 484 | C22 | — | 12 | 1 | no | — |
| 485 | C22 | — | 16 | 4 | no | — |
| 486 | C22 | C2×#938 | 1 | 1 | yes | characteristic |
| 487 | C22 | #5806 | 1 | 2 | yes | — |
| 488 | C4 | — | 2 | 2 | no | — |
| 489 | C4 | — | 4 | 4 | no | — |
| 490 | C4 | — | 12 | 1 | no | — |
| 491 | C4 | — | 12 | 1 | no | — |
| 492 | C4 | — | 12 | 1 | no | — |
| 493 | C4 | D4×#67 | 1 | 2 | yes | — |
| 494 | C4 | #5805 | 1 | 1 | yes | characteristic |
| 495 | C4 | C2×#953 | 1 | 1 | yes | characteristic |
| 496 | C3 | #24288 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 497 | C2 | — | 2 | 1 | no | — |
| 498 | C2 | — | 2 | 2 | no | — |
| 499 | C2 | — | 4 | 2 | no | — |
| 500 | C2 | — | 8 | 4 | no | — |
| 501 | C2 | — | 24 | 1 | no | — |
| 502 | C2 | #84053 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #5805 |
| commutator, cosocle | C22×C3×C4 | C24 |
| Frattini | C22×C4 | C23×D3 |
| Fitting | C3×#3584 | C2 |
| socle | C2×C3 | #2408 |
| hypercenter | #3584 | D3 |
| hypocenter | C3 | #24288 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #3584 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3584 | C2 | semidirect |
| 2. #84121 | C2 | semidirect |
| 3. #83928 | C2 | semidirect |
| 4. #83787 | C2 | semidirect |
| 5. #72145 | C2 | semidirect |
| 6. #72121 | C2 | semidirect |
| 7. #72117 | C2 | semidirect |
| 8. #72085 | C2 | semidirect |
| 9. #72014 | C2 | semidirect |
| 21. #5822 | C22 | semidirect |
| 22. C2×#901 | C22 | semidirect |
| 28. #5237 | C22 | semidirect |
| 29. C2×#837 | C22 | semidirect |
| 31. #5234 | C22 | semidirect |
| 35. #5129 | C22 | semidirect |
| 37. #5118 | C22 | semidirect |
| 39. C2×#819 | C22 | semidirect |
| 40. C2×#815 | C22 | semidirect |
| 41. C3⋊#338 | C22 | semidirect |
| 42. #3584 | D3 | semidirect |
| 73. C2×C3×Q16 | D4 | semidirect |
| 75. C2×C3×SD16 | D4 | semidirect |
| 77. C2×C3×D8 | D4 | semidirect |
| 249. C2×Q16 | #67 | semidirect |
| 251. C2×SD16 | #67 | semidirect |
| 253. C2×D8 | #67 | semidirect |
| 496. C3 | #24288 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3584 | 2 |
| #84121 | 2 |
| #83928 | 2 |
| #83787 | 2 |
| #72145 | 2 |
| #72121 | 2 |
| #72117 | 2 |
| #72085 | 2 |
| #72014 | 2 |
| #24288 | 3 |