Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1322 conjugacy classes of subgroups.
The subgroup lattice has 496 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 | #84125 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #83915 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #83795 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72146 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #72118 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #72115 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #72086 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #72012 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #24286 | — | 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 | #5828 | C22 | 1 | 1 | yes | characteristic |
| 21 | #5828 | C22 | 1 | 4 | yes | complemented |
| 22 | C22×ΓL(1,𝔽25) | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #5244 | — | 2 | 4 | no | — |
| 24 | #5243 | — | 2 | 4 | no | — |
| 25 | #5240 | C22 | 1 | 2 | yes | — |
| 26 | #5240 | C22 | 1 | 4 | yes | complemented |
| 27 | #5236 | C22 | 1 | 4 | yes | complemented |
| 28 | #5234 | C22 | 1 | 1 | yes | characteristic |
| 29 | #5233 | C22 | 1 | 4 | yes | complemented |
| 30 | #5189 | — | 2 | 2 | no | — |
| 31 | #5174 | — | 2 | 4 | no | — |
| 32 | #5159 | — | 2 | 2 | no | — |
| 33 | #5128 | — | 2 | 2 | no | — |
| 34 | #5126 | C22 | 1 | 1 | yes | characteristic, complemented |
| 35 | #5119 | C22 | 1 | 1 | yes | characteristic, complemented |
| 36 | #5111 | — | 2 | 2 | no | — |
| 37 | #5106 | C22 | 1 | 2 | yes | complemented |
| 38 | #5101 | C22 | 1 | 1 | yes | characteristic, complemented |
| 39 | #4557 | C22 | 1 | 2 | yes | complemented |
| 40 | #3584 | D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | #3054 | — | 3 | 2 | no | — |
| 42 | #3025 | — | 3 | 2 | no | — |
| 43 | #2930 | — | 3 | 1 | no | — |
| 44 | #2053 | — | 3 | 2 | no | — |
| 45 | #2010 | — | 3 | 2 | no | — |
| 46 | #1976 | — | 3 | 2 | no | — |
| 47 | #1812 | — | 3 | 1 | no | — |
| 48 | #1794 | — | 3 | 2 | no | — |
| 49 | C3×2−1+4 | — | 2 | 4 | no | — |
| 50 | C3×2−1+4 | — | 2 | 2 | no | — |
| 51 | C3×H(5,𝔽2) | — | 2 | 4 | no | — |
| 52 | C3×H(5,𝔽2) | — | 2 | 4 | no | — |
| 53 | C3×H(5,𝔽2) | — | 2 | 2 | no | — |
| 54 | C2×C3×#41 | — | 2 | 4 | no | — |
| 55 | C2×C3×#41 | — | 4 | 4 | no | — |
| 56 | C2×C3×#41 | C23 | 1 | 4 | yes | — |
| 57 | C2×C3×#41 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22×C4×D3 | C23 | 1 | 1 | yes | characteristic |
| 59 | C3×#137 | — | 2 | 8 | no | — |
| 60 | C3×#137 | — | 2 | 4 | no | — |
| 61 | C3×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 62 | C3×AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 63 | C3×#135 | — | 2 | 4 | no | — |
| 64 | C3×#135 | — | 2 | 4 | no | — |
| 65 | C3×#135 | — | 2 | 2 | no | — |
| 66 | C3×#135 | — | 2 | 4 | no | — |
| 67 | C3×#135 | — | 2 | 2 | no | — |
| 68 | C3×#135 | — | 2 | 2 | no | — |
| 69 | C3×#135 | — | 2 | 2 | no | — |
| 70 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 71 | C2×C3×Q16 | D4 | 1 | 2 | yes | complemented |
| 72 | C2×C3×SD16 | — | 2 | 4 | no | — |
| 73 | C2×C3×SD16 | D4 | 1 | 4 | yes | complemented |
| 74 | C2×C3×D8 | — | 2 | 2 | no | — |
| 75 | C2×C3×D8 | D4 | 1 | 2 | yes | complemented |
| 76 | C3×#131 | — | 2 | 2 | no | — |
| 77 | C3×#131 | — | 2 | 2 | no | — |
| 78 | C2×C3×M16 | D4 | 1 | 2 | yes | complemented |
| 79 | C2×C3×M16 | C23 | 1 | 1 | yes | characteristic |
| 80 | C22×C3×C8 | D4 | 1 | 2 | yes | complemented |
| 81 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 82 | #946 | — | 2 | 1 | no | — |
| 83 | #946 | — | 2 | 4 | no | — |
| 84 | #938 | — | 2 | 1 | no | — |
| 85 | #938 | — | 2 | 4 | no | — |
| 86 | #937 | — | 2 | 4 | no | — |
| 87 | #937 | D4 | 1 | 2 | yes | — |
| 88 | #935 | — | 2 | 4 | no | — |
| 89 | #935 | D4 | 1 | 2 | yes | — |
| 90 | C4×#67 | — | 2 | 2 | no | — |
| 91 | C4×#67 | — | 4 | 4 | no | — |
| 92 | #926 | C23 | 1 | 4 | yes | — |
| 93 | #926 | C23 | 1 | 1 | yes | characteristic |
| 94 | C2×#208 | C23 | 1 | 1 | yes | characteristic |
| 95 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 96 | C2×ΓL(1,𝔽25) | — | 2 | 1 | no | — |
| 97 | C2×ΓL(1,𝔽25) | — | 2 | 2 | no | — |
| 98 | C2×ΓL(1,𝔽25) | — | 2 | 2 | no | — |
| 99 | C2×ΓL(1,𝔽25) | — | 2 | 1 | no | — |
| 100 | #837 | — | 4 | 4 | no | — |
| 101 | #835 | — | 2 | 4 | no | — |
| 102 | #834 | — | 2 | 4 | no | — |
| 103 | #834 | — | 2 | 8 | no | — |
| 104 | #832 | — | 2 | 4 | no | — |
| 105 | #830 | — | 2 | 2 | no | — |
| 106 | (C2×D3)⋊C8 | — | 2 | 2 | no | — |
| 107 | #818 | — | 2 | 2 | no | — |
| 108 | #817 | — | 2 | 2 | no | — |
| 109 | #815 | — | 2 | 1 | no | — |
| 110 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 111 | C8×Dic3 | — | 2 | 1 | no | — |
| 112 | #585 | C2×D3 | 1 | 2 | yes | — |
| 113 | #578 | — | 2 | 2 | no | — |
| 114 | #577 | — | 2 | 4 | no | — |
| 115 | #576 | — | 2 | 2 | no | — |
| 116 | #575 | — | 2 | 2 | no | — |
| 117 | #575 | C2×D3 | 1 | 2 | yes | — |
| 118 | C2×#135 | — | 2 | 2 | no | — |
| 119 | C2×#135 | C2×D3 | 1 | 2 | yes | — |
| 120 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 121 | C22×M16 | — | 3 | 1 | no | — |
| 122 | #525 | — | 3 | 1 | no | — |
| 123 | #525 | — | 3 | 4 | no | — |
| 124 | #470 | — | 6 | 2 | no | — |
| 125 | #469 | — | 6 | 2 | no | — |
| 126 | #468 | — | 6 | 4 | no | — |
| 127 | #458 | — | 6 | 4 | no | — |
| 128 | B(4,𝔽2) | — | 6 | 4 | no | — |
| 129 | #436 | — | 6 | 2 | no | — |
| 130 | #435 | — | 6 | 2 | no | — |
| 131 | #427 | — | 3 | 2 | no | — |
| 132 | #424 | — | 3 | 1 | no | — |
| 133 | #421 | — | 3 | 4 | no | — |
| 134 | #416 | — | 3 | 4 | no | — |
| 135 | #410 | — | 3 | 2 | no | — |
| 136 | #410 | — | 3 | 4 | no | — |
| 137 | #408 | — | 3 | 1 | no | — |
| 138 | #407 | — | 3 | 1 | no | — |
| 139 | #405 | — | 3 | 1 | no | — |
| 140 | H(3,ℤ/4) | — | 3 | 2 | no | — |
| 141 | C23×D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 142 | C3×#41 | — | 2 | 2 | no | — |
| 143 | C3×#41 | — | 2 | 2 | no | — |
| 144 | C3×#41 | — | 2 | 8 | no | — |
| 145 | C3×#41 | — | 2 | 4 | no | — |
| 146 | C3×#41 | — | 2 | 4 | no | — |
| 147 | C3×#41 | — | 2 | 4 | no | — |
| 148 | C3×#41 | — | 2 | 4 | no | — |
| 149 | C3×#41 | — | 4 | 8 | no | — |
| 150 | C3×#41 | — | 4 | 4 | no | — |
| 151 | C3×#41 | — | 4 | 4 | no | — |
| 152 | C3×#41 | — | 4 | 4 | no | — |
| 153 | C3×#41 | — | 4 | 4 | no | — |
| 154 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 155 | C2×C3×Q8 | — | 4 | 4 | no | — |
| 156 | C2×C3×Q8 | C2×D4 | 1 | 4 | yes | — |
| 157 | C2×C3×Q8 | #41 | 1 | 1 | yes | characteristic |
| 158 | C2×C3×D4 | — | 2 | 4 | no | — |
| 159 | C2×C3×D4 | — | 2 | 4 | no | — |
| 160 | C2×C3×D4 | — | 4 | 4 | no | — |
| 161 | C2×C3×D4 | — | 4 | 8 | no | — |
| 162 | C2×C3×D4 | — | 4 | 4 | no | — |
| 163 | C2×C3×D4 | C2×D4 | 1 | 2 | yes | — |
| 164 | C2×C3×D4 | C2×D4 | 1 | 4 | yes | — |
| 165 | C2×C3×D4 | #41 | 1 | 1 | yes | characteristic |
| 166 | C22×C3×C4 | — | 2 | 4 | no | — |
| 167 | C22×C3×C4 | — | 4 | 4 | no | — |
| 168 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 169 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 170 | C2×#67 | — | 2 | 2 | no | — |
| 171 | C2×#67 | — | 4 | 4 | no | — |
| 172 | C22×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C4×D3 | — | 2 | 1 | no | — |
| 174 | C2×C4×D3 | — | 2 | 2 | no | — |
| 175 | C2×C4×D3 | — | 2 | 1 | no | — |
| 176 | C2×C4×D3 | — | 2 | 2 | no | — |
| 177 | C3×Q16 | — | 2 | 4 | no | — |
| 178 | C3×Q16 | — | 2 | 2 | no | — |
| 179 | C3×Q16 | — | 2 | 2 | no | — |
| 180 | C3×SD16 | — | 2 | 8 | no | — |
| 181 | C3×SD16 | — | 2 | 4 | no | — |
| 182 | C3×SD16 | — | 2 | 4 | no | — |
| 183 | C3×D8 | — | 2 | 4 | no | — |
| 184 | C3×D8 | — | 2 | 2 | no | — |
| 185 | C3×D8 | — | 2 | 2 | no | — |
| 186 | C3×M16 | — | 2 | 2 | no | — |
| 187 | C3×M16 | — | 2 | 4 | no | — |
| 188 | C2×C3×C8 | — | 2 | 2 | no | — |
| 189 | C2×C3×C8 | — | 2 | 2 | no | — |
| 190 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 191 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 192 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 194 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 195 | #217 | — | 2 | 2 | no | — |
| 196 | #217 | — | 2 | 4 | no | — |
| 197 | #217 | — | 4 | 4 | no | — |
| 198 | #217 | C2×D4 | 1 | 2 | yes | — |
| 199 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 200 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 201 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 202 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 203 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 204 | Dic3⋊C4 | — | 4 | 4 | no | — |
| 205 | C4×Dic3 | — | 2 | 1 | no | — |
| 206 | C4×Dic3 | — | 2 | 4 | no | — |
| 207 | #208 | — | 2 | 2 | no | — |
| 208 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 209 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 210 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 211 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 212 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 213 | ΓL(1,𝔽25) | — | 4 | 2 | no | — |
| 214 | 2−1+4 | — | 2 | 4 | no | — |
| 215 | 2−1+4 | — | 2 | 2 | no | — |
| 216 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 217 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 218 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 219 | C2×#41 | — | 2 | 4 | no | — |
| 220 | C2×#41 | — | 4 | 4 | no | — |
| 221 | C2×#41 | C22×D3 | 1 | 4 | yes | — |
| 222 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 223 | C23×C4 | — | 3 | 1 | no | — |
| 224 | #137 | — | 2 | 8 | no | — |
| 225 | #137 | — | 2 | 4 | no | — |
| 226 | AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 227 | AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 228 | #135 | — | 2 | 4 | no | — |
| 229 | #135 | — | 2 | 4 | no | — |
| 230 | #135 | — | 2 | 2 | no | — |
| 231 | #135 | — | 2 | 4 | no | — |
| 232 | #135 | — | 2 | 2 | no | — |
| 233 | #135 | — | 2 | 2 | no | — |
| 234 | #135 | — | 2 | 2 | no | — |
| 235 | C2×Q16 | — | 2 | 2 | no | — |
| 236 | C2×Q16 | #67 | 1 | 2 | yes | complemented |
| 237 | C2×SD16 | — | 2 | 4 | no | — |
| 238 | C2×SD16 | #67 | 1 | 4 | yes | complemented |
| 239 | C2×D8 | — | 2 | 2 | no | — |
| 240 | C2×D8 | #67 | 1 | 2 | yes | complemented |
| 241 | #131 | — | 2 | 2 | no | — |
| 242 | #131 | — | 2 | 2 | no | — |
| 243 | C2×M16 | — | 3 | 1 | no | — |
| 244 | C2×M16 | — | 6 | 1 | no | — |
| 245 | C2×M16 | — | 6 | 1 | no | — |
| 246 | C2×M16 | — | 6 | 2 | no | — |
| 247 | C2×M16 | — | 6 | 2 | no | — |
| 248 | C2×M16 | #67 | 1 | 2 | yes | complemented |
| 249 | C2×M16 | C22×D3 | 1 | 1 | yes | characteristic |
| 250 | C22×C8 | — | 3 | 1 | no | — |
| 251 | C22×C8 | #67 | 1 | 2 | yes | complemented |
| 252 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 253 | #123 | — | 3 | 2 | no | — |
| 254 | #123 | — | 6 | 4 | no | — |
| 255 | C22⋊Q8 | — | 6 | 1 | no | — |
| 256 | C22⋊Q8 | — | 6 | 4 | no | — |
| 257 | #121 | — | 6 | 1 | no | — |
| 258 | #121 | — | 6 | 4 | no | — |
| 259 | #120 | — | 3 | 2 | no | — |
| 260 | #120 | — | 6 | 4 | no | — |
| 261 | C4×D4 | — | 6 | 2 | no | — |
| 262 | C4×D4 | — | 12 | 4 | no | — |
| 263 | #117 | — | 3 | 1 | no | — |
| 264 | #117 | — | 3 | 4 | no | — |
| 265 | #107 | — | 6 | 2 | no | — |
| 266 | #106 | — | 6 | 2 | no | — |
| 267 | C4⋊C8 | — | 6 | 2 | no | — |
| 268 | #104 | — | 12 | 4 | no | — |
| 269 | #103 | — | 6 | 4 | no | — |
| 270 | #102 | — | 6 | 4 | no | — |
| 271 | #99 | — | 6 | 8 | no | — |
| 272 | #99 | — | 6 | 4 | no | — |
| 273 | C22⋊C8 | — | 6 | 2 | no | — |
| 274 | C22⋊C8 | — | 6 | 2 | no | — |
| 275 | #97 | — | 6 | 1 | no | — |
| 276 | C4×C8 | — | 6 | 1 | no | — |
| 277 | C23×C3 | — | 2 | 4 | no | — |
| 278 | C23×C3 | — | 4 | 4 | no | — |
| 279 | C23×C3 | #121 | 1 | 2 | yes | — |
| 280 | C23×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 281 | C22×D3 | — | 2 | 1 | no | — |
| 282 | C22×D3 | — | 2 | 2 | no | — |
| 283 | C22×D3 | — | 8 | 1 | no | — |
| 284 | C3×Q8 | — | 2 | 2 | no | — |
| 285 | C3×Q8 | — | 2 | 4 | no | — |
| 286 | C3×Q8 | — | 2 | 4 | no | — |
| 287 | C3×Q8 | — | 4 | 4 | no | — |
| 288 | C3×D4 | — | 2 | 4 | no | — |
| 289 | C3×D4 | — | 2 | 4 | no | — |
| 290 | C3×D4 | — | 2 | 2 | no | — |
| 291 | C3×D4 | — | 2 | 4 | no | — |
| 292 | C3×D4 | — | 4 | 4 | no | — |
| 293 | C3×D4 | — | 4 | 8 | no | — |
| 294 | C3×D4 | — | 4 | 4 | no | — |
| 295 | C3×D4 | — | 4 | 4 | no | — |
| 296 | C2×C3×C4 | — | 2 | 2 | no | — |
| 297 | C2×C3×C4 | — | 2 | 2 | no | — |
| 298 | C2×C3×C4 | — | 2 | 4 | no | — |
| 299 | C2×C3×C4 | — | 4 | 8 | no | — |
| 300 | C2×C3×C4 | — | 4 | 4 | no | — |
| 301 | C2×C3×C4 | — | 4 | 4 | no | — |
| 302 | C2×C3×C4 | — | 4 | 4 | no | — |
| 303 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 304 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 305 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 306 | C2×C3×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 307 | C2×C3×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 308 | #67 | — | 8 | 2 | no | — |
| 309 | #67 | — | 8 | 8 | no | — |
| 310 | C2×Dic3 | — | 2 | 2 | no | — |
| 311 | C2×Dic3 | — | 2 | 1 | no | — |
| 312 | C2×Dic3 | — | 2 | 2 | no | — |
| 313 | C2×Dic3 | — | 4 | 4 | no | — |
| 314 | C4×D3 | — | 4 | 1 | no | — |
| 315 | C4×D3 | — | 4 | 1 | no | — |
| 316 | C4×D3 | — | 4 | 2 | no | — |
| 317 | C3×C8 | — | 2 | 2 | no | — |
| 318 | C3×C8 | #120 | 1 | 1 | yes | characteristic |
| 319 | C3×C8 | #120 | 1 | 1 | yes | characteristic |
| 320 | C3×C8 | #120 | 1 | 2 | yes | — |
| 321 | C3⋊C8 | — | 4 | 1 | no | — |
| 322 | C3⋊C8 | — | 4 | 1 | no | — |
| 323 | C24 | — | 3 | 1 | no | — |
| 324 | #41 | — | 2 | 2 | no | — |
| 325 | #41 | — | 2 | 2 | no | — |
| 326 | #41 | — | 2 | 8 | no | — |
| 327 | #41 | — | 2 | 4 | no | — |
| 328 | #41 | — | 2 | 4 | no | — |
| 329 | #41 | — | 2 | 4 | no | — |
| 330 | #41 | — | 2 | 4 | no | — |
| 331 | #41 | — | 4 | 8 | no | — |
| 332 | #41 | — | 4 | 4 | no | — |
| 333 | #41 | — | 4 | 4 | no | — |
| 334 | #41 | — | 4 | 4 | no | — |
| 335 | #41 | — | 4 | 4 | no | — |
| 336 | C2×Q8 | — | 2 | 4 | no | — |
| 337 | C2×Q8 | — | 4 | 4 | no | — |
| 338 | C2×Q8 | #237 | 1 | 1 | yes | characteristic |
| 339 | C2×Q8 | C2×#67 | 1 | 4 | yes | — |
| 340 | C2×D4 | — | 2 | 4 | no | — |
| 341 | C2×D4 | — | 2 | 4 | no | — |
| 342 | C2×D4 | — | 4 | 4 | no | — |
| 343 | C2×D4 | — | 4 | 8 | no | — |
| 344 | C2×D4 | — | 4 | 4 | no | — |
| 345 | C2×D4 | — | 6 | 2 | no | — |
| 346 | C2×D4 | — | 12 | 4 | no | — |
| 347 | C2×D4 | D3×D4 | 1 | 2 | yes | — |
| 348 | C2×D4 | #237 | 1 | 1 | yes | characteristic |
| 349 | C2×D4 | C2×#67 | 1 | 4 | yes | — |
| 350 | C22×C4 | — | 2 | 4 | no | — |
| 351 | C22×C4 | — | 3 | 1 | no | — |
| 352 | C22×C4 | — | 4 | 4 | no | — |
| 353 | C22×C4 | — | 6 | 1 | no | — |
| 354 | C22×C4 | — | 6 | 2 | no | — |
| 355 | C22×C4 | — | 6 | 1 | no | — |
| 356 | C22×C4 | — | 6 | 2 | no | — |
| 357 | C22×C4 | C2×#67 | 1 | 2 | yes | — |
| 358 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 359 | Q16 | — | 2 | 4 | no | — |
| 360 | Q16 | — | 2 | 2 | no | — |
| 361 | Q16 | — | 2 | 2 | no | — |
| 362 | SD16 | — | 2 | 8 | no | — |
| 363 | SD16 | — | 2 | 4 | no | — |
| 364 | SD16 | — | 2 | 4 | no | — |
| 365 | D8 | — | 2 | 4 | no | — |
| 366 | D8 | — | 2 | 2 | no | — |
| 367 | D8 | — | 2 | 2 | no | — |
| 368 | M16 | — | 2 | 2 | no | — |
| 369 | M16 | — | 2 | 4 | no | — |
| 370 | M16 | — | 6 | 2 | no | — |
| 371 | M16 | — | 12 | 1 | no | — |
| 372 | M16 | — | 12 | 1 | no | — |
| 373 | M16 | — | 12 | 2 | no | — |
| 374 | C2×C8 | — | 2 | 2 | no | — |
| 375 | C2×C8 | — | 2 | 2 | no | — |
| 376 | C2×C8 | — | 6 | 1 | no | — |
| 377 | C2×C8 | — | 6 | 1 | no | — |
| 378 | C2×C8 | — | 6 | 2 | no | — |
| 379 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 380 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 381 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 382 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 383 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 384 | C4⋊C4 | — | 6 | 2 | no | — |
| 385 | C4⋊C4 | — | 6 | 1 | no | — |
| 386 | C4⋊C4 | — | 6 | 4 | no | — |
| 387 | C4⋊C4 | — | 12 | 4 | no | — |
| 388 | C22⋊C4 | — | 3 | 2 | no | — |
| 389 | C22⋊C4 | — | 6 | 2 | no | — |
| 390 | C22⋊C4 | — | 6 | 2 | no | — |
| 391 | C22⋊C4 | — | 6 | 4 | no | — |
| 392 | C22⋊C4 | — | 12 | 4 | no | — |
| 393 | C22⋊C4 | — | 12 | 4 | no | — |
| 394 | C42 | — | 6 | 1 | no | — |
| 395 | C42 | — | 6 | 4 | no | — |
| 396 | C22×C3 | — | 2 | 2 | no | — |
| 397 | C22×C3 | — | 4 | 1 | no | — |
| 398 | C22×C3 | — | 4 | 2 | no | — |
| 399 | C22×C3 | — | 4 | 4 | no | — |
| 400 | C22×C3 | — | 8 | 4 | no | — |
| 401 | C22×C3 | — | 16 | 4 | no | — |
| 402 | C22×C3 | C2×#121 | 1 | 1 | yes | characteristic |
| 403 | C22×C3 | D42 | 1 | 2 | yes | — |
| 404 | C2×D3 | — | 4 | 1 | no | — |
| 405 | C2×D3 | — | 4 | 4 | no | — |
| 406 | C2×D3 | — | 4 | 1 | no | — |
| 407 | C2×D3 | — | 4 | 1 | no | — |
| 408 | C3×C4 | — | 2 | 2 | no | — |
| 409 | C3×C4 | — | 4 | 4 | no | — |
| 410 | C3×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 411 | C3×C4 | #535 | 1 | 1 | yes | characteristic |
| 412 | C3×C4 | #546 | 1 | 2 | yes | — |
| 413 | Dic3 | — | 4 | 2 | no | — |
| 414 | Dic3 | — | 4 | 1 | no | — |
| 415 | Dic3 | — | 8 | 4 | no | — |
| 416 | C23 | — | 2 | 4 | no | — |
| 417 | C23 | — | 4 | 4 | no | — |
| 418 | C23 | — | 6 | 1 | no | — |
| 419 | C23 | — | 6 | 2 | no | — |
| 420 | C23 | — | 24 | 1 | no | — |
| 421 | C23 | #938 | 1 | 2 | yes | — |
| 422 | C23 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 423 | Q8 | — | 2 | 2 | no | — |
| 424 | Q8 | — | 2 | 4 | no | — |
| 425 | Q8 | — | 2 | 4 | no | — |
| 426 | Q8 | — | 4 | 4 | no | — |
| 427 | D4 | — | 2 | 4 | no | — |
| 428 | D4 | — | 2 | 2 | no | — |
| 429 | D4 | — | 2 | 4 | no | — |
| 430 | D4 | — | 2 | 4 | no | — |
| 431 | D4 | — | 4 | 4 | no | — |
| 432 | D4 | — | 4 | 8 | no | — |
| 433 | D4 | — | 4 | 4 | no | — |
| 434 | D4 | — | 4 | 4 | no | — |
| 435 | D4 | — | 24 | 2 | no | — |
| 436 | D4 | — | 24 | 8 | no | — |
| 437 | C2×C4 | — | 2 | 2 | no | — |
| 438 | C2×C4 | — | 2 | 2 | no | — |
| 439 | C2×C4 | — | 2 | 4 | no | — |
| 440 | C2×C4 | — | 4 | 8 | no | — |
| 441 | C2×C4 | — | 4 | 4 | no | — |
| 442 | C2×C4 | — | 4 | 4 | no | — |
| 443 | C2×C4 | — | 4 | 4 | no | — |
| 444 | C2×C4 | — | 6 | 2 | no | — |
| 445 | C2×C4 | — | 6 | 1 | no | — |
| 446 | C2×C4 | — | 6 | 2 | no | — |
| 447 | C2×C4 | — | 12 | 1 | no | — |
| 448 | C2×C4 | — | 12 | 2 | no | — |
| 449 | C2×C4 | — | 12 | 1 | no | — |
| 450 | C2×C4 | — | 12 | 4 | no | — |
| 451 | C2×C4 | #938 | 1 | 2 | yes | — |
| 452 | C2×C4 | C2×#237 | 1 | 1 | yes | characteristic |
| 453 | C2×C4 | #1004 | 1 | 2 | yes | — |
| 454 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 455 | C2×C4 | C22×#67 | 1 | 2 | yes | — |
| 456 | C8 | — | 2 | 2 | no | — |
| 457 | C8 | — | 12 | 1 | no | — |
| 458 | C8 | — | 12 | 1 | no | — |
| 459 | C8 | #953 | 1 | 1 | yes | characteristic |
| 460 | C8 | #953 | 1 | 1 | yes | characteristic |
| 461 | C8 | #953 | 1 | 2 | yes | — |
| 462 | C2×C3 | — | 2 | 1 | no | — |
| 463 | C2×C3 | — | 2 | 2 | no | — |
| 464 | C2×C3 | — | 4 | 2 | no | — |
| 465 | C2×C3 | — | 8 | 4 | no | — |
| 466 | C2×C3 | #2408 | 1 | 1 | yes | characteristic |
| 467 | D3 | — | 8 | 1 | no | — |
| 468 | C22 | — | 2 | 2 | no | — |
| 469 | C22 | — | 4 | 1 | no | — |
| 470 | C22 | — | 4 | 2 | no | — |
| 471 | C22 | — | 4 | 4 | no | — |
| 472 | C22 | — | 8 | 4 | no | — |
| 473 | C22 | — | 12 | 1 | no | — |
| 474 | C22 | — | 12 | 4 | no | — |
| 475 | C22 | — | 12 | 1 | no | — |
| 476 | C22 | — | 12 | 1 | no | — |
| 477 | C22 | — | 16 | 4 | no | — |
| 478 | C22 | C2×#938 | 1 | 1 | yes | characteristic |
| 479 | C22 | D4×#67 | 1 | 2 | yes | — |
| 480 | C4 | — | 2 | 2 | no | — |
| 481 | C4 | — | 4 | 4 | no | — |
| 482 | C4 | — | 12 | 2 | no | — |
| 483 | C4 | — | 12 | 1 | no | — |
| 484 | C4 | — | 24 | 4 | no | — |
| 485 | C4 | #5805 | 1 | 1 | yes | characteristic |
| 486 | C4 | #5806 | 1 | 2 | yes | — |
| 487 | C4 | C2×#953 | 1 | 1 | yes | characteristic |
| 488 | C3 | #24286 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 489 | C2 | — | 2 | 1 | no | — |
| 490 | C2 | — | 2 | 2 | no | — |
| 491 | C2 | — | 4 | 2 | no | — |
| 492 | C2 | — | 8 | 4 | no | — |
| 493 | C2 | — | 24 | 1 | no | — |
| 494 | 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 | #24286 |
| 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. #84125 | C2 | semidirect |
| 3. #83915 | C2 | semidirect |
| 4. #83795 | C2 | semidirect |
| 5. #72146 | C2 | semidirect |
| 6. #72118 | C2 | semidirect |
| 7. #72115 | C2 | semidirect |
| 8. #72086 | C2 | semidirect |
| 9. #72012 | C2 | semidirect |
| 21. #5828 | C22 | semidirect |
| 22. C22×ΓL(1,𝔽25) | C22 | semidirect |
| 26. #5240 | C22 | semidirect |
| 27. #5236 | C22 | semidirect |
| 29. #5233 | C22 | semidirect |
| 34. #5126 | C22 | semidirect |
| 35. #5119 | C22 | semidirect |
| 37. #5106 | C22 | semidirect |
| 38. #5101 | C22 | semidirect |
| 39. #4557 | C22 | semidirect |
| 40. #3584 | D3 | semidirect |
| 71. C2×C3×Q16 | D4 | semidirect |
| 73. C2×C3×SD16 | D4 | semidirect |
| 75. C2×C3×D8 | D4 | semidirect |
| 78. C2×C3×M16 | D4 | semidirect |
| 80. C22×C3×C8 | D4 | semidirect |
| 236. C2×Q16 | #67 | semidirect |
| 238. C2×SD16 | #67 | semidirect |
| 240. C2×D8 | #67 | semidirect |
| 248. C2×M16 | #67 | semidirect |
| 251. C22×C8 | #67 | semidirect |
| 488. C3 | #24286 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3584 | 2 |
| #84125 | 2 |
| #83915 | 2 |
| #83795 | 2 |
| #72146 | 2 |
| #72118 | 2 |
| #72115 | 2 |
| #72086 | 2 |
| #72012 | 2 |
| #24286 | 3 |