Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2598 conjugacy classes of subgroups.
The subgroup lattice has 304 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#5766 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C2×C3×#494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#5192 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C2×#5095 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #81808 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 6 | C2×#4700 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#3293 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C2×#1007 | C22 | 1 | 4 | yes | — |
| 9 | C22×C3×Q16 | C22 | 1 | 4 | yes | — |
| 10 | C2×C3×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 11 | C2×#942 | C22 | 1 | 8 | yes | — |
| 12 | #5766 | — | 2 | 16 | no | — |
| 13 | #5766 | C22 | 1 | 32 | yes | complemented |
| 14 | C2×#919 | — | 2 | 4 | no | — |
| 15 | C2×#918 | — | 2 | 8 | no | — |
| 16 | C2×D3×Q16 | — | 2 | 4 | no | — |
| 17 | C2×#904 | C22 | 1 | 2 | yes | — |
| 18 | C2×#901 | C22 | 1 | 4 | yes | — |
| 19 | C3×#494 | C22 | 1 | 8 | yes | complemented |
| 20 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#836 | C22 | 1 | 4 | yes | — |
| 22 | #5192 | C22 | 1 | 32 | yes | complemented |
| 23 | C2×#819 | C22 | 1 | 2 | yes | — |
| 24 | #5095 | C22 | 1 | 32 | yes | complemented |
| 25 | C2×#805 | C22 | 1 | 4 | yes | — |
| 26 | #4700 | C22 | 1 | 16 | yes | complemented |
| 27 | C2×#578 | — | 3 | 4 | no | — |
| 28 | #3293 | — | 3 | 16 | no | — |
| 29 | C2×#494 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×#473 | — | 3 | 4 | no | — |
| 31 | C2×#456 | — | 3 | 4 | no | — |
| 32 | C2×#443 | — | 3 | 2 | no | — |
| 33 | C22×C3×Q8 | C23 | 1 | 4 | yes | — |
| 34 | #1007 | — | 2 | 16 | no | — |
| 35 | #1007 | — | 4 | 4 | no | — |
| 36 | #1007 | C23 | 1 | 16 | yes | — |
| 37 | C2×#239 | — | 2 | 8 | no | — |
| 38 | C2×D3×Q8 | — | 2 | 8 | no | — |
| 39 | C2×#235 | — | 4 | 4 | no | — |
| 40 | C2×#235 | C23 | 1 | 4 | yes | — |
| 41 | C2×C3×Q16 | — | 2 | 4 | no | — |
| 42 | C2×C3×Q16 | — | 2 | 4 | no | — |
| 43 | C2×C3×Q16 | — | 2 | 8 | no | — |
| 44 | C2×C3×Q16 | C23 | 1 | 16 | yes | — |
| 45 | C22×C3×C8 | C23 | 1 | 2 | yes | — |
| 46 | C3×C4⋊Q8 | — | 4 | 2 | no | — |
| 47 | C3×C4⋊Q8 | C23 | 1 | 8 | yes | — |
| 48 | C2×C3×C4⋊C4 | — | 4 | 2 | no | — |
| 49 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#216 | — | 2 | 8 | no | — |
| 51 | #942 | — | 2 | 16 | no | — |
| 52 | #942 | C23 | 1 | 32 | yes | — |
| 53 | C2×#215 | — | 2 | 8 | no | — |
| 54 | C2×#208 | C23 | 1 | 4 | yes | — |
| 55 | #919 | — | 2 | 32 | no | — |
| 56 | #918 | — | 2 | 64 | no | — |
| 57 | D3×Q16 | — | 2 | 32 | no | — |
| 58 | C2×Dic12 | D4 | 1 | 2 | yes | — |
| 59 | #904 | D4 | 1 | 8 | yes | — |
| 60 | #904 | C23 | 1 | 8 | yes | — |
| 61 | C2×D24 | D4 | 1 | 2 | yes | — |
| 62 | C2×#204 | D4 | 1 | 4 | yes | — |
| 63 | #901 | — | 2 | 8 | no | — |
| 64 | #901 | C23 | 1 | 16 | yes | — |
| 65 | C2×ΓL(1,𝔽25) | — | 2 | 4 | no | — |
| 66 | C2×D3×C8 | — | 2 | 4 | no | — |
| 67 | C3×C4×C8 | C23 | 1 | 4 | yes | — |
| 68 | #836 | C23 | 1 | 16 | yes | — |
| 69 | #819 | C23 | 1 | 8 | yes | — |
| 70 | #805 | C23 | 1 | 16 | yes | — |
| 71 | C2×2−1+4 | — | 3 | 4 | no | — |
| 72 | #578 | — | 3 | 32 | no | — |
| 73 | #578 | — | 6 | 16 | no | — |
| 74 | C2×#137 | — | 3 | 8 | no | — |
| 75 | C2×#137 | — | 6 | 8 | no | — |
| 76 | C2×#135 | — | 3 | 2 | no | — |
| 77 | C2×#135 | — | 6 | 4 | no | — |
| 78 | C22×Q16 | — | 6 | 4 | no | — |
| 79 | C22×Q16 | C2×D3 | 1 | 4 | yes | — |
| 80 | C2×#131 | — | 3 | 4 | no | — |
| 81 | C2×C4⋊Q8 | C2×D3 | 1 | 2 | yes | — |
| 82 | #494 | C2×D3 | 1 | 8 | yes | complemented |
| 83 | #473 | — | 3 | 32 | no | — |
| 84 | #456 | — | 3 | 32 | no | — |
| 85 | #443 | — | 3 | 16 | no | — |
| 86 | C2×#108 | — | 3 | 2 | no | — |
| 87 | C2×#104 | — | 3 | 4 | no | — |
| 88 | C2×#101 | — | 3 | 4 | no | — |
| 89 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 90 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 91 | C2×C3×Q8 | — | 2 | 8 | no | — |
| 92 | C2×C3×Q8 | — | 2 | 8 | no | — |
| 93 | C2×C3×Q8 | C24 | 1 | 8 | yes | — |
| 94 | C22×C3×C4 | — | 2 | 1 | no | — |
| 95 | C22×C3×C4 | — | 2 | 4 | no | — |
| 96 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 97 | C2×#67 | — | 2 | 4 | no | — |
| 98 | #239 | — | 2 | 32 | no | — |
| 99 | #239 | — | 4 | 16 | no | — |
| 100 | D3×Q8 | — | 2 | 32 | no | — |
| 101 | D3×Q8 | — | 4 | 16 | no | — |
| 102 | #235 | — | 2 | 4 | no | — |
| 103 | #235 | — | 2 | 4 | no | — |
| 104 | #235 | — | 4 | 16 | no | — |
| 105 | #235 | — | 4 | 16 | no | — |
| 106 | #235 | C2×D4 | 1 | 8 | yes | — |
| 107 | #235 | C24 | 1 | 8 | yes | — |
| 108 | C2×D12 | — | 4 | 4 | no | — |
| 109 | C2×D12 | C2×D4 | 1 | 4 | yes | — |
| 110 | C2×C4×D3 | — | 2 | 4 | no | — |
| 111 | C2×C4×D3 | — | 4 | 8 | no | — |
| 112 | C2×Dic6 | — | 4 | 4 | no | — |
| 113 | C2×Dic6 | C2×D4 | 1 | 4 | yes | — |
| 114 | C3×Q16 | — | 2 | 16 | no | — |
| 115 | C3×Q16 | — | 2 | 16 | no | — |
| 116 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 117 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 118 | C2×C3×C8 | C2×D4 | 1 | 4 | yes | — |
| 119 | C2×C3×C8 | C24 | 1 | 4 | yes | — |
| 120 | C3×C4⋊C4 | — | 4 | 8 | no | — |
| 121 | C3×C42 | — | 2 | 1 | no | — |
| 122 | C3×C42 | C24 | 1 | 2 | yes | — |
| 123 | #216 | — | 2 | 32 | no | — |
| 124 | #215 | — | 2 | 32 | no | — |
| 125 | #208 | — | 2 | 4 | no | — |
| 126 | #208 | C24 | 1 | 8 | yes | — |
| 127 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 128 | Dic12 | C2×D4 | 1 | 8 | yes | — |
| 129 | D24 | C2×D4 | 1 | 8 | yes | — |
| 130 | #204 | C2×D4 | 1 | 16 | yes | — |
| 131 | ΓL(1,𝔽25) | — | 2 | 16 | no | — |
| 132 | D3×C8 | — | 2 | 16 | no | — |
| 133 | 2−1+4 | — | 3 | 16 | no | — |
| 134 | 2−1+4 | — | 6 | 16 | no | — |
| 135 | 2−1+4 | — | 12 | 4 | no | — |
| 136 | C2×#41 | — | 3 | 4 | no | — |
| 137 | C2×#41 | — | 6 | 8 | no | — |
| 138 | C2×#41 | — | 12 | 4 | no | — |
| 139 | C22×Q8 | — | 6 | 8 | no | — |
| 140 | C22×Q8 | C22×D3 | 1 | 4 | yes | — |
| 141 | #137 | — | 3 | 32 | no | — |
| 142 | #137 | — | 6 | 64 | no | — |
| 143 | #137 | — | 6 | 16 | no | — |
| 144 | #135 | — | 3 | 8 | no | — |
| 145 | #135 | — | 3 | 8 | no | — |
| 146 | #135 | — | 6 | 32 | no | — |
| 147 | C2×Q16 | — | 2 | 4 | no | — |
| 148 | C2×Q16 | — | 2 | 4 | no | — |
| 149 | C2×Q16 | — | 2 | 8 | no | — |
| 150 | C2×Q16 | — | 3 | 2 | no | — |
| 151 | C2×Q16 | — | 6 | 32 | no | — |
| 152 | C2×Q16 | — | 6 | 8 | no | — |
| 153 | C2×Q16 | C22×D3 | 1 | 16 | yes | — |
| 154 | C2×SD16 | — | 3 | 4 | no | — |
| 155 | C2×SD16 | — | 6 | 8 | no | — |
| 156 | C2×D8 | — | 3 | 2 | no | — |
| 157 | #131 | — | 3 | 16 | no | — |
| 158 | #131 | — | 6 | 8 | no | — |
| 159 | C2×M16 | — | 3 | 4 | no | — |
| 160 | C2×M16 | — | 6 | 4 | no | — |
| 161 | C22×C8 | — | 6 | 4 | no | — |
| 162 | C22×C8 | C22×D3 | 1 | 2 | yes | — |
| 163 | C4⋊Q8 | — | 4 | 2 | no | — |
| 164 | C4⋊Q8 | C22×D3 | 1 | 8 | yes | — |
| 165 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 166 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 167 | #108 | — | 3 | 8 | no | — |
| 168 | #104 | — | 3 | 16 | no | — |
| 169 | #101 | — | 3 | 16 | no | — |
| 170 | C4×C8 | C22×D3 | 1 | 4 | yes | — |
| 171 | C23×C3 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 172 | C22×D3 | — | 2 | 4 | no | — |
| 173 | C3×Q8 | — | 2 | 16 | no | — |
| 174 | C3×Q8 | — | 4 | 8 | no | — |
| 175 | C2×C3×C4 | — | 2 | 4 | no | — |
| 176 | C2×C3×C4 | — | 2 | 2 | no | — |
| 177 | C2×C3×C4 | — | 2 | 8 | no | — |
| 178 | C2×C3×C4 | — | 4 | 4 | no | — |
| 179 | C2×C3×C4 | — | 4 | 4 | no | — |
| 180 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 181 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 182 | C2×C3×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 183 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 184 | #67 | — | 2 | 8 | no | — |
| 185 | #67 | — | 2 | 8 | no | — |
| 186 | C2×Dic3 | — | 2 | 4 | no | — |
| 187 | D12 | — | 2 | 4 | no | — |
| 188 | D12 | — | 4 | 16 | no | — |
| 189 | D12 | C22×D4 | 1 | 8 | yes | — |
| 190 | C4×D3 | — | 2 | 8 | no | — |
| 191 | C4×D3 | — | 2 | 8 | no | — |
| 192 | C4×D3 | — | 4 | 32 | no | — |
| 193 | Dic6 | — | 2 | 4 | no | — |
| 194 | Dic6 | — | 4 | 16 | no | — |
| 195 | Dic6 | C22×D4 | 1 | 8 | yes | — |
| 196 | C3×C8 | C22×D4 | 1 | 4 | yes | — |
| 197 | C3×C8 | C22×D4 | 1 | 4 | yes | — |
| 198 | C3⋊C8 | — | 2 | 8 | no | — |
| 199 | #41 | — | 3 | 8 | no | — |
| 200 | #41 | — | 3 | 8 | no | — |
| 201 | #41 | — | 6 | 4 | no | — |
| 202 | #41 | — | 6 | 4 | no | — |
| 203 | #41 | — | 6 | 32 | no | — |
| 204 | #41 | — | 12 | 16 | no | — |
| 205 | #41 | — | 12 | 16 | no | — |
| 206 | #41 | — | 12 | 16 | no | — |
| 207 | C2×Q8 | — | 2 | 4 | no | — |
| 208 | C2×Q8 | — | 2 | 8 | no | — |
| 209 | C2×Q8 | — | 2 | 8 | no | — |
| 210 | C2×Q8 | — | 3 | 4 | no | — |
| 211 | C2×Q8 | — | 6 | 32 | no | — |
| 212 | C2×Q8 | — | 12 | 4 | no | — |
| 213 | C2×Q8 | — | 12 | 16 | no | — |
| 214 | C2×Q8 | C23×D3 | 1 | 8 | yes | — |
| 215 | C2×D4 | — | 3 | 4 | no | — |
| 216 | C2×D4 | — | 6 | 4 | no | — |
| 217 | C2×D4 | — | 12 | 4 | no | — |
| 218 | C22×C4 | — | 2 | 1 | no | — |
| 219 | C22×C4 | — | 2 | 4 | no | — |
| 220 | C22×C4 | — | 6 | 4 | no | — |
| 221 | C22×C4 | — | 12 | 8 | no | — |
| 222 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 2 | 16 | no | — |
| 224 | Q16 | — | 2 | 16 | no | — |
| 225 | Q16 | — | 3 | 8 | no | — |
| 226 | Q16 | — | 6 | 32 | no | — |
| 227 | SD16 | — | 3 | 16 | no | — |
| 228 | SD16 | — | 6 | 32 | no | — |
| 229 | D8 | — | 3 | 8 | no | — |
| 230 | M16 | — | 3 | 8 | no | — |
| 231 | M16 | — | 6 | 4 | no | — |
| 232 | M16 | — | 6 | 16 | no | — |
| 233 | C2×C8 | — | 6 | 4 | no | — |
| 234 | C2×C8 | — | 6 | 16 | no | — |
| 235 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 236 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 237 | C2×C8 | D3×D4 | 1 | 4 | yes | — |
| 238 | C2×C8 | C23×D3 | 1 | 4 | yes | — |
| 239 | C4⋊C4 | — | 4 | 8 | no | — |
| 240 | C42 | — | 2 | 1 | no | — |
| 241 | C42 | C23×D3 | 1 | 2 | yes | — |
| 242 | C22×C3 | — | 2 | 2 | no | — |
| 243 | C22×C3 | D42 | 1 | 1 | yes | characteristic |
| 244 | C22×C3 | D42 | 1 | 1 | yes | characteristic |
| 245 | C22×C3 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 246 | C2×D3 | — | 2 | 8 | no | — |
| 247 | C2×D3 | — | 4 | 8 | no | — |
| 248 | C3×C4 | — | 2 | 4 | no | — |
| 249 | C3×C4 | — | 4 | 8 | no | — |
| 250 | C3×C4 | D42 | 1 | 2 | yes | — |
| 251 | C3×C4 | C23×D4 | 1 | 2 | yes | — |
| 252 | Dic3 | — | 2 | 8 | no | — |
| 253 | C23 | — | 6 | 4 | no | — |
| 254 | C23 | #1004 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 2 | 16 | no | — |
| 256 | Q8 | — | 3 | 8 | no | — |
| 257 | Q8 | — | 4 | 8 | no | — |
| 258 | Q8 | — | 6 | 4 | no | — |
| 259 | Q8 | — | 12 | 16 | no | — |
| 260 | D4 | — | 3 | 8 | no | — |
| 261 | D4 | — | 6 | 8 | no | — |
| 262 | D4 | — | 6 | 8 | no | — |
| 263 | D4 | — | 6 | 4 | no | — |
| 264 | D4 | — | 12 | 16 | no | — |
| 265 | C2×C4 | — | 2 | 4 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 8 | no | — |
| 268 | C2×C4 | — | 4 | 4 | no | — |
| 269 | C2×C4 | — | 4 | 4 | no | — |
| 270 | C2×C4 | — | 6 | 4 | no | — |
| 271 | C2×C4 | — | 6 | 8 | no | — |
| 272 | C2×C4 | — | 6 | 8 | no | — |
| 273 | C2×C4 | — | 12 | 32 | no | — |
| 274 | C2×C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 275 | C2×C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 276 | C2×C4 | #1004 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C24×D3 | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 6 | 8 | no | — |
| 279 | C8 | C2×D3×D4 | 1 | 4 | yes | — |
| 280 | C8 | C2×D3×D4 | 1 | 4 | yes | — |
| 281 | C2×C3 | — | 2 | 1 | no | — |
| 282 | C2×C3 | — | 2 | 1 | no | — |
| 283 | C2×C3 | #3293 | 1 | 2 | yes | complemented |
| 284 | C2×C3 | C2×D42 | 1 | 1 | yes | characteristic |
| 285 | D3 | — | 4 | 8 | no | — |
| 286 | C22 | — | 2 | 2 | no | — |
| 287 | C22 | — | 6 | 8 | no | — |
| 288 | C22 | — | 12 | 8 | no | — |
| 289 | C22 | #5717 | 1 | 1 | yes | characteristic |
| 290 | C22 | #5717 | 1 | 1 | yes | characteristic |
| 291 | C22 | C2×#1004 | 1 | 1 | yes | characteristic |
| 292 | C4 | — | 2 | 4 | no | — |
| 293 | C4 | — | 4 | 8 | no | — |
| 294 | C4 | — | 6 | 8 | no | — |
| 295 | C4 | #5717 | 1 | 2 | yes | — |
| 296 | C4 | C22×D3×D4 | 1 | 2 | yes | — |
| 297 | C3 | C2×#3293 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 298 | C2 | — | 2 | 1 | no | — |
| 299 | C2 | — | 2 | 1 | no | — |
| 300 | C2 | — | 12 | 8 | no | — |
| 301 | C2 | #81808 | 1 | 2 | yes | direct product factor |
| 302 | C2 | C2×#5717 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5766 | C2 | semidirect |
| 2. C2×C3×#494 | C2 | semidirect |
| 3. C2×#5192 | C2 | semidirect |
| 4. C2×#5095 | C2 | semidirect |
| 5. #81808 | C2 | direct (N × H) |
| 6. C2×#4700 | C2 | semidirect |
| 13. #5766 | C22 | semidirect |
| 19. C3×#494 | C22 | semidirect |
| 22. #5192 | C22 | semidirect |
| 24. #5095 | C22 | semidirect |
| 26. #4700 | C22 | semidirect |
| 29. C2×#494 | D3 | semidirect |
| 82. #494 | C2×D3 | semidirect |
| 283. C2×C3 | #3293 | semidirect |
| 297. C3 | C2×#3293 | semidirect |
| 301. C2 | #81808 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5766 | 2 |
| C2×C3×#494 | 2 |
| C2×#5192 | 2 |
| C2×#5095 | 2 |
| #81808 | 2 |
| C2×#4700 | 2 |
| C2×#3293 | 3 |