Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1184 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×#3483 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #83322 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #71942 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #71930 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #71820 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #33567 | — | 3 | 1 | no | maximal, Sylow |
| 7 | #5964 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×Q82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×#555 | — | 2 | 3 | no | — |
| 10 | C3×#555 | C22 | 1 | 3 | yes | complemented |
| 11 | C3×#550 | — | 2 | 3 | no | — |
| 12 | C3×#532 | — | 2 | 3 | no | — |
| 13 | C3×#532 | C22 | 1 | 3 | yes | — |
| 14 | C3×C4×#41 | — | 2 | 3 | no | — |
| 15 | #5822 | — | 2 | 3 | no | — |
| 16 | #5237 | C22 | 1 | 3 | yes | — |
| 17 | #5095 | C22 | 1 | 6 | yes | — |
| 18 | #5077 | — | 2 | 3 | no | — |
| 19 | #5062 | C22 | 1 | 6 | yes | — |
| 20 | #5061 | — | 2 | 3 | no | — |
| 21 | C2×#805 | — | 2 | 3 | no | — |
| 22 | #4548 | C22 | 1 | 6 | yes | complemented |
| 23 | #4542 | C22 | 1 | 6 | yes | complemented |
| 24 | #3483 | D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | #3018 | — | 3 | 3 | no | — |
| 26 | #2205 | — | 3 | 2 | no | — |
| 27 | #2204 | — | 3 | 6 | no | — |
| 28 | #2132 | — | 3 | 3 | no | — |
| 29 | C2×C3×#41 | — | 2 | 3 | no | — |
| 30 | C2×C3×#41 | — | 2 | 1 | no | — |
| 31 | C2×C3×#41 | D4 | 1 | 3 | yes | — |
| 32 | C2×C3×#41 | C23 | 1 | 1 | yes | characteristic |
| 33 | #1010 | — | 4 | 1 | no | — |
| 34 | #1009 | — | 4 | 1 | no | — |
| 35 | D3×#41 | — | 4 | 2 | no | — |
| 36 | #1007 | C23 | 1 | 2 | yes | — |
| 37 | #1004 | — | 2 | 3 | no | — |
| 38 | C2×#235 | — | 2 | 3 | no | — |
| 39 | C3×C4⋊Q8 | — | 2 | 3 | no | — |
| 40 | C3×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 41 | C3×#127 | — | 2 | 3 | no | — |
| 42 | C3×#127 | D4 | 1 | 3 | yes | — |
| 43 | C3×#124 | — | 2 | 3 | no | — |
| 44 | C3×#124 | — | 2 | 3 | no | — |
| 45 | C3×#124 | D4 | 1 | 3 | yes | — |
| 46 | C3×#124 | C23 | 1 | 3 | yes | — |
| 47 | C3×#121 | — | 2 | 6 | no | — |
| 48 | C3×#121 | — | 2 | 6 | no | — |
| 49 | C3×#121 | — | 2 | 6 | no | — |
| 50 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 51 | C3×C4×Q8 | — | 2 | 3 | no | — |
| 52 | C3×C4×D4 | — | 2 | 3 | no | — |
| 53 | C3×C4×D4 | — | 2 | 6 | no | — |
| 54 | C3×C4×D4 | — | 2 | 6 | no | — |
| 55 | C3×C4×D4 | — | 2 | 3 | no | — |
| 56 | C3×#117 | — | 2 | 3 | no | — |
| 57 | C3×#117 | — | 2 | 6 | no | — |
| 58 | C2×C3×C42 | — | 2 | 3 | no | — |
| 59 | C2×C3×C42 | D4 | 1 | 3 | yes | — |
| 60 | #951 | — | 4 | 3 | no | — |
| 61 | #950 | — | 4 | 6 | no | — |
| 62 | #949 | — | 4 | 3 | no | — |
| 63 | #942 | — | 2 | 6 | no | — |
| 64 | #932 | — | 2 | 6 | no | — |
| 65 | C2×#208 | — | 2 | 3 | no | — |
| 66 | #836 | C23 | 1 | 6 | yes | — |
| 67 | #833 | — | 2 | 3 | no | — |
| 68 | #805 | — | 2 | 6 | no | — |
| 69 | #805 | — | 2 | 6 | no | — |
| 70 | #585 | — | 3 | 1 | no | — |
| 71 | #575 | — | 6 | 3 | no | — |
| 72 | Q82 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 73 | #555 | — | 2 | 3 | no | — |
| 74 | #555 | C2×D3 | 1 | 3 | yes | complemented |
| 75 | #550 | — | 2 | 3 | no | — |
| 76 | #532 | — | 2 | 3 | no | — |
| 77 | #532 | C2×D3 | 1 | 3 | yes | — |
| 78 | C4×#41 | — | 2 | 3 | no | — |
| 79 | #456 | — | 3 | 6 | no | — |
| 80 | #455 | — | 6 | 3 | no | — |
| 81 | #454 | — | 3 | 6 | no | — |
| 82 | #453 | — | 6 | 3 | no | — |
| 83 | C2×#104 | — | 6 | 3 | no | — |
| 84 | #413 | — | 3 | 3 | no | — |
| 85 | #356 | — | 3 | 6 | no | — |
| 86 | #355 | — | 3 | 6 | no | — |
| 87 | C3×#41 | — | 4 | 1 | no | — |
| 88 | C3×#41 | — | 4 | 1 | no | — |
| 89 | C3×#41 | — | 4 | 3 | no | — |
| 90 | C3×#41 | — | 4 | 3 | no | — |
| 91 | C3×#41 | — | 4 | 12 | no | — |
| 92 | C3×#41 | — | 8 | 1 | no | — |
| 93 | C3×#41 | — | 8 | 1 | no | — |
| 94 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 95 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 96 | C2×C3×Q8 | C2×D4 | 1 | 3 | yes | — |
| 97 | C2×C3×Q8 | C24 | 1 | 1 | yes | characteristic |
| 98 | C2×C3×D4 | — | 2 | 3 | no | — |
| 99 | C2×C3×D4 | — | 2 | 6 | no | — |
| 100 | C2×C3×D4 | — | 2 | 3 | no | — |
| 101 | C2×C3×D4 | — | 2 | 3 | no | — |
| 102 | C2×C3×D4 | C2×D4 | 1 | 3 | yes | — |
| 103 | C2×C3×D4 | C2×D4 | 1 | 3 | yes | — |
| 104 | C22×C3×C4 | — | 2 | 6 | no | — |
| 105 | C22×C3×C4 | — | 2 | 3 | no | — |
| 106 | C22×C3×C4 | — | 2 | 3 | no | — |
| 107 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 108 | C2×#67 | — | 4 | 3 | no | — |
| 109 | #239 | — | 4 | 2 | no | — |
| 110 | D3×Q8 | — | 4 | 2 | no | — |
| 111 | #237 | — | 4 | 6 | no | — |
| 112 | D3×D4 | — | 4 | 6 | no | — |
| 113 | #235 | — | 2 | 6 | no | — |
| 114 | #235 | — | 2 | 6 | no | — |
| 115 | #235 | — | 4 | 3 | no | — |
| 116 | #235 | — | 4 | 1 | no | — |
| 117 | C2×D12 | — | 2 | 3 | no | — |
| 118 | C2×C4×D3 | — | 4 | 3 | no | — |
| 119 | C2×Dic6 | — | 2 | 3 | no | — |
| 120 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 121 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 122 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 123 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 124 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 125 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 126 | C3×C42 | — | 2 | 3 | no | — |
| 127 | C3×C42 | — | 2 | 3 | no | — |
| 128 | C3×C42 | — | 2 | 3 | no | — |
| 129 | C3×C42 | C2×D4 | 1 | 3 | yes | — |
| 130 | C3×C42 | C2×D4 | 1 | 3 | yes | — |
| 131 | #216 | — | 4 | 6 | no | — |
| 132 | #215 | — | 4 | 6 | no | — |
| 133 | #214 | — | 4 | 6 | no | — |
| 134 | #213 | — | 4 | 6 | no | — |
| 135 | #208 | — | 2 | 6 | no | — |
| 136 | #208 | — | 4 | 3 | no | — |
| 137 | C2×C3⋊C8 | — | 4 | 3 | no | — |
| 138 | 2−1+4 | — | 3 | 2 | no | — |
| 139 | 2−1+4 | — | 12 | 1 | no | — |
| 140 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 141 | H(5,𝔽2) | — | 12 | 1 | no | — |
| 142 | C2×#41 | — | 2 | 3 | no | — |
| 143 | C2×#41 | — | 2 | 1 | no | — |
| 144 | C2×#41 | — | 6 | 3 | no | — |
| 145 | C2×#41 | — | 12 | 2 | no | — |
| 146 | C2×#41 | #67 | 1 | 3 | yes | — |
| 147 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 148 | #137 | — | 6 | 6 | no | — |
| 149 | #137 | — | 12 | 3 | no | — |
| 150 | AGL(1,ℤ/8) | — | 6 | 6 | no | — |
| 151 | AGL(1,ℤ/8) | — | 12 | 3 | no | — |
| 152 | #135 | — | 12 | 6 | no | — |
| 153 | C2×M16 | — | 6 | 3 | no | — |
| 154 | C4⋊Q8 | — | 2 | 3 | no | — |
| 155 | C4⋊Q8 | C22×D3 | 1 | 3 | yes | — |
| 156 | #127 | — | 2 | 3 | no | — |
| 157 | #127 | #67 | 1 | 3 | yes | — |
| 158 | #124 | — | 2 | 3 | no | — |
| 159 | #124 | — | 2 | 3 | no | — |
| 160 | #124 | #67 | 1 | 3 | yes | — |
| 161 | #124 | C22×D3 | 1 | 3 | yes | — |
| 162 | #121 | — | 2 | 6 | no | — |
| 163 | #121 | — | 2 | 6 | no | — |
| 164 | #121 | — | 2 | 6 | no | — |
| 165 | C4×Q8 | — | 2 | 1 | no | — |
| 166 | C4×Q8 | — | 2 | 3 | no | — |
| 167 | C4×D4 | — | 2 | 3 | no | — |
| 168 | C4×D4 | — | 2 | 6 | no | — |
| 169 | C4×D4 | — | 2 | 6 | no | — |
| 170 | C4×D4 | — | 2 | 3 | no | — |
| 171 | #117 | — | 2 | 3 | no | — |
| 172 | #117 | — | 2 | 6 | no | — |
| 173 | C2×C42 | — | 2 | 3 | no | — |
| 174 | C2×C42 | #67 | 1 | 3 | yes | — |
| 175 | #104 | — | 6 | 6 | no | — |
| 176 | #104 | — | 6 | 6 | no | — |
| 177 | #101 | — | 3 | 6 | no | — |
| 178 | #100 | — | 6 | 3 | no | — |
| 179 | C23×C3 | — | 2 | 3 | no | — |
| 180 | C23×C3 | #120 | 1 | 3 | yes | — |
| 181 | C22×D3 | — | 4 | 3 | no | — |
| 182 | C3×Q8 | — | 2 | 4 | no | — |
| 183 | C3×Q8 | — | 4 | 1 | no | — |
| 184 | C3×Q8 | — | 4 | 3 | no | — |
| 185 | C3×Q8 | — | 8 | 1 | no | — |
| 186 | C3×D4 | — | 4 | 3 | no | — |
| 187 | C3×D4 | — | 4 | 3 | no | — |
| 188 | C3×D4 | — | 4 | 12 | no | — |
| 189 | C3×D4 | — | 4 | 6 | no | — |
| 190 | C3×D4 | — | 8 | 3 | no | — |
| 191 | C3×D4 | — | 8 | 3 | no | — |
| 192 | C2×C3×C4 | — | 2 | 3 | no | — |
| 193 | C2×C3×C4 | — | 2 | 3 | no | — |
| 194 | C2×C3×C4 | — | 2 | 3 | no | — |
| 195 | C2×C3×C4 | — | 2 | 3 | no | — |
| 196 | C2×C3×C4 | — | 2 | 3 | no | — |
| 197 | C2×C3×C4 | — | 4 | 6 | no | — |
| 198 | C2×C3×C4 | — | 4 | 12 | no | — |
| 199 | C2×C3×C4 | — | 4 | 3 | no | — |
| 200 | C2×C3×C4 | — | 4 | 3 | no | — |
| 201 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 202 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 203 | #67 | — | 4 | 6 | no | — |
| 204 | #67 | — | 4 | 2 | no | — |
| 205 | C2×Dic3 | — | 4 | 3 | no | — |
| 206 | D12 | — | 2 | 6 | no | — |
| 207 | D12 | — | 4 | 1 | no | — |
| 208 | C4×D3 | — | 4 | 6 | no | — |
| 209 | C4×D3 | — | 4 | 2 | no | — |
| 210 | Dic6 | — | 2 | 6 | no | — |
| 211 | Dic6 | — | 4 | 1 | no | — |
| 212 | C3⋊C8 | — | 4 | 6 | no | — |
| 213 | #41 | — | 4 | 1 | no | — |
| 214 | #41 | — | 4 | 1 | no | — |
| 215 | #41 | — | 4 | 3 | no | — |
| 216 | #41 | — | 4 | 3 | no | — |
| 217 | #41 | — | 4 | 12 | no | — |
| 218 | #41 | — | 6 | 6 | no | — |
| 219 | #41 | — | 6 | 6 | no | — |
| 220 | #41 | — | 8 | 1 | no | — |
| 221 | #41 | — | 8 | 1 | no | — |
| 222 | #41 | — | 12 | 3 | no | — |
| 223 | #41 | — | 12 | 6 | no | — |
| 224 | #41 | — | 12 | 2 | no | — |
| 225 | #41 | — | 12 | 1 | no | — |
| 226 | C2×Q8 | — | 2 | 1 | no | — |
| 227 | C2×Q8 | — | 2 | 1 | no | — |
| 228 | C2×Q8 | — | 6 | 3 | no | — |
| 229 | C2×Q8 | — | 12 | 2 | no | — |
| 230 | C2×Q8 | C2×#67 | 1 | 3 | yes | — |
| 231 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 232 | C2×D4 | — | 2 | 3 | no | — |
| 233 | C2×D4 | — | 2 | 6 | no | — |
| 234 | C2×D4 | — | 2 | 3 | no | — |
| 235 | C2×D4 | — | 2 | 3 | no | — |
| 236 | C2×D4 | — | 6 | 3 | no | — |
| 237 | C2×D4 | — | 12 | 6 | no | — |
| 238 | C2×D4 | — | 12 | 3 | no | — |
| 239 | C2×D4 | C2×#67 | 1 | 3 | yes | — |
| 240 | C2×D4 | C2×#67 | 1 | 3 | yes | — |
| 241 | C22×C4 | — | 2 | 6 | no | — |
| 242 | C22×C4 | — | 2 | 3 | no | — |
| 243 | C22×C4 | — | 2 | 3 | no | — |
| 244 | C22×C4 | — | 12 | 3 | no | — |
| 245 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 246 | Q16 | — | 12 | 6 | no | — |
| 247 | SD16 | — | 12 | 6 | no | — |
| 248 | SD16 | — | 12 | 6 | no | — |
| 249 | D8 | — | 12 | 6 | no | — |
| 250 | M16 | — | 6 | 6 | no | — |
| 251 | M16 | — | 12 | 3 | no | — |
| 252 | C2×C8 | — | 12 | 3 | no | — |
| 253 | C4⋊C4 | — | 2 | 3 | no | — |
| 254 | C4⋊C4 | — | 2 | 3 | no | — |
| 255 | C4⋊C4 | — | 2 | 6 | no | — |
| 256 | C22⋊C4 | — | 2 | 6 | no | — |
| 257 | C22⋊C4 | — | 2 | 6 | no | — |
| 258 | C22⋊C4 | — | 2 | 6 | no | — |
| 259 | C42 | — | 2 | 3 | no | — |
| 260 | C42 | — | 2 | 3 | no | — |
| 261 | C42 | — | 2 | 3 | no | — |
| 262 | C42 | C2×#67 | 1 | 3 | yes | — |
| 263 | C42 | C2×#67 | 1 | 3 | yes | — |
| 264 | C22×C3 | — | 2 | 3 | no | — |
| 265 | C22×C3 | — | 4 | 3 | no | — |
| 266 | C22×C3 | — | 4 | 6 | no | — |
| 267 | C22×C3 | — | 4 | 3 | no | — |
| 268 | C22×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 269 | C2×D3 | — | 4 | 2 | no | — |
| 270 | C2×D3 | — | 16 | 3 | no | — |
| 271 | C3×C4 | — | 2 | 6 | no | — |
| 272 | C3×C4 | — | 2 | 3 | no | — |
| 273 | C3×C4 | — | 4 | 3 | no | — |
| 274 | C3×C4 | — | 4 | 3 | no | — |
| 275 | C3×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 276 | C3×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 277 | Dic3 | — | 4 | 2 | no | — |
| 278 | C23 | — | 2 | 3 | no | — |
| 279 | C23 | — | 12 | 3 | no | — |
| 280 | C23 | #953 | 1 | 3 | yes | — |
| 281 | Q8 | — | 2 | 4 | no | — |
| 282 | Q8 | — | 4 | 1 | no | — |
| 283 | Q8 | — | 4 | 3 | no | — |
| 284 | Q8 | — | 6 | 6 | no | — |
| 285 | Q8 | — | 8 | 1 | no | — |
| 286 | Q8 | — | 12 | 1 | no | — |
| 287 | D4 | — | 4 | 3 | no | — |
| 288 | D4 | — | 4 | 3 | no | — |
| 289 | D4 | — | 4 | 12 | no | — |
| 290 | D4 | — | 4 | 6 | no | — |
| 291 | D4 | — | 6 | 6 | no | — |
| 292 | D4 | — | 8 | 3 | no | — |
| 293 | D4 | — | 8 | 3 | no | — |
| 294 | D4 | — | 12 | 6 | no | — |
| 295 | D4 | — | 12 | 2 | no | — |
| 296 | D4 | — | 12 | 1 | no | — |
| 297 | C2×C4 | — | 2 | 3 | no | — |
| 298 | C2×C4 | — | 2 | 3 | no | — |
| 299 | C2×C4 | — | 2 | 3 | no | — |
| 300 | C2×C4 | — | 2 | 3 | no | — |
| 301 | C2×C4 | — | 2 | 3 | no | — |
| 302 | C2×C4 | — | 4 | 6 | no | — |
| 303 | C2×C4 | — | 4 | 12 | no | — |
| 304 | C2×C4 | — | 4 | 3 | no | — |
| 305 | C2×C4 | — | 4 | 3 | no | — |
| 306 | C2×C4 | — | 12 | 6 | no | — |
| 307 | C2×C4 | — | 12 | 3 | no | — |
| 308 | C2×C4 | — | 12 | 2 | no | — |
| 309 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C22×#67 | 1 | 3 | yes | — |
| 311 | C8 | — | 12 | 6 | no | — |
| 312 | C2×C3 | — | 2 | 1 | no | — |
| 313 | C2×C3 | — | 4 | 3 | no | — |
| 314 | C2×C3 | — | 8 | 3 | no | — |
| 315 | C2×C3 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 316 | D3 | — | 8 | 2 | no | — |
| 317 | C22 | — | 2 | 3 | no | — |
| 318 | C22 | — | 4 | 3 | no | — |
| 319 | C22 | — | 4 | 6 | no | — |
| 320 | C22 | — | 4 | 3 | no | — |
| 321 | C22 | — | 12 | 2 | no | — |
| 322 | C22 | — | 48 | 3 | no | — |
| 323 | C22 | C2×#953 | 1 | 1 | yes | characteristic |
| 324 | C4 | — | 2 | 6 | no | — |
| 325 | C4 | — | 2 | 3 | no | — |
| 326 | C4 | — | 4 | 3 | no | — |
| 327 | C4 | — | 4 | 3 | no | — |
| 328 | C4 | — | 12 | 2 | no | — |
| 329 | C4 | #5244 | 1 | 1 | yes | characteristic |
| 330 | C4 | #5244 | 1 | 1 | yes | characteristic |
| 331 | C3 | #33567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 4 | 3 | no | — |
| 334 | C2 | — | 8 | 3 | no | — |
| 335 | C2 | — | 24 | 2 | no | — |
| 336 | C2 | C2×#5244 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3483 | C2 | semidirect |
| 2. #83322 | C2 | semidirect |
| 3. #71942 | C2 | semidirect |
| 4. #71930 | C2 | semidirect |
| 5. #71820 | C2 | semidirect |
| 8. C3×Q82 | C22 | semidirect |
| 10. C3×#555 | C22 | semidirect |
| 22. #4548 | C22 | semidirect |
| 23. #4542 | C22 | semidirect |
| 24. #3483 | D3 | semidirect |
| 72. Q82 | C2×D3 | semidirect |
| 74. #555 | C2×D3 | semidirect |
| 331. C3 | #33567 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3483 | 2 |
| #83322 | 2 |
| #71942 | 2 |
| #71930 | 2 |
| #71820 | 2 |
| #33567 | 3 |