Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1047 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 151 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 | C2×C4×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#5424 | C2 | 1 | 2 | yes | maximal |
| 3 | #85334 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×#5391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#5389 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#5386 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#3076 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×Dic6 | — | 4 | 1 | no | maximal |
| 10 | C4×#41⋊C3 | C22 | 1 | 4 | yes | — |
| 11 | C2×C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 13 | #5424 | C22 | 1 | 8 | yes | — |
| 14 | C2×#859 | C22 | 1 | 2 | yes | — |
| 15 | C2×#859 | C22 | 1 | 2 | yes | — |
| 16 | #5391 | C22 | 1 | 4 | yes | complemented |
| 17 | #5389 | C22 | 1 | 8 | yes | complemented |
| 18 | #5386 | C22 | 1 | 4 | yes | complemented |
| 19 | C2×#533 | — | 3 | 1 | no | — |
| 20 | C2×C4×#41 | D3 | 1 | 1 | yes | characteristic |
| 21 | #3076 | — | 3 | 8 | no | — |
| 22 | #3076 | — | 3 | 8 | no | — |
| 23 | C2×#479 | — | 3 | 2 | no | — |
| 24 | C2×#477 | — | 3 | 1 | no | — |
| 25 | C2×#475 | — | 3 | 1 | no | — |
| 26 | C2×#478 | — | 3 | 2 | no | — |
| 27 | C2×#476 | — | 3 | 1 | no | — |
| 28 | C2×#474 | — | 3 | 1 | no | — |
| 29 | C2×#427 | — | 3 | 2 | no | — |
| 30 | C22×C4⋊C8 | — | 3 | 1 | no | — |
| 31 | C2×#416 | — | 3 | 2 | no | — |
| 32 | C22×Dic6 | — | 4 | 1 | no | — |
| 33 | C2×#41⋊C3 | Q8 | 1 | 4 | yes | complemented |
| 34 | C2×#41⋊C3 | C23 | 1 | 2 | yes | — |
| 35 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C3×C42 | — | 4 | 1 | no | — |
| 37 | C2×C4⋊Dic3 | — | 4 | 1 | no | — |
| 38 | C2×Dic3⋊C4 | — | 4 | 2 | no | — |
| 39 | C2×C4×Dic3 | — | 4 | 2 | no | — |
| 40 | C4×Dic6 | — | 4 | 8 | no | — |
| 41 | C4×SL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 42 | C4×SL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 43 | #859 | C23 | 1 | 4 | yes | — |
| 44 | #859 | C23 | 1 | 4 | yes | — |
| 45 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 46 | C23×C8 | — | 3 | 2 | no | — |
| 47 | #533 | — | 3 | 8 | no | — |
| 48 | #533 | — | 6 | 4 | no | — |
| 49 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 50 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 51 | C2×#125 | — | 3 | 2 | no | — |
| 52 | C2×C22⋊Q8 | — | 6 | 1 | no | — |
| 53 | C2×C22⋊Q8 | — | 6 | 1 | no | — |
| 54 | C4×#41 | — | 3 | 4 | no | — |
| 55 | C4×#41 | C2×D3 | 1 | 4 | yes | — |
| 56 | C2×C4×Q8 | — | 6 | 1 | no | — |
| 57 | C2×C4×Q8 | — | 6 | 1 | no | — |
| 58 | C2×C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C2×C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C2×C4×D4 | — | 3 | 1 | no | — |
| 61 | C2×C4×D4 | — | 3 | 1 | no | — |
| 62 | C2×#117 | — | 3 | 1 | no | — |
| 63 | C2×#117 | — | 3 | 2 | no | — |
| 64 | C22×C42 | — | 3 | 1 | no | — |
| 65 | #479 | — | 3 | 8 | no | — |
| 66 | #479 | — | 3 | 8 | no | — |
| 67 | #478 | — | 3 | 16 | no | — |
| 68 | #477 | — | 3 | 4 | no | — |
| 69 | #477 | — | 3 | 4 | no | — |
| 70 | #476 | — | 3 | 8 | no | — |
| 71 | #475 | — | 3 | 4 | no | — |
| 72 | #475 | — | 3 | 4 | no | — |
| 73 | #474 | — | 3 | 8 | no | — |
| 74 | #427 | — | 3 | 16 | no | — |
| 75 | C2×#107 | — | 3 | 2 | no | — |
| 76 | C2×#107 | — | 3 | 2 | no | — |
| 77 | C2×#106 | — | 3 | 2 | no | — |
| 78 | C2×#106 | — | 3 | 2 | no | — |
| 79 | C2×C4⋊C8 | — | 3 | 2 | no | — |
| 80 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 81 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 82 | C2×C4⋊C8 | — | 3 | 8 | no | — |
| 83 | #416 | — | 3 | 8 | no | — |
| 84 | #416 | — | 3 | 8 | no | — |
| 85 | C2×#103 | — | 3 | 2 | no | — |
| 86 | C2×#103 | — | 3 | 2 | no | — |
| 87 | C2×#102 | — | 3 | 2 | no | — |
| 88 | C2×#102 | — | 3 | 2 | no | — |
| 89 | C22×C3×C4 | — | 4 | 1 | no | — |
| 90 | C22×C3×C4 | — | 4 | 1 | no | — |
| 91 | C22×C3×C4 | — | 4 | 1 | no | — |
| 92 | C22×Dic3 | — | 4 | 2 | no | — |
| 93 | C22×Dic3 | — | 4 | 2 | no | — |
| 94 | C2×Dic6 | — | 4 | 4 | no | — |
| 95 | C2×Dic6 | — | 4 | 8 | no | — |
| 96 | #41⋊C3 | C2×Q8 | 1 | 4 | yes | complemented |
| 97 | C2×SL(2,𝔽3) | C2×Q8 | 1 | 2 | yes | — |
| 98 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 99 | C3×C42 | — | 4 | 4 | no | — |
| 100 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 101 | Dic3⋊C4 | — | 4 | 8 | no | — |
| 102 | C4×Dic3 | — | 4 | 8 | no | — |
| 103 | C2×#41 | — | 3 | 2 | no | — |
| 104 | C2×#41 | — | 3 | 4 | no | — |
| 105 | C2×#41 | Dic6 | 1 | 4 | yes | complemented |
| 106 | C2×#41 | C22×D3 | 1 | 2 | yes | — |
| 107 | C22×Q8 | — | 6 | 1 | no | — |
| 108 | C22×Q8 | — | 6 | 1 | no | — |
| 109 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 110 | C22×D4 | — | 3 | 1 | no | — |
| 111 | C23×C4 | — | 3 | 1 | no | — |
| 112 | C23×C4 | — | 3 | 1 | no | — |
| 113 | C23×C4 | — | 3 | 1 | no | — |
| 114 | C22×C8 | — | 3 | 2 | no | — |
| 115 | C22×C8 | — | 3 | 2 | no | — |
| 116 | C22×C8 | — | 3 | 8 | no | — |
| 117 | C22×C8 | — | 6 | 8 | no | — |
| 118 | C4⋊Q8 | — | 3 | 4 | no | — |
| 119 | C4⋊Q8 | — | 3 | 4 | no | — |
| 120 | C4⋊Q8 | — | 6 | 2 | no | — |
| 121 | C4⋊Q8 | — | 6 | 2 | no | — |
| 122 | #125 | — | 3 | 8 | no | — |
| 123 | #125 | — | 6 | 4 | no | — |
| 124 | C22⋊Q8 | — | 6 | 8 | no | — |
| 125 | C22⋊Q8 | — | 6 | 8 | no | — |
| 126 | C4×Q8 | — | 3 | 2 | no | — |
| 127 | C4×Q8 | — | 3 | 2 | no | — |
| 128 | C4×Q8 | — | 6 | 8 | no | — |
| 129 | C4×Q8 | — | 6 | 8 | no | — |
| 130 | C4×Q8 | C22×D3 | 1 | 2 | yes | — |
| 131 | C4×Q8 | C22×D3 | 1 | 2 | yes | — |
| 132 | C4×D4 | — | 3 | 4 | no | — |
| 133 | C4×D4 | — | 3 | 4 | no | — |
| 134 | C4×D4 | — | 6 | 2 | no | — |
| 135 | C4×D4 | — | 6 | 2 | no | — |
| 136 | #117 | — | 3 | 4 | no | — |
| 137 | #117 | — | 3 | 8 | no | — |
| 138 | #117 | — | 6 | 2 | no | — |
| 139 | #117 | — | 6 | 4 | no | — |
| 140 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 141 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 142 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 143 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 144 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 145 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 146 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 147 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 148 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 149 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 150 | C2×C42 | — | 3 | 1 | no | — |
| 151 | C2×C42 | — | 3 | 1 | no | — |
| 152 | C2×C42 | — | 3 | 1 | no | — |
| 153 | C2×C42 | — | 3 | 4 | no | — |
| 154 | C2×C42 | — | 3 | 4 | no | — |
| 155 | C2×C42 | — | 6 | 2 | no | — |
| 156 | C2×C42 | S4 | 1 | 1 | yes | characteristic |
| 157 | #107 | — | 3 | 8 | no | — |
| 158 | #107 | — | 3 | 8 | no | — |
| 159 | #106 | — | 3 | 8 | no | — |
| 160 | #106 | — | 3 | 8 | no | — |
| 161 | C4⋊C8 | — | 3 | 4 | no | — |
| 162 | C4⋊C8 | — | 3 | 4 | no | — |
| 163 | C4⋊C8 | — | 3 | 8 | no | — |
| 164 | #103 | — | 3 | 4 | no | — |
| 165 | #103 | — | 3 | 4 | no | — |
| 166 | #103 | — | 3 | 4 | no | — |
| 167 | #103 | — | 3 | 4 | no | — |
| 168 | #102 | — | 3 | 8 | no | — |
| 169 | #102 | — | 3 | 8 | no | — |
| 170 | C23×C3 | — | 4 | 1 | no | — |
| 171 | C2×C3×C4 | — | 4 | 4 | no | — |
| 172 | C2×C3×C4 | — | 4 | 4 | no | — |
| 173 | C2×C3×C4 | — | 4 | 2 | no | — |
| 174 | C2×C3×C4 | — | 4 | 2 | no | — |
| 175 | C2×C3×C4 | — | 4 | 2 | no | — |
| 176 | C2×C3×C4 | — | 8 | 2 | no | — |
| 177 | C2×Dic3 | — | 4 | 4 | no | — |
| 178 | C2×Dic3 | — | 4 | 4 | no | — |
| 179 | C2×Dic3 | — | 4 | 8 | no | — |
| 180 | C2×Dic3 | — | 8 | 4 | no | — |
| 181 | Dic6 | — | 4 | 16 | no | — |
| 182 | SL(2,𝔽3) | C22×Q8 | 1 | 1 | yes | characteristic |
| 183 | C24 | — | 3 | 1 | no | — |
| 184 | #41 | — | 3 | 8 | no | — |
| 185 | #41 | — | 3 | 4 | no | — |
| 186 | #41 | — | 6 | 4 | no | — |
| 187 | #41 | C2×Dic6 | 1 | 4 | yes | complemented |
| 188 | C2×Q8 | — | 3 | 1 | no | — |
| 189 | C2×Q8 | — | 3 | 2 | no | — |
| 190 | C2×Q8 | — | 6 | 4 | no | — |
| 191 | C2×Q8 | — | 6 | 4 | no | — |
| 192 | C2×Q8 | — | 12 | 4 | no | — |
| 193 | C2×Q8 | — | 12 | 4 | no | — |
| 194 | C2×Q8 | C2×Dic6 | 1 | 2 | yes | — |
| 195 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 196 | C2×D4 | — | 3 | 2 | no | — |
| 197 | C2×D4 | — | 3 | 4 | no | — |
| 198 | C2×D4 | — | 6 | 1 | no | — |
| 199 | C2×D4 | — | 6 | 2 | no | — |
| 200 | C22×C4 | — | 3 | 1 | no | — |
| 201 | C22×C4 | — | 3 | 4 | no | — |
| 202 | C22×C4 | — | 3 | 4 | no | — |
| 203 | C22×C4 | — | 3 | 2 | no | — |
| 204 | C22×C4 | — | 3 | 2 | no | — |
| 205 | C22×C4 | — | 3 | 1 | no | — |
| 206 | C22×C4 | — | 3 | 1 | no | — |
| 207 | C22×C4 | — | 3 | 4 | no | — |
| 208 | C22×C4 | — | 3 | 4 | no | — |
| 209 | C22×C4 | — | 6 | 4 | no | — |
| 210 | C22×C4 | — | 6 | 2 | no | — |
| 211 | C22×C4 | — | 6 | 2 | no | — |
| 212 | C22×C4 | — | 12 | 2 | no | — |
| 213 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 214 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 215 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | — | 3 | 4 | no | — |
| 217 | C2×C8 | — | 3 | 4 | no | — |
| 218 | C2×C8 | — | 6 | 4 | no | — |
| 219 | C2×C8 | — | 6 | 4 | no | — |
| 220 | C2×C8 | — | 6 | 16 | no | — |
| 221 | C4⋊C4 | — | 3 | 2 | no | — |
| 222 | C4⋊C4 | — | 3 | 2 | no | — |
| 223 | C4⋊C4 | — | 3 | 4 | no | — |
| 224 | C4⋊C4 | — | 3 | 4 | no | — |
| 225 | C4⋊C4 | — | 6 | 1 | no | — |
| 226 | C4⋊C4 | — | 6 | 1 | no | — |
| 227 | C4⋊C4 | — | 6 | 2 | no | — |
| 228 | C4⋊C4 | — | 6 | 2 | no | — |
| 229 | C4⋊C4 | — | 6 | 8 | no | — |
| 230 | C4⋊C4 | — | 6 | 4 | no | — |
| 231 | C4⋊C4 | — | 6 | 8 | no | — |
| 232 | C4⋊C4 | — | 6 | 4 | no | — |
| 233 | C22⋊C4 | — | 6 | 4 | no | — |
| 234 | C22⋊C4 | — | 6 | 8 | no | — |
| 235 | C42 | — | 3 | 2 | no | — |
| 236 | C42 | — | 3 | 2 | no | — |
| 237 | C42 | — | 3 | 4 | no | — |
| 238 | C42 | — | 3 | 2 | no | — |
| 239 | C42 | — | 3 | 2 | no | — |
| 240 | C42 | — | 6 | 8 | no | — |
| 241 | C42 | C2×S4 | 1 | 4 | yes | — |
| 242 | C22×C3 | — | 4 | 2 | no | — |
| 243 | C22×C3 | — | 4 | 2 | no | — |
| 244 | C22×C3 | — | 4 | 2 | no | — |
| 245 | C22×C3 | — | 4 | 1 | no | — |
| 246 | C3×C4 | — | 4 | 4 | no | — |
| 247 | C3×C4 | — | 4 | 4 | no | — |
| 248 | C3×C4 | — | 8 | 2 | no | — |
| 249 | Dic3 | — | 4 | 8 | no | — |
| 250 | Dic3 | — | 8 | 4 | no | — |
| 251 | C23 | — | 3 | 4 | no | — |
| 252 | C23 | — | 3 | 2 | no | — |
| 253 | C23 | — | 6 | 2 | no | — |
| 254 | C23 | — | 6 | 2 | no | — |
| 255 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 256 | Q8 | — | 3 | 2 | no | — |
| 257 | Q8 | — | 3 | 1 | no | — |
| 258 | Q8 | — | 6 | 1 | no | — |
| 259 | Q8 | — | 12 | 8 | no | — |
| 260 | Q8 | — | 12 | 8 | no | — |
| 261 | Q8 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 262 | D4 | — | 3 | 4 | no | — |
| 263 | D4 | — | 6 | 2 | no | — |
| 264 | D4 | — | 6 | 4 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 3 | 2 | no | — |
| 267 | C2×C4 | — | 3 | 2 | no | — |
| 268 | C2×C4 | — | 3 | 1 | no | — |
| 269 | C2×C4 | — | 3 | 1 | no | — |
| 270 | C2×C4 | — | 3 | 4 | no | — |
| 271 | C2×C4 | — | 3 | 4 | no | — |
| 272 | C2×C4 | — | 3 | 8 | no | — |
| 273 | C2×C4 | — | 3 | 2 | no | — |
| 274 | C2×C4 | — | 3 | 2 | no | — |
| 275 | C2×C4 | — | 6 | 2 | no | — |
| 276 | C2×C4 | — | 6 | 2 | no | — |
| 277 | C2×C4 | — | 6 | 8 | no | — |
| 278 | C2×C4 | — | 6 | 4 | no | — |
| 279 | C2×C4 | — | 6 | 4 | no | — |
| 280 | C2×C4 | — | 12 | 4 | no | — |
| 281 | C2×C4 | — | 12 | 4 | no | — |
| 282 | C2×C4 | — | 12 | 4 | no | — |
| 283 | C2×C4 | A4⋊Q8 | 1 | 4 | yes | — |
| 284 | C2×C4 | #985 | 1 | 4 | yes | — |
| 285 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 286 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 287 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 288 | C8 | — | 6 | 4 | no | — |
| 289 | C8 | — | 6 | 4 | no | — |
| 290 | C2×C3 | — | 4 | 4 | no | — |
| 291 | C2×C3 | — | 4 | 1 | no | — |
| 292 | C2×C3 | — | 4 | 1 | no | — |
| 293 | C2×C3 | — | 4 | 1 | no | — |
| 294 | C22 | — | 3 | 4 | no | — |
| 295 | C22 | — | 6 | 8 | no | — |
| 296 | C22 | — | 6 | 2 | no | — |
| 297 | C22 | — | 6 | 2 | no | — |
| 298 | C22 | C2×A4⋊Q8 | 1 | 2 | yes | — |
| 299 | C22 | C2×#985 | 1 | 2 | yes | — |
| 300 | C22 | #5923 | 1 | 2 | yes | — |
| 301 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 302 | C4 | — | 2 | 2 | no | — |
| 303 | C4 | — | 3 | 1 | no | — |
| 304 | C4 | — | 3 | 1 | no | — |
| 305 | C4 | — | 3 | 2 | no | — |
| 306 | C4 | — | 6 | 2 | no | — |
| 307 | C4 | — | 6 | 2 | no | — |
| 308 | C4 | — | 12 | 4 | no | — |
| 309 | C4 | — | 12 | 4 | no | — |
| 310 | C4 | C2×A4⋊Q8 | 1 | 4 | yes | — |
| 311 | C4 | C2×#985 | 1 | 4 | yes | — |
| 312 | C3 | — | 4 | 1 | no | Sylow |
| 313 | C2 | — | 6 | 4 | no | — |
| 314 | C2 | #85334 | 1 | 4 | yes | direct product factor |
| 315 | C2 | C22×A4⋊Q8 | 1 | 1 | yes | characteristic |
| 316 | C2 | C22×#985 | 1 | 1 | yes | characteristic |
| 317 | C2 | C2×#5923 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #85334 | C2 | direct (N × H) |
| 4. C2×#5391 | C2 | semidirect |
| 5. C2×#5389 | C2 | semidirect |
| 6. C2×#5386 | C2 | semidirect |
| 16. #5391 | C22 | semidirect |
| 17. #5389 | C22 | semidirect |
| 18. #5386 | C22 | semidirect |
| 33. C2×#41⋊C3 | Q8 | semidirect |
| 96. #41⋊C3 | C2×Q8 | semidirect |
| 105. C2×#41 | Dic6 | semidirect |
| 187. #41 | C2×Dic6 | semidirect |
| 314. C2 | #85334 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#41⋊C3 | 2 |
| C2×#5424 | 2 |
| #85334 | 2 |
| C2×#5391 | 2 |
| C2×#5389 | 2 |
| C2×#5386 | 2 |
| C2×#3076 | 3 |
| C2×C4×Dic6 | 4 |