Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
294 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 45 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 | A4×#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #73104 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×A4⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #12683 | — | 3 | 1 | no | maximal, Sylow |
| 5 | A4×#41 | C4 | 1 | 1 | yes | characteristic |
| 6 | A4×M16 | — | 2 | 1 | no | — |
| 7 | A4×M16 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×C8×A4 | — | 2 | 1 | no | — |
| 9 | C2×C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 10 | A4⋊C16 | — | 2 | 1 | no | — |
| 11 | A4⋊C16 | — | 2 | 1 | no | — |
| 12 | #4561 | — | 4 | 1 | no | maximal |
| 13 | C22×#131 | D3 | 1 | 1 | yes | characteristic |
| 14 | C2×#350 | — | 3 | 1 | no | — |
| 15 | C2×#350 | — | 3 | 1 | no | — |
| 16 | C2×#350 | — | 3 | 1 | no | — |
| 17 | C2×#350 | — | 3 | 1 | no | — |
| 18 | #2112 | — | 3 | 1 | no | — |
| 19 | C2×C22⋊C16 | — | 3 | 1 | no | — |
| 20 | Q8×A4 | C8 | 1 | 1 | yes | characteristic |
| 21 | D4×A4 | — | 2 | 1 | no | — |
| 22 | D4×A4 | C8 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×A4 | — | 2 | 1 | no | — |
| 24 | C2×C4×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C3×#131 | — | 4 | 1 | no | — |
| 26 | C8×A4 | — | 2 | 1 | no | — |
| 27 | C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 28 | C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 29 | #812 | — | 4 | 1 | no | — |
| 30 | C2×C3⋊C16 | — | 4 | 1 | no | — |
| 31 | C22×#41 | Dic3 | 1 | 1 | yes | characteristic |
| 32 | C2×#131 | — | 3 | 1 | no | — |
| 33 | C2×#131 | — | 3 | 1 | no | — |
| 34 | C2×#131 | — | 3 | 1 | no | — |
| 35 | C2×#131 | — | 3 | 1 | no | — |
| 36 | C2×#131 | — | 6 | 1 | no | — |
| 37 | C2×#131 | — | 6 | 1 | no | — |
| 38 | C22×M16 | — | 2 | 1 | no | — |
| 39 | C22×M16 | Dic3 | 1 | 1 | yes | characteristic |
| 40 | C23×C8 | — | 2 | 1 | no | — |
| 41 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 42 | C2×M32 | — | 3 | 1 | no | — |
| 43 | C2×M32 | — | 3 | 1 | no | — |
| 44 | C22×C16 | — | 3 | 1 | no | — |
| 45 | C22×C16 | — | 3 | 1 | no | — |
| 46 | #350 | — | 6 | 2 | no | — |
| 47 | #350 | — | 6 | 2 | no | — |
| 48 | #350 | — | 6 | 1 | no | — |
| 49 | #350 | — | 6 | 1 | no | — |
| 50 | #350 | — | 6 | 1 | no | — |
| 51 | #350 | — | 6 | 1 | no | — |
| 52 | C22⋊C16 | — | 6 | 1 | no | — |
| 53 | C22⋊C16 | — | 6 | 1 | no | — |
| 54 | C22⋊C16 | — | 6 | 1 | no | — |
| 55 | C22⋊C16 | — | 6 | 1 | no | — |
| 56 | C22×A4 | — | 2 | 1 | no | — |
| 57 | C22×A4 | M16 | 1 | 1 | yes | characteristic |
| 58 | C3×#41 | — | 4 | 1 | no | — |
| 59 | C4×A4 | — | 2 | 1 | no | — |
| 60 | C4×A4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C4×A4 | C2×C8 | 1 | 1 | yes | characteristic |
| 62 | C3×M16 | — | 4 | 1 | no | — |
| 63 | C3×M16 | — | 8 | 1 | no | — |
| 64 | C2×C3×C8 | — | 4 | 1 | no | — |
| 65 | C2×C3×C8 | — | 8 | 1 | no | — |
| 66 | C3⋊C16 | — | 8 | 1 | no | — |
| 67 | C3⋊C16 | — | 8 | 1 | no | — |
| 68 | C2×#41 | — | 3 | 1 | no | — |
| 69 | C2×#41 | — | 3 | 1 | no | — |
| 70 | C2×#41 | — | 3 | 1 | no | — |
| 71 | C2×#41 | — | 3 | 1 | no | — |
| 72 | C2×#41 | — | 6 | 1 | no | — |
| 73 | C2×#41 | — | 6 | 1 | no | — |
| 74 | C22×Q8 | C3⋊C8 | 1 | 1 | yes | characteristic |
| 75 | C22×D4 | — | 2 | 1 | no | — |
| 76 | C22×D4 | C3⋊C8 | 1 | 1 | yes | characteristic |
| 77 | C23×C4 | — | 2 | 1 | no | — |
| 78 | C23×C4 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 79 | #131 | — | 3 | 2 | no | — |
| 80 | #131 | — | 3 | 2 | no | — |
| 81 | #131 | — | 3 | 1 | no | — |
| 82 | #131 | — | 3 | 1 | no | — |
| 83 | #131 | — | 3 | 1 | no | — |
| 84 | #131 | — | 6 | 1 | no | — |
| 85 | #131 | — | 6 | 1 | no | — |
| 86 | #131 | — | 6 | 1 | no | — |
| 87 | #131 | — | 6 | 2 | no | — |
| 88 | #131 | — | 6 | 2 | no | — |
| 89 | #131 | S4 | 1 | 1 | yes | characteristic |
| 90 | C2×M16 | — | 3 | 1 | no | — |
| 91 | C2×M16 | — | 3 | 1 | no | — |
| 92 | C2×M16 | — | 6 | 1 | no | — |
| 93 | C2×M16 | — | 6 | 1 | no | — |
| 94 | C2×M16 | — | 6 | 1 | no | — |
| 95 | C2×M16 | — | 6 | 1 | no | — |
| 96 | C2×M16 | — | 6 | 1 | no | — |
| 97 | C22×C8 | — | 2 | 1 | no | — |
| 98 | C22×C8 | — | 3 | 1 | no | — |
| 99 | C22×C8 | — | 3 | 1 | no | — |
| 100 | C22×C8 | — | 3 | 1 | no | — |
| 101 | C22×C8 | — | 3 | 1 | no | — |
| 102 | C22×C8 | — | 6 | 1 | no | — |
| 103 | C22×C8 | — | 6 | 1 | no | — |
| 104 | C22×C8 | — | 6 | 1 | no | — |
| 105 | C22×C8 | — | 6 | 1 | no | — |
| 106 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 107 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 108 | M32 | — | 6 | 1 | no | — |
| 109 | M32 | — | 6 | 1 | no | — |
| 110 | M32 | — | 6 | 1 | no | — |
| 111 | M32 | — | 6 | 1 | no | — |
| 112 | C2×C16 | — | 6 | 1 | no | — |
| 113 | C2×C16 | — | 6 | 1 | no | — |
| 114 | C2×C16 | — | 6 | 1 | no | — |
| 115 | C2×C16 | — | 6 | 1 | no | — |
| 116 | C2×C16 | — | 6 | 1 | no | — |
| 117 | C2×C16 | — | 6 | 1 | no | — |
| 118 | C2×C16 | — | 6 | 1 | no | — |
| 119 | C2×C16 | — | 6 | 1 | no | — |
| 120 | C2×A4 | — | 2 | 1 | no | — |
| 121 | C2×A4 | — | 4 | 1 | no | — |
| 122 | C2×A4 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 123 | C3×Q8 | — | 4 | 1 | no | — |
| 124 | C3×D4 | — | 4 | 1 | no | — |
| 125 | C3×D4 | — | 8 | 1 | no | — |
| 126 | C2×C3×C4 | — | 4 | 1 | no | — |
| 127 | C2×C3×C4 | — | 8 | 1 | no | — |
| 128 | C3×C8 | — | 4 | 1 | no | — |
| 129 | C3×C8 | — | 4 | 1 | no | — |
| 130 | C3×C8 | — | 8 | 1 | no | — |
| 131 | C24 | — | 2 | 1 | no | — |
| 132 | C24 | #208 | 1 | 1 | yes | characteristic |
| 133 | #41 | — | 3 | 2 | no | — |
| 134 | #41 | — | 3 | 2 | no | — |
| 135 | #41 | — | 3 | 1 | no | — |
| 136 | #41 | — | 3 | 1 | no | — |
| 137 | #41 | — | 3 | 1 | no | — |
| 138 | #41 | — | 6 | 1 | no | — |
| 139 | #41 | — | 6 | 1 | no | — |
| 140 | #41 | — | 6 | 1 | no | — |
| 141 | #41 | — | 6 | 2 | no | — |
| 142 | #41 | — | 6 | 2 | no | — |
| 143 | #41 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 144 | C2×Q8 | — | 3 | 1 | no | — |
| 145 | C2×Q8 | — | 3 | 1 | no | — |
| 146 | C2×Q8 | — | 6 | 1 | no | — |
| 147 | C2×D4 | — | 3 | 1 | no | — |
| 148 | C2×D4 | — | 3 | 1 | no | — |
| 149 | C2×D4 | — | 6 | 1 | no | — |
| 150 | C2×D4 | — | 6 | 1 | no | — |
| 151 | C2×D4 | — | 6 | 1 | no | — |
| 152 | C2×D4 | — | 6 | 1 | no | — |
| 153 | C2×D4 | — | 6 | 1 | no | — |
| 154 | C22×C4 | — | 2 | 1 | no | — |
| 155 | C22×C4 | — | 3 | 1 | no | — |
| 156 | C22×C4 | — | 3 | 1 | no | — |
| 157 | C22×C4 | — | 3 | 1 | no | — |
| 158 | C22×C4 | — | 3 | 1 | no | — |
| 159 | C22×C4 | — | 6 | 1 | no | — |
| 160 | C22×C4 | — | 6 | 1 | no | — |
| 161 | C22×C4 | — | 6 | 1 | no | — |
| 162 | C22×C4 | — | 6 | 1 | no | — |
| 163 | C22×C4 | C2×C3⋊C8 | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | #217 | 1 | 1 | yes | characteristic |
| 165 | M16 | — | 2 | 1 | no | — |
| 166 | M16 | — | 3 | 2 | no | — |
| 167 | M16 | — | 3 | 1 | no | — |
| 168 | M16 | — | 6 | 1 | no | — |
| 169 | M16 | — | 6 | 1 | no | — |
| 170 | M16 | — | 6 | 1 | no | — |
| 171 | M16 | — | 6 | 2 | no | — |
| 172 | M16 | — | 6 | 1 | no | — |
| 173 | M16 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | — | 2 | 1 | no | — |
| 175 | C2×C8 | — | 3 | 1 | no | — |
| 176 | C2×C8 | — | 3 | 1 | no | — |
| 177 | C2×C8 | — | 3 | 1 | no | — |
| 178 | C2×C8 | — | 3 | 1 | no | — |
| 179 | C2×C8 | — | 3 | 1 | no | — |
| 180 | C2×C8 | — | 3 | 1 | no | — |
| 181 | C2×C8 | — | 3 | 1 | no | — |
| 182 | C2×C8 | — | 6 | 1 | no | — |
| 183 | C2×C8 | — | 6 | 1 | no | — |
| 184 | C2×C8 | — | 6 | 1 | no | — |
| 185 | C2×C8 | — | 6 | 1 | no | — |
| 186 | C2×C8 | — | 6 | 2 | no | — |
| 187 | C2×C8 | — | 6 | 1 | no | — |
| 188 | C2×C8 | — | 6 | 1 | no | — |
| 189 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 190 | C16 | — | 12 | 1 | no | — |
| 191 | C16 | — | 12 | 1 | no | — |
| 192 | C16 | — | 12 | 1 | no | — |
| 193 | C16 | — | 12 | 1 | no | — |
| 194 | C22×C3 | — | 4 | 1 | no | — |
| 195 | C22×C3 | — | 8 | 1 | no | — |
| 196 | A4 | #350 | 1 | 1 | yes | characteristic, complemented |
| 197 | C3×C4 | — | 4 | 1 | no | — |
| 198 | C3×C4 | — | 4 | 1 | no | — |
| 199 | C3×C4 | — | 8 | 1 | no | — |
| 200 | C23 | — | 2 | 1 | no | — |
| 201 | C23 | — | 3 | 1 | no | — |
| 202 | C23 | — | 3 | 1 | no | — |
| 203 | C23 | — | 4 | 1 | no | — |
| 204 | C23 | — | 6 | 1 | no | — |
| 205 | C23 | — | 6 | 1 | no | — |
| 206 | C23 | — | 6 | 1 | no | — |
| 207 | C23 | — | 12 | 1 | no | — |
| 208 | C23 | #830 | 1 | 1 | yes | characteristic |
| 209 | Q8 | — | 3 | 2 | no | — |
| 210 | Q8 | — | 3 | 1 | no | — |
| 211 | Q8 | — | 6 | 1 | no | — |
| 212 | Q8 | A4⋊C8 | 1 | 1 | yes | characteristic |
| 213 | D4 | — | 2 | 1 | no | — |
| 214 | D4 | — | 3 | 2 | no | — |
| 215 | D4 | — | 3 | 1 | no | — |
| 216 | D4 | — | 6 | 1 | no | — |
| 217 | D4 | — | 6 | 1 | no | — |
| 218 | D4 | — | 6 | 1 | no | — |
| 219 | D4 | — | 6 | 2 | no | — |
| 220 | D4 | — | 6 | 1 | no | — |
| 221 | D4 | A4⋊C8 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | — | 2 | 1 | no | — |
| 223 | C2×C4 | — | 3 | 1 | no | — |
| 224 | C2×C4 | — | 3 | 1 | no | — |
| 225 | C2×C4 | — | 3 | 1 | no | — |
| 226 | C2×C4 | — | 3 | 1 | no | — |
| 227 | C2×C4 | — | 3 | 1 | no | — |
| 228 | C2×C4 | — | 3 | 1 | no | — |
| 229 | C2×C4 | — | 3 | 1 | no | — |
| 230 | C2×C4 | — | 6 | 1 | no | — |
| 231 | C2×C4 | — | 6 | 1 | no | — |
| 232 | C2×C4 | — | 6 | 1 | no | — |
| 233 | C2×C4 | — | 6 | 1 | no | — |
| 234 | C2×C4 | — | 6 | 2 | no | — |
| 235 | C2×C4 | — | 6 | 1 | no | — |
| 236 | C2×C4 | — | 6 | 1 | no | — |
| 237 | C2×C4 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 238 | C8 | — | 2 | 1 | no | — |
| 239 | C8 | — | 3 | 1 | no | — |
| 240 | C8 | — | 3 | 1 | no | — |
| 241 | C8 | — | 6 | 1 | no | — |
| 242 | C8 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 243 | C8 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 244 | C2×C3 | — | 4 | 1 | no | — |
| 245 | C2×C3 | — | 8 | 1 | no | — |
| 246 | C2×C3 | — | 16 | 1 | no | — |
| 247 | C22 | — | 2 | 1 | no | — |
| 248 | C22 | — | 3 | 1 | no | — |
| 249 | C22 | — | 3 | 1 | no | — |
| 250 | C22 | — | 3 | 1 | no | — |
| 251 | C22 | — | 6 | 1 | no | — |
| 252 | C22 | — | 6 | 1 | no | — |
| 253 | C22 | — | 6 | 1 | no | — |
| 254 | C22 | — | 6 | 1 | no | — |
| 255 | C22 | — | 6 | 1 | no | — |
| 256 | C22 | — | 12 | 1 | no | — |
| 257 | C22 | — | 12 | 1 | no | — |
| 258 | C22 | — | 12 | 1 | no | — |
| 259 | C22 | — | 12 | 1 | no | — |
| 260 | C22 | #4561 | 1 | 1 | yes | characteristic, complemented |
| 261 | C22 | #5409 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 3 | 1 | no | — |
| 264 | C4 | — | 3 | 1 | no | — |
| 265 | C4 | — | 6 | 1 | no | — |
| 266 | C4 | C2×A4⋊C8 | 1 | 1 | yes | characteristic |
| 267 | C4 | #5432 | 1 | 1 | yes | characteristic |
| 268 | C3 | — | 4 | 1 | no | Sylow |
| 269 | C2 | — | 2 | 1 | no | — |
| 270 | C2 | — | 3 | 1 | no | — |
| 271 | C2 | — | 3 | 1 | no | — |
| 272 | C2 | — | 4 | 1 | no | — |
| 273 | C2 | — | 6 | 1 | no | — |
| 274 | C2 | — | 12 | 1 | no | — |
| 275 | C2 | #73141 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #73104 | C2 | semidirect |
| 3. C2×A4⋊C16 | C2 | semidirect |
| 196. A4 | #350 | semidirect |
| 260. C22 | #4561 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#131 | 2 |
| #73104 | 2 |
| C2×A4⋊C16 | 2 |
| #12683 | 3 |
| #4561 | 4 |