Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
915 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 129 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 | Q8×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #85415 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #85337 | C2 | 1 | 3 | yes | maximal |
| 5 | C4×#984 | C2 | 1 | 3 | yes | maximal |
| 6 | Q8×#137 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#984 | C22 | 1 | 1 | yes | characteristic |
| 8 | Q8×Dic6 | — | 4 | 1 | no | maximal |
| 9 | Q8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 10 | #557⋊C3 | C22 | 1 | 3 | yes | — |
| 11 | C4×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 12 | #5426 | C22 | 1 | 3 | yes | — |
| 13 | #5391 | C22 | 1 | 6 | yes | — |
| 14 | #5388 | C22 | 1 | 6 | yes | — |
| 15 | C4×#226 | C22 | 1 | 6 | yes | — |
| 16 | #5386 | C22 | 1 | 6 | yes | — |
| 17 | Q8×#41 | D3 | 1 | 1 | yes | characteristic |
| 18 | C2×Q82 | — | 3 | 1 | no | — |
| 19 | #3387 | — | 3 | 6 | no | — |
| 20 | #3385 | — | 3 | 6 | no | — |
| 21 | Q8×Q16 | — | 3 | 2 | no | — |
| 22 | Q8×SD16 | — | 3 | 2 | no | — |
| 23 | #3165 | — | 3 | 3 | no | — |
| 24 | #3079 | — | 3 | 3 | no | — |
| 25 | #3078 | — | 3 | 3 | no | — |
| 26 | Q8×M16 | — | 3 | 1 | no | — |
| 27 | C4×#137 | — | 3 | 3 | no | — |
| 28 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 29 | #984 | Q8 | 1 | 4 | yes | direct product factor |
| 30 | C2×#226 | C23 | 1 | 2 | yes | — |
| 31 | C3×C4×Q8 | — | 4 | 1 | no | — |
| 32 | Q8×Dic3 | — | 4 | 2 | no | — |
| 33 | #888 | — | 4 | 6 | no | — |
| 34 | #869 | — | 4 | 3 | no | — |
| 35 | C4×Dic6 | — | 4 | 3 | no | — |
| 36 | C4×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 37 | C4×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 38 | #859 | C23 | 1 | 6 | yes | — |
| 39 | C2×#137 | — | 3 | 1 | no | — |
| 40 | Q82 | — | 3 | 2 | no | — |
| 41 | Q82 | — | 3 | 6 | no | — |
| 42 | Q82 | — | 6 | 3 | no | — |
| 43 | Q82 | — | 6 | 1 | no | — |
| 44 | Q82 | C2×D3 | 1 | 1 | yes | characteristic |
| 45 | #557 | C2×D3 | 1 | 3 | yes | — |
| 46 | #554 | — | 3 | 3 | no | — |
| 47 | D4×Q8 | — | 3 | 1 | no | — |
| 48 | #533 | — | 3 | 3 | no | — |
| 49 | C2×C4⋊Q8 | — | 3 | 3 | no | — |
| 50 | C2×C4⋊Q8 | — | 6 | 3 | no | — |
| 51 | C4×#41 | C2×D3 | 1 | 3 | yes | — |
| 52 | C2×C4×Q8 | — | 3 | 1 | no | — |
| 53 | C2×C4×Q8 | — | 3 | 3 | no | — |
| 54 | C2×C4×Q8 | — | 6 | 1 | no | — |
| 55 | #501 | — | 3 | 12 | no | — |
| 56 | #500 | — | 3 | 6 | no | — |
| 57 | #498 | — | 3 | 6 | no | — |
| 58 | #479 | — | 3 | 12 | no | — |
| 59 | #477 | — | 3 | 6 | no | — |
| 60 | #477 | — | 3 | 6 | no | — |
| 61 | #476 | — | 3 | 6 | no | — |
| 62 | #475 | — | 3 | 6 | no | — |
| 63 | #475 | — | 3 | 6 | no | — |
| 64 | #474 | — | 3 | 6 | no | — |
| 65 | #446 | — | 3 | 6 | no | — |
| 66 | C8×Q8 | — | 3 | 2 | no | — |
| 67 | #441 | — | 3 | 6 | no | — |
| 68 | #440 | — | 3 | 6 | no | — |
| 69 | C4×Q16 | — | 3 | 6 | no | — |
| 70 | C4×SD16 | — | 3 | 6 | no | — |
| 71 | #428 | — | 3 | 6 | no | — |
| 72 | #428 | — | 3 | 3 | no | — |
| 73 | #423 | — | 3 | 3 | no | — |
| 74 | #419 | — | 3 | 3 | no | — |
| 75 | #417 | — | 3 | 3 | no | — |
| 76 | C4×M16 | — | 3 | 3 | no | — |
| 77 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 78 | C2×Dic6 | — | 4 | 1 | no | — |
| 79 | C2×Dic6 | — | 4 | 6 | no | — |
| 80 | #41⋊C3 | C2×Q8 | 1 | 2 | yes | — |
| 81 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 82 | #226 | C2×Q8 | 1 | 4 | yes | complemented |
| 83 | C3×C4⋊C4 | — | 4 | 3 | no | — |
| 84 | C3×C42 | — | 4 | 3 | no | — |
| 85 | C4⋊Dic3 | — | 4 | 6 | no | — |
| 86 | C4⋊Dic3 | — | 4 | 3 | no | — |
| 87 | Dic3⋊C4 | — | 4 | 6 | no | — |
| 88 | C4×Dic3 | — | 4 | 6 | no | — |
| 89 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 90 | C22×Q8 | — | 3 | 1 | no | — |
| 91 | C22×Q8 | — | 3 | 1 | no | — |
| 92 | C22×Q8 | — | 6 | 3 | no | — |
| 93 | #137 | — | 3 | 4 | no | — |
| 94 | #137 | — | 3 | 4 | no | — |
| 95 | C2×Q16 | — | 3 | 2 | no | — |
| 96 | C2×SD16 | — | 3 | 2 | no | — |
| 97 | C2×M16 | — | 3 | 1 | no | — |
| 98 | C2×M16 | — | 3 | 3 | no | — |
| 99 | C4⋊Q8 | — | 3 | 3 | no | — |
| 100 | C4⋊Q8 | — | 3 | 3 | no | — |
| 101 | C4⋊Q8 | — | 3 | 6 | no | — |
| 102 | C4⋊Q8 | — | 3 | 6 | no | — |
| 103 | C4⋊Q8 | — | 6 | 3 | no | — |
| 104 | C4⋊Q8 | — | 6 | 12 | no | — |
| 105 | C4⋊Q8 | — | 6 | 6 | no | — |
| 106 | C4⋊Q8 | — | 6 | 6 | no | — |
| 107 | C4⋊Q8 | — | 6 | 3 | no | — |
| 108 | #125 | — | 3 | 6 | no | — |
| 109 | C22⋊Q8 | — | 6 | 3 | no | — |
| 110 | C22⋊Q8 | — | 6 | 3 | no | — |
| 111 | C4×Q8 | — | 3 | 1 | no | — |
| 112 | C4×Q8 | — | 3 | 3 | no | — |
| 113 | C4×Q8 | — | 3 | 3 | no | — |
| 114 | C4×Q8 | — | 3 | 6 | no | — |
| 115 | C4×Q8 | — | 3 | 6 | no | — |
| 116 | C4×Q8 | — | 6 | 1 | no | — |
| 117 | C4×Q8 | — | 6 | 3 | no | — |
| 118 | C4×Q8 | — | 6 | 6 | no | — |
| 119 | C4×Q8 | — | 6 | 2 | no | — |
| 120 | C4×Q8 | — | 6 | 3 | no | — |
| 121 | C4×Q8 | S4 | 1 | 1 | yes | characteristic |
| 122 | C4×Q8 | C22×D3 | 1 | 3 | yes | — |
| 123 | C4×Q8 | C22×D3 | 1 | 3 | yes | — |
| 124 | C4×D4 | — | 3 | 3 | no | — |
| 125 | C4×D4 | — | 3 | 3 | no | — |
| 126 | #117 | — | 3 | 3 | no | — |
| 127 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 128 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 129 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 130 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 131 | C2×C42 | — | 3 | 3 | no | — |
| 132 | C2×C42 | — | 6 | 3 | no | — |
| 133 | #107 | — | 3 | 6 | no | — |
| 134 | #107 | — | 3 | 12 | no | — |
| 135 | #106 | — | 3 | 6 | no | — |
| 136 | #106 | — | 3 | 12 | no | — |
| 137 | C4⋊C8 | — | 3 | 6 | no | — |
| 138 | C4⋊C8 | — | 3 | 6 | no | — |
| 139 | #103 | — | 3 | 6 | no | — |
| 140 | #103 | — | 3 | 6 | no | — |
| 141 | #103 | — | 3 | 6 | no | — |
| 142 | #102 | — | 3 | 6 | no | — |
| 143 | #97 | — | 3 | 6 | no | — |
| 144 | C4×C8 | — | 3 | 6 | no | — |
| 145 | C3×Q8 | — | 4 | 4 | no | — |
| 146 | C2×C3×C4 | — | 4 | 3 | no | — |
| 147 | C2×C3×C4 | — | 4 | 1 | no | — |
| 148 | C2×C3×C4 | — | 4 | 3 | no | — |
| 149 | C2×Dic3 | — | 4 | 2 | no | — |
| 150 | C2×Dic3 | — | 4 | 6 | no | — |
| 151 | Dic6 | — | 4 | 4 | no | — |
| 152 | Dic6 | — | 16 | 6 | no | — |
| 153 | SL(2,𝔽3) | C22×Q8 | 1 | 1 | yes | characteristic |
| 154 | #41 | — | 3 | 2 | no | — |
| 155 | #41 | D3×Q8 | 1 | 2 | yes | — |
| 156 | C2×Q8 | — | 3 | 3 | no | — |
| 157 | C2×Q8 | — | 3 | 2 | no | — |
| 158 | C2×Q8 | — | 3 | 4 | no | — |
| 159 | C2×Q8 | — | 6 | 1 | no | — |
| 160 | C2×Q8 | — | 6 | 2 | no | — |
| 161 | C2×Q8 | — | 6 | 6 | no | — |
| 162 | C2×Q8 | — | 6 | 4 | no | — |
| 163 | C2×Q8 | — | 6 | 1 | no | — |
| 164 | C2×Q8 | — | 6 | 6 | no | — |
| 165 | C2×Q8 | — | 24 | 6 | no | — |
| 166 | C2×Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 167 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 168 | C2×D4 | — | 3 | 1 | no | — |
| 169 | C22×C4 | — | 3 | 1 | no | — |
| 170 | C22×C4 | — | 3 | 3 | no | — |
| 171 | C22×C4 | — | 3 | 3 | no | — |
| 172 | C22×C4 | — | 6 | 1 | no | — |
| 173 | C22×C4 | — | 6 | 3 | no | — |
| 174 | Q16 | — | 3 | 4 | no | — |
| 175 | Q16 | — | 3 | 4 | no | — |
| 176 | SD16 | — | 3 | 8 | no | — |
| 177 | M16 | — | 3 | 4 | no | — |
| 178 | M16 | — | 6 | 6 | no | — |
| 179 | C2×C8 | — | 3 | 2 | no | — |
| 180 | C2×C8 | — | 3 | 6 | no | — |
| 181 | C4⋊C4 | — | 3 | 3 | no | — |
| 182 | C4⋊C4 | — | 3 | 3 | no | — |
| 183 | C4⋊C4 | — | 3 | 6 | no | — |
| 184 | C4⋊C4 | — | 3 | 3 | no | — |
| 185 | C4⋊C4 | — | 3 | 6 | no | — |
| 186 | C4⋊C4 | — | 6 | 3 | no | — |
| 187 | C4⋊C4 | — | 6 | 6 | no | — |
| 188 | C4⋊C4 | — | 6 | 6 | no | — |
| 189 | C4⋊C4 | — | 6 | 3 | no | — |
| 190 | C4⋊C4 | — | 6 | 6 | no | — |
| 191 | C4⋊C4 | — | 6 | 6 | no | — |
| 192 | C4⋊C4 | C2×S4 | 1 | 3 | yes | — |
| 193 | C22⋊C4 | — | 6 | 3 | no | — |
| 194 | C42 | — | 3 | 3 | no | — |
| 195 | C42 | — | 3 | 3 | no | — |
| 196 | C42 | — | 3 | 3 | no | — |
| 197 | C42 | — | 6 | 6 | no | — |
| 198 | C42 | — | 6 | 6 | no | — |
| 199 | C42 | C2×S4 | 1 | 3 | yes | — |
| 200 | C22×C3 | — | 4 | 1 | no | — |
| 201 | C3×C4 | — | 4 | 2 | no | — |
| 202 | C3×C4 | — | 4 | 6 | no | — |
| 203 | C3×C4 | — | 8 | 3 | no | — |
| 204 | Dic3 | — | 4 | 4 | no | — |
| 205 | Dic3 | — | 8 | 6 | no | — |
| 206 | C23 | — | 3 | 1 | no | — |
| 207 | Q8 | — | 3 | 1 | no | — |
| 208 | Q8 | — | 3 | 4 | no | — |
| 209 | Q8 | — | 6 | 2 | no | — |
| 210 | Q8 | — | 6 | 4 | no | — |
| 211 | Q8 | — | 6 | 6 | no | — |
| 212 | Q8 | — | 24 | 6 | no | — |
| 213 | Q8 | — | 24 | 1 | no | — |
| 214 | Q8 | — | 24 | 6 | no | — |
| 215 | Q8 | #984 | 1 | 4 | yes | direct product factor |
| 216 | Q8 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 217 | D4 | — | 3 | 2 | no | — |
| 218 | D4 | — | 6 | 1 | no | — |
| 219 | C2×C4 | — | 3 | 1 | no | — |
| 220 | C2×C4 | — | 3 | 2 | no | — |
| 221 | C2×C4 | — | 3 | 2 | no | — |
| 222 | C2×C4 | — | 3 | 3 | no | — |
| 223 | C2×C4 | — | 3 | 3 | no | — |
| 224 | C2×C4 | — | 6 | 4 | no | — |
| 225 | C2×C4 | — | 6 | 2 | no | — |
| 226 | C2×C4 | — | 6 | 6 | no | — |
| 227 | C2×C4 | — | 6 | 6 | no | — |
| 228 | C2×C4 | — | 12 | 3 | no | — |
| 229 | C2×C4 | — | 12 | 6 | no | — |
| 230 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 231 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 232 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 233 | C8 | — | 3 | 4 | no | — |
| 234 | C8 | — | 6 | 6 | no | — |
| 235 | C2×C3 | — | 4 | 1 | no | — |
| 236 | C2×C3 | — | 4 | 1 | no | — |
| 237 | C2×C3 | — | 4 | 1 | no | — |
| 238 | C22 | — | 3 | 2 | no | — |
| 239 | C22 | — | 6 | 1 | no | — |
| 240 | C22 | — | 6 | 1 | no | — |
| 241 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 3 | no | — |
| 243 | C4 | — | 3 | 1 | no | — |
| 244 | C4 | — | 3 | 1 | no | — |
| 245 | C4 | — | 6 | 4 | no | — |
| 246 | C4 | — | 6 | 3 | no | — |
| 247 | C4 | — | 6 | 3 | no | — |
| 248 | C4 | — | 12 | 6 | no | — |
| 249 | C4 | Q8×S4 | 1 | 2 | yes | — |
| 250 | C4 | C2×#984 | 1 | 6 | yes | — |
| 251 | C3 | — | 4 | 1 | no | Sylow |
| 252 | C2 | — | 6 | 2 | no | — |
| 253 | C2 | C2×Q8×S4 | 1 | 1 | yes | characteristic |
| 254 | C2 | C22×#984 | 1 | 1 | yes | characteristic |
| 255 | C2 | #87498 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×#226 | C2 | semidirect |
| 3. #85415 | C2 | semidirect |
| 29. #984 | Q8 | direct (N × H) |
| 82. #226 | C2×Q8 | semidirect |
| 215. Q8 | #984 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#41⋊C3 | 2 |
| Q8×#226 | 2 |
| #85415 | 2 |
| #85337 | 2 |
| C4×#984 | 2 |
| Q8×#137 | 3 |
| Q8×Dic6 | 4 |