Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
362 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 41 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 | Q16×S4 | C2 | 1 | 2 | yes | maximal |
| 2 | A4×Q32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #73136 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #73037 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C16×S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | D4×Q32 | — | 3 | 1 | no | maximal, Sylow |
| 7 | Q8×S4 | — | 2 | 2 | no | — |
| 8 | A4×Q16 | C22 | 1 | 2 | yes | — |
| 9 | #5416 | — | 2 | 2 | no | — |
| 10 | C8×S4 | C22 | 1 | 1 | yes | characteristic |
| 11 | #5398 | C22 | 1 | 2 | yes | — |
| 12 | D3×Q32 | — | 4 | 1 | no | maximal |
| 13 | A4×C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | A4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×Q32 | — | 3 | 1 | no | — |
| 16 | C22×Q32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | D4×Q16 | — | 3 | 2 | no | — |
| 18 | #2247 | — | 3 | 1 | no | — |
| 19 | #2216 | — | 3 | 1 | no | — |
| 20 | #2216 | — | 3 | 1 | no | — |
| 21 | #2209 | — | 3 | 2 | no | — |
| 22 | #2188 | — | 3 | 2 | no | — |
| 23 | #2188 | — | 3 | 2 | no | — |
| 24 | C4×Q32 | — | 3 | 1 | no | — |
| 25 | D4×C16 | — | 3 | 1 | no | — |
| 26 | Q8×A4 | — | 2 | 2 | no | — |
| 27 | C4×S4 | — | 4 | 2 | no | — |
| 28 | C4×S4 | D4 | 1 | 1 | yes | characteristic |
| 29 | A4⋊Q8 | — | 2 | 2 | no | — |
| 30 | A4⋊Q8 | — | 4 | 2 | no | — |
| 31 | D3×Q16 | — | 4 | 2 | no | — |
| 32 | C8×A4 | C23 | 1 | 1 | yes | characteristic |
| 33 | A4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 34 | C3×Q32 | — | 4 | 1 | no | — |
| 35 | #829 | — | 4 | 2 | no | — |
| 36 | Dic24 | — | 4 | 1 | no | — |
| 37 | D3×C16 | — | 4 | 1 | no | — |
| 38 | C22×Q16 | — | 3 | 2 | no | — |
| 39 | C22×Q16 | C2×D3 | 1 | 2 | yes | — |
| 40 | D4×Q8 | — | 6 | 2 | no | — |
| 41 | C2×Q32 | — | 3 | 1 | no | — |
| 42 | C2×Q32 | — | 3 | 2 | no | — |
| 43 | C2×Q32 | — | 3 | 1 | no | — |
| 44 | C2×Q32 | — | 3 | 2 | no | — |
| 45 | C2×Q32 | — | 3 | 1 | no | — |
| 46 | C2×Q32 | — | 3 | 2 | no | — |
| 47 | C2×Q32 | — | 6 | 2 | no | — |
| 48 | C2×Q32 | — | 6 | 1 | no | — |
| 49 | C2×Q32 | — | 6 | 1 | no | — |
| 50 | C22×C16 | — | 3 | 1 | no | — |
| 51 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | #494 | — | 3 | 2 | no | — |
| 53 | #467 | — | 3 | 2 | no | — |
| 54 | #467 | — | 3 | 2 | no | — |
| 55 | #462 | — | 6 | 2 | no | — |
| 56 | #451 | — | 6 | 2 | no | — |
| 57 | #451 | — | 6 | 2 | no | — |
| 58 | C4×Q16 | — | 3 | 2 | no | — |
| 59 | C8×D4 | — | 3 | 1 | no | — |
| 60 | #366 | — | 3 | 1 | no | — |
| 61 | C4⋊C16 | — | 3 | 1 | no | — |
| 62 | #358 | — | 3 | 2 | no | — |
| 63 | #358 | — | 3 | 2 | no | — |
| 64 | #358 | — | 3 | 2 | no | — |
| 65 | C22⋊C16 | — | 3 | 1 | no | — |
| 66 | C22⋊C16 | — | 3 | 1 | no | — |
| 67 | C4×C16 | — | 3 | 1 | no | — |
| 68 | C2×S4 | D8 | 1 | 1 | yes | characteristic |
| 69 | D3×Q8 | — | 8 | 2 | no | — |
| 70 | C4×A4 | — | 4 | 2 | no | — |
| 71 | C4×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | SL(2,ℤ/4) | — | 4 | 2 | no | — |
| 73 | SL(2,ℤ/4) | D8 | 1 | 1 | yes | characteristic |
| 74 | C3×Q16 | — | 4 | 2 | no | — |
| 75 | #216 | — | 8 | 2 | no | — |
| 76 | Dic12 | — | 4 | 2 | no | — |
| 77 | D3×C8 | — | 4 | 1 | no | — |
| 78 | C3×C16 | — | 4 | 1 | no | — |
| 79 | C3⋊C16 | — | 4 | 1 | no | — |
| 80 | C22×Q8 | — | 2 | 2 | no | — |
| 81 | C22×Q8 | — | 6 | 2 | no | — |
| 82 | C2×Q16 | — | 3 | 2 | no | — |
| 83 | C2×Q16 | — | 3 | 2 | no | — |
| 84 | C2×Q16 | — | 3 | 2 | no | — |
| 85 | C2×Q16 | — | 6 | 2 | no | — |
| 86 | C2×Q16 | — | 6 | 2 | no | — |
| 87 | C2×Q16 | — | 6 | 2 | no | — |
| 88 | C2×Q16 | — | 6 | 2 | no | — |
| 89 | C2×Q16 | — | 6 | 2 | no | — |
| 90 | C2×Q16 | — | 12 | 2 | no | — |
| 91 | C22×C8 | — | 3 | 1 | no | — |
| 92 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C4⋊Q8 | — | 6 | 2 | no | — |
| 94 | C4⋊Q8 | — | 12 | 2 | no | — |
| 95 | C22⋊Q8 | — | 6 | 2 | no | — |
| 96 | C22⋊Q8 | — | 6 | 2 | no | — |
| 97 | C22⋊Q8 | — | 12 | 2 | no | — |
| 98 | C22⋊Q8 | — | 12 | 2 | no | — |
| 99 | C4×Q8 | — | 6 | 2 | no | — |
| 100 | C4×D4 | — | 3 | 1 | no | — |
| 101 | C4×D4 | — | 12 | 2 | no | — |
| 102 | Q32 | — | 3 | 2 | no | — |
| 103 | Q32 | — | 3 | 1 | no | — |
| 104 | Q32 | — | 6 | 1 | no | — |
| 105 | Q32 | — | 6 | 1 | no | — |
| 106 | Q32 | — | 6 | 2 | no | — |
| 107 | Q32 | — | 6 | 2 | no | — |
| 108 | Q32 | — | 6 | 1 | no | — |
| 109 | Q32 | — | 12 | 2 | no | — |
| 110 | Q32 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 111 | C2×C16 | — | 3 | 1 | no | — |
| 112 | C2×C16 | — | 3 | 1 | no | — |
| 113 | C2×C16 | — | 3 | 1 | no | — |
| 114 | C2×C16 | — | 3 | 1 | no | — |
| 115 | C2×C16 | — | 6 | 1 | no | — |
| 116 | C2×C16 | — | 6 | 1 | no | — |
| 117 | #107 | — | 3 | 2 | no | — |
| 118 | C4⋊C8 | — | 3 | 1 | no | — |
| 119 | #103 | — | 6 | 2 | no | — |
| 120 | #103 | — | 6 | 2 | no | — |
| 121 | #103 | — | 6 | 2 | no | — |
| 122 | C22⋊C8 | — | 3 | 1 | no | — |
| 123 | C22⋊C8 | — | 3 | 1 | no | — |
| 124 | C4×C8 | — | 3 | 1 | no | — |
| 125 | C2×A4 | C2×D8 | 1 | 1 | yes | characteristic |
| 126 | S4 | Q32 | 1 | 2 | yes | direct product factor |
| 127 | C3×Q8 | — | 8 | 2 | no | — |
| 128 | C4×D3 | — | 4 | 1 | no | — |
| 129 | C4×D3 | — | 16 | 2 | no | — |
| 130 | Dic6 | — | 8 | 2 | no | — |
| 131 | Dic6 | — | 16 | 2 | no | — |
| 132 | C3×C8 | — | 4 | 1 | no | — |
| 133 | C3⋊C8 | — | 4 | 1 | no | — |
| 134 | C2×Q8 | — | 6 | 2 | no | — |
| 135 | C2×Q8 | — | 6 | 2 | no | — |
| 136 | C2×Q8 | — | 6 | 2 | no | — |
| 137 | C2×Q8 | — | 12 | 2 | no | — |
| 138 | C2×Q8 | — | 12 | 2 | no | — |
| 139 | C2×Q8 | — | 12 | 2 | no | — |
| 140 | C2×Q8 | — | 12 | 2 | no | — |
| 141 | C2×Q8 | — | 24 | 2 | no | — |
| 142 | C2×D4 | — | 3 | 1 | no | — |
| 143 | C22×C4 | — | 3 | 1 | no | — |
| 144 | C22×C4 | — | 4 | 2 | no | — |
| 145 | C22×C4 | — | 12 | 2 | no | — |
| 146 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 147 | Q16 | — | 3 | 2 | no | — |
| 148 | Q16 | — | 6 | 2 | no | — |
| 149 | Q16 | — | 6 | 2 | no | — |
| 150 | Q16 | — | 6 | 2 | no | — |
| 151 | Q16 | — | 6 | 2 | no | — |
| 152 | Q16 | — | 12 | 2 | no | — |
| 153 | Q16 | — | 12 | 2 | no | — |
| 154 | Q16 | — | 24 | 2 | no | — |
| 155 | Q16 | C2×S4 | 1 | 2 | yes | — |
| 156 | C2×C8 | — | 3 | 1 | no | — |
| 157 | C2×C8 | — | 3 | 1 | no | — |
| 158 | C2×C8 | — | 3 | 1 | no | — |
| 159 | C2×C8 | — | 3 | 1 | no | — |
| 160 | C2×C8 | — | 6 | 1 | no | — |
| 161 | C2×C8 | — | 6 | 1 | no | — |
| 162 | C4⋊C4 | — | 3 | 1 | no | — |
| 163 | C4⋊C4 | — | 6 | 2 | no | — |
| 164 | C4⋊C4 | — | 12 | 2 | no | — |
| 165 | C4⋊C4 | — | 12 | 2 | no | — |
| 166 | C4⋊C4 | — | 12 | 2 | no | — |
| 167 | C4⋊C4 | — | 12 | 2 | no | — |
| 168 | C4⋊C4 | — | 12 | 2 | no | — |
| 169 | C22⋊C4 | — | 3 | 1 | no | — |
| 170 | C22⋊C4 | — | 3 | 1 | no | — |
| 171 | C22⋊C4 | — | 12 | 2 | no | — |
| 172 | C22⋊C4 | — | 12 | 2 | no | — |
| 173 | C42 | — | 3 | 1 | no | — |
| 174 | C42 | — | 12 | 2 | no | — |
| 175 | C16 | — | 3 | 1 | no | — |
| 176 | C16 | — | 6 | 1 | no | — |
| 177 | C16 | — | 6 | 1 | no | — |
| 178 | C16 | C2×S4 | 1 | 1 | yes | characteristic |
| 179 | C2×D3 | — | 4 | 1 | no | — |
| 180 | A4 | C2×Q32 | 1 | 1 | yes | characteristic, complemented |
| 181 | C3×C4 | — | 4 | 1 | no | — |
| 182 | C3×C4 | — | 16 | 2 | no | — |
| 183 | Dic3 | — | 4 | 1 | no | — |
| 184 | Dic3 | — | 16 | 2 | no | — |
| 185 | C23 | — | 3 | 1 | no | — |
| 186 | C23 | D3×D8 | 1 | 1 | yes | characteristic |
| 187 | Q8 | — | 2 | 2 | no | — |
| 188 | Q8 | — | 6 | 2 | no | — |
| 189 | Q8 | — | 12 | 2 | no | — |
| 190 | Q8 | — | 12 | 2 | no | — |
| 191 | Q8 | — | 12 | 2 | no | — |
| 192 | Q8 | — | 12 | 2 | no | — |
| 193 | Q8 | — | 24 | 2 | no | — |
| 194 | Q8 | — | 24 | 2 | no | — |
| 195 | D4 | — | 3 | 2 | no | — |
| 196 | D4 | — | 3 | 2 | no | — |
| 197 | C2×C4 | — | 3 | 1 | no | — |
| 198 | C2×C4 | — | 3 | 1 | no | — |
| 199 | C2×C4 | — | 3 | 1 | no | — |
| 200 | C2×C4 | — | 3 | 1 | no | — |
| 201 | C2×C4 | — | 3 | 1 | no | — |
| 202 | C2×C4 | — | 6 | 1 | no | — |
| 203 | C2×C4 | — | 6 | 1 | no | — |
| 204 | C2×C4 | — | 12 | 2 | no | — |
| 205 | C2×C4 | — | 12 | 2 | no | — |
| 206 | C2×C4 | — | 12 | 2 | no | — |
| 207 | C2×C4 | — | 12 | 2 | no | — |
| 208 | C2×C4 | — | 24 | 2 | no | — |
| 209 | C2×C4 | — | 24 | 2 | no | — |
| 210 | C8 | — | 3 | 1 | no | — |
| 211 | C8 | — | 6 | 1 | no | — |
| 212 | C8 | — | 6 | 1 | no | — |
| 213 | C8 | C22×S4 | 1 | 1 | yes | characteristic |
| 214 | C2×C3 | — | 4 | 1 | no | — |
| 215 | D3 | — | 4 | 2 | no | — |
| 216 | C22 | — | 3 | 1 | no | — |
| 217 | C22 | — | 3 | 1 | no | — |
| 218 | C22 | — | 3 | 2 | no | — |
| 219 | C22 | — | 6 | 1 | no | — |
| 220 | C22 | — | 6 | 1 | no | — |
| 221 | C22 | D3×Q32 | 1 | 1 | yes | characteristic, complemented |
| 222 | C4 | — | 3 | 2 | no | — |
| 223 | C4 | — | 3 | 1 | no | — |
| 224 | C4 | — | 4 | 2 | no | — |
| 225 | C4 | — | 6 | 1 | no | — |
| 226 | C4 | — | 6 | 1 | no | — |
| 227 | C4 | — | 12 | 2 | no | — |
| 228 | C4 | — | 24 | 2 | no | — |
| 229 | C4 | — | 24 | 2 | no | — |
| 230 | C4 | D4×S4 | 1 | 1 | yes | characteristic |
| 231 | C3 | — | 4 | 1 | no | Sylow |
| 232 | C2 | — | 3 | 1 | no | — |
| 233 | C2 | — | 3 | 1 | no | — |
| 234 | C2 | — | 6 | 2 | no | — |
| 235 | C2 | D8×S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×Q32 | C2 | semidirect |
| 3. #73136 | C2 | semidirect |
| 4. #73037 | C2 | semidirect |
| 16. C22×Q32 | D3 | semidirect |
| 110. Q32 | S4 | direct (N × H) |
| 126. S4 | Q32 | direct (N × H) |
| 180. A4 | C2×Q32 | semidirect |
| 221. C22 | D3×Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q16×S4 | 2 |
| A4×Q32 | 2 |
| #73136 | 2 |
| #73037 | 2 |
| C16×S4 | 2 |
| D4×Q32 | 3 |
| D3×Q32 | 4 |