Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
569 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 569 conjugacy classes of subgroups — too many to draw. Only its 103 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 | C22×#131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#5426 | C2 | 1 | 2 | yes | maximal |
| 3 | #73125 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×#1815 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#131⋊C3 | C4 | 1 | 4 | yes | complemented |
| 7 | C2×#131⋊C3 | C22 | 1 | 2 | yes | — |
| 8 | #5426 | — | 2 | 2 | no | — |
| 9 | #5426 | C22 | 1 | 4 | yes | — |
| 10 | C2×#859 | — | 2 | 2 | no | — |
| 11 | C2×#817 | — | 4 | 1 | no | maximal |
| 12 | C22×#131 | D3 | 1 | 1 | yes | characteristic |
| 13 | C22×#108 | — | 3 | 1 | no | — |
| 14 | C22×#106 | — | 3 | 1 | no | — |
| 15 | C2×#417 | — | 3 | 2 | no | — |
| 16 | #1815 | — | 3 | 4 | no | — |
| 17 | #1815 | — | 3 | 4 | no | — |
| 18 | C2×#343 | — | 3 | 2 | no | — |
| 19 | C2×#41⋊C3 | C2×C4 | 1 | 2 | yes | — |
| 20 | C2×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×#41⋊C3 | Q8 | 1 | 2 | yes | — |
| 22 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 23 | C22×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×C3×C8 | — | 4 | 1 | no | — |
| 25 | C2×C4⋊Dic3 | — | 4 | 2 | no | — |
| 26 | #131⋊C3 | C2×C4 | 1 | 4 | yes | complemented |
| 27 | #859 | — | 2 | 4 | no | — |
| 28 | #817 | — | 4 | 4 | no | — |
| 29 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | C2×#131 | — | 3 | 2 | no | — |
| 31 | C2×#131 | — | 3 | 4 | no | — |
| 32 | C2×#131 | Dic3 | 1 | 4 | yes | complemented |
| 33 | C2×#131 | C2×D3 | 1 | 2 | yes | — |
| 34 | C22×M16 | — | 3 | 1 | no | — |
| 35 | C22×M16 | — | 3 | 2 | no | — |
| 36 | C23×C8 | — | 3 | 1 | no | — |
| 37 | C22×C4⋊C4 | — | 3 | 2 | no | — |
| 38 | C2×#108 | — | 3 | 4 | no | — |
| 39 | C2×#108 | — | 6 | 4 | no | — |
| 40 | C2×#106 | — | 3 | 4 | no | — |
| 41 | C2×#106 | — | 6 | 1 | no | — |
| 42 | C2×#106 | — | 6 | 2 | no | — |
| 43 | C2×#106 | — | 6 | 1 | no | — |
| 44 | #417 | — | 3 | 4 | no | — |
| 45 | #417 | — | 3 | 4 | no | — |
| 46 | #417 | — | 6 | 2 | no | — |
| 47 | #417 | — | 6 | 2 | no | — |
| 48 | C2×#103 | — | 6 | 2 | no | — |
| 49 | C2×#102 | — | 6 | 2 | no | — |
| 50 | #343 | — | 3 | 8 | no | — |
| 51 | C22×C3×C4 | — | 4 | 1 | no | — |
| 52 | C22×Dic3 | — | 8 | 2 | no | — |
| 53 | #41⋊C3 | C4⋊C4 | 1 | 2 | yes | — |
| 54 | #41⋊C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | #41⋊C3 | C2×Q8 | 1 | 1 | yes | characteristic |
| 56 | C2×SL(2,𝔽3) | C4⋊C4 | 1 | 2 | yes | — |
| 57 | C2×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C3×C8 | — | 4 | 4 | no | — |
| 59 | C2×C3×C8 | — | 4 | 2 | no | — |
| 60 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 61 | C4⋊Dic3 | — | 8 | 2 | no | — |
| 62 | C2×#41 | — | 3 | 2 | no | — |
| 63 | C2×#41 | — | 3 | 1 | no | — |
| 64 | C2×#41 | — | 3 | 2 | no | — |
| 65 | C2×#41 | — | 3 | 1 | no | — |
| 66 | C2×#41 | Dic6 | 1 | 2 | yes | — |
| 67 | C2×#41 | D12 | 1 | 1 | yes | characteristic |
| 68 | C2×#41 | C2×Dic3 | 1 | 2 | yes | — |
| 69 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 70 | C22×Q8 | D12 | 1 | 1 | yes | characteristic |
| 71 | C22×D4 | — | 3 | 1 | no | — |
| 72 | C23×C4 | — | 3 | 1 | no | — |
| 73 | C23×C4 | — | 6 | 2 | no | — |
| 74 | #131 | — | 3 | 8 | no | — |
| 75 | #131 | — | 3 | 4 | no | — |
| 76 | #131 | — | 6 | 4 | no | — |
| 77 | #131 | C2×Dic3 | 1 | 4 | yes | complemented |
| 78 | C2×M16 | — | 3 | 4 | no | — |
| 79 | C2×M16 | — | 3 | 2 | no | — |
| 80 | C2×M16 | — | 3 | 4 | no | — |
| 81 | C2×M16 | — | 6 | 2 | no | — |
| 82 | C2×M16 | — | 6 | 1 | no | — |
| 83 | C2×M16 | — | 6 | 2 | no | — |
| 84 | C2×M16 | — | 6 | 4 | no | — |
| 85 | C2×M16 | — | 6 | 4 | no | — |
| 86 | C22×C8 | — | 3 | 1 | no | — |
| 87 | C22×C8 | — | 3 | 4 | no | — |
| 88 | C22×C8 | — | 3 | 4 | no | — |
| 89 | C22×C8 | — | 3 | 2 | no | — |
| 90 | C22×C8 | — | 3 | 2 | no | — |
| 91 | C22×C8 | — | 6 | 2 | no | — |
| 92 | C22×C8 | S4 | 1 | 1 | yes | characteristic |
| 93 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 94 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 95 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 96 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 97 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 98 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 99 | #108 | — | 6 | 8 | no | — |
| 100 | #106 | — | 6 | 4 | no | — |
| 101 | #106 | — | 6 | 4 | no | — |
| 102 | #103 | — | 6 | 4 | no | — |
| 103 | #103 | — | 6 | 4 | no | — |
| 104 | #102 | — | 6 | 4 | no | — |
| 105 | #102 | — | 6 | 4 | no | — |
| 106 | C23×C3 | — | 4 | 1 | no | — |
| 107 | C2×C3×C4 | — | 4 | 2 | no | — |
| 108 | C2×C3×C4 | — | 4 | 1 | no | — |
| 109 | C2×C3×C4 | — | 4 | 2 | no | — |
| 110 | C2×C3×C4 | — | 4 | 1 | no | — |
| 111 | C2×Dic3 | — | 8 | 4 | no | — |
| 112 | C2×Dic3 | — | 16 | 4 | no | — |
| 113 | SL(2,𝔽3) | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C3×C8 | — | 4 | 4 | no | — |
| 115 | C24 | — | 3 | 1 | no | — |
| 116 | #41 | — | 3 | 2 | no | — |
| 117 | #41 | — | 3 | 1 | no | — |
| 118 | #41 | — | 3 | 2 | no | — |
| 119 | #41 | — | 3 | 2 | no | — |
| 120 | #41 | — | 3 | 2 | no | — |
| 121 | #41 | — | 3 | 2 | no | — |
| 122 | #41 | — | 3 | 1 | no | — |
| 123 | #41 | — | 6 | 1 | no | — |
| 124 | #41 | — | 6 | 1 | no | — |
| 125 | #41 | — | 6 | 2 | no | — |
| 126 | #41 | C4⋊Dic3 | 1 | 2 | yes | — |
| 127 | #41 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 128 | #41 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 129 | C2×Q8 | — | 3 | 1 | no | — |
| 130 | C2×Q8 | — | 3 | 2 | no | — |
| 131 | C2×Q8 | C4⋊Dic3 | 1 | 2 | yes | — |
| 132 | C2×Q8 | C2×D12 | 1 | 1 | yes | characteristic |
| 133 | C2×D4 | — | 3 | 1 | no | — |
| 134 | C2×D4 | — | 3 | 1 | no | — |
| 135 | C2×D4 | — | 3 | 2 | no | — |
| 136 | C2×D4 | — | 3 | 2 | no | — |
| 137 | C2×D4 | — | 6 | 1 | no | — |
| 138 | C2×D4 | — | 6 | 2 | no | — |
| 139 | C22×C4 | — | 3 | 1 | no | — |
| 140 | C22×C4 | — | 3 | 2 | no | — |
| 141 | C22×C4 | — | 3 | 2 | no | — |
| 142 | C22×C4 | — | 3 | 1 | no | — |
| 143 | C22×C4 | — | 3 | 1 | no | — |
| 144 | C22×C4 | — | 3 | 2 | no | — |
| 145 | C22×C4 | — | 3 | 2 | no | — |
| 146 | C22×C4 | — | 3 | 1 | no | — |
| 147 | C22×C4 | — | 3 | 1 | no | — |
| 148 | C22×C4 | — | 6 | 4 | no | — |
| 149 | C22×C4 | — | 12 | 2 | no | — |
| 150 | C22×C4 | — | 12 | 2 | no | — |
| 151 | C22×C4 | — | 24 | 4 | no | — |
| 152 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 153 | M16 | — | 3 | 4 | no | — |
| 154 | M16 | — | 6 | 2 | no | — |
| 155 | M16 | — | 6 | 4 | no | — |
| 156 | M16 | — | 6 | 4 | no | — |
| 157 | M16 | — | 6 | 4 | no | — |
| 158 | M16 | — | 12 | 4 | no | — |
| 159 | C2×C8 | — | 3 | 4 | no | — |
| 160 | C2×C8 | — | 3 | 2 | no | — |
| 161 | C2×C8 | — | 3 | 4 | no | — |
| 162 | C2×C8 | — | 3 | 4 | no | — |
| 163 | C2×C8 | — | 3 | 8 | no | — |
| 164 | C2×C8 | — | 6 | 4 | no | — |
| 165 | C2×C8 | — | 12 | 4 | no | — |
| 166 | C2×C8 | SL(2,ℤ/4) | 1 | 4 | yes | — |
| 167 | C2×C8 | C2×S4 | 1 | 2 | yes | — |
| 168 | C4⋊C4 | — | 6 | 4 | no | — |
| 169 | C4⋊C4 | — | 6 | 4 | no | — |
| 170 | C4⋊C4 | — | 12 | 2 | no | — |
| 171 | C4⋊C4 | — | 12 | 2 | no | — |
| 172 | C22×C3 | — | 4 | 1 | no | — |
| 173 | C22×C3 | — | 4 | 4 | no | — |
| 174 | C22×C3 | — | 4 | 2 | no | — |
| 175 | C3×C4 | — | 4 | 1 | no | — |
| 176 | C3×C4 | — | 4 | 2 | no | — |
| 177 | C3×C4 | — | 4 | 1 | no | — |
| 178 | Dic3 | — | 16 | 4 | no | — |
| 179 | C23 | — | 3 | 1 | no | — |
| 180 | C23 | — | 3 | 1 | no | — |
| 181 | C23 | — | 3 | 2 | no | — |
| 182 | C23 | — | 3 | 2 | no | — |
| 183 | C23 | — | 6 | 4 | no | — |
| 184 | C23 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 185 | Q8 | — | 3 | 2 | no | — |
| 186 | Q8 | — | 3 | 1 | no | — |
| 187 | Q8 | — | 6 | 1 | no | — |
| 188 | Q8 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 189 | D4 | — | 3 | 2 | no | — |
| 190 | D4 | — | 3 | 1 | no | — |
| 191 | D4 | — | 3 | 1 | no | — |
| 192 | D4 | — | 6 | 1 | no | — |
| 193 | D4 | — | 6 | 1 | no | — |
| 194 | D4 | — | 6 | 2 | no | — |
| 195 | D4 | — | 6 | 2 | no | — |
| 196 | C2×C4 | — | 3 | 1 | no | — |
| 197 | C2×C4 | — | 3 | 1 | no | — |
| 198 | C2×C4 | — | 3 | 2 | no | — |
| 199 | C2×C4 | — | 3 | 2 | no | — |
| 200 | C2×C4 | — | 3 | 1 | no | — |
| 201 | C2×C4 | — | 3 | 1 | no | — |
| 202 | C2×C4 | — | 3 | 2 | no | — |
| 203 | C2×C4 | — | 3 | 2 | no | — |
| 204 | C2×C4 | — | 3 | 2 | no | — |
| 205 | C2×C4 | — | 3 | 2 | no | — |
| 206 | C2×C4 | — | 3 | 2 | no | — |
| 207 | C2×C4 | — | 3 | 1 | no | — |
| 208 | C2×C4 | — | 3 | 1 | no | — |
| 209 | C2×C4 | — | 3 | 2 | no | — |
| 210 | C2×C4 | — | 12 | 4 | no | — |
| 211 | C2×C4 | — | 24 | 4 | no | — |
| 212 | C2×C4 | — | 24 | 4 | no | — |
| 213 | C2×C4 | — | 24 | 4 | no | — |
| 214 | C2×C4 | A4⋊Q8 | 1 | 2 | yes | — |
| 215 | C2×C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | C2×SL(2,ℤ/4) | 1 | 2 | yes | — |
| 217 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 218 | C8 | — | 3 | 4 | no | — |
| 219 | C8 | — | 12 | 4 | no | — |
| 220 | C8 | C2×SL(2,ℤ/4) | 1 | 4 | yes | — |
| 221 | C2×C3 | — | 4 | 4 | no | — |
| 222 | C2×C3 | — | 4 | 2 | no | — |
| 223 | C2×C3 | — | 4 | 1 | no | — |
| 224 | C22 | — | 3 | 1 | no | — |
| 225 | C22 | — | 3 | 1 | no | — |
| 226 | C22 | — | 3 | 2 | no | — |
| 227 | C22 | — | 6 | 4 | no | — |
| 228 | C22 | — | 6 | 4 | no | — |
| 229 | C22 | — | 6 | 2 | no | — |
| 230 | C22 | — | 6 | 2 | no | — |
| 231 | C22 | #5406 | 1 | 4 | yes | — |
| 232 | C22 | C4⋊SL(2,ℤ/4) | 1 | 2 | yes | — |
| 233 | C22 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 234 | C4 | — | 3 | 1 | no | — |
| 235 | C4 | — | 3 | 1 | no | — |
| 236 | C4 | — | 3 | 2 | no | — |
| 237 | C4 | — | 24 | 4 | no | — |
| 238 | C4 | C4⋊SL(2,ℤ/4) | 1 | 2 | yes | — |
| 239 | C4 | C2×A4⋊Q8 | 1 | 1 | yes | characteristic |
| 240 | C4 | C22×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 241 | C3 | — | 4 | 1 | no | Sylow |
| 242 | C2 | — | 6 | 1 | no | — |
| 243 | C2 | — | 6 | 1 | no | — |
| 244 | C2 | — | 6 | 2 | no | — |
| 245 | C2 | #73125 | 1 | 4 | yes | direct product factor |
| 246 | C2 | C2×#5406 | 1 | 2 | yes | — |
| 247 | C2 | C2×C4⋊SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #73125 | C2 | direct (N × H) |
| 6. C2×#131⋊C3 | C4 | semidirect |
| 26. #131⋊C3 | C2×C4 | semidirect |
| 32. C2×#131 | Dic3 | semidirect |
| 77. #131 | C2×Dic3 | semidirect |
| 245. C2 | #73125 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131⋊C3 | 2 |
| C2×#5426 | 2 |
| #73125 | 2 |
| C2×#1815 | 3 |
| C2×#817 | 4 |