← C3×#9853 #10628460
Subgroups — C3×#9853 #10628460
Order 768.
422 conjugacy classes of subgroups.
The subgroup lattice has 226 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#3170 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1994 | C2 | 1 | 2 | yes | maximal |
| 3 | C3×#1744 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C3×#1566 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C3×#1493 | C2 | 1 | 2 | yes | maximal |
| 6 | #9853 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C3×#533 | C22 | 1 | 2 | yes | — |
| 8 | C3×#500 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×#494 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×#487 | C4 | 1 | 2 | yes | complemented |
| 11 | C3×#467 | — | 2 | 2 | no | — |
| 12 | C3×C4×Q16 | — | 2 | 2 | no | — |
| 13 | C3×#423 | C22 | 1 | 2 | yes | — |
| 14 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×C43 | C22 | 1 | 1 | yes | characteristic |
| 17 | C3×#339 | — | 2 | 4 | no | — |
| 18 | C3×#336 | — | 2 | 2 | no | — |
| 19 | C3×#334 | — | 2 | 2 | no | — |
| 20 | C3×Q8⋊C8 | — | 2 | 4 | no | — |
| 21 | #3170 | C2×C3 | 1 | 1 | yes | characteristic |
| 22 | #1994 | C2×C3 | 1 | 2 | yes | — |
| 23 | #1744 | C2×C3 | 1 | 1 | yes | characteristic |
| 24 | #1566 | C2×C3 | 1 | 1 | yes | characteristic |
| 25 | #1493 | C2×C3 | 1 | 2 | yes | — |
| 26 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 27 | C2×C3×M16 | — | 2 | 2 | no | — |
| 28 | C22×C3×C8 | — | 2 | 1 | no | — |
| 29 | C22×C3×C8 | D4 | 1 | 2 | yes | — |
| 30 | C3×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 31 | C3×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 32 | C3×#125 | — | 2 | 2 | no | — |
| 33 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 34 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 35 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 36 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 37 | C3×#117 | — | 2 | 2 | no | — |
| 38 | C2×C3×C42 | — | 2 | 1 | no | — |
| 39 | C2×C3×C42 | — | 2 | 1 | no | — |
| 40 | C2×C3×C42 | D4 | 1 | 2 | yes | — |
| 41 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 42 | C3×#107 | — | 2 | 2 | no | — |
| 43 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 44 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 45 | C3×#103 | — | 2 | 4 | no | — |
| 46 | C3×C4×C8 | — | 2 | 1 | no | — |
| 47 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C3×C4×C8 | D4 | 1 | 2 | yes | — |
| 50 | C3×C4×C8 | D4 | 1 | 2 | yes | — |
| 51 | #533 | C22×C3 | 1 | 2 | yes | — |
| 52 | #500 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 53 | #494 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 54 | #487 | C3×C4 | 1 | 2 | yes | complemented |
| 55 | #467 | — | 2 | 2 | no | — |
| 56 | C4×Q16 | — | 2 | 2 | no | — |
| 57 | #423 | C22×C3 | 1 | 2 | yes | — |
| 58 | C2×C4×C8 | C22×C3 | 1 | 1 | yes | characteristic |
| 59 | C2×C4×C8 | C22×C3 | 1 | 1 | yes | characteristic |
| 60 | C43 | C22×C3 | 1 | 1 | yes | characteristic |
| 61 | #339 | — | 2 | 4 | no | — |
| 62 | #336 | — | 2 | 2 | no | — |
| 63 | #334 | — | 2 | 2 | no | — |
| 64 | Q8⋊C8 | — | 2 | 4 | no | — |
| 65 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 66 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 67 | C22×C3×C4 | — | 2 | 2 | no | — |
| 68 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 71 | C3×Q16 | — | 4 | 4 | no | — |
| 72 | C3×M16 | — | 4 | 4 | no | — |
| 73 | C2×C3×C8 | — | 2 | 2 | no | — |
| 74 | C2×C3×C8 | — | 4 | 2 | no | — |
| 75 | C2×C3×C8 | — | 4 | 2 | no | — |
| 76 | C2×C3×C8 | — | 4 | 2 | no | — |
| 77 | C2×C3×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 78 | C2×C3×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 79 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 80 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 82 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 83 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 84 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 85 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 86 | C3×C42 | — | 2 | 4 | no | — |
| 87 | C3×C42 | — | 2 | 4 | no | — |
| 88 | C3×C42 | — | 4 | 2 | no | — |
| 89 | C3×C42 | — | 4 | 2 | no | — |
| 90 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 94 | C2×Q16 | — | 2 | 2 | no | — |
| 95 | C2×M16 | — | 2 | 2 | no | — |
| 96 | C22×C8 | — | 2 | 1 | no | — |
| 97 | C22×C8 | C3×D4 | 1 | 2 | yes | — |
| 98 | C4⋊Q8 | C2×C3×C4 | 1 | 2 | yes | — |
| 99 | C4⋊Q8 | C2×C3×C4 | 1 | 2 | yes | — |
| 100 | #125 | — | 2 | 2 | no | — |
| 101 | C22⋊Q8 | — | 2 | 2 | no | — |
| 102 | C22⋊Q8 | — | 2 | 2 | no | — |
| 103 | C4×Q8 | — | 2 | 2 | no | — |
| 104 | C4×Q8 | — | 2 | 2 | no | — |
| 105 | #117 | — | 2 | 2 | no | — |
| 106 | C2×C42 | — | 2 | 1 | no | — |
| 107 | C2×C42 | — | 2 | 1 | no | — |
| 108 | C2×C42 | C3×D4 | 1 | 2 | yes | — |
| 109 | C2×C42 | C23×C3 | 1 | 1 | yes | characteristic |
| 110 | #107 | — | 2 | 2 | no | — |
| 111 | C4⋊C8 | — | 2 | 2 | no | — |
| 112 | C4⋊C8 | — | 2 | 2 | no | — |
| 113 | #103 | — | 2 | 4 | no | — |
| 114 | C4×C8 | — | 2 | 1 | no | — |
| 115 | C4×C8 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 116 | C4×C8 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 117 | C4×C8 | C3×D4 | 1 | 2 | yes | — |
| 118 | C4×C8 | C3×D4 | 1 | 2 | yes | — |
| 119 | C23×C3 | #121 | 1 | 1 | yes | characteristic |
| 120 | C3×Q8 | — | 4 | 4 | no | — |
| 121 | C3×Q8 | — | 4 | 4 | no | — |
| 122 | C2×C3×C4 | — | 2 | 2 | no | — |
| 123 | C2×C3×C4 | — | 2 | 2 | no | — |
| 124 | C2×C3×C4 | — | 2 | 4 | no | — |
| 125 | C2×C3×C4 | — | 4 | 2 | no | — |
| 126 | C2×C3×C4 | — | 4 | 2 | no | — |
| 127 | C2×C3×C4 | — | 4 | 2 | no | — |
| 128 | C2×C3×C4 | — | 4 | 2 | no | — |
| 129 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 130 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 131 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 132 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 133 | C2×C3×C4 | #124 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 135 | C2×C3×C4 | #137 | 1 | 4 | yes | — |
| 136 | C3×C8 | — | 2 | 2 | no | — |
| 137 | C3×C8 | — | 2 | 2 | no | — |
| 138 | C3×C8 | — | 4 | 2 | no | — |
| 139 | C3×C8 | — | 8 | 2 | no | — |
| 140 | C2×Q8 | — | 2 | 2 | no | — |
| 141 | C2×Q8 | — | 2 | 2 | no | — |
| 142 | C22×C4 | — | 2 | 2 | no | — |
| 143 | C22×C4 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 144 | C22×C4 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 145 | C22×C4 | C3×#41 | 1 | 1 | yes | characteristic |
| 146 | Q16 | — | 4 | 4 | no | — |
| 147 | M16 | — | 4 | 4 | no | — |
| 148 | C2×C8 | — | 2 | 2 | no | — |
| 149 | C2×C8 | — | 4 | 2 | no | — |
| 150 | C2×C8 | — | 4 | 2 | no | — |
| 151 | C2×C8 | — | 4 | 2 | no | — |
| 152 | C2×C8 | C3×C22⋊C4 | 1 | 2 | yes | — |
| 153 | C2×C8 | C3×C22⋊C4 | 1 | 2 | yes | — |
| 154 | C4⋊C4 | — | 2 | 2 | no | — |
| 155 | C4⋊C4 | — | 2 | 2 | no | — |
| 156 | C4⋊C4 | — | 2 | 2 | no | — |
| 157 | C4⋊C4 | — | 2 | 2 | no | — |
| 158 | C4⋊C4 | — | 4 | 2 | no | — |
| 159 | C4⋊C4 | — | 4 | 2 | no | — |
| 160 | C22⋊C4 | — | 4 | 2 | no | — |
| 161 | C42 | — | 2 | 4 | no | — |
| 162 | C42 | — | 2 | 4 | no | — |
| 163 | C42 | — | 4 | 2 | no | — |
| 164 | C42 | — | 4 | 2 | no | — |
| 165 | C42 | C22×C3×C4 | 1 | 1 | yes | characteristic |
| 166 | C42 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 167 | C42 | C2×C3×D4 | 1 | 1 | yes | characteristic |
| 168 | C42 | C3×#41 | 1 | 1 | yes | characteristic |
| 169 | C22×C3 | — | 2 | 1 | no | — |
| 170 | C22×C3 | — | 2 | 1 | no | — |
| 171 | C22×C3 | #390 | 1 | 1 | yes | characteristic |
| 172 | C22×C3 | #470 | 1 | 2 | yes | — |
| 173 | C3×C4 | — | 2 | 1 | no | — |
| 174 | C3×C4 | — | 2 | 1 | no | — |
| 175 | C3×C4 | — | 4 | 4 | no | — |
| 176 | C3×C4 | — | 8 | 2 | no | — |
| 177 | C3×C4 | — | 8 | 2 | no | — |
| 178 | C3×C4 | #421 | 1 | 2 | yes | — |
| 179 | C3×C4 | #441 | 1 | 2 | yes | — |
| 180 | C3×C4 | #443 | 1 | 2 | yes | — |
| 181 | C3×C4 | #497 | 1 | 2 | yes | — |
| 182 | C23 | C3×#121 | 1 | 1 | yes | characteristic |
| 183 | Q8 | — | 4 | 4 | no | — |
| 184 | Q8 | — | 4 | 4 | no | — |
| 185 | C2×C4 | — | 2 | 2 | no | — |
| 186 | C2×C4 | — | 2 | 2 | no | — |
| 187 | C2×C4 | — | 2 | 4 | no | — |
| 188 | C2×C4 | — | 4 | 2 | no | — |
| 189 | C2×C4 | — | 4 | 2 | no | — |
| 190 | C2×C4 | — | 4 | 2 | no | — |
| 191 | C2×C4 | — | 4 | 2 | no | — |
| 192 | C2×C4 | C2×C3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 193 | C2×C4 | C3×C4×D4 | 1 | 1 | yes | characteristic |
| 194 | C2×C4 | C3×C4×D4 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | C3×#121 | 1 | 1 | yes | characteristic |
| 196 | C2×C4 | C3×#124 | 1 | 1 | yes | characteristic |
| 197 | C2×C4 | C3×#127 | 1 | 1 | yes | characteristic |
| 198 | C2×C4 | C3×#137 | 1 | 4 | yes | — |
| 199 | C8 | — | 2 | 2 | no | — |
| 200 | C8 | — | 2 | 2 | no | — |
| 201 | C8 | — | 4 | 2 | no | — |
| 202 | C8 | — | 8 | 2 | no | — |
| 203 | C2×C3 | — | 2 | 2 | no | — |
| 204 | C2×C3 | #1963 | 1 | 1 | yes | characteristic |
| 205 | C2×C3 | #1971 | 1 | 1 | yes | characteristic |
| 206 | C2×C3 | #1974 | 1 | 1 | yes | characteristic |
| 207 | C22 | — | 2 | 1 | no | — |
| 208 | C22 | — | 2 | 1 | no | — |
| 209 | C22 | C3×#390 | 1 | 1 | yes | characteristic |
| 210 | C22 | C3×#470 | 1 | 2 | yes | — |
| 211 | C4 | — | 2 | 1 | no | — |
| 212 | C4 | — | 2 | 1 | no | — |
| 213 | C4 | — | 4 | 4 | no | — |
| 214 | C4 | — | 8 | 2 | no | — |
| 215 | C4 | — | 8 | 2 | no | — |
| 216 | C4 | C3×#421 | 1 | 2 | yes | — |
| 217 | C4 | C3×#441 | 1 | 2 | yes | — |
| 218 | C4 | C3×#443 | 1 | 2 | yes | — |
| 219 | C4 | C3×#497 | 1 | 2 | yes | — |
| 220 | C3 | #9853 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 221 | C2 | — | 2 | 2 | no | — |
| 222 | C2 | C3×#1963 | 1 | 1 | yes | characteristic |
| 223 | C2 | C3×#1971 | 1 | 1 | yes | characteristic |
| 224 | C2 | C3×#1974 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta