Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
665 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 74 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 | C3×#1007⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×C3⋊#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×#983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#64201 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×D3×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C3×#64197 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C3×#64196 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C3×#5927 | — | 3 | 1 | no | maximal |
| 9 | #10596440 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 10 | C3×#5779 | — | 3 | 1 | no | maximal |
| 11 | C2×C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#3805 | — | 4 | 1 | no | maximal |
| 13 | C3×D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×C3⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×C3⋊#226 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×C3⋊#226 | C22 | 1 | 1 | yes | characteristic |
| 17 | C32×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 18 | C32×#226 | C22 | 1 | 1 | yes | characteristic |
| 19 | C3×#1010 | — | 3 | 1 | no | — |
| 20 | C3×#1007 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | C3×2−1+4⋊C3 | — | 3 | 1 | no | — |
| 22 | #1007⋊C3 | — | 2 | 1 | no | — |
| 23 | #1007⋊C3 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C3⋊#226 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 25 | C3×#985 | — | 3 | 1 | no | — |
| 26 | C3×#984 | — | 3 | 1 | no | — |
| 27 | C3×#983 | — | 2 | 1 | no | — |
| 28 | C3×#983 | — | 3 | 1 | no | — |
| 29 | C3×#983 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | C3×#983 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×C3×#226 | — | 3 | 1 | no | — |
| 32 | C2×C3×#226 | — | 3 | 1 | no | — |
| 33 | C2×C3×#226 | — | 6 | 1 | no | — |
| 34 | #64201 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 35 | D3×#226 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 36 | #64197 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 37 | #64196 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 38 | C32×#137 | — | 3 | 1 | no | — |
| 39 | C3×#950 | — | 3 | 1 | no | — |
| 40 | C2×C3×#216 | — | 3 | 1 | no | — |
| 41 | C3×#918 | — | 3 | 1 | no | — |
| 42 | C3×#918 | — | 3 | 1 | no | — |
| 43 | C3×D3×Q16 | — | 3 | 1 | no | — |
| 44 | C3×D3×Q16 | — | 3 | 1 | no | — |
| 45 | C3×#916 | — | 3 | 1 | no | — |
| 46 | C3×#916 | — | 3 | 1 | no | — |
| 47 | C3×#915 | — | 3 | 1 | no | — |
| 48 | C3×#915 | — | 3 | 1 | no | — |
| 49 | C3×#909 | — | 3 | 1 | no | — |
| 50 | C3×#907 | — | 3 | 1 | no | — |
| 51 | C2×C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 52 | C32×#67 | — | 4 | 1 | no | — |
| 53 | C32×#67 | — | 4 | 1 | no | — |
| 54 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 55 | C3×#627 | — | 4 | 1 | no | — |
| 56 | C3×#625 | — | 4 | 1 | no | — |
| 57 | C3×D3×Dic3 | — | 4 | 1 | no | — |
| 58 | C3×D3×Dic3 | — | 4 | 1 | no | — |
| 59 | #5927 | — | 3 | 1 | no | — |
| 60 | C3×#578 | — | 9 | 1 | no | — |
| 61 | #5779 | — | 3 | 1 | no | — |
| 62 | C32×#41 | — | 3 | 1 | no | — |
| 63 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 64 | C3×#239 | — | 3 | 1 | no | — |
| 65 | C3×#239 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 67 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 68 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 69 | C3×D3×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 70 | C3×#237 | — | 3 | 1 | no | — |
| 71 | C3×#237 | — | 3 | 1 | no | — |
| 72 | C3×#237 | — | 6 | 1 | no | — |
| 73 | C3×#237 | — | 6 | 1 | no | — |
| 74 | C3×#237 | — | 12 | 1 | no | — |
| 75 | C3×#235 | — | 3 | 1 | no | — |
| 76 | C3×#235 | — | 3 | 1 | no | — |
| 77 | C3×#235 | — | 6 | 1 | no | — |
| 78 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 79 | C2×C3×Dic6 | — | 6 | 1 | no | — |
| 80 | C3×#41⋊C3 | — | 3 | 1 | no | — |
| 81 | C2×C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 82 | C2×C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 83 | C2×C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 84 | C2×C3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 85 | C2×C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 86 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×SL(2,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic |
| 88 | #3805 | — | 4 | 1 | no | — |
| 89 | D3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 90 | D3×SL(2,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic |
| 91 | C3⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 92 | C3⋊(#41⋊C3) | C22×C3 | 1 | 1 | yes | characteristic |
| 93 | C3⋊#226 | C22×C3 | 1 | 1 | yes | characteristic |
| 94 | C3⋊#226 | C22×C3 | 1 | 1 | yes | characteristic |
| 95 | C3×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 96 | C3×GL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 97 | C3×GL(2,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic |
| 98 | C3×#226 | — | 2 | 1 | no | — |
| 99 | C3×#226 | — | 3 | 1 | no | — |
| 100 | C3×#226 | — | 3 | 1 | no | — |
| 101 | C3×#226 | — | 6 | 1 | no | — |
| 102 | C3×#226 | — | 6 | 1 | no | — |
| 103 | C3×#226 | C2×D3 | 1 | 1 | yes | characteristic |
| 104 | C3×#226 | C22×C3 | 1 | 1 | yes | characteristic |
| 105 | C32×Q16 | — | 3 | 1 | no | — |
| 106 | C32×Q16 | — | 3 | 1 | no | — |
| 107 | C32×SD16 | — | 3 | 1 | no | — |
| 108 | C32×SD16 | — | 3 | 1 | no | — |
| 109 | C32×M16 | — | 3 | 1 | no | — |
| 110 | C3×#216 | — | 3 | 1 | no | — |
| 111 | C3×#216 | — | 3 | 1 | no | — |
| 112 | C3×#216 | — | 3 | 1 | no | — |
| 113 | C3×#216 | — | 3 | 1 | no | — |
| 114 | C3×#216 | — | 3 | 1 | no | — |
| 115 | C3×#215 | — | 3 | 1 | no | — |
| 116 | C3×#214 | — | 3 | 1 | no | — |
| 117 | C3×#213 | — | 3 | 1 | no | — |
| 118 | C2×C3×C3⋊C8 | — | 3 | 1 | no | — |
| 119 | C3×Dic12 | — | 3 | 1 | no | — |
| 120 | C3×Dic12 | — | 3 | 1 | no | — |
| 121 | C3×#204 | — | 3 | 1 | no | — |
| 122 | C3×#204 | — | 3 | 1 | no | — |
| 123 | C3×ΓL(1,𝔽25) | — | 3 | 1 | no | — |
| 124 | C3×ΓL(1,𝔽25) | — | 3 | 1 | no | — |
| 125 | C3×D3×C8 | — | 3 | 1 | no | — |
| 126 | C3×D3×C8 | — | 3 | 1 | no | — |
| 127 | C22×C33 | — | 4 | 1 | no | — |
| 128 | C2×C32×D3 | — | 4 | 1 | no | — |
| 129 | C2×C32×D3 | — | 4 | 1 | no | — |
| 130 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 131 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 132 | C32×Dic3 | — | 4 | 1 | no | — |
| 133 | C32×Dic3 | — | 4 | 1 | no | — |
| 134 | C3×2−1+4 | — | 3 | 1 | no | — |
| 135 | C3×2−1+4 | — | 9 | 1 | no | — |
| 136 | #1010 | — | 3 | 1 | no | — |
| 137 | #1007 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 138 | 2−1+4⋊C3 | — | 3 | 1 | no | — |
| 139 | 2−1+4⋊C3 | — | 6 | 1 | no | — |
| 140 | #985 | — | 3 | 1 | no | — |
| 141 | #984 | — | 3 | 1 | no | — |
| 142 | #983 | — | 3 | 1 | no | — |
| 143 | #983 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 144 | C2×#226 | — | 3 | 1 | no | — |
| 145 | C2×#226 | — | 3 | 1 | no | — |
| 146 | C2×#226 | — | 6 | 1 | no | — |
| 147 | C3×#137 | — | 3 | 1 | no | — |
| 148 | C3×#137 | — | 3 | 1 | no | — |
| 149 | C3×#137 | — | 6 | 1 | no | — |
| 150 | C3×#137 | — | 9 | 1 | no | — |
| 151 | C3×#137 | — | 9 | 1 | no | — |
| 152 | C3×#137 | — | 9 | 1 | no | — |
| 153 | C3×#137 | — | 9 | 1 | no | — |
| 154 | C3×#137 | — | 9 | 1 | no | — |
| 155 | C3×#135 | — | 9 | 1 | no | — |
| 156 | C3×#135 | — | 9 | 1 | no | — |
| 157 | C3×#135 | — | 9 | 1 | no | — |
| 158 | C2×C3×Q16 | — | 9 | 1 | no | — |
| 159 | C2×C3×Q16 | — | 9 | 1 | no | — |
| 160 | C2×C3×Q16 | — | 9 | 1 | no | — |
| 161 | C3×#131 | — | 9 | 1 | no | — |
| 162 | #950 | — | 3 | 1 | no | — |
| 163 | C2×#216 | — | 3 | 1 | no | — |
| 164 | #918 | — | 3 | 1 | no | — |
| 165 | #918 | — | 3 | 1 | no | — |
| 166 | D3×Q16 | — | 3 | 1 | no | — |
| 167 | D3×Q16 | — | 3 | 1 | no | — |
| 168 | #916 | — | 3 | 1 | no | — |
| 169 | #916 | — | 3 | 1 | no | — |
| 170 | #915 | — | 3 | 1 | no | — |
| 171 | #915 | — | 3 | 1 | no | — |
| 172 | #909 | — | 3 | 1 | no | — |
| 173 | #907 | — | 3 | 1 | no | — |
| 174 | C32×Q8 | — | 3 | 1 | no | — |
| 175 | C32×Q8 | — | 3 | 1 | no | — |
| 176 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 177 | C32×D4 | — | 3 | 1 | no | — |
| 178 | C32×D4 | — | 6 | 1 | no | — |
| 179 | C32×D4 | — | 12 | 1 | no | — |
| 180 | C2×C32×C4 | — | 3 | 1 | no | — |
| 181 | C2×C32×C4 | — | 6 | 1 | no | — |
| 182 | C2×C3⋊Dic3 | — | 4 | 1 | no | — |
| 183 | C3×#67 | — | 3 | 1 | no | — |
| 184 | C3×#67 | — | 4 | 1 | no | — |
| 185 | C3×#67 | — | 4 | 1 | no | — |
| 186 | C3×#67 | — | 4 | 1 | no | — |
| 187 | C3×#67 | — | 6 | 1 | no | — |
| 188 | C3×#67 | — | 6 | 1 | no | — |
| 189 | C3×#67 | — | 8 | 1 | no | — |
| 190 | C3×#67 | — | 8 | 1 | no | — |
| 191 | C3×#67 | — | 12 | 1 | no | — |
| 192 | C3×#67 | S4 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 194 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 195 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 196 | C2×C3×Dic3 | — | 12 | 1 | no | — |
| 197 | C2×C3×Dic3 | — | 12 | 1 | no | — |
| 198 | C2×C3×Dic3 | — | 24 | 1 | no | — |
| 199 | C3×D12 | — | 3 | 1 | no | — |
| 200 | C3×D12 | — | 3 | 1 | no | — |
| 201 | C3×C4×D3 | — | 3 | 1 | no | — |
| 202 | C3×C4×D3 | — | 3 | 1 | no | — |
| 203 | C3×C4×D3 | — | 6 | 1 | no | — |
| 204 | C3×C4×D3 | — | 6 | 1 | no | — |
| 205 | C3×C4×D3 | — | 6 | 1 | no | — |
| 206 | C3×C4×D3 | — | 12 | 1 | no | — |
| 207 | C3×Dic6 | — | 3 | 1 | no | — |
| 208 | C3×Dic6 | — | 3 | 1 | no | — |
| 209 | C3×Dic6 | — | 3 | 1 | no | — |
| 210 | C3×Dic6 | — | 3 | 1 | no | — |
| 211 | C3×Dic6 | — | 6 | 1 | no | — |
| 212 | C3×Dic6 | — | 6 | 1 | no | — |
| 213 | C3×Dic6 | — | 6 | 1 | no | — |
| 214 | C3×Dic6 | — | 12 | 1 | no | — |
| 215 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 216 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 217 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 218 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 219 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 220 | C3×SL(2,𝔽3) | C23×C3 | 1 | 1 | yes | characteristic |
| 221 | #627 | — | 4 | 1 | no | — |
| 222 | #625 | — | 4 | 1 | no | — |
| 223 | D3×Dic3 | — | 4 | 1 | no | — |
| 224 | D3×Dic3 | — | 4 | 1 | no | — |
| 225 | C32×C8 | — | 3 | 1 | no | — |
| 226 | C32×C8 | — | 3 | 1 | no | — |
| 227 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 228 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 229 | #578 | — | 9 | 1 | no | Sylow |
| 230 | C2×C33 | — | 4 | 1 | no | — |
| 231 | C2×C33 | — | 8 | 1 | no | — |
| 232 | C32×D3 | — | 8 | 1 | no | — |
| 233 | C32×D3 | — | 8 | 1 | no | — |
| 234 | C3×#41 | — | 3 | 1 | no | — |
| 235 | C3×#41 | — | 3 | 1 | no | — |
| 236 | C3×#41 | — | 3 | 1 | no | — |
| 237 | C3×#41 | — | 6 | 1 | no | — |
| 238 | C3×#41 | — | 9 | 1 | no | — |
| 239 | C3×#41 | — | 9 | 1 | no | — |
| 240 | C3×#41 | — | 9 | 1 | no | — |
| 241 | C3×#41 | — | 9 | 1 | no | — |
| 242 | C3×#41 | — | 9 | 1 | no | — |
| 243 | C3×#41 | — | 18 | 1 | no | — |
| 244 | C3×#41 | — | 18 | 1 | no | — |
| 245 | C3×#41 | — | 18 | 1 | no | — |
| 246 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 247 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 248 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 249 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 250 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 251 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 252 | C2×C3×Q8 | — | 18 | 1 | no | — |
| 253 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 254 | C2×C3×Q8 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 255 | #239 | — | 3 | 1 | no | — |
| 256 | #239 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 257 | D3×Q8 | — | 3 | 1 | no | — |
| 258 | D3×Q8 | — | 3 | 1 | no | — |
| 259 | D3×Q8 | — | 3 | 1 | no | — |
| 260 | D3×Q8 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 261 | #237 | — | 3 | 1 | no | — |
| 262 | #237 | — | 3 | 1 | no | — |
| 263 | #237 | — | 6 | 1 | no | — |
| 264 | #237 | — | 6 | 1 | no | — |
| 265 | #237 | — | 12 | 1 | no | — |
| 266 | #235 | — | 3 | 1 | no | — |
| 267 | #235 | — | 3 | 1 | no | — |
| 268 | #235 | — | 6 | 1 | no | — |
| 269 | C2×Dic6 | — | 3 | 1 | no | — |
| 270 | C2×Dic6 | — | 6 | 1 | no | — |
| 271 | #41⋊C3 | — | 3 | 1 | no | — |
| 272 | #41⋊C3 | — | 6 | 1 | no | — |
| 273 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 274 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 275 | C2×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 276 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 277 | C2×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 278 | C2×SL(2,𝔽3) | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 279 | GL(2,𝔽3) | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 280 | #226 | — | 3 | 1 | no | — |
| 281 | #226 | — | 3 | 1 | no | — |
| 282 | #226 | — | 6 | 1 | no | — |
| 283 | #226 | — | 6 | 1 | no | — |
| 284 | #226 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 285 | C3×Q16 | — | 3 | 1 | no | — |
| 286 | C3×Q16 | — | 3 | 1 | no | — |
| 287 | C3×Q16 | — | 3 | 1 | no | — |
| 288 | C3×Q16 | — | 3 | 1 | no | — |
| 289 | C3×Q16 | — | 6 | 1 | no | — |
| 290 | C3×Q16 | — | 6 | 1 | no | — |
| 291 | C3×Q16 | — | 9 | 1 | no | — |
| 292 | C3×Q16 | — | 9 | 1 | no | — |
| 293 | C3×Q16 | — | 9 | 1 | no | — |
| 294 | C3×Q16 | — | 9 | 1 | no | — |
| 295 | C3×Q16 | — | 9 | 1 | no | — |
| 296 | C3×Q16 | — | 9 | 1 | no | — |
| 297 | C3×Q16 | — | 9 | 1 | no | — |
| 298 | C3×SD16 | — | 3 | 1 | no | — |
| 299 | C3×SD16 | — | 3 | 1 | no | — |
| 300 | C3×SD16 | — | 3 | 1 | no | — |
| 301 | C3×SD16 | — | 3 | 1 | no | — |
| 302 | C3×SD16 | — | 6 | 1 | no | — |
| 303 | C3×SD16 | — | 6 | 1 | no | — |
| 304 | C3×SD16 | — | 9 | 1 | no | — |
| 305 | C3×SD16 | — | 9 | 1 | no | — |
| 306 | C3×SD16 | — | 9 | 1 | no | — |
| 307 | C3×SD16 | — | 9 | 1 | no | — |
| 308 | C3×D8 | — | 9 | 1 | no | — |
| 309 | C3×M16 | — | 3 | 1 | no | — |
| 310 | C3×M16 | — | 3 | 1 | no | — |
| 311 | C3×M16 | — | 6 | 1 | no | — |
| 312 | C3×M16 | — | 9 | 1 | no | — |
| 313 | C3×M16 | — | 9 | 1 | no | — |
| 314 | C2×C3×C8 | — | 9 | 1 | no | — |
| 315 | C2×C3×C8 | — | 9 | 1 | no | — |
| 316 | C2×C3×C8 | — | 9 | 1 | no | — |
| 317 | #216 | — | 3 | 1 | no | — |
| 318 | #216 | — | 3 | 1 | no | — |
| 319 | #216 | — | 3 | 1 | no | — |
| 320 | #216 | — | 3 | 1 | no | — |
| 321 | #216 | — | 3 | 1 | no | — |
| 322 | #215 | — | 3 | 1 | no | — |
| 323 | #214 | — | 3 | 1 | no | — |
| 324 | #213 | — | 3 | 1 | no | — |
| 325 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 326 | Dic12 | — | 3 | 1 | no | — |
| 327 | Dic12 | — | 3 | 1 | no | — |
| 328 | #204 | — | 3 | 1 | no | — |
| 329 | #204 | — | 3 | 1 | no | — |
| 330 | ΓL(1,𝔽25) | — | 3 | 1 | no | — |
| 331 | ΓL(1,𝔽25) | — | 3 | 1 | no | — |
| 332 | D3×C8 | — | 3 | 1 | no | — |
| 333 | D3×C8 | — | 3 | 1 | no | — |
| 334 | C22×C32 | — | 4 | 1 | no | — |
| 335 | C22×C32 | — | 4 | 1 | no | — |
| 336 | C22×C32 | — | 6 | 1 | no | — |
| 337 | C22×C32 | — | 8 | 1 | no | — |
| 338 | C22×C32 | — | 8 | 1 | no | — |
| 339 | C22×C32 | — | 8 | 1 | no | — |
| 340 | C22×C32 | — | 12 | 1 | no | — |
| 341 | C22×C32 | — | 16 | 1 | no | — |
| 342 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 343 | C2×C3×D3 | — | 3 | 1 | no | — |
| 344 | C2×C3×D3 | — | 4 | 1 | no | — |
| 345 | C2×C3×D3 | — | 4 | 1 | no | — |
| 346 | C2×C3×D3 | — | 4 | 1 | no | — |
| 347 | C2×C3×D3 | — | 8 | 1 | no | — |
| 348 | C2×C3×D3 | — | 8 | 1 | no | — |
| 349 | C2×C3×D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 350 | C32×C4 | — | 3 | 1 | no | — |
| 351 | C32×C4 | — | 3 | 1 | no | — |
| 352 | C32×C4 | — | 6 | 1 | no | — |
| 353 | C32×C4 | — | 12 | 1 | no | — |
| 354 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 355 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 356 | C3×Dic3 | — | 3 | 1 | no | — |
| 357 | C3×Dic3 | — | 4 | 1 | no | — |
| 358 | C3×Dic3 | — | 4 | 1 | no | — |
| 359 | C3×Dic3 | — | 4 | 1 | no | — |
| 360 | C3×Dic3 | — | 6 | 1 | no | — |
| 361 | C3×Dic3 | — | 6 | 1 | no | — |
| 362 | C3×Dic3 | — | 8 | 1 | no | — |
| 363 | C3×Dic3 | — | 8 | 1 | no | — |
| 364 | C3×Dic3 | — | 12 | 1 | no | — |
| 365 | C3×Dic3 | — | 12 | 1 | no | — |
| 366 | C3×Dic3 | — | 24 | 1 | no | — |
| 367 | C3×Dic3 | — | 24 | 1 | no | — |
| 368 | C3×Dic3 | C2×S4 | 1 | 1 | yes | characteristic |
| 369 | 2−1+4 | — | 3 | 1 | no | — |
| 370 | 2−1+4 | — | 9 | 1 | no | — |
| 371 | #137 | — | 3 | 1 | no | — |
| 372 | #137 | — | 9 | 1 | no | — |
| 373 | #137 | — | 9 | 1 | no | — |
| 374 | #137 | — | 9 | 1 | no | — |
| 375 | #137 | — | 9 | 1 | no | — |
| 376 | #137 | — | 9 | 1 | no | — |
| 377 | #135 | — | 9 | 1 | no | — |
| 378 | #135 | — | 9 | 1 | no | — |
| 379 | #135 | — | 9 | 1 | no | — |
| 380 | C2×Q16 | — | 9 | 1 | no | — |
| 381 | C2×Q16 | — | 9 | 1 | no | — |
| 382 | C2×Q16 | — | 9 | 1 | no | — |
| 383 | #131 | — | 9 | 1 | no | — |
| 384 | C33 | — | 4 | 1 | no | Sylow |
| 385 | C3×Q8 | — | 2 | 1 | no | — |
| 386 | C3×Q8 | — | 3 | 1 | no | — |
| 387 | C3×Q8 | — | 3 | 1 | no | — |
| 388 | C3×Q8 | — | 3 | 1 | no | — |
| 389 | C3×Q8 | — | 3 | 1 | no | — |
| 390 | C3×Q8 | — | 6 | 1 | no | — |
| 391 | C3×Q8 | — | 6 | 1 | no | — |
| 392 | C3×Q8 | — | 9 | 1 | no | — |
| 393 | C3×Q8 | — | 9 | 1 | no | — |
| 394 | C3×Q8 | — | 9 | 1 | no | — |
| 395 | C3×Q8 | — | 9 | 1 | no | — |
| 396 | C3×Q8 | — | 18 | 1 | no | — |
| 397 | C3×Q8 | — | 18 | 1 | no | — |
| 398 | C3×Q8 | — | 18 | 1 | no | — |
| 399 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 400 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 401 | C3×D4 | — | 3 | 1 | no | — |
| 402 | C3×D4 | — | 3 | 1 | no | — |
| 403 | C3×D4 | — | 3 | 1 | no | — |
| 404 | C3×D4 | — | 6 | 1 | no | — |
| 405 | C3×D4 | — | 6 | 1 | no | — |
| 406 | C3×D4 | — | 6 | 1 | no | — |
| 407 | C3×D4 | — | 9 | 1 | no | — |
| 408 | C3×D4 | — | 9 | 1 | no | — |
| 409 | C3×D4 | — | 9 | 1 | no | — |
| 410 | C3×D4 | — | 12 | 1 | no | — |
| 411 | C3×D4 | — | 12 | 1 | no | — |
| 412 | C3×D4 | — | 18 | 1 | no | — |
| 413 | C3×D4 | — | 18 | 1 | no | — |
| 414 | C3×D4 | — | 24 | 1 | no | — |
| 415 | C2×C3×C4 | — | 3 | 1 | no | — |
| 416 | C2×C3×C4 | — | 3 | 1 | no | — |
| 417 | C2×C3×C4 | — | 6 | 1 | no | — |
| 418 | C2×C3×C4 | — | 6 | 1 | no | — |
| 419 | C2×C3×C4 | — | 6 | 1 | no | — |
| 420 | C2×C3×C4 | — | 9 | 1 | no | — |
| 421 | C2×C3×C4 | — | 9 | 1 | no | — |
| 422 | C2×C3×C4 | — | 12 | 1 | no | — |
| 423 | C2×C3×C4 | — | 18 | 1 | no | — |
| 424 | C2×C3×C4 | — | 18 | 1 | no | — |
| 425 | C2×C3×C4 | — | 18 | 1 | no | — |
| 426 | C2×C3×C4 | — | 18 | 1 | no | — |
| 427 | C2×C3×C4 | — | 18 | 1 | no | — |
| 428 | C2×C3×C4 | — | 18 | 1 | no | — |
| 429 | #67 | — | 3 | 1 | no | — |
| 430 | #67 | — | 4 | 1 | no | — |
| 431 | #67 | — | 6 | 1 | no | — |
| 432 | #67 | — | 6 | 1 | no | — |
| 433 | #67 | — | 12 | 1 | no | — |
| 434 | #67 | C3×S4 | 1 | 1 | yes | characteristic |
| 435 | C2×Dic3 | — | 6 | 1 | no | — |
| 436 | C2×Dic3 | — | 6 | 1 | no | — |
| 437 | C2×Dic3 | — | 6 | 1 | no | — |
| 438 | C2×Dic3 | — | 12 | 1 | no | — |
| 439 | C2×Dic3 | — | 12 | 1 | no | — |
| 440 | C2×Dic3 | — | 24 | 1 | no | — |
| 441 | D12 | — | 3 | 1 | no | — |
| 442 | D12 | — | 3 | 1 | no | — |
| 443 | C4×D3 | — | 3 | 1 | no | — |
| 444 | C4×D3 | — | 3 | 1 | no | — |
| 445 | C4×D3 | — | 6 | 1 | no | — |
| 446 | C4×D3 | — | 6 | 1 | no | — |
| 447 | C4×D3 | — | 6 | 1 | no | — |
| 448 | C4×D3 | — | 12 | 1 | no | — |
| 449 | Dic6 | — | 3 | 1 | no | — |
| 450 | Dic6 | — | 3 | 1 | no | — |
| 451 | Dic6 | — | 3 | 1 | no | — |
| 452 | Dic6 | — | 3 | 1 | no | — |
| 453 | Dic6 | — | 6 | 1 | no | — |
| 454 | Dic6 | — | 6 | 1 | no | — |
| 455 | Dic6 | — | 6 | 1 | no | — |
| 456 | Dic6 | — | 12 | 1 | no | — |
| 457 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 458 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 459 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 460 | SL(2,𝔽3) | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 461 | C3×C8 | — | 3 | 1 | no | — |
| 462 | C3×C8 | — | 3 | 1 | no | — |
| 463 | C3×C8 | — | 3 | 1 | no | — |
| 464 | C3×C8 | — | 3 | 1 | no | — |
| 465 | C3×C8 | — | 6 | 1 | no | — |
| 466 | C3×C8 | — | 6 | 1 | no | — |
| 467 | C3×C8 | — | 9 | 1 | no | — |
| 468 | C3×C8 | — | 9 | 1 | no | — |
| 469 | C3⋊C8 | — | 3 | 1 | no | — |
| 470 | C3⋊C8 | — | 3 | 1 | no | — |
| 471 | C2×C32 | — | 2 | 1 | no | — |
| 472 | C2×C32 | — | 4 | 1 | no | — |
| 473 | C2×C32 | — | 4 | 1 | no | — |
| 474 | C2×C32 | — | 8 | 1 | no | — |
| 475 | C2×C32 | — | 8 | 1 | no | — |
| 476 | C2×C32 | — | 8 | 1 | no | — |
| 477 | C2×C32 | — | 8 | 2 | no | — |
| 478 | C2×C32 | — | 8 | 1 | no | — |
| 479 | C2×C32 | — | 8 | 1 | no | — |
| 480 | C2×C32 | — | 12 | 1 | no | — |
| 481 | C2×C32 | — | 16 | 1 | no | — |
| 482 | C2×C32 | — | 16 | 1 | no | — |
| 483 | C2×C32 | — | 16 | 1 | no | — |
| 484 | C2×C32 | — | 16 | 2 | no | — |
| 485 | C2×C32 | — | 24 | 1 | no | — |
| 486 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 487 | C3×D3 | — | 2 | 1 | no | — |
| 488 | C3×D3 | — | 6 | 1 | no | — |
| 489 | C3×D3 | — | 8 | 1 | no | — |
| 490 | C3×D3 | — | 8 | 1 | no | — |
| 491 | C3×D3 | — | 8 | 1 | no | — |
| 492 | C3×D3 | — | 16 | 1 | no | — |
| 493 | C3×D3 | — | 16 | 1 | no | — |
| 494 | #41 | — | 3 | 1 | no | — |
| 495 | #41 | — | 3 | 1 | no | — |
| 496 | #41 | — | 9 | 1 | no | — |
| 497 | #41 | — | 9 | 1 | no | — |
| 498 | #41 | — | 9 | 1 | no | — |
| 499 | #41 | — | 9 | 1 | no | — |
| 500 | #41 | — | 9 | 1 | no | — |
| 501 | #41 | — | 18 | 1 | no | — |
| 502 | #41 | — | 18 | 1 | no | — |
| 503 | #41 | — | 18 | 1 | no | — |
| 504 | C2×Q8 | — | 3 | 1 | no | — |
| 505 | C2×Q8 | — | 9 | 1 | no | — |
| 506 | C2×Q8 | — | 9 | 1 | no | — |
| 507 | C2×Q8 | — | 9 | 1 | no | — |
| 508 | C2×Q8 | — | 9 | 1 | no | — |
| 509 | C2×Q8 | — | 18 | 1 | no | — |
| 510 | C2×Q8 | C3×D32 | 1 | 1 | yes | characteristic |
| 511 | Q16 | — | 3 | 1 | no | — |
| 512 | Q16 | — | 3 | 1 | no | — |
| 513 | Q16 | — | 9 | 1 | no | — |
| 514 | Q16 | — | 9 | 1 | no | — |
| 515 | Q16 | — | 9 | 1 | no | — |
| 516 | Q16 | — | 9 | 1 | no | — |
| 517 | Q16 | — | 9 | 1 | no | — |
| 518 | Q16 | — | 9 | 1 | no | — |
| 519 | Q16 | — | 9 | 1 | no | — |
| 520 | SD16 | — | 3 | 1 | no | — |
| 521 | SD16 | — | 3 | 1 | no | — |
| 522 | SD16 | — | 9 | 1 | no | — |
| 523 | SD16 | — | 9 | 1 | no | — |
| 524 | SD16 | — | 9 | 1 | no | — |
| 525 | SD16 | — | 9 | 1 | no | — |
| 526 | D8 | — | 9 | 1 | no | — |
| 527 | M16 | — | 3 | 1 | no | — |
| 528 | M16 | — | 9 | 1 | no | — |
| 529 | M16 | — | 9 | 1 | no | — |
| 530 | C2×C8 | — | 9 | 1 | no | — |
| 531 | C2×C8 | — | 9 | 1 | no | — |
| 532 | C2×C8 | — | 9 | 1 | no | — |
| 533 | C22×C3 | — | 2 | 1 | no | — |
| 534 | C22×C3 | — | 3 | 1 | no | — |
| 535 | C22×C3 | — | 4 | 1 | no | — |
| 536 | C22×C3 | — | 6 | 1 | no | — |
| 537 | C22×C3 | — | 6 | 1 | no | — |
| 538 | C22×C3 | — | 8 | 1 | no | — |
| 539 | C22×C3 | — | 8 | 1 | no | — |
| 540 | C22×C3 | — | 9 | 1 | no | — |
| 541 | C22×C3 | — | 12 | 1 | no | — |
| 542 | C22×C3 | — | 12 | 1 | no | — |
| 543 | C22×C3 | — | 16 | 1 | no | — |
| 544 | C22×C3 | — | 24 | 1 | no | — |
| 545 | C22×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 546 | C22×C3 | C2×C3×S4 | 1 | 1 | yes | characteristic |
| 547 | C2×D3 | — | 3 | 1 | no | — |
| 548 | C2×D3 | — | 4 | 1 | no | — |
| 549 | C2×D3 | C2×C3×S4 | 1 | 1 | yes | characteristic |
| 550 | C3×C4 | — | 3 | 1 | no | — |
| 551 | C3×C4 | — | 3 | 1 | no | — |
| 552 | C3×C4 | — | 3 | 1 | no | — |
| 553 | C3×C4 | — | 3 | 1 | no | — |
| 554 | C3×C4 | — | 3 | 1 | no | — |
| 555 | C3×C4 | — | 6 | 1 | no | — |
| 556 | C3×C4 | — | 6 | 1 | no | — |
| 557 | C3×C4 | — | 6 | 1 | no | — |
| 558 | C3×C4 | — | 6 | 1 | no | — |
| 559 | C3×C4 | — | 9 | 1 | no | — |
| 560 | C3×C4 | — | 12 | 1 | no | — |
| 561 | C3×C4 | — | 12 | 1 | no | — |
| 562 | C3×C4 | — | 18 | 1 | no | — |
| 563 | C3×C4 | — | 18 | 1 | no | — |
| 564 | C3×C4 | — | 24 | 1 | no | — |
| 565 | Dic3 | — | 3 | 1 | no | — |
| 566 | Dic3 | — | 4 | 1 | no | — |
| 567 | Dic3 | — | 6 | 1 | no | — |
| 568 | Dic3 | — | 6 | 1 | no | — |
| 569 | Dic3 | — | 12 | 1 | no | — |
| 570 | Dic3 | — | 12 | 1 | no | — |
| 571 | Dic3 | — | 24 | 1 | no | — |
| 572 | Dic3 | — | 24 | 1 | no | — |
| 573 | Dic3 | C2×C3×S4 | 1 | 1 | yes | characteristic |
| 574 | C32 | — | 4 | 1 | no | — |
| 575 | C32 | — | 4 | 1 | no | — |
| 576 | C32 | — | 8 | 1 | no | — |
| 577 | C32 | — | 8 | 1 | no | — |
| 578 | C32 | — | 8 | 1 | no | — |
| 579 | C32 | — | 16 | 1 | no | — |
| 580 | C32 | #5927 | 1 | 1 | yes | characteristic, complemented |
| 581 | Q8 | — | 3 | 1 | no | — |
| 582 | Q8 | — | 3 | 1 | no | — |
| 583 | Q8 | — | 9 | 1 | no | — |
| 584 | Q8 | — | 9 | 1 | no | — |
| 585 | Q8 | — | 9 | 1 | no | — |
| 586 | Q8 | — | 9 | 1 | no | — |
| 587 | Q8 | — | 18 | 1 | no | — |
| 588 | Q8 | — | 18 | 1 | no | — |
| 589 | Q8 | — | 18 | 1 | no | — |
| 590 | Q8 | C2×C3×D32 | 1 | 1 | yes | characteristic |
| 591 | D4 | — | 3 | 1 | no | — |
| 592 | D4 | — | 3 | 1 | no | — |
| 593 | D4 | — | 6 | 1 | no | — |
| 594 | D4 | — | 9 | 1 | no | — |
| 595 | D4 | — | 9 | 1 | no | — |
| 596 | D4 | — | 9 | 1 | no | — |
| 597 | D4 | — | 18 | 1 | no | — |
| 598 | D4 | — | 18 | 1 | no | — |
| 599 | C2×C4 | — | 3 | 1 | no | — |
| 600 | C2×C4 | — | 6 | 1 | no | — |
| 601 | C2×C4 | — | 9 | 1 | no | — |
| 602 | C2×C4 | — | 9 | 1 | no | — |
| 603 | C2×C4 | — | 18 | 1 | no | — |
| 604 | C2×C4 | — | 18 | 1 | no | — |
| 605 | C2×C4 | — | 18 | 1 | no | — |
| 606 | C2×C4 | — | 18 | 1 | no | — |
| 607 | C2×C4 | — | 18 | 1 | no | — |
| 608 | C2×C4 | — | 18 | 1 | no | — |
| 609 | C8 | — | 3 | 1 | no | — |
| 610 | C8 | — | 3 | 1 | no | — |
| 611 | C8 | — | 9 | 1 | no | — |
| 612 | C8 | — | 9 | 1 | no | — |
| 613 | C2×C3 | — | 2 | 1 | no | — |
| 614 | C2×C3 | — | 2 | 1 | no | — |
| 615 | C2×C3 | — | 2 | 1 | no | — |
| 616 | C2×C3 | — | 4 | 1 | no | — |
| 617 | C2×C3 | — | 4 | 1 | no | — |
| 618 | C2×C3 | — | 6 | 1 | no | — |
| 619 | C2×C3 | — | 8 | 1 | no | — |
| 620 | C2×C3 | — | 8 | 2 | no | — |
| 621 | C2×C3 | — | 8 | 1 | no | — |
| 622 | C2×C3 | — | 8 | 1 | no | — |
| 623 | C2×C3 | — | 12 | 1 | no | — |
| 624 | C2×C3 | — | 12 | 1 | no | — |
| 625 | C2×C3 | — | 16 | 1 | no | — |
| 626 | C2×C3 | — | 16 | 2 | no | — |
| 627 | C2×C3 | — | 16 | 1 | no | — |
| 628 | C2×C3 | — | 18 | 1 | no | — |
| 629 | C2×C3 | — | 24 | 1 | no | — |
| 630 | C2×C3 | — | 24 | 1 | no | — |
| 631 | C2×C3 | — | 48 | 1 | no | — |
| 632 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 633 | C2×C3 | C22×C3×S4 | 1 | 1 | yes | characteristic |
| 634 | D3 | — | 2 | 1 | no | — |
| 635 | D3 | — | 6 | 1 | no | — |
| 636 | D3 | — | 8 | 1 | no | — |
| 637 | C22 | — | 3 | 1 | no | — |
| 638 | C22 | — | 6 | 1 | no | — |
| 639 | C22 | — | 9 | 1 | no | — |
| 640 | C22 | C3×D3×S4 | 1 | 1 | yes | characteristic |
| 641 | C4 | — | 3 | 1 | no | — |
| 642 | C4 | — | 3 | 1 | no | — |
| 643 | C4 | — | 3 | 1 | no | — |
| 644 | C4 | — | 6 | 1 | no | — |
| 645 | C4 | — | 9 | 1 | no | — |
| 646 | C4 | — | 18 | 1 | no | — |
| 647 | C4 | — | 18 | 1 | no | — |
| 648 | C3 | — | 2 | 1 | no | — |
| 649 | C3 | — | 4 | 1 | no | — |
| 650 | C3 | — | 8 | 1 | no | — |
| 651 | C3 | — | 8 | 1 | no | — |
| 652 | C3 | — | 16 | 1 | no | — |
| 653 | C3 | #10596440 | 1 | 1 | yes | characteristic, direct product factor |
| 654 | C3 | C3×#5927 | 1 | 1 | yes | characteristic, complemented |
| 655 | C2 | — | 2 | 1 | no | — |
| 656 | C2 | — | 6 | 1 | no | — |
| 657 | C2 | — | 12 | 1 | no | — |
| 658 | C2 | — | 18 | 1 | no | — |
| 659 | C2 | C2×C3×D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23×C3 |
| Frattini | C2 | C2×C3×D3×S4 |
| Fitting | C2×C32×Q8 | C2×D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C22×C3 | D3×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#1007⋊C3 | C2 | semidirect |
| 2. C2×C3×C3⋊#226 | C2 | semidirect |
| 3. C32×#983 | C2 | semidirect |
| 4. C3×#64201 | C2 | semidirect |
| 5. C3×D3×#226 | C2 | semidirect |
| 6. C3×#64197 | C2 | semidirect |
| 7. C3×#64196 | C2 | semidirect |
| 9. #10596440 | C3 | direct (N × H) |
| 20. C3×#1007 | D3 | semidirect |
| 23. #1007⋊C3 | C2×C3 | semidirect |
| 24. C2×C3⋊#226 | C2×C3 | semidirect |
| 29. C3×#983 | D3 | semidirect |
| 30. C3×#983 | C2×C3 | semidirect |
| 34. #64201 | C2×C3 | semidirect |
| 35. D3×#226 | C2×C3 | semidirect |
| 36. #64197 | C2×C3 | semidirect |
| 37. #64196 | C2×C3 | semidirect |
| 137. #1007 | C3×D3 | semidirect |
| 143. #983 | C3×D3 | semidirect |
| 580. C32 | #5927 | semidirect |
| 653. C3 | #10596440 | direct (N × H) |
| 654. C3 | C3×#5927 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1007⋊C3 | 2 |
| C2×C3×C3⋊#226 | 2 |
| C32×#983 | 2 |
| C3×#64201 | 2 |
| C3×D3×#226 | 2 |
| C3×#64197 | 2 |
| C3×#64196 | 2 |
| C3×#5927 | 3 |
| #10596440 | 3 |
| C3×#5779 | 3 |
| C3×#3805 | 4 |