Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
924 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 92 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 | #11724493 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#1010⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×D3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D32×SL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #64315⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D3×#1010 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | D3×2−1+4⋊C3 | — | 3 | 1 | no | maximal |
| 8 | C2×#1010⋊C3 | — | 3 | 1 | no | maximal |
| 9 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×D3×D12 | — | 4 | 1 | no | maximal |
| 11 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C3×D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×D3×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 14 | C3×C3⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #64367 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×#1010 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C3×D3×#41 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C3×2−1+4⋊C3 | — | 3 | 1 | no | — |
| 19 | C2×C3×#41⋊C3 | — | 3 | 1 | no | — |
| 20 | D32×Q8 | C2×C3 | 1 | 2 | yes | complemented |
| 21 | #64315 | C2×C3 | 1 | 2 | yes | complemented |
| 22 | D3×#237 | — | 3 | 2 | no | — |
| 23 | #64309 | — | 3 | 2 | no | — |
| 24 | D3×#235 | — | 3 | 1 | no | — |
| 25 | #64298 | — | 3 | 1 | no | — |
| 26 | #64297 | — | 3 | 1 | no | — |
| 27 | C2×D3×Dic6 | — | 3 | 1 | no | — |
| 28 | #1010⋊C3 | — | 2 | 1 | no | — |
| 29 | #1010⋊C3 | — | 3 | 1 | no | — |
| 30 | #1010⋊C3 | — | 3 | 1 | no | — |
| 31 | #1010⋊C3 | — | 3 | 2 | no | — |
| 32 | #1010⋊C3 | — | 6 | 1 | no | — |
| 33 | #1010⋊C3 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 34 | D3×#41⋊C3 | — | 2 | 1 | no | — |
| 35 | D3×#41⋊C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 36 | #1007⋊C3 | — | 3 | 2 | no | — |
| 37 | C2×D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 38 | C2×D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 39 | C2×C3×D32 | — | 4 | 2 | no | — |
| 40 | C3×C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 41 | C32×D12 | — | 4 | 1 | no | — |
| 42 | C32×C4×D3 | — | 4 | 1 | no | — |
| 43 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 44 | C3×#626 | — | 4 | 2 | no | — |
| 45 | D3×2−1+4 | — | 3 | 1 | no | — |
| 46 | C2×#1010 | — | 3 | 1 | no | — |
| 47 | C2×2−1+4⋊C3 | — | 9 | 1 | no | — |
| 48 | C32×#41 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 49 | Q8×C3⋊D3 | C22×C3 | 1 | 2 | yes | complemented |
| 50 | #3830 | — | 3 | 2 | no | — |
| 51 | #3828 | — | 3 | 1 | no | — |
| 52 | C2×#634 | — | 3 | 1 | no | — |
| 53 | C3×#239 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 54 | C3×D3×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 55 | C3×D3×Q8 | C22×C3 | 1 | 2 | yes | complemented |
| 56 | C3×#237 | — | 3 | 2 | no | — |
| 57 | C3×#237 | — | 3 | 1 | no | — |
| 58 | C3×D3×D4 | — | 3 | 1 | no | — |
| 59 | C3×D3×D4 | — | 12 | 1 | no | — |
| 60 | C3×#235 | — | 3 | 1 | no | — |
| 61 | C3×#235 | — | 3 | 1 | no | — |
| 62 | C2×C3×D12 | — | 12 | 1 | no | — |
| 63 | C2×C3×C4×D3 | — | 3 | 1 | no | — |
| 64 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 65 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 66 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 67 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 68 | C3×#41⋊C3 | — | 3 | 1 | no | — |
| 69 | C3×#41⋊C3 | — | 4 | 1 | no | — |
| 70 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 71 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×C3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 73 | C2×C3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 74 | D3×#67 | — | 3 | 2 | no | — |
| 75 | C2×#627 | — | 3 | 2 | no | — |
| 76 | #3805 | — | 3 | 4 | no | — |
| 77 | #3804 | — | 3 | 4 | no | — |
| 78 | #3804 | — | 3 | 2 | no | — |
| 79 | C2×D3×Dic3 | — | 3 | 2 | no | — |
| 80 | D3×D12 | A4 | 1 | 1 | yes | characteristic |
| 81 | C4×D32 | — | 3 | 2 | no | — |
| 82 | #3799 | — | 3 | 1 | no | — |
| 83 | #3798 | — | 3 | 2 | no | — |
| 84 | #3797 | — | 3 | 2 | no | — |
| 85 | #3796 | — | 3 | 2 | no | — |
| 86 | #3795 | — | 3 | 1 | no | — |
| 87 | #3795 | — | 3 | 2 | no | — |
| 88 | D3×Dic6 | — | 3 | 2 | no | — |
| 89 | D3×Dic6 | — | 3 | 1 | no | — |
| 90 | D3×Dic6 | — | 3 | 2 | no | — |
| 91 | D3×Dic6 | — | 3 | 2 | no | — |
| 92 | D3×Dic6 | — | 3 | 2 | no | — |
| 93 | D3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 94 | D3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 95 | D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 96 | D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 97 | D3×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 98 | D3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 99 | D3×SL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 100 | C3⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 101 | C3⋊(#41⋊C3) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 102 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 103 | C2×C32×D3 | — | 4 | 1 | no | — |
| 104 | C2×C32×D3 | — | 4 | 2 | no | — |
| 105 | C3×D32 | — | 8 | 4 | no | — |
| 106 | C33×C4 | — | 4 | 1 | no | — |
| 107 | C32×Dic3 | — | 4 | 1 | no | — |
| 108 | C3×2−1+4 | — | 3 | 1 | no | — |
| 109 | C2×C3×#41 | — | 3 | 1 | no | — |
| 110 | #1010 | — | 3 | 1 | no | — |
| 111 | #1010 | — | 3 | 1 | no | — |
| 112 | #1010 | — | 3 | 2 | no | — |
| 113 | #1010 | — | 6 | 1 | no | — |
| 114 | #1010 | — | 9 | 1 | no | — |
| 115 | #1010 | — | 9 | 1 | no | — |
| 116 | #1010 | — | 9 | 1 | no | — |
| 117 | #1010 | — | 9 | 2 | no | — |
| 118 | #1010 | — | 9 | 2 | no | — |
| 119 | #1010 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 120 | D3×#41 | — | 9 | 1 | no | — |
| 121 | D3×#41 | — | 9 | 2 | no | — |
| 122 | D3×#41 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 123 | #1007 | — | 3 | 2 | no | — |
| 124 | #1007 | — | 9 | 1 | no | — |
| 125 | C2×D3×Q8 | — | 3 | 2 | no | — |
| 126 | C2×D3×Q8 | — | 3 | 2 | no | — |
| 127 | C2×D3×Q8 | — | 9 | 1 | no | — |
| 128 | C2×#237 | — | 9 | 2 | no | — |
| 129 | C2×#235 | — | 9 | 1 | no | — |
| 130 | C22×Dic6 | — | 9 | 1 | no | — |
| 131 | 2−1+4⋊C3 | — | 3 | 1 | no | — |
| 132 | 2−1+4⋊C3 | — | 6 | 1 | no | — |
| 133 | 2−1+4⋊C3 | — | 9 | 2 | no | — |
| 134 | 2−1+4⋊C3 | — | 9 | 1 | no | — |
| 135 | C2×#41⋊C3 | — | 3 | 1 | no | — |
| 136 | C2×#41⋊C3 | — | 6 | 1 | no | — |
| 137 | C22×SL(2,𝔽3) | — | 9 | 2 | no | — |
| 138 | C22×C3×D3 | — | 3 | 1 | no | — |
| 139 | C22×C3×D3 | — | 12 | 2 | no | — |
| 140 | C22×C3×D3 | — | 12 | 2 | no | — |
| 141 | C2×D32 | C2×A4 | 1 | 2 | yes | — |
| 142 | C32×Q8 | C23×C3 | 1 | 1 | yes | characteristic |
| 143 | C32×D4 | — | 3 | 1 | no | — |
| 144 | C32×D4 | — | 12 | 1 | no | — |
| 145 | C2×C32×C4 | — | 3 | 1 | no | — |
| 146 | C2×C32×C4 | — | 12 | 1 | no | — |
| 147 | #638 | — | 3 | 2 | no | — |
| 148 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 149 | C4⋊(C3⋊D3) | C2×A4 | 1 | 1 | yes | characteristic |
| 150 | C4×C3⋊D3 | — | 3 | 2 | no | — |
| 151 | #634 | — | 3 | 1 | no | — |
| 152 | #634 | — | 3 | 2 | no | — |
| 153 | C3×#67 | — | 3 | 2 | no | — |
| 154 | C3×#67 | — | 3 | 2 | no | — |
| 155 | C3×#67 | — | 12 | 2 | no | — |
| 156 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 157 | C2×C3×Dic3 | — | 3 | 1 | no | — |
| 158 | C3×D12 | — | 3 | 1 | no | — |
| 159 | C3×D12 | — | 4 | 1 | no | — |
| 160 | C3×D12 | — | 8 | 1 | no | — |
| 161 | C3×D12 | — | 12 | 1 | no | — |
| 162 | C3×D12 | — | 12 | 1 | no | — |
| 163 | C3×D12 | — | 12 | 2 | no | — |
| 164 | C3×D12 | — | 24 | 1 | no | — |
| 165 | C3×D12 | C2×A4 | 1 | 1 | yes | characteristic |
| 166 | C3×C4×D3 | — | 3 | 1 | no | — |
| 167 | C3×C4×D3 | — | 3 | 2 | no | — |
| 168 | C3×C4×D3 | — | 3 | 1 | no | — |
| 169 | C3×C4×D3 | — | 3 | 1 | no | — |
| 170 | C3×C4×D3 | — | 4 | 1 | no | — |
| 171 | C3×C4×D3 | — | 8 | 1 | no | — |
| 172 | C3×C4×D3 | C2×A4 | 1 | 1 | yes | characteristic |
| 173 | C3×Dic6 | — | 3 | 1 | no | — |
| 174 | C3×Dic6 | — | 3 | 2 | no | — |
| 175 | C3×Dic6 | — | 3 | 1 | no | — |
| 176 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 177 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 178 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 179 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 180 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 181 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 182 | #627 | — | 3 | 4 | no | — |
| 183 | #627 | — | 3 | 2 | no | — |
| 184 | #626 | — | 3 | 2 | no | — |
| 185 | #626 | C2×A4 | 1 | 2 | yes | — |
| 186 | #625 | — | 3 | 2 | no | — |
| 187 | #624 | — | 3 | 2 | no | — |
| 188 | D3×Dic3 | — | 3 | 4 | no | — |
| 189 | D3×Dic3 | — | 3 | 2 | no | — |
| 190 | D3×Dic3 | — | 3 | 2 | no | — |
| 191 | C2×2−1+4 | — | 9 | 1 | no | Sylow |
| 192 | C2×C33 | — | 4 | 1 | no | — |
| 193 | C3×C3⋊D3 | — | 8 | 2 | no | — |
| 194 | C32×D3 | — | 4 | 2 | no | — |
| 195 | C32×D3 | — | 8 | 2 | no | — |
| 196 | C3×#41 | — | 2 | 1 | no | — |
| 197 | C3×#41 | — | 3 | 1 | no | — |
| 198 | C3×#41 | — | 9 | 1 | no | — |
| 199 | C3×#41 | — | 9 | 1 | no | — |
| 200 | C3×#41 | — | 9 | 2 | no | — |
| 201 | C3×#41 | — | 9 | 1 | no | — |
| 202 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 203 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 204 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 205 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 206 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 207 | C2×C3×D4 | — | 9 | 1 | no | — |
| 208 | C2×C3×D4 | — | 36 | 1 | no | — |
| 209 | C22×C3×C4 | — | 9 | 1 | no | — |
| 210 | C2×#67 | — | 9 | 2 | no | — |
| 211 | C22×Dic3 | — | 9 | 2 | no | — |
| 212 | #239 | — | 9 | 1 | no | — |
| 213 | #239 | — | 9 | 2 | no | — |
| 214 | #239 | C2×C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 215 | D3×Q8 | — | 3 | 2 | no | — |
| 216 | D3×Q8 | — | 3 | 2 | no | — |
| 217 | D3×Q8 | — | 6 | 2 | no | — |
| 218 | D3×Q8 | — | 9 | 2 | no | — |
| 219 | D3×Q8 | — | 9 | 2 | no | — |
| 220 | D3×Q8 | — | 9 | 2 | no | — |
| 221 | D3×Q8 | — | 9 | 2 | no | — |
| 222 | D3×Q8 | — | 9 | 2 | no | — |
| 223 | D3×Q8 | — | 9 | 1 | no | — |
| 224 | D3×Q8 | — | 9 | 2 | no | — |
| 225 | D3×Q8 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 226 | D3×Q8 | C2×C3×D3 | 1 | 2 | yes | complemented |
| 227 | #237 | — | 3 | 2 | no | — |
| 228 | #237 | — | 3 | 1 | no | — |
| 229 | #237 | — | 9 | 2 | no | — |
| 230 | #237 | — | 9 | 1 | no | — |
| 231 | #237 | — | 9 | 2 | no | — |
| 232 | #237 | — | 9 | 2 | no | — |
| 233 | #237 | — | 9 | 4 | no | — |
| 234 | #237 | — | 9 | 4 | no | — |
| 235 | #237 | — | 9 | 4 | no | — |
| 236 | #237 | — | 9 | 2 | no | — |
| 237 | #237 | — | 9 | 2 | no | — |
| 238 | #237 | — | 18 | 2 | no | — |
| 239 | D3×D4 | — | 3 | 1 | no | — |
| 240 | D3×D4 | — | 3 | 1 | no | — |
| 241 | D3×D4 | — | 9 | 2 | no | — |
| 242 | #235 | — | 3 | 1 | no | — |
| 243 | #235 | — | 3 | 1 | no | — |
| 244 | #235 | — | 9 | 1 | no | — |
| 245 | #235 | — | 9 | 2 | no | — |
| 246 | #235 | — | 9 | 2 | no | — |
| 247 | #235 | — | 9 | 2 | no | — |
| 248 | #235 | — | 9 | 1 | no | — |
| 249 | #235 | — | 9 | 1 | no | — |
| 250 | #235 | — | 9 | 2 | no | — |
| 251 | #235 | — | 9 | 4 | no | — |
| 252 | #235 | — | 9 | 1 | no | — |
| 253 | #235 | — | 18 | 1 | no | — |
| 254 | C2×D12 | — | 3 | 1 | no | — |
| 255 | C2×C4×D3 | — | 3 | 1 | no | — |
| 256 | C2×C4×D3 | — | 9 | 2 | no | — |
| 257 | C2×C4×D3 | — | 9 | 2 | no | — |
| 258 | C2×C4×D3 | — | 9 | 2 | no | — |
| 259 | C2×Dic6 | — | 3 | 1 | no | — |
| 260 | C2×Dic6 | — | 9 | 1 | no | — |
| 261 | C2×Dic6 | — | 9 | 2 | no | — |
| 262 | C2×Dic6 | — | 9 | 2 | no | — |
| 263 | C2×Dic6 | — | 9 | 2 | no | — |
| 264 | C2×Dic6 | — | 9 | 2 | no | — |
| 265 | C2×Dic6 | — | 9 | 2 | no | — |
| 266 | C2×Dic6 | — | 9 | 2 | no | — |
| 267 | C2×Dic6 | — | 9 | 1 | no | — |
| 268 | C2×Dic6 | — | 9 | 1 | no | — |
| 269 | C2×Dic6 | — | 18 | 1 | no | — |
| 270 | #41⋊C3 | — | 2 | 1 | no | — |
| 271 | #41⋊C3 | — | 2 | 1 | no | — |
| 272 | #41⋊C3 | — | 3 | 1 | no | — |
| 273 | #41⋊C3 | — | 4 | 1 | no | — |
| 274 | #41⋊C3 | — | 6 | 1 | no | — |
| 275 | #41⋊C3 | D32 | 1 | 1 | yes | characteristic, complemented |
| 276 | C2×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 277 | C2×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 278 | C2×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 279 | C2×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 280 | C2×SL(2,𝔽3) | — | 9 | 2 | no | — |
| 281 | C22×C32 | — | 3 | 1 | no | — |
| 282 | C22×C32 | — | 12 | 1 | no | — |
| 283 | C22×C32 | — | 12 | 2 | no | — |
| 284 | C2×C3⋊D3 | C22×A4 | 1 | 2 | yes | — |
| 285 | C2×C3×D3 | — | 3 | 1 | no | — |
| 286 | C2×C3×D3 | — | 4 | 1 | no | — |
| 287 | C2×C3×D3 | — | 4 | 2 | no | — |
| 288 | C2×C3×D3 | — | 6 | 2 | no | — |
| 289 | C2×C3×D3 | — | 8 | 1 | no | — |
| 290 | C2×C3×D3 | — | 8 | 2 | no | — |
| 291 | C2×C3×D3 | — | 12 | 2 | no | — |
| 292 | C2×C3×D3 | — | 12 | 2 | no | — |
| 293 | C2×C3×D3 | — | 24 | 2 | no | — |
| 294 | C2×C3×D3 | — | 24 | 4 | no | — |
| 295 | C2×C3×D3 | — | 24 | 4 | no | — |
| 296 | C2×C3×D3 | C22×A4 | 1 | 2 | yes | — |
| 297 | C2×C3×D3 | C22×A4 | 1 | 1 | yes | characteristic |
| 298 | D32 | — | 2 | 4 | no | — |
| 299 | C32×C4 | — | 3 | 1 | no | — |
| 300 | C32×C4 | — | 4 | 1 | no | — |
| 301 | C32×C4 | — | 4 | 1 | no | — |
| 302 | C32×C4 | — | 8 | 1 | no | — |
| 303 | C32×C4 | — | 8 | 1 | no | — |
| 304 | C32×C4 | — | 8 | 1 | no | — |
| 305 | C32×C4 | — | 12 | 1 | no | — |
| 306 | C32×C4 | — | 16 | 1 | no | — |
| 307 | C32×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 308 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 309 | C3×Dic3 | — | 3 | 2 | no | — |
| 310 | C3×Dic3 | — | 3 | 1 | no | — |
| 311 | C3×Dic3 | — | 4 | 1 | no | — |
| 312 | C3×Dic3 | — | 8 | 1 | no | — |
| 313 | C3×Dic3 | C22×A4 | 1 | 1 | yes | characteristic |
| 314 | 2−1+4 | — | 3 | 1 | no | — |
| 315 | 2−1+4 | — | 9 | 2 | no | — |
| 316 | 2−1+4 | — | 9 | 1 | no | — |
| 317 | 2−1+4 | — | 27 | 1 | no | — |
| 318 | 2−1+4 | — | 27 | 1 | no | — |
| 319 | 2−1+4 | — | 27 | 2 | no | — |
| 320 | C2×#41 | — | 3 | 1 | no | — |
| 321 | C2×#41 | — | 27 | 1 | no | — |
| 322 | C2×#41 | — | 27 | 2 | no | — |
| 323 | C22×Q8 | — | 9 | 2 | no | — |
| 324 | C22×Q8 | — | 27 | 1 | no | — |
| 325 | C33 | — | 4 | 1 | no | Sylow |
| 326 | C23×C3 | — | 9 | 1 | no | — |
| 327 | C23×C3 | — | 36 | 2 | no | — |
| 328 | C22×D3 | — | 3 | 2 | no | — |
| 329 | C22×D3 | — | 3 | 2 | no | — |
| 330 | C22×D3 | — | 3 | 1 | no | — |
| 331 | C3×Q8 | — | 2 | 1 | no | — |
| 332 | C3×Q8 | — | 9 | 1 | no | — |
| 333 | C3×Q8 | — | 9 | 2 | no | — |
| 334 | C3×Q8 | — | 9 | 1 | no | — |
| 335 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 336 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 337 | C3×D4 | — | 3 | 1 | no | — |
| 338 | C3×D4 | — | 3 | 1 | no | — |
| 339 | C3×D4 | — | 3 | 1 | no | — |
| 340 | C3×D4 | — | 6 | 1 | no | — |
| 341 | C3×D4 | — | 9 | 1 | no | — |
| 342 | C3×D4 | — | 9 | 2 | no | — |
| 343 | C3×D4 | — | 9 | 2 | no | — |
| 344 | C3×D4 | — | 12 | 1 | no | — |
| 345 | C3×D4 | — | 24 | 1 | no | — |
| 346 | C3×D4 | — | 36 | 1 | no | — |
| 347 | C3×D4 | — | 36 | 2 | no | — |
| 348 | C2×C3×C4 | — | 3 | 1 | no | — |
| 349 | C2×C3×C4 | — | 3 | 1 | no | — |
| 350 | C2×C3×C4 | — | 3 | 1 | no | — |
| 351 | C2×C3×C4 | — | 6 | 1 | no | — |
| 352 | C2×C3×C4 | — | 9 | 1 | no | — |
| 353 | C2×C3×C4 | — | 9 | 1 | no | — |
| 354 | C2×C3×C4 | — | 9 | 2 | no | — |
| 355 | C2×C3×C4 | — | 9 | 1 | no | — |
| 356 | C2×C3×C4 | — | 9 | 2 | no | — |
| 357 | C2×C3×C4 | — | 9 | 1 | no | — |
| 358 | C2×C3×C4 | — | 12 | 1 | no | — |
| 359 | C2×C3×C4 | — | 24 | 1 | no | — |
| 360 | #67 | — | 3 | 2 | no | — |
| 361 | #67 | — | 3 | 2 | no | — |
| 362 | #67 | — | 3 | 2 | no | — |
| 363 | #67 | — | 9 | 2 | no | — |
| 364 | #67 | — | 9 | 2 | no | — |
| 365 | #67 | — | 9 | 2 | no | — |
| 366 | #67 | — | 9 | 2 | no | — |
| 367 | #67 | — | 9 | 2 | no | — |
| 368 | #67 | — | 18 | 2 | no | — |
| 369 | C2×Dic3 | — | 3 | 2 | no | — |
| 370 | C2×Dic3 | — | 3 | 1 | no | — |
| 371 | C2×Dic3 | — | 9 | 2 | no | — |
| 372 | C2×Dic3 | — | 9 | 2 | no | — |
| 373 | C2×Dic3 | — | 9 | 4 | no | — |
| 374 | C2×Dic3 | — | 9 | 2 | no | — |
| 375 | C2×Dic3 | — | 9 | 2 | no | — |
| 376 | C2×Dic3 | — | 18 | 2 | no | — |
| 377 | D12 | — | 3 | 1 | no | — |
| 378 | D12 | — | 3 | 2 | no | — |
| 379 | D12 | — | 3 | 1 | no | — |
| 380 | D12 | — | 3 | 1 | no | — |
| 381 | D12 | — | 6 | 1 | no | — |
| 382 | D12 | — | 9 | 2 | no | — |
| 383 | D12 | D3×A4 | 1 | 1 | yes | characteristic |
| 384 | C4×D3 | — | 3 | 1 | no | — |
| 385 | C4×D3 | — | 3 | 2 | no | — |
| 386 | C4×D3 | — | 3 | 1 | no | — |
| 387 | C4×D3 | — | 3 | 1 | no | — |
| 388 | C4×D3 | — | 9 | 2 | no | — |
| 389 | C4×D3 | — | 9 | 2 | no | — |
| 390 | C4×D3 | — | 9 | 2 | no | — |
| 391 | C4×D3 | — | 9 | 2 | no | — |
| 392 | C4×D3 | — | 9 | 4 | no | — |
| 393 | C4×D3 | — | 9 | 2 | no | — |
| 394 | C4×D3 | — | 9 | 2 | no | — |
| 395 | C4×D3 | — | 18 | 2 | no | — |
| 396 | C4×D3 | D3×A4 | 1 | 1 | yes | characteristic |
| 397 | Dic6 | — | 3 | 1 | no | — |
| 398 | Dic6 | — | 3 | 2 | no | — |
| 399 | Dic6 | — | 3 | 1 | no | — |
| 400 | Dic6 | — | 9 | 2 | no | — |
| 401 | Dic6 | — | 9 | 4 | no | — |
| 402 | Dic6 | — | 9 | 4 | no | — |
| 403 | Dic6 | — | 9 | 1 | no | — |
| 404 | Dic6 | — | 9 | 1 | no | — |
| 405 | Dic6 | — | 9 | 2 | no | — |
| 406 | Dic6 | — | 9 | 2 | no | — |
| 407 | Dic6 | — | 9 | 2 | no | — |
| 408 | Dic6 | — | 18 | 1 | no | — |
| 409 | Dic6 | — | 18 | 2 | no | — |
| 410 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 411 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 412 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 413 | SL(2,𝔽3) | C2×D32 | 1 | 1 | yes | characteristic |
| 414 | C2×C32 | — | 4 | 1 | no | — |
| 415 | C2×C32 | — | 4 | 1 | no | — |
| 416 | C2×C32 | — | 6 | 1 | no | — |
| 417 | C2×C32 | — | 8 | 1 | no | — |
| 418 | C2×C32 | — | 8 | 1 | no | — |
| 419 | C2×C32 | — | 8 | 1 | no | — |
| 420 | C2×C32 | — | 12 | 2 | no | — |
| 421 | C2×C32 | — | 16 | 1 | no | — |
| 422 | C2×C32 | — | 24 | 2 | no | — |
| 423 | C2×C32 | C23×A4 | 1 | 1 | yes | characteristic |
| 424 | C3⋊D3 | — | 2 | 2 | no | — |
| 425 | C3×D3 | — | 2 | 2 | no | — |
| 426 | C3×D3 | — | 4 | 2 | no | — |
| 427 | C3×D3 | — | 6 | 1 | no | — |
| 428 | C3×D3 | — | 8 | 2 | no | — |
| 429 | C3×D3 | — | 8 | 2 | no | — |
| 430 | C3×D3 | — | 16 | 2 | no | — |
| 431 | C3×D3 | — | 24 | 2 | no | — |
| 432 | C3×D3 | — | 24 | 2 | no | — |
| 433 | C3×D3 | — | 48 | 2 | no | — |
| 434 | C3×D3 | 2−1+4⋊C3 | 1 | 2 | yes | complemented |
| 435 | #41 | — | 3 | 1 | no | — |
| 436 | #41 | — | 9 | 1 | no | — |
| 437 | #41 | — | 9 | 1 | no | — |
| 438 | #41 | — | 9 | 2 | no | — |
| 439 | #41 | — | 9 | 1 | no | — |
| 440 | #41 | — | 27 | 1 | no | — |
| 441 | #41 | — | 27 | 2 | no | — |
| 442 | #41 | — | 27 | 1 | no | — |
| 443 | #41 | — | 27 | 2 | no | — |
| 444 | #41 | — | 27 | 1 | no | — |
| 445 | #41 | — | 27 | 4 | no | — |
| 446 | #41 | — | 27 | 2 | no | — |
| 447 | #41 | — | 27 | 2 | no | — |
| 448 | #41 | — | 27 | 4 | no | — |
| 449 | #41 | — | 27 | 2 | no | — |
| 450 | #41 | C3×D32 | 1 | 1 | yes | characteristic, complemented |
| 451 | C2×Q8 | — | 3 | 1 | no | — |
| 452 | C2×Q8 | — | 3 | 2 | no | — |
| 453 | C2×Q8 | — | 9 | 1 | no | — |
| 454 | C2×Q8 | — | 9 | 2 | no | — |
| 455 | C2×Q8 | — | 27 | 1 | no | — |
| 456 | C2×Q8 | — | 27 | 1 | no | — |
| 457 | C2×Q8 | — | 27 | 2 | no | — |
| 458 | C2×Q8 | — | 27 | 2 | no | — |
| 459 | C2×Q8 | — | 27 | 2 | no | — |
| 460 | C2×Q8 | — | 27 | 2 | no | — |
| 461 | C2×Q8 | — | 27 | 2 | no | — |
| 462 | C2×Q8 | — | 27 | 2 | no | — |
| 463 | C2×D4 | — | 9 | 1 | no | — |
| 464 | C2×D4 | — | 9 | 1 | no | — |
| 465 | C2×D4 | — | 27 | 2 | no | — |
| 466 | C22×C4 | — | 9 | 1 | no | — |
| 467 | C22×C4 | — | 27 | 2 | no | — |
| 468 | C22×C4 | — | 27 | 2 | no | — |
| 469 | C22×C3 | — | 3 | 1 | no | — |
| 470 | C22×C3 | — | 3 | 1 | no | — |
| 471 | C22×C3 | — | 3 | 1 | no | — |
| 472 | C22×C3 | — | 3 | 2 | no | — |
| 473 | C22×C3 | — | 6 | 1 | no | — |
| 474 | C22×C3 | — | 9 | 1 | no | — |
| 475 | C22×C3 | — | 12 | 1 | no | — |
| 476 | C22×C3 | — | 12 | 2 | no | — |
| 477 | C22×C3 | — | 18 | 2 | no | — |
| 478 | C22×C3 | — | 24 | 1 | no | — |
| 479 | C22×C3 | — | 24 | 2 | no | — |
| 480 | C22×C3 | — | 36 | 2 | no | — |
| 481 | C22×C3 | — | 72 | 4 | no | — |
| 482 | C2×D3 | — | 3 | 1 | no | — |
| 483 | C2×D3 | — | 3 | 2 | no | — |
| 484 | C2×D3 | — | 3 | 2 | no | — |
| 485 | C2×D3 | — | 6 | 2 | no | — |
| 486 | C2×D3 | — | 6 | 4 | no | — |
| 487 | C2×D3 | — | 6 | 4 | no | — |
| 488 | C2×D3 | — | 6 | 2 | no | — |
| 489 | C2×D3 | C2×D3×A4 | 1 | 2 | yes | — |
| 490 | C2×D3 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 491 | C3×C4 | — | 2 | 1 | no | — |
| 492 | C3×C4 | — | 3 | 1 | no | — |
| 493 | C3×C4 | — | 3 | 1 | no | — |
| 494 | C3×C4 | — | 3 | 1 | no | — |
| 495 | C3×C4 | — | 4 | 1 | no | — |
| 496 | C3×C4 | — | 6 | 1 | no | — |
| 497 | C3×C4 | — | 8 | 1 | no | — |
| 498 | C3×C4 | — | 8 | 1 | no | — |
| 499 | C3×C4 | — | 9 | 1 | no | — |
| 500 | C3×C4 | — | 9 | 2 | no | — |
| 501 | C3×C4 | — | 12 | 1 | no | — |
| 502 | C3×C4 | — | 16 | 1 | no | — |
| 503 | C3×C4 | — | 24 | 1 | no | — |
| 504 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 505 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 506 | Dic3 | — | 3 | 2 | no | — |
| 507 | Dic3 | — | 3 | 1 | no | — |
| 508 | Dic3 | — | 9 | 2 | no | — |
| 509 | Dic3 | — | 9 | 2 | no | — |
| 510 | Dic3 | — | 18 | 2 | no | — |
| 511 | Dic3 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 512 | C32 | — | 4 | 1 | no | — |
| 513 | C32 | — | 4 | 1 | no | — |
| 514 | C32 | — | 8 | 1 | no | — |
| 515 | C32 | — | 8 | 1 | no | — |
| 516 | C32 | — | 8 | 1 | no | — |
| 517 | C32 | — | 16 | 1 | no | — |
| 518 | C32 | C2×2−1+4⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 519 | C23 | — | 9 | 1 | no | — |
| 520 | C23 | — | 9 | 2 | no | — |
| 521 | Q8 | — | 9 | 2 | no | — |
| 522 | Q8 | — | 9 | 1 | no | — |
| 523 | Q8 | — | 9 | 1 | no | — |
| 524 | Q8 | — | 27 | 2 | no | — |
| 525 | Q8 | — | 27 | 2 | no | — |
| 526 | Q8 | — | 27 | 4 | no | — |
| 527 | Q8 | — | 27 | 1 | no | — |
| 528 | Q8 | C2×C3×D32 | 1 | 1 | yes | characteristic |
| 529 | D4 | — | 3 | 1 | no | — |
| 530 | D4 | — | 3 | 1 | no | — |
| 531 | D4 | — | 9 | 2 | no | — |
| 532 | D4 | — | 9 | 1 | no | — |
| 533 | D4 | — | 9 | 2 | no | — |
| 534 | D4 | — | 9 | 2 | no | — |
| 535 | D4 | — | 9 | 1 | no | — |
| 536 | D4 | — | 27 | 2 | no | — |
| 537 | D4 | — | 27 | 2 | no | — |
| 538 | D4 | — | 27 | 2 | no | — |
| 539 | C2×C4 | — | 3 | 1 | no | — |
| 540 | C2×C4 | — | 3 | 1 | no | — |
| 541 | C2×C4 | — | 9 | 1 | no | — |
| 542 | C2×C4 | — | 9 | 1 | no | — |
| 543 | C2×C4 | — | 9 | 1 | no | — |
| 544 | C2×C4 | — | 9 | 2 | no | — |
| 545 | C2×C4 | — | 9 | 2 | no | — |
| 546 | C2×C4 | — | 9 | 1 | no | — |
| 547 | C2×C4 | — | 27 | 2 | no | — |
| 548 | C2×C4 | — | 27 | 2 | no | — |
| 549 | C2×C4 | — | 27 | 2 | no | — |
| 550 | C2×C4 | — | 27 | 2 | no | — |
| 551 | C2×C4 | — | 27 | 4 | no | — |
| 552 | C2×C4 | — | 27 | 2 | no | — |
| 553 | C2×C3 | — | 2 | 1 | no | — |
| 554 | C2×C3 | — | 3 | 2 | no | — |
| 555 | C2×C3 | — | 4 | 1 | no | — |
| 556 | C2×C3 | — | 6 | 2 | no | — |
| 557 | C2×C3 | — | 6 | 1 | no | — |
| 558 | C2×C3 | — | 6 | 1 | no | — |
| 559 | C2×C3 | — | 8 | 1 | no | — |
| 560 | C2×C3 | — | 8 | 1 | no | — |
| 561 | C2×C3 | — | 12 | 2 | no | — |
| 562 | C2×C3 | — | 12 | 1 | no | — |
| 563 | C2×C3 | — | 16 | 1 | no | — |
| 564 | C2×C3 | — | 18 | 1 | no | — |
| 565 | C2×C3 | — | 24 | 2 | no | — |
| 566 | C2×C3 | — | 24 | 2 | no | — |
| 567 | C2×C3 | — | 48 | 2 | no | — |
| 568 | C2×C3 | — | 72 | 2 | no | — |
| 569 | C2×C3 | C22×D3×A4 | 1 | 1 | yes | characteristic |
| 570 | C2×C3 | C22×D3×A4 | 1 | 1 | yes | characteristic |
| 571 | D3 | — | 2 | 2 | no | — |
| 572 | D3 | — | 6 | 1 | no | — |
| 573 | D3 | — | 6 | 2 | no | — |
| 574 | D3 | — | 6 | 2 | no | — |
| 575 | D3 | — | 12 | 2 | no | — |
| 576 | D3 | #1010⋊C3 | 1 | 2 | yes | direct product factor |
| 577 | C22 | — | 3 | 1 | no | — |
| 578 | C22 | — | 3 | 1 | no | — |
| 579 | C22 | — | 3 | 2 | no | — |
| 580 | C22 | — | 9 | 1 | no | — |
| 581 | C22 | — | 9 | 2 | no | — |
| 582 | C22 | — | 18 | 4 | no | — |
| 583 | C22 | — | 18 | 2 | no | — |
| 584 | C4 | — | 3 | 1 | no | — |
| 585 | C4 | — | 3 | 1 | no | — |
| 586 | C4 | — | 9 | 1 | no | — |
| 587 | C4 | — | 9 | 2 | no | — |
| 588 | C4 | — | 27 | 2 | no | — |
| 589 | C4 | D32×A4 | 1 | 1 | yes | characteristic |
| 590 | C3 | — | 2 | 1 | no | — |
| 591 | C3 | — | 4 | 1 | no | — |
| 592 | C3 | — | 8 | 1 | no | — |
| 593 | C3 | — | 8 | 1 | no | — |
| 594 | C3 | — | 16 | 1 | no | — |
| 595 | C3 | C2×#1010⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 596 | C3 | D3×2−1+4⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 597 | C2 | — | 3 | 2 | no | — |
| 598 | C2 | — | 6 | 2 | no | — |
| 599 | C2 | — | 6 | 1 | no | — |
| 600 | C2 | — | 18 | 2 | no | — |
| 601 | C2 | — | 18 | 1 | no | — |
| 602 | C2 | C2×D32×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Fitting | C32×#41 | C22×C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | C4 | D32×A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724493 | C2 | semidirect |
| 2. C3×#1010⋊C3 | C2 | semidirect |
| 3. C3×D3×#41⋊C3 | C2 | semidirect |
| 4. D32×SL(2,𝔽3) | C2 | semidirect |
| 5. #64315⋊C3 | C2 | semidirect |
| 6. D3×#1010 | C3 | semidirect |
| 9. C32×#41⋊C3 | C22 | semidirect |
| 11. C3⋊D3×SL(2,𝔽3) | C22 | semidirect |
| 13. C3×D3×SL(2,𝔽3) | C22 | semidirect |
| 14. C3×C3⋊(#41⋊C3) | C22 | semidirect |
| 15. #64367 | C2×C3 | semidirect |
| 16. C3×#1010 | C2×C3 | semidirect |
| 17. C3×D3×#41 | C2×C3 | semidirect |
| 20. D32×Q8 | C2×C3 | semidirect |
| 21. #64315 | C2×C3 | semidirect |
| 33. #1010⋊C3 | D3 | direct (N × H) |
| 35. D3×#41⋊C3 | D3 | semidirect |
| 48. C32×#41 | C22×C3 | semidirect |
| 49. Q8×C3⋊D3 | C22×C3 | semidirect |
| 53. C3×#239 | C22×C3 | semidirect |
| 55. C3×D3×Q8 | C22×C3 | semidirect |
| 70. C3×#41⋊C3 | C2×D3 | semidirect |
| 71. C3×#41⋊C3 | C2×D3 | semidirect |
| 99. D3×SL(2,𝔽3) | C2×D3 | semidirect |
| 101. C3⋊(#41⋊C3) | C2×D3 | semidirect |
| 119. #1010 | C3×D3 | semidirect |
| 122. D3×#41 | C3×D3 | semidirect |
| 202. C3×#41 | C2×C3×D3 | semidirect |
| 203. C3×#41 | C2×C3×D3 | semidirect |
| 214. #239 | C2×C3×D3 | semidirect |
| 226. D3×Q8 | C2×C3×D3 | semidirect |
| 275. #41⋊C3 | D32 | semidirect |
| 434. C3×D3 | 2−1+4⋊C3 | semidirect |
| 450. #41 | C3×D32 | semidirect |
| 518. C32 | C2×2−1+4⋊C3 | semidirect |
| 576. D3 | #1010⋊C3 | direct (N × H) |
| 595. C3 | C2×#1010⋊C3 | semidirect |
| 596. C3 | D3×2−1+4⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724493 | 2 |
| C3×#1010⋊C3 | 2 |
| C3×D3×#41⋊C3 | 2 |
| D32×SL(2,𝔽3) | 2 |
| #64315⋊C3 | 2 |
| D3×#1010 | 3 |
| D3×2−1+4⋊C3 | 3 |
| C2×#1010⋊C3 | 3 |
| C3×D3×D12 | 4 |