Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1022 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 79 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 | C2×C3⋊D3×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724468 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×#983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3⋊D3×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11724182 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11724181 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C3⋊D3×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D3×#983 | — | 3 | 4 | no | maximal |
| 9 | C3⋊D3×#137 | — | 3 | 1 | no | maximal |
| 10 | C2×C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3⋊D3×#67 | — | 4 | 1 | no | maximal |
| 12 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 13 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3⋊(C3⋊GL(2,𝔽3)) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C3⋊(C3⋊#226) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C32×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C32×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #41×C3⋊D3 | — | 3 | 1 | no | — |
| 19 | C2×Q8×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C2×D3×SL(2,𝔽3) | — | 3 | 4 | no | — |
| 21 | #64264 | — | 3 | 4 | no | — |
| 22 | #64264 | — | 6 | 4 | no | — |
| 23 | C3×#983 | D3 | 1 | 4 | yes | complemented |
| 24 | D3×GL(2,𝔽3) | — | 3 | 4 | no | — |
| 25 | #64202 | — | 3 | 4 | no | — |
| 26 | #64201 | — | 3 | 4 | no | — |
| 27 | D3×#226 | — | 3 | 4 | no | — |
| 28 | C32×#137 | — | 3 | 1 | no | — |
| 29 | #64160 | — | 3 | 1 | no | — |
| 30 | #64151 | — | 3 | 1 | no | — |
| 31 | #64127 | — | 3 | 1 | no | — |
| 32 | #64127 | — | 3 | 1 | no | — |
| 33 | Q16×C3⋊D3 | — | 3 | 1 | no | — |
| 34 | Q16×C3⋊D3 | — | 3 | 1 | no | — |
| 35 | #64124 | — | 3 | 1 | no | — |
| 36 | #64124 | — | 3 | 1 | no | — |
| 37 | SD16×C3⋊D3 | — | 3 | 1 | no | — |
| 38 | SD16×C3⋊D3 | — | 3 | 1 | no | — |
| 39 | #64118 | — | 3 | 1 | no | — |
| 40 | M16×C3⋊D3 | — | 3 | 1 | no | — |
| 41 | C22×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 42 | C2×D3×C3⋊D3 | — | 4 | 1 | no | — |
| 43 | #6312 | — | 4 | 1 | no | — |
| 44 | C32×#67 | — | 4 | 1 | no | — |
| 45 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 46 | #6292 | — | 4 | 1 | no | — |
| 47 | #6290 | — | 4 | 1 | no | — |
| 48 | Dic3×C3⋊D3 | — | 4 | 1 | no | — |
| 49 | C2×#983 | — | 9 | 1 | no | — |
| 50 | D3×#137 | — | 9 | 4 | no | — |
| 51 | C32×#41 | — | 3 | 1 | no | — |
| 52 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 53 | #3832 | — | 3 | 1 | no | — |
| 54 | Q8×C3⋊D3 | — | 3 | 1 | no | — |
| 55 | Q8×C3⋊D3 | — | 3 | 1 | no | — |
| 56 | Q8×C3⋊D3 | — | 3 | 1 | no | — |
| 57 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 59 | #3830 | — | 3 | 1 | no | — |
| 60 | #3830 | — | 6 | 1 | no | — |
| 61 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 62 | D4×C3⋊D3 | — | 6 | 1 | no | — |
| 63 | #3828 | — | 3 | 1 | no | — |
| 64 | #3828 | — | 6 | 1 | no | — |
| 65 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 66 | C2×C4×C3⋊D3 | — | 6 | 1 | no | — |
| 67 | C2×#634 | — | 3 | 1 | no | — |
| 68 | C2×C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 69 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 4 | yes | complemented |
| 70 | D3×#67 | — | 12 | 4 | no | — |
| 71 | D3×SL(2,𝔽3) | — | 3 | 4 | no | — |
| 72 | D3×SL(2,𝔽3) | — | 3 | 4 | no | — |
| 73 | C3⋊GL(2,𝔽3) | — | 3 | 4 | no | — |
| 74 | C3⋊GL(2,𝔽3) | — | 6 | 4 | no | — |
| 75 | C3⋊#226 | — | 3 | 4 | no | — |
| 76 | C3⋊#226 | — | 6 | 4 | no | — |
| 77 | C3×GL(2,𝔽3) | C2×D3 | 1 | 4 | yes | complemented |
| 78 | C3×#226 | C2×D3 | 1 | 4 | yes | complemented |
| 79 | C32×Q16 | — | 3 | 1 | no | — |
| 80 | C32×Q16 | — | 3 | 1 | no | — |
| 81 | C32×SD16 | — | 3 | 1 | no | — |
| 82 | C32×SD16 | — | 3 | 1 | no | — |
| 83 | C32×M16 | — | 3 | 1 | no | — |
| 84 | #3756 | — | 3 | 1 | no | — |
| 85 | #3756 | — | 3 | 1 | no | — |
| 86 | #3756 | — | 3 | 1 | no | — |
| 87 | #3756 | — | 3 | 1 | no | — |
| 88 | #3755 | — | 3 | 1 | no | — |
| 89 | #3755 | — | 3 | 1 | no | — |
| 90 | #3754 | — | 3 | 1 | no | — |
| 91 | #3754 | — | 3 | 1 | no | — |
| 92 | #3748 | — | 3 | 1 | no | — |
| 93 | #3746 | — | 3 | 1 | no | — |
| 94 | #3746 | — | 3 | 1 | no | — |
| 95 | #3744 | — | 3 | 1 | no | — |
| 96 | #3744 | — | 3 | 1 | no | — |
| 97 | #3743 | — | 3 | 1 | no | — |
| 98 | #3743 | — | 3 | 1 | no | — |
| 99 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 100 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 101 | C22×C33 | — | 4 | 1 | no | — |
| 102 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 103 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 104 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 105 | C2×C3×C3⋊D3 | — | 8 | 2 | no | — |
| 106 | C2×C32×D3 | — | 4 | 1 | no | — |
| 107 | D3×C3⋊D3 | — | 8 | 2 | no | — |
| 108 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 109 | C32×Dic3 | — | 4 | 1 | no | — |
| 110 | D3×#41 | — | 9 | 4 | no | — |
| 111 | C2×D3×Q8 | — | 3 | 4 | no | — |
| 112 | C22×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 113 | #983 | — | 9 | 1 | no | — |
| 114 | #983 | — | 9 | 1 | no | — |
| 115 | #983 | — | 9 | 1 | no | — |
| 116 | #983 | — | 18 | 4 | no | — |
| 117 | #983 | C3⋊D3 | 1 | 1 | yes | characteristic, direct product factor |
| 118 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 119 | C2×#226 | — | 9 | 1 | no | — |
| 120 | C3×#137 | — | 3 | 4 | no | — |
| 121 | #951 | — | 9 | 4 | no | — |
| 122 | #942 | — | 9 | 4 | no | — |
| 123 | #918 | — | 9 | 4 | no | — |
| 124 | #918 | — | 9 | 4 | no | — |
| 125 | D3×Q16 | — | 9 | 4 | no | — |
| 126 | D3×Q16 | — | 9 | 4 | no | — |
| 127 | #915 | — | 9 | 4 | no | — |
| 128 | #915 | — | 9 | 4 | no | — |
| 129 | D3×SD16 | — | 9 | 4 | no | — |
| 130 | D3×SD16 | — | 9 | 4 | no | — |
| 131 | #909 | — | 9 | 4 | no | — |
| 132 | D3×M16 | — | 9 | 4 | no | — |
| 133 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 134 | C22×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 135 | C22×C3×D3 | — | 12 | 4 | no | — |
| 136 | C2×D32 | — | 12 | 4 | no | — |
| 137 | C32×Q8 | — | 3 | 1 | no | — |
| 138 | C32×Q8 | — | 3 | 1 | no | — |
| 139 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 140 | C32×D4 | — | 3 | 1 | no | — |
| 141 | C32×D4 | — | 6 | 1 | no | — |
| 142 | C2×C32×C4 | — | 3 | 1 | no | — |
| 143 | C2×C32×C4 | — | 6 | 1 | no | — |
| 144 | #638 | — | 6 | 1 | no | — |
| 145 | #638 | — | 6 | 1 | no | — |
| 146 | #638 | — | 6 | 1 | no | — |
| 147 | #638 | — | 12 | 4 | no | — |
| 148 | #638 | — | 24 | 4 | no | — |
| 149 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 150 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 151 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 152 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 153 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 154 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 155 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 156 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 157 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 158 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 159 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 160 | #634 | — | 3 | 1 | no | — |
| 161 | #634 | — | 3 | 1 | no | — |
| 162 | #634 | — | 3 | 1 | no | — |
| 163 | #634 | — | 6 | 1 | no | — |
| 164 | #634 | — | 6 | 1 | no | — |
| 165 | C3×#67 | — | 4 | 4 | no | — |
| 166 | C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 167 | C3×SL(2,𝔽3) | C22×D3 | 1 | 4 | yes | — |
| 168 | #626 | — | 12 | 4 | no | — |
| 169 | #625 | — | 12 | 4 | no | — |
| 170 | D3×Dic3 | — | 12 | 4 | no | — |
| 171 | C32×C8 | — | 3 | 1 | no | — |
| 172 | C32×C8 | — | 3 | 1 | no | — |
| 173 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 174 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 175 | C2×#137 | — | 27 | 1 | no | Sylow |
| 176 | C2×C33 | — | 4 | 1 | no | — |
| 177 | C2×C33 | — | 8 | 1 | no | — |
| 178 | C3⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 179 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 180 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 181 | C32×D3 | — | 8 | 1 | no | — |
| 182 | C3×#41 | — | 3 | 4 | no | — |
| 183 | C2×C3×Q8 | D32 | 1 | 4 | yes | complemented |
| 184 | C2×#67 | — | 36 | 1 | no | — |
| 185 | #239 | — | 9 | 4 | no | — |
| 186 | D3×Q8 | — | 3 | 4 | no | — |
| 187 | D3×Q8 | — | 3 | 4 | no | — |
| 188 | D3×Q8 | — | 9 | 4 | no | — |
| 189 | D3×Q8 | — | 9 | 4 | no | — |
| 190 | D3×Q8 | — | 9 | 4 | no | — |
| 191 | #237 | — | 9 | 4 | no | — |
| 192 | #237 | — | 18 | 4 | no | — |
| 193 | D3×D4 | — | 9 | 4 | no | — |
| 194 | D3×D4 | — | 18 | 4 | no | — |
| 195 | #235 | — | 9 | 4 | no | — |
| 196 | #235 | — | 18 | 4 | no | — |
| 197 | C2×C4×D3 | — | 9 | 4 | no | — |
| 198 | C2×C4×D3 | — | 18 | 4 | no | — |
| 199 | C2×Dic6 | — | 9 | 4 | no | — |
| 200 | C2×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 201 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 202 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 203 | C2×SL(2,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 204 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 205 | GL(2,𝔽3) | — | 18 | 4 | no | — |
| 206 | GL(2,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 207 | #226 | — | 9 | 1 | no | — |
| 208 | #226 | — | 18 | 4 | no | — |
| 209 | #226 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 210 | C3×Q16 | — | 3 | 4 | no | — |
| 211 | C3×Q16 | — | 3 | 4 | no | — |
| 212 | C3×SD16 | — | 3 | 4 | no | — |
| 213 | C3×SD16 | — | 3 | 4 | no | — |
| 214 | C3×M16 | — | 3 | 4 | no | — |
| 215 | #216 | — | 9 | 4 | no | — |
| 216 | #216 | — | 9 | 4 | no | — |
| 217 | #216 | — | 9 | 4 | no | — |
| 218 | #216 | — | 9 | 4 | no | — |
| 219 | #215 | — | 9 | 4 | no | — |
| 220 | #215 | — | 9 | 4 | no | — |
| 221 | #214 | — | 9 | 4 | no | — |
| 222 | #214 | — | 9 | 4 | no | — |
| 223 | #208 | — | 9 | 4 | no | — |
| 224 | Dic12 | — | 9 | 4 | no | — |
| 225 | Dic12 | — | 9 | 4 | no | — |
| 226 | #204 | — | 9 | 4 | no | — |
| 227 | #204 | — | 9 | 4 | no | — |
| 228 | ΓL(1,𝔽25) | — | 9 | 4 | no | — |
| 229 | ΓL(1,𝔽25) | — | 9 | 4 | no | — |
| 230 | D3×C8 | — | 9 | 4 | no | — |
| 231 | D3×C8 | — | 9 | 4 | no | — |
| 232 | C22×C32 | — | 4 | 4 | no | — |
| 233 | C22×C32 | — | 6 | 1 | no | — |
| 234 | C22×C32 | — | 8 | 4 | no | — |
| 235 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 236 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 237 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 238 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 239 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 240 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 241 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 242 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 243 | C2×C3×D3 | — | 4 | 4 | no | — |
| 244 | C2×C3×D3 | — | 12 | 4 | no | — |
| 245 | C2×C3×D3 | — | 12 | 4 | no | — |
| 246 | C2×C3×D3 | — | 24 | 8 | no | — |
| 247 | D32 | — | 24 | 8 | no | — |
| 248 | C32×C4 | — | 3 | 1 | no | — |
| 249 | C32×C4 | — | 3 | 1 | no | — |
| 250 | C32×C4 | — | 6 | 1 | no | — |
| 251 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 252 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 253 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 254 | C3⋊Dic3 | — | 12 | 4 | no | — |
| 255 | C3⋊Dic3 | — | 24 | 4 | no | — |
| 256 | C3×Dic3 | — | 4 | 4 | no | — |
| 257 | C2×#41 | — | 27 | 1 | no | — |
| 258 | C22×Q8 | — | 9 | 1 | no | — |
| 259 | #137 | — | 3 | 1 | no | — |
| 260 | #137 | — | 27 | 1 | no | — |
| 261 | #137 | — | 27 | 1 | no | — |
| 262 | #137 | — | 27 | 1 | no | — |
| 263 | #137 | — | 27 | 1 | no | — |
| 264 | #137 | — | 27 | 1 | no | — |
| 265 | #137 | — | 27 | 1 | no | — |
| 266 | #137 | — | 27 | 1 | no | — |
| 267 | C2×Q16 | — | 27 | 1 | no | — |
| 268 | C2×Q16 | — | 27 | 1 | no | — |
| 269 | C2×SD16 | — | 27 | 1 | no | — |
| 270 | C2×SD16 | — | 27 | 1 | no | — |
| 271 | C2×M16 | — | 27 | 1 | no | — |
| 272 | C33 | — | 4 | 1 | no | Sylow |
| 273 | C23×C3 | — | 36 | 1 | no | — |
| 274 | C22×D3 | — | 3 | 4 | no | — |
| 275 | C22×D3 | — | 18 | 4 | no | — |
| 276 | C22×D3 | — | 36 | 1 | no | — |
| 277 | C3×Q8 | — | 3 | 4 | no | — |
| 278 | C3×Q8 | — | 3 | 4 | no | — |
| 279 | C3×Q8 | C2×D32 | 1 | 4 | yes | — |
| 280 | C3×D4 | — | 3 | 4 | no | — |
| 281 | C3×D4 | — | 6 | 4 | no | — |
| 282 | C2×C3×C4 | — | 3 | 4 | no | — |
| 283 | C2×C3×C4 | — | 6 | 4 | no | — |
| 284 | #67 | — | 4 | 1 | no | — |
| 285 | #67 | — | 18 | 4 | no | — |
| 286 | #67 | — | 18 | 4 | no | — |
| 287 | #67 | — | 18 | 4 | no | — |
| 288 | #67 | — | 36 | 1 | no | — |
| 289 | #67 | — | 36 | 1 | no | — |
| 290 | #67 | — | 36 | 1 | no | — |
| 291 | #67 | — | 72 | 4 | no | — |
| 292 | C2×Dic3 | — | 9 | 4 | no | — |
| 293 | C2×Dic3 | — | 18 | 4 | no | — |
| 294 | C2×Dic3 | — | 36 | 1 | no | — |
| 295 | D12 | — | 9 | 4 | no | — |
| 296 | D12 | — | 18 | 4 | no | — |
| 297 | C4×D3 | — | 9 | 4 | no | — |
| 298 | C4×D3 | — | 9 | 4 | no | — |
| 299 | C4×D3 | — | 9 | 4 | no | — |
| 300 | C4×D3 | — | 9 | 4 | no | — |
| 301 | C4×D3 | — | 18 | 4 | no | — |
| 302 | C4×D3 | — | 18 | 4 | no | — |
| 303 | C4×D3 | — | 18 | 4 | no | — |
| 304 | Dic6 | — | 9 | 4 | no | — |
| 305 | Dic6 | — | 9 | 4 | no | — |
| 306 | Dic6 | — | 9 | 4 | no | — |
| 307 | Dic6 | — | 18 | 4 | no | — |
| 308 | Dic6 | — | 18 | 4 | no | — |
| 309 | SL(2,𝔽3) | — | 2 | 4 | no | — |
| 310 | SL(2,𝔽3) | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 311 | C3×C8 | — | 3 | 4 | no | — |
| 312 | C3×C8 | — | 3 | 4 | no | — |
| 313 | C3⋊C8 | — | 9 | 4 | no | — |
| 314 | C3⋊C8 | — | 9 | 4 | no | — |
| 315 | C2×C32 | — | 2 | 1 | no | — |
| 316 | C2×C32 | — | 4 | 4 | no | — |
| 317 | C2×C32 | — | 8 | 4 | no | — |
| 318 | C2×C32 | — | 8 | 4 | no | — |
| 319 | C2×C32 | — | 12 | 1 | no | — |
| 320 | C2×C32 | — | 16 | 4 | no | — |
| 321 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 322 | C3⋊D3 | — | 2 | 1 | no | — |
| 323 | C3⋊D3 | — | 12 | 1 | no | — |
| 324 | C3⋊D3 | — | 24 | 4 | no | — |
| 325 | C3⋊D3 | — | 48 | 4 | no | — |
| 326 | C3⋊D3 | #983 | 1 | 2 | yes | direct product factor |
| 327 | C3×D3 | — | 8 | 4 | no | — |
| 328 | C3×D3 | — | 12 | 8 | no | — |
| 329 | C3×D3 | — | 24 | 4 | no | — |
| 330 | #41 | — | 3 | 1 | no | — |
| 331 | #41 | — | 27 | 1 | no | — |
| 332 | #41 | — | 27 | 1 | no | — |
| 333 | #41 | — | 27 | 1 | no | — |
| 334 | #41 | — | 54 | 1 | no | — |
| 335 | #41 | — | 54 | 1 | no | — |
| 336 | C2×Q8 | — | 9 | 1 | no | — |
| 337 | C2×Q8 | — | 9 | 1 | no | — |
| 338 | C2×Q8 | — | 27 | 1 | no | — |
| 339 | C2×Q8 | — | 27 | 1 | no | — |
| 340 | C2×Q8 | — | 27 | 1 | no | — |
| 341 | C2×Q8 | — | 27 | 1 | no | — |
| 342 | C2×Q8 | D3×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 343 | C2×D4 | — | 27 | 1 | no | — |
| 344 | C2×D4 | — | 54 | 1 | no | — |
| 345 | C22×C4 | — | 27 | 1 | no | — |
| 346 | C22×C4 | — | 54 | 1 | no | — |
| 347 | Q16 | — | 3 | 1 | no | — |
| 348 | Q16 | — | 3 | 1 | no | — |
| 349 | Q16 | — | 27 | 1 | no | — |
| 350 | Q16 | — | 27 | 1 | no | — |
| 351 | Q16 | — | 27 | 1 | no | — |
| 352 | Q16 | — | 27 | 1 | no | — |
| 353 | Q16 | — | 27 | 1 | no | — |
| 354 | Q16 | — | 27 | 1 | no | — |
| 355 | SD16 | — | 3 | 1 | no | — |
| 356 | SD16 | — | 3 | 1 | no | — |
| 357 | SD16 | — | 27 | 1 | no | — |
| 358 | SD16 | — | 27 | 1 | no | — |
| 359 | SD16 | — | 27 | 1 | no | — |
| 360 | SD16 | — | 27 | 1 | no | — |
| 361 | SD16 | — | 27 | 1 | no | — |
| 362 | SD16 | — | 27 | 1 | no | — |
| 363 | M16 | — | 3 | 1 | no | — |
| 364 | M16 | — | 27 | 1 | no | — |
| 365 | M16 | — | 27 | 1 | no | — |
| 366 | M16 | — | 27 | 1 | no | — |
| 367 | C2×C8 | — | 27 | 1 | no | — |
| 368 | C2×C8 | — | 27 | 1 | no | — |
| 369 | C22×C3 | — | 4 | 1 | no | — |
| 370 | C22×C3 | — | 6 | 4 | no | — |
| 371 | C22×C3 | — | 8 | 4 | no | — |
| 372 | C22×C3 | — | 36 | 1 | no | — |
| 373 | C22×C3 | — | 36 | 1 | no | — |
| 374 | C22×C3 | — | 72 | 2 | no | — |
| 375 | C22×C3 | D3×S4 | 1 | 4 | yes | — |
| 376 | C2×D3 | — | 3 | 4 | no | — |
| 377 | C2×D3 | — | 3 | 4 | no | — |
| 378 | C2×D3 | — | 4 | 1 | no | — |
| 379 | C2×D3 | — | 6 | 8 | no | — |
| 380 | C2×D3 | — | 18 | 4 | no | — |
| 381 | C2×D3 | — | 36 | 1 | no | — |
| 382 | C2×D3 | — | 36 | 8 | no | — |
| 383 | C2×D3 | — | 72 | 4 | no | — |
| 384 | C2×D3 | — | 72 | 2 | no | — |
| 385 | C3×C4 | — | 3 | 4 | no | — |
| 386 | C3×C4 | — | 3 | 4 | no | — |
| 387 | C3×C4 | — | 6 | 4 | no | — |
| 388 | Dic3 | — | 4 | 1 | no | — |
| 389 | Dic3 | — | 9 | 4 | no | — |
| 390 | Dic3 | — | 9 | 4 | no | — |
| 391 | Dic3 | — | 18 | 4 | no | — |
| 392 | Dic3 | — | 36 | 1 | no | — |
| 393 | Dic3 | — | 72 | 4 | no | — |
| 394 | C32 | — | 4 | 4 | no | — |
| 395 | C32 | — | 8 | 4 | no | — |
| 396 | C32 | C2×#983 | 1 | 1 | yes | characteristic, complemented |
| 397 | C23 | — | 9 | 1 | no | — |
| 398 | C23 | — | 54 | 1 | no | — |
| 399 | Q8 | — | 3 | 1 | no | — |
| 400 | Q8 | — | 3 | 1 | no | — |
| 401 | Q8 | — | 27 | 1 | no | — |
| 402 | Q8 | — | 27 | 1 | no | — |
| 403 | Q8 | — | 27 | 1 | no | — |
| 404 | Q8 | — | 54 | 1 | no | — |
| 405 | Q8 | — | 54 | 1 | no | — |
| 406 | Q8 | C2×D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 407 | D4 | — | 3 | 1 | no | — |
| 408 | D4 | — | 6 | 1 | no | — |
| 409 | D4 | — | 27 | 1 | no | — |
| 410 | D4 | — | 54 | 1 | no | — |
| 411 | D4 | — | 54 | 1 | no | — |
| 412 | D4 | — | 54 | 1 | no | — |
| 413 | D4 | — | 54 | 1 | no | — |
| 414 | C2×C4 | — | 3 | 1 | no | — |
| 415 | C2×C4 | — | 6 | 1 | no | — |
| 416 | C2×C4 | — | 27 | 1 | no | — |
| 417 | C2×C4 | — | 27 | 1 | no | — |
| 418 | C2×C4 | — | 27 | 1 | no | — |
| 419 | C2×C4 | — | 27 | 1 | no | — |
| 420 | C2×C4 | — | 27 | 1 | no | — |
| 421 | C2×C4 | — | 54 | 1 | no | — |
| 422 | C2×C4 | — | 54 | 1 | no | — |
| 423 | C2×C4 | — | 54 | 1 | no | — |
| 424 | C2×C4 | — | 54 | 1 | no | — |
| 425 | C8 | — | 3 | 1 | no | — |
| 426 | C8 | — | 3 | 1 | no | — |
| 427 | C8 | — | 27 | 1 | no | — |
| 428 | C8 | — | 27 | 1 | no | — |
| 429 | C2×C3 | — | 2 | 4 | no | — |
| 430 | C2×C3 | — | 4 | 1 | no | — |
| 431 | C2×C3 | — | 8 | 1 | no | — |
| 432 | C2×C3 | — | 8 | 4 | no | — |
| 433 | C2×C3 | — | 12 | 4 | no | — |
| 434 | C2×C3 | — | 16 | 4 | no | — |
| 435 | C2×C3 | — | 36 | 2 | no | — |
| 436 | C2×C3 | — | 72 | 1 | no | — |
| 437 | C2×C3 | C2×D3×S4 | 1 | 4 | yes | — |
| 438 | D3 | — | 3 | 8 | no | — |
| 439 | D3 | — | 6 | 4 | no | — |
| 440 | D3 | — | 8 | 1 | no | — |
| 441 | D3 | — | 36 | 4 | no | — |
| 442 | D3 | — | 72 | 1 | no | — |
| 443 | D3 | — | 144 | 4 | no | — |
| 444 | C22 | — | 6 | 1 | no | — |
| 445 | C22 | — | 9 | 1 | no | — |
| 446 | C22 | — | 9 | 1 | no | — |
| 447 | C22 | — | 18 | 2 | no | — |
| 448 | C22 | — | 54 | 1 | no | — |
| 449 | C22 | — | 108 | 2 | no | — |
| 450 | C22 | C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| 451 | C4 | — | 3 | 1 | no | — |
| 452 | C4 | — | 3 | 1 | no | — |
| 453 | C4 | — | 6 | 1 | no | — |
| 454 | C4 | — | 27 | 1 | no | — |
| 455 | C4 | — | 27 | 1 | no | — |
| 456 | C4 | — | 54 | 1 | no | — |
| 457 | C3 | — | 4 | 1 | no | — |
| 458 | C3 | — | 8 | 4 | no | — |
| 459 | C3 | D3×#983 | 1 | 4 | yes | complemented |
| 460 | C2 | — | 2 | 1 | no | — |
| 461 | C2 | — | 9 | 2 | no | — |
| 462 | C2 | — | 12 | 1 | no | — |
| 463 | C2 | — | 18 | 1 | no | — |
| 464 | C2 | — | 108 | 1 | no | — |
| 465 | C2 | C2×C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D3×S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C2×C32×Q8 | C2×D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C22 | C3⋊D3×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3⋊D3×SL(2,𝔽3) | C2 | semidirect |
| 2. #11724468 | C2 | semidirect |
| 3. C32×#983 | C2 | semidirect |
| 4. C3⋊D3×GL(2,𝔽3) | C2 | semidirect |
| 5. #11724182 | C2 | semidirect |
| 6. #11724181 | C2 | semidirect |
| 7. C3⋊D3×#226 | C2 | semidirect |
| 10. C2×C32×SL(2,𝔽3) | C22 | semidirect |
| 13. C3⋊D3×SL(2,𝔽3) | C22 | semidirect |
| 14. C3⋊(C3⋊GL(2,𝔽3)) | C22 | semidirect |
| 15. C3⋊(C3⋊#226) | C22 | semidirect |
| 16. C32×GL(2,𝔽3) | C22 | semidirect |
| 17. C32×#226 | C22 | semidirect |
| 19. C2×Q8×C3⋊D3 | D3 | semidirect |
| 23. C3×#983 | D3 | semidirect |
| 52. C2×C32×Q8 | C2×D3 | semidirect |
| 58. Q8×C3⋊D3 | C2×D3 | semidirect |
| 69. C2×C3×SL(2,𝔽3) | C2×D3 | semidirect |
| 77. C3×GL(2,𝔽3) | C2×D3 | semidirect |
| 78. C3×#226 | C2×D3 | semidirect |
| 117. #983 | C3⋊D3 | direct (N × H) |
| 183. C2×C3×Q8 | D32 | semidirect |
| 203. C2×SL(2,𝔽3) | C2×C3⋊D3 | semidirect |
| 206. GL(2,𝔽3) | C2×C3⋊D3 | semidirect |
| 209. #226 | C2×C3⋊D3 | semidirect |
| 326. C3⋊D3 | #983 | direct (N × H) |
| 342. C2×Q8 | D3×C3⋊D3 | semidirect |
| 396. C32 | C2×#983 | semidirect |
| 459. C3 | D3×#983 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊D3×SL(2,𝔽3) | 2 |
| #11724468 | 2 |
| C32×#983 | 2 |
| C3⋊D3×GL(2,𝔽3) | 2 |
| #11724182 | 2 |
| #11724181 | 2 |
| C3⋊D3×#226 | 2 |
| D3×#983 | 3 |
| C3⋊D3×#137 | 3 |
| C3⋊D3×#67 | 4 |