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 | C3⋊D3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724470 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3⋊D3×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3⋊D3×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11724178 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11724177 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D3×#985 | — | 3 | 4 | no | maximal |
| 9 | C3⋊D3×#135 | — | 3 | 1 | no | maximal |
| 10 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4×D3×C3⋊D3 | — | 4 | 1 | no | maximal |
| 12 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 13 | C3⋊(C3⋊(#41⋊C3)) | 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 | #41×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | D3×#41⋊C3 | — | 3 | 4 | no | — |
| 21 | #64266 | — | 3 | 4 | no | — |
| 22 | #64266 | — | 6 | 4 | no | — |
| 23 | C3×#985 | D3 | 1 | 4 | yes | complemented |
| 24 | D3×GL(2,𝔽3) | — | 3 | 4 | no | — |
| 25 | D3×#226 | — | 3 | 4 | no | — |
| 26 | #64198 | — | 3 | 4 | no | — |
| 27 | #64197 | — | 3 | 4 | no | — |
| 28 | C32×#135 | — | 3 | 1 | no | — |
| 29 | #64159 | — | 3 | 1 | no | — |
| 30 | #64159 | — | 3 | 1 | no | — |
| 31 | #64128 | — | 3 | 1 | no | — |
| 32 | Q16×C3⋊D3 | — | 3 | 1 | no | — |
| 33 | #64125 | — | 3 | 1 | no | — |
| 34 | #64125 | — | 3 | 1 | no | — |
| 35 | SD16×C3⋊D3 | — | 3 | 1 | no | — |
| 36 | SD16×C3⋊D3 | — | 3 | 1 | no | — |
| 37 | #64121 | — | 3 | 1 | no | — |
| 38 | D8×C3⋊D3 | — | 3 | 1 | no | — |
| 39 | #64113 | — | 3 | 1 | no | — |
| 40 | C2×C8×C3⋊D3 | — | 3 | 1 | no | — |
| 41 | C2×D3×C3⋊D3 | — | 4 | 1 | no | — |
| 42 | C4×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 43 | C3×C4×C3⋊D3 | — | 4 | 1 | no | — |
| 44 | C32×C4×D3 | — | 4 | 1 | no | — |
| 45 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 46 | #6289 | — | 4 | 1 | no | — |
| 47 | Dic3×C3⋊D3 | — | 4 | 1 | no | — |
| 48 | D3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 49 | C2×#985 | — | 9 | 1 | no | — |
| 50 | D3×#135 | — | 9 | 4 | no | — |
| 51 | C32×#41 | — | 3 | 1 | no | — |
| 52 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 53 | #3832 | — | 3 | 1 | no | — |
| 54 | #3832 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 55 | Q8×C3⋊D3 | — | 3 | 1 | no | — |
| 56 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | #3830 | — | 3 | 1 | no | — |
| 58 | #3830 | — | 3 | 1 | no | — |
| 59 | #3830 | — | 6 | 1 | no | — |
| 60 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 61 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 62 | D4×C3⋊D3 | — | 6 | 1 | no | — |
| 63 | #3828 | — | 3 | 1 | no | — |
| 64 | #3828 | — | 3 | 1 | no | — |
| 65 | #3828 | — | 6 | 1 | no | — |
| 66 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 67 | C2×C4×C3⋊D3 | — | 6 | 1 | no | — |
| 68 | C3×#41⋊C3 | — | 2 | 4 | no | — |
| 69 | C3×#41⋊C3 | C2×D3 | 1 | 4 | yes | complemented |
| 70 | C4×D32 | — | 12 | 4 | no | — |
| 71 | D3×SL(2,𝔽3) | — | 3 | 4 | no | — |
| 72 | C3⋊(#41⋊C3) | — | 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×SD16 | — | 3 | 1 | no | — |
| 81 | C32×SD16 | — | 3 | 1 | no | — |
| 82 | C32×D8 | — | 3 | 1 | no | — |
| 83 | C2×C32×C8 | — | 3 | 1 | no | — |
| 84 | #3756 | — | 3 | 1 | no | — |
| 85 | #3756 | — | 3 | 1 | no | — |
| 86 | #3755 | — | 3 | 1 | no | — |
| 87 | #3755 | — | 3 | 1 | no | — |
| 88 | #3754 | — | 3 | 1 | no | — |
| 89 | #3754 | — | 3 | 1 | no | — |
| 90 | #3753 | — | 3 | 1 | no | — |
| 91 | #3753 | — | 3 | 1 | no | — |
| 92 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 93 | #3746 | — | 3 | 1 | no | — |
| 94 | #3745 | — | 3 | 1 | no | — |
| 95 | #3744 | — | 3 | 1 | no | — |
| 96 | #3744 | — | 3 | 1 | no | — |
| 97 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 98 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 99 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 100 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 101 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 102 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 103 | C2×C32×D3 | — | 4 | 1 | no | — |
| 104 | D3×C3⋊D3 | — | 4 | 4 | no | — |
| 105 | C33×C4 | — | 4 | 1 | no | — |
| 106 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 107 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 108 | C32×Dic3 | — | 4 | 1 | no | — |
| 109 | D3×#41 | — | 3 | 4 | no | — |
| 110 | D3×#41 | — | 9 | 4 | no | — |
| 111 | C2×#41⋊C3 | — | 9 | 1 | no | — |
| 112 | #985 | — | 9 | 1 | no | — |
| 113 | #985 | — | 9 | 1 | no | — |
| 114 | #985 | — | 9 | 1 | no | — |
| 115 | #985 | — | 18 | 4 | no | — |
| 116 | #985 | C3⋊D3 | 1 | 1 | yes | characteristic, direct product factor |
| 117 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 118 | C2×#226 | — | 9 | 1 | no | — |
| 119 | C3×#135 | — | 3 | 4 | no | — |
| 120 | #950 | — | 9 | 4 | no | — |
| 121 | #950 | — | 9 | 4 | no | — |
| 122 | #919 | — | 9 | 4 | no | — |
| 123 | D3×Q16 | — | 9 | 4 | no | — |
| 124 | #916 | — | 9 | 4 | no | — |
| 125 | #916 | — | 9 | 4 | no | — |
| 126 | D3×SD16 | — | 9 | 4 | no | — |
| 127 | D3×SD16 | — | 9 | 4 | no | — |
| 128 | #912 | — | 9 | 4 | no | — |
| 129 | D3×D8 | — | 9 | 4 | no | — |
| 130 | #904 | — | 9 | 4 | no | — |
| 131 | C2×D3×C8 | — | 9 | 4 | no | — |
| 132 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 133 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 134 | C2×D32 | — | 12 | 4 | no | — |
| 135 | C32×Q8 | — | 3 | 1 | no | — |
| 136 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 137 | C32×D4 | — | 3 | 1 | no | — |
| 138 | C32×D4 | — | 3 | 1 | no | — |
| 139 | C32×D4 | — | 6 | 1 | no | — |
| 140 | C2×C32×C4 | — | 3 | 1 | no | — |
| 141 | C2×C32×C4 | — | 6 | 1 | no | — |
| 142 | #638 | — | 6 | 1 | no | — |
| 143 | #638 | — | 6 | 1 | no | — |
| 144 | #638 | — | 6 | 1 | no | — |
| 145 | #638 | — | 6 | 1 | no | — |
| 146 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 147 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 148 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 149 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 150 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 151 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 152 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 153 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 154 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 155 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 156 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 157 | C4×C3⋊D3 | — | 12 | 4 | no | — |
| 158 | C4×C3⋊D3 | — | 24 | 4 | no | — |
| 159 | C4×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 160 | #634 | — | 3 | 1 | no | — |
| 161 | #634 | — | 3 | 1 | no | — |
| 162 | #634 | — | 6 | 1 | no | — |
| 163 | C3×C4×D3 | — | 4 | 4 | no | — |
| 164 | C3×C4×D3 | — | 12 | 4 | no | — |
| 165 | C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 166 | C3×SL(2,𝔽3) | C22×D3 | 1 | 4 | yes | — |
| 167 | #624 | — | 12 | 4 | no | — |
| 168 | D3×Dic3 | — | 12 | 4 | no | — |
| 169 | D3×Dic3 | — | 12 | 4 | no | — |
| 170 | C32×C8 | — | 3 | 1 | no | — |
| 171 | C32×C8 | — | 3 | 1 | no | — |
| 172 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 173 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 174 | C2×#135 | — | 27 | 1 | no | Sylow |
| 175 | C2×C33 | — | 4 | 1 | no | — |
| 176 | C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 177 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 178 | C32×D3 | — | 4 | 2 | no | — |
| 179 | C3×#41 | — | 3 | 4 | no | — |
| 180 | C3×#41 | D32 | 1 | 4 | yes | complemented |
| 181 | #239 | — | 3 | 4 | no | — |
| 182 | #239 | — | 9 | 4 | no | — |
| 183 | D3×Q8 | — | 3 | 4 | no | — |
| 184 | D3×Q8 | — | 9 | 4 | no | — |
| 185 | #237 | — | 9 | 4 | no | — |
| 186 | #237 | — | 9 | 4 | no | — |
| 187 | #237 | — | 18 | 4 | no | — |
| 188 | D3×D4 | — | 9 | 4 | no | — |
| 189 | D3×D4 | — | 9 | 4 | no | — |
| 190 | D3×D4 | — | 18 | 4 | no | — |
| 191 | #235 | — | 9 | 4 | no | — |
| 192 | #235 | — | 9 | 4 | no | — |
| 193 | #235 | — | 18 | 4 | no | — |
| 194 | C2×C4×D3 | — | 9 | 4 | no | — |
| 195 | C2×C4×D3 | — | 18 | 4 | no | — |
| 196 | C2×C4×D3 | — | 36 | 1 | no | — |
| 197 | #41⋊C3 | — | 2 | 4 | no | — |
| 198 | #41⋊C3 | — | 9 | 1 | no | — |
| 199 | #41⋊C3 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 200 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 201 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 202 | GL(2,𝔽3) | — | 18 | 4 | no | — |
| 203 | GL(2,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 204 | #226 | — | 9 | 1 | no | — |
| 205 | #226 | — | 18 | 4 | no | — |
| 206 | #226 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 207 | C3×Q16 | — | 3 | 4 | no | — |
| 208 | C3×SD16 | — | 3 | 4 | no | — |
| 209 | C3×SD16 | — | 3 | 4 | no | — |
| 210 | C3×D8 | — | 3 | 4 | no | — |
| 211 | C2×C3×C8 | — | 3 | 4 | no | — |
| 212 | #216 | — | 9 | 4 | no | — |
| 213 | #216 | — | 9 | 4 | no | — |
| 214 | #215 | — | 9 | 4 | no | — |
| 215 | #215 | — | 9 | 4 | no | — |
| 216 | #214 | — | 9 | 4 | no | — |
| 217 | #214 | — | 9 | 4 | no | — |
| 218 | #213 | — | 9 | 4 | no | — |
| 219 | #213 | — | 9 | 4 | no | — |
| 220 | C2×C3⋊C8 | — | 9 | 4 | no | — |
| 221 | Dic12 | — | 9 | 4 | no | — |
| 222 | D24 | — | 9 | 4 | no | — |
| 223 | #204 | — | 9 | 4 | no | — |
| 224 | #204 | — | 9 | 4 | no | — |
| 225 | D3×C8 | — | 9 | 4 | no | — |
| 226 | D3×C8 | — | 9 | 4 | no | — |
| 227 | D3×C8 | — | 9 | 4 | no | — |
| 228 | D3×C8 | — | 9 | 4 | no | — |
| 229 | C22×C32 | — | 3 | 1 | no | — |
| 230 | C22×C32 | — | 6 | 1 | no | — |
| 231 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 232 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 233 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 234 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 235 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 236 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 237 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 238 | C2×C3×D3 | — | 4 | 4 | no | — |
| 239 | C2×C3×D3 | — | 12 | 4 | no | — |
| 240 | D32 | — | 12 | 16 | no | — |
| 241 | C32×C4 | — | 3 | 1 | no | — |
| 242 | C32×C4 | — | 4 | 4 | no | — |
| 243 | C32×C4 | — | 6 | 1 | no | — |
| 244 | C32×C4 | — | 8 | 4 | no | — |
| 245 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 246 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 247 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 248 | C3⋊Dic3 | — | 12 | 4 | no | — |
| 249 | C3⋊Dic3 | — | 24 | 4 | no | — |
| 250 | C3⋊Dic3 | C2×S4 | 1 | 1 | yes | characteristic |
| 251 | C3×Dic3 | — | 4 | 4 | no | — |
| 252 | C3×Dic3 | — | 12 | 4 | no | — |
| 253 | C2×#41 | — | 9 | 1 | no | — |
| 254 | C2×#41 | — | 27 | 1 | no | — |
| 255 | #135 | — | 3 | 1 | no | — |
| 256 | #135 | — | 27 | 1 | no | — |
| 257 | #135 | — | 27 | 1 | no | — |
| 258 | #135 | — | 27 | 1 | no | — |
| 259 | #135 | — | 27 | 1 | no | — |
| 260 | #135 | — | 27 | 1 | no | — |
| 261 | #135 | — | 27 | 1 | no | — |
| 262 | #135 | — | 27 | 1 | no | — |
| 263 | C2×Q16 | — | 27 | 1 | no | — |
| 264 | C2×SD16 | — | 27 | 1 | no | — |
| 265 | C2×SD16 | — | 27 | 1 | no | — |
| 266 | C2×D8 | — | 27 | 1 | no | — |
| 267 | C22×C8 | — | 27 | 1 | no | — |
| 268 | C33 | — | 4 | 1 | no | Sylow |
| 269 | C22×D3 | — | 9 | 4 | no | — |
| 270 | C22×D3 | — | 18 | 4 | no | — |
| 271 | C22×D3 | — | 36 | 1 | no | — |
| 272 | C3×Q8 | — | 3 | 4 | no | — |
| 273 | C3×Q8 | C2×D32 | 1 | 4 | yes | — |
| 274 | C3×D4 | — | 3 | 4 | no | — |
| 275 | C3×D4 | — | 3 | 4 | no | — |
| 276 | C3×D4 | — | 6 | 4 | no | — |
| 277 | C2×C3×C4 | — | 3 | 4 | no | — |
| 278 | C2×C3×C4 | — | 6 | 4 | no | — |
| 279 | C2×C3×C4 | — | 36 | 1 | no | — |
| 280 | #67 | — | 18 | 4 | no | — |
| 281 | #67 | — | 18 | 4 | no | — |
| 282 | #67 | — | 18 | 4 | no | — |
| 283 | #67 | — | 18 | 4 | no | — |
| 284 | C2×Dic3 | — | 9 | 4 | no | — |
| 285 | C2×Dic3 | — | 18 | 4 | no | — |
| 286 | C2×Dic3 | — | 36 | 1 | no | — |
| 287 | D12 | — | 9 | 4 | no | — |
| 288 | D12 | — | 9 | 4 | no | — |
| 289 | D12 | — | 18 | 4 | no | — |
| 290 | C4×D3 | — | 3 | 4 | no | — |
| 291 | C4×D3 | — | 4 | 1 | no | — |
| 292 | C4×D3 | — | 9 | 4 | no | — |
| 293 | C4×D3 | — | 9 | 4 | no | — |
| 294 | C4×D3 | — | 9 | 4 | no | — |
| 295 | C4×D3 | — | 18 | 4 | no | — |
| 296 | C4×D3 | — | 18 | 4 | no | — |
| 297 | C4×D3 | — | 18 | 4 | no | — |
| 298 | C4×D3 | — | 36 | 1 | no | — |
| 299 | C4×D3 | — | 36 | 1 | no | — |
| 300 | C4×D3 | — | 36 | 1 | no | — |
| 301 | C4×D3 | — | 72 | 4 | no | — |
| 302 | Dic6 | — | 9 | 4 | no | — |
| 303 | Dic6 | — | 9 | 4 | no | — |
| 304 | Dic6 | — | 18 | 4 | no | — |
| 305 | SL(2,𝔽3) | — | 2 | 4 | no | — |
| 306 | SL(2,𝔽3) | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 307 | C3×C8 | — | 3 | 4 | no | — |
| 308 | C3×C8 | — | 3 | 4 | no | — |
| 309 | C3⋊C8 | — | 9 | 4 | no | — |
| 310 | C3⋊C8 | — | 9 | 4 | no | — |
| 311 | C2×C32 | — | 4 | 4 | no | — |
| 312 | C2×C32 | — | 6 | 1 | no | — |
| 313 | C2×C32 | — | 8 | 4 | no | — |
| 314 | C2×C32 | — | 12 | 1 | no | — |
| 315 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 316 | C3⋊D3 | — | 6 | 1 | no | — |
| 317 | C3⋊D3 | — | 12 | 1 | no | — |
| 318 | C3⋊D3 | — | 12 | 8 | no | — |
| 319 | C3⋊D3 | — | 24 | 8 | no | — |
| 320 | C3⋊D3 | #985 | 1 | 2 | yes | direct product factor |
| 321 | C3×D3 | — | 4 | 8 | no | — |
| 322 | C3×D3 | — | 12 | 8 | no | — |
| 323 | #41 | — | 3 | 1 | no | — |
| 324 | #41 | — | 9 | 1 | no | — |
| 325 | #41 | — | 27 | 1 | no | — |
| 326 | #41 | — | 27 | 1 | no | — |
| 327 | #41 | — | 27 | 1 | no | — |
| 328 | #41 | — | 27 | 1 | no | — |
| 329 | #41 | — | 27 | 1 | no | — |
| 330 | #41 | — | 54 | 1 | no | — |
| 331 | #41 | — | 54 | 1 | no | — |
| 332 | #41 | D3×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 333 | C2×Q8 | — | 9 | 1 | no | — |
| 334 | C2×Q8 | — | 27 | 1 | no | — |
| 335 | C2×D4 | — | 27 | 1 | no | — |
| 336 | C2×D4 | — | 27 | 1 | no | — |
| 337 | C2×D4 | — | 54 | 1 | no | — |
| 338 | C22×C4 | — | 27 | 1 | no | — |
| 339 | C22×C4 | — | 54 | 1 | no | — |
| 340 | Q16 | — | 3 | 1 | no | — |
| 341 | Q16 | — | 27 | 1 | no | — |
| 342 | Q16 | — | 27 | 1 | no | — |
| 343 | Q16 | — | 27 | 1 | no | — |
| 344 | SD16 | — | 3 | 1 | no | — |
| 345 | SD16 | — | 3 | 1 | no | — |
| 346 | SD16 | — | 27 | 1 | no | — |
| 347 | SD16 | — | 27 | 1 | no | — |
| 348 | SD16 | — | 27 | 1 | no | — |
| 349 | SD16 | — | 27 | 1 | no | — |
| 350 | SD16 | — | 27 | 1 | no | — |
| 351 | SD16 | — | 27 | 1 | no | — |
| 352 | D8 | — | 3 | 1 | no | — |
| 353 | D8 | — | 27 | 1 | no | — |
| 354 | D8 | — | 27 | 1 | no | — |
| 355 | D8 | — | 27 | 1 | no | — |
| 356 | C2×C8 | — | 3 | 1 | no | — |
| 357 | C2×C8 | — | 27 | 1 | no | — |
| 358 | C2×C8 | — | 27 | 1 | no | — |
| 359 | C2×C8 | — | 27 | 1 | no | — |
| 360 | C2×C8 | — | 27 | 1 | no | — |
| 361 | C2×C8 | — | 27 | 1 | no | — |
| 362 | C22×C3 | — | 3 | 4 | no | — |
| 363 | C22×C3 | — | 6 | 4 | no | — |
| 364 | C22×C3 | — | 36 | 1 | no | — |
| 365 | C2×D3 | — | 3 | 4 | no | — |
| 366 | C2×D3 | — | 4 | 1 | no | — |
| 367 | C2×D3 | — | 9 | 4 | no | — |
| 368 | C2×D3 | — | 18 | 8 | no | — |
| 369 | C2×D3 | — | 18 | 4 | no | — |
| 370 | C2×D3 | — | 36 | 1 | no | — |
| 371 | C2×D3 | — | 36 | 8 | no | — |
| 372 | C2×D3 | — | 36 | 4 | no | — |
| 373 | C2×D3 | — | 72 | 4 | no | — |
| 374 | C3×C4 | — | 3 | 4 | no | — |
| 375 | C3×C4 | — | 4 | 1 | no | — |
| 376 | C3×C4 | — | 6 | 4 | no | — |
| 377 | C3×C4 | — | 8 | 4 | no | — |
| 378 | C3×C4 | — | 36 | 1 | no | — |
| 379 | C3×C4 | D3×S4 | 1 | 4 | yes | — |
| 380 | Dic3 | — | 3 | 4 | no | — |
| 381 | Dic3 | — | 4 | 1 | no | — |
| 382 | Dic3 | — | 9 | 4 | no | — |
| 383 | Dic3 | — | 18 | 4 | no | — |
| 384 | Dic3 | — | 36 | 1 | no | — |
| 385 | Dic3 | — | 72 | 4 | no | — |
| 386 | C32 | — | 4 | 4 | no | — |
| 387 | C32 | — | 8 | 4 | no | — |
| 388 | C32 | C2×#985 | 1 | 1 | yes | characteristic, complemented |
| 389 | C23 | — | 27 | 1 | no | — |
| 390 | C23 | — | 54 | 1 | no | — |
| 391 | Q8 | — | 3 | 1 | no | — |
| 392 | Q8 | — | 27 | 1 | no | — |
| 393 | Q8 | — | 27 | 1 | no | — |
| 394 | Q8 | — | 54 | 1 | no | — |
| 395 | Q8 | C2×D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 396 | D4 | — | 3 | 1 | no | — |
| 397 | D4 | — | 3 | 1 | no | — |
| 398 | D4 | — | 6 | 1 | no | — |
| 399 | D4 | — | 27 | 1 | no | — |
| 400 | D4 | — | 27 | 1 | no | — |
| 401 | D4 | — | 54 | 1 | no | — |
| 402 | D4 | — | 54 | 1 | no | — |
| 403 | D4 | — | 54 | 1 | no | — |
| 404 | D4 | — | 54 | 1 | no | — |
| 405 | D4 | — | 54 | 1 | no | — |
| 406 | C2×C4 | — | 3 | 1 | no | — |
| 407 | C2×C4 | — | 6 | 1 | no | — |
| 408 | C2×C4 | — | 9 | 1 | no | — |
| 409 | C2×C4 | — | 27 | 1 | no | — |
| 410 | C2×C4 | — | 27 | 1 | no | — |
| 411 | C2×C4 | — | 27 | 1 | no | — |
| 412 | C2×C4 | — | 27 | 1 | no | — |
| 413 | C2×C4 | — | 54 | 1 | no | — |
| 414 | C2×C4 | — | 54 | 1 | no | — |
| 415 | C2×C4 | — | 54 | 1 | no | — |
| 416 | C2×C4 | — | 54 | 1 | no | — |
| 417 | C8 | — | 3 | 1 | no | — |
| 418 | C8 | — | 3 | 1 | no | — |
| 419 | C8 | — | 27 | 1 | no | — |
| 420 | C8 | — | 27 | 1 | no | — |
| 421 | C2×C3 | — | 4 | 1 | no | — |
| 422 | C2×C3 | — | 6 | 4 | no | — |
| 423 | C2×C3 | — | 8 | 4 | no | — |
| 424 | C2×C3 | — | 12 | 4 | no | — |
| 425 | C2×C3 | — | 36 | 2 | no | — |
| 426 | C2×C3 | C2×D3×S4 | 1 | 4 | yes | — |
| 427 | D3 | — | 3 | 8 | no | — |
| 428 | D3 | — | 4 | 2 | no | — |
| 429 | D3 | — | 18 | 4 | no | — |
| 430 | D3 | — | 36 | 2 | no | — |
| 431 | D3 | — | 36 | 4 | no | — |
| 432 | D3 | — | 72 | 8 | no | — |
| 433 | C22 | — | 3 | 1 | no | — |
| 434 | C22 | — | 6 | 1 | no | — |
| 435 | C22 | — | 9 | 1 | no | — |
| 436 | C22 | — | 27 | 1 | no | — |
| 437 | C22 | — | 54 | 2 | no | — |
| 438 | C22 | — | 54 | 1 | no | — |
| 439 | C22 | — | 108 | 2 | no | — |
| 440 | C4 | — | 3 | 1 | no | — |
| 441 | C4 | — | 6 | 1 | no | — |
| 442 | C4 | — | 9 | 1 | no | — |
| 443 | C4 | — | 27 | 1 | no | — |
| 444 | C4 | — | 54 | 1 | no | — |
| 445 | C4 | C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| 446 | C3 | — | 4 | 1 | no | — |
| 447 | C3 | — | 8 | 4 | no | — |
| 448 | C3 | D3×#985 | 1 | 4 | yes | complemented |
| 449 | C2 | — | 6 | 1 | no | — |
| 450 | C2 | — | 9 | 2 | no | — |
| 451 | C2 | — | 12 | 1 | no | — |
| 452 | C2 | — | 54 | 1 | no | — |
| 453 | C2 | — | 108 | 1 | no | — |
| 454 | C2 | C2×C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊D3×S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Frattini | C2 | C2×C3⋊D3×S4 |
| Fitting | C32×#41 | C2×D3 |
| socle | C2×C32 | C22×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊D3×#41⋊C3 | C2 | semidirect |
| 2. #11724470 | C2 | semidirect |
| 3. C32×#985 | C2 | semidirect |
| 4. C3⋊D3×GL(2,𝔽3) | C2 | semidirect |
| 5. C3⋊D3×#226 | C2 | semidirect |
| 6. #11724178 | C2 | semidirect |
| 7. #11724177 | C2 | semidirect |
| 10. C32×#41⋊C3 | C22 | semidirect |
| 13. C3⋊(C3⋊(#41⋊C3)) | 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. #41×C3⋊D3 | D3 | semidirect |
| 23. C3×#985 | D3 | semidirect |
| 52. C32×#41 | C2×D3 | semidirect |
| 54. #3832 | C2×D3 | semidirect |
| 69. C3×#41⋊C3 | C2×D3 | semidirect |
| 77. C3×GL(2,𝔽3) | C2×D3 | semidirect |
| 78. C3×#226 | C2×D3 | semidirect |
| 116. #985 | C3⋊D3 | direct (N × H) |
| 180. C3×#41 | D32 | semidirect |
| 199. #41⋊C3 | C2×C3⋊D3 | semidirect |
| 203. GL(2,𝔽3) | C2×C3⋊D3 | semidirect |
| 206. #226 | C2×C3⋊D3 | semidirect |
| 320. C3⋊D3 | #985 | direct (N × H) |
| 332. #41 | D3×C3⋊D3 | semidirect |
| 388. C32 | C2×#985 | semidirect |
| 448. C3 | D3×#985 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×#41⋊C3 | 2 |
| #11724470 | 2 |
| C32×#985 | 2 |
| C3⋊D3×GL(2,𝔽3) | 2 |
| C3⋊D3×#226 | 2 |
| #11724178 | 2 |
| #11724177 | 2 |
| D3×#985 | 3 |
| C3⋊D3×#135 | 3 |
| C4×D3×C3⋊D3 | 4 |