Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1001 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 66 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 | D32×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724211 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D3×C3⋊GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3⋊D3×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×D3×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×D3×GL(2,𝔽3) | — | 3 | 2 | no | maximal |
| 7 | D32×SD16 | — | 3 | 1 | no | maximal |
| 8 | C2×D33 | — | 4 | 1 | no | maximal |
| 9 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×D3×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 11 | C3⋊(C3⋊GL(2,𝔽3)) | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C3×C3⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 13 | C32×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | D32×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | D32×D4 | — | 3 | 1 | no | — |
| 16 | C2×D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 17 | C2×C3⋊GL(2,𝔽3) | — | 3 | 2 | no | — |
| 18 | C2×C3×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 19 | D3×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 20 | D3×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 21 | D3×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 22 | D3×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 23 | D3×GL(2,𝔽3) | — | 6 | 1 | no | — |
| 24 | D3×GL(2,𝔽3) | D3 | 1 | 2 | yes | direct product factor |
| 25 | SD16×C3⋊D3 | — | 3 | 1 | no | — |
| 26 | C3×D3×SD16 | — | 3 | 2 | no | — |
| 27 | #63940 | — | 3 | 1 | no | — |
| 28 | D3×#215 | — | 3 | 2 | no | — |
| 29 | #63930 | — | 3 | 1 | no | — |
| 30 | D3×#214 | — | 3 | 2 | no | — |
| 31 | #63796 | — | 3 | 1 | no | — |
| 32 | D3×#204 | — | 3 | 2 | no | — |
| 33 | D32×C8 | — | 3 | 1 | no | — |
| 34 | C2×#1113 | — | 4 | 1 | no | — |
| 35 | C2×D3×C3⋊D3 | — | 4 | 2 | no | — |
| 36 | C2×D3×C3⋊D3 | — | 4 | 1 | no | — |
| 37 | C2×C3×D32 | — | 4 | 2 | no | — |
| 38 | C2×C3×D32 | — | 4 | 1 | no | — |
| 39 | D33 | — | 4 | 8 | no | — |
| 40 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 41 | C22×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 42 | C2×D3×SD16 | — | 9 | 2 | no | — |
| 43 | C22×D32 | — | 6 | 1 | no | — |
| 44 | C22×D32 | — | 12 | 2 | no | — |
| 45 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 46 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 47 | C3×D3×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 48 | C3×D3×D4 | — | 3 | 2 | no | — |
| 49 | C2×C3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 50 | #3811 | — | 6 | 1 | no | — |
| 51 | D3×#67 | — | 6 | 2 | no | — |
| 52 | #3802 | — | 3 | 1 | no | — |
| 53 | D3×D12 | — | 3 | 2 | no | — |
| 54 | C4×D32 | — | 3 | 1 | no | — |
| 55 | #3797 | — | 3 | 1 | no | — |
| 56 | D3×Dic6 | — | 3 | 2 | no | — |
| 57 | D3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 58 | D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 59 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 60 | D3×SL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 61 | C3⋊GL(2,𝔽3) | — | 3 | 2 | no | — |
| 62 | C3⋊GL(2,𝔽3) | — | 6 | 2 | no | — |
| 63 | C3⋊GL(2,𝔽3) | — | 6 | 1 | no | — |
| 64 | C3⋊GL(2,𝔽3) | — | 12 | 1 | no | — |
| 65 | C3⋊GL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 66 | C3×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 67 | C3×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 68 | C3×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 69 | C3×GL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 70 | C32×SD16 | — | 3 | 1 | no | — |
| 71 | #3755 | — | 3 | 1 | no | — |
| 72 | #3754 | — | 3 | 1 | no | — |
| 73 | #3744 | — | 3 | 1 | no | — |
| 74 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 75 | C3×#215 | — | 3 | 2 | no | — |
| 76 | C3×#214 | — | 3 | 2 | no | — |
| 77 | C3×#204 | — | 3 | 2 | no | — |
| 78 | C3×D3×C8 | — | 3 | 2 | no | — |
| 79 | #3717 | — | 3 | 2 | no | — |
| 80 | #3716 | — | 3 | 2 | no | — |
| 81 | #3715 | — | 3 | 2 | no | — |
| 82 | #3710 | — | 3 | 1 | no | — |
| 83 | D3×C3⋊C8 | — | 3 | 2 | no | — |
| 84 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 85 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 86 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 87 | C2×C32×D3 | — | 4 | 1 | no | — |
| 88 | C2×C32×D3 | — | 4 | 2 | no | — |
| 89 | #1113 | — | 4 | 4 | no | — |
| 90 | D3×C3⋊D3 | — | 4 | 4 | no | — |
| 91 | D3×C3⋊D3 | — | 4 | 8 | no | — |
| 92 | C3×D32 | — | 4 | 8 | no | — |
| 93 | C3×D32 | — | 4 | 4 | no | — |
| 94 | C2×D3×Q8 | — | 3 | 2 | no | — |
| 95 | C2×D3×D4 | — | 9 | 2 | no | — |
| 96 | C22×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 97 | C2×GL(2,𝔽3) | — | 3 | 2 | no | — |
| 98 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 99 | C2×GL(2,𝔽3) | — | 9 | 2 | no | — |
| 100 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 101 | C2×GL(2,𝔽3) | — | 18 | 2 | no | — |
| 102 | C2×C3×SD16 | — | 9 | 2 | no | — |
| 103 | C2×#215 | — | 9 | 2 | no | — |
| 104 | C2×#214 | — | 9 | 2 | no | — |
| 105 | D3×SD16 | — | 3 | 2 | no | — |
| 106 | D3×SD16 | — | 9 | 2 | no | — |
| 107 | D3×SD16 | — | 9 | 2 | no | — |
| 108 | D3×SD16 | — | 9 | 2 | no | — |
| 109 | D3×SD16 | — | 9 | 2 | no | — |
| 110 | D3×SD16 | — | 9 | 2 | no | — |
| 111 | D3×SD16 | — | 9 | 2 | no | — |
| 112 | D3×SD16 | — | 9 | 2 | no | — |
| 113 | D3×SD16 | — | 18 | 1 | no | — |
| 114 | C2×#204 | — | 9 | 2 | no | — |
| 115 | C2×D3×C8 | — | 9 | 2 | no | — |
| 116 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 117 | C22×C3⋊D3 | — | 12 | 2 | no | — |
| 118 | C22×C3×D3 | — | 6 | 2 | no | — |
| 119 | C22×C3×D3 | — | 12 | 2 | no | — |
| 120 | C22×C3×D3 | — | 12 | 2 | no | — |
| 121 | C2×D32 | — | 4 | 2 | no | — |
| 122 | C2×D32 | — | 6 | 1 | no | — |
| 123 | C2×D32 | — | 6 | 2 | no | — |
| 124 | C2×D32 | — | 12 | 2 | no | — |
| 125 | C2×D32 | — | 12 | 2 | no | — |
| 126 | C2×D32 | — | 12 | 2 | no | — |
| 127 | C2×D32 | — | 12 | 16 | no | — |
| 128 | C2×D32 | — | 12 | 4 | no | — |
| 129 | C2×D32 | — | 24 | 2 | no | — |
| 130 | C2×D32 | — | 24 | 1 | no | — |
| 131 | C2×D32 | S4 | 1 | 1 | yes | characteristic |
| 132 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic, complemented |
| 133 | C32×D4 | — | 3 | 1 | no | — |
| 134 | #638 | — | 6 | 1 | no | — |
| 135 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 136 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 137 | #634 | — | 3 | 1 | no | — |
| 138 | C3×#67 | — | 6 | 2 | no | — |
| 139 | C3×D12 | — | 3 | 2 | no | — |
| 140 | C3×C4×D3 | — | 3 | 2 | no | — |
| 141 | C3×Dic6 | — | 3 | 2 | no | — |
| 142 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 143 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 144 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 145 | C3×SL(2,𝔽3) | C22×D3 | 1 | 2 | yes | complemented |
| 146 | #627 | — | 6 | 1 | no | — |
| 147 | #626 | — | 6 | 2 | no | — |
| 148 | #625 | — | 6 | 1 | no | — |
| 149 | #624 | — | 3 | 1 | no | — |
| 150 | D3×Dic3 | — | 3 | 2 | no | — |
| 151 | C32×C8 | — | 3 | 1 | no | — |
| 152 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 153 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 154 | C22×SD16 | — | 27 | 1 | no | Sylow |
| 155 | C2×C33 | — | 4 | 1 | no | — |
| 156 | C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 157 | C3×C3⋊D3 | — | 4 | 4 | no | — |
| 158 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 159 | C32×D3 | — | 4 | 2 | no | — |
| 160 | C32×D3 | — | 4 | 4 | no | — |
| 161 | C23×D3 | — | 18 | 2 | no | — |
| 162 | C23×D3 | — | 36 | 1 | no | — |
| 163 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 164 | C2×C3×D4 | — | 9 | 2 | no | — |
| 165 | C2×#67 | — | 18 | 2 | no | — |
| 166 | D3×Q8 | — | 3 | 2 | no | — |
| 167 | D3×Q8 | — | 6 | 1 | no | — |
| 168 | D3×Q8 | — | 9 | 2 | no | — |
| 169 | D3×Q8 | — | 9 | 2 | no | — |
| 170 | D3×Q8 | D32 | 1 | 2 | yes | complemented |
| 171 | D3×D4 | — | 3 | 2 | no | — |
| 172 | D3×D4 | — | 9 | 2 | no | — |
| 173 | D3×D4 | — | 9 | 2 | no | — |
| 174 | D3×D4 | — | 9 | 2 | no | — |
| 175 | D3×D4 | — | 18 | 1 | no | — |
| 176 | D3×D4 | — | 18 | 2 | no | — |
| 177 | D3×D4 | — | 18 | 2 | no | — |
| 178 | C2×D12 | — | 9 | 2 | no | — |
| 179 | C2×C4×D3 | — | 9 | 2 | no | — |
| 180 | C2×Dic6 | — | 9 | 2 | no | — |
| 181 | C2×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 182 | C2×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 183 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 184 | GL(2,𝔽3) | — | 3 | 2 | no | — |
| 185 | GL(2,𝔽3) | — | 6 | 2 | no | — |
| 186 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 187 | GL(2,𝔽3) | — | 18 | 2 | no | — |
| 188 | GL(2,𝔽3) | — | 36 | 1 | no | — |
| 189 | GL(2,𝔽3) | D32 | 1 | 1 | yes | characteristic, direct product factor |
| 190 | C3×SD16 | — | 3 | 2 | no | — |
| 191 | C3×SD16 | — | 6 | 1 | no | — |
| 192 | C3×SD16 | — | 9 | 2 | no | — |
| 193 | C3×SD16 | — | 9 | 2 | no | — |
| 194 | C3×SD16 | — | 9 | 2 | no | — |
| 195 | C2×C3×C8 | — | 9 | 2 | no | — |
| 196 | #215 | — | 3 | 2 | no | — |
| 197 | #215 | — | 9 | 2 | no | — |
| 198 | #215 | — | 9 | 2 | no | — |
| 199 | #215 | — | 9 | 2 | no | — |
| 200 | #215 | — | 18 | 1 | no | — |
| 201 | #214 | — | 3 | 2 | no | — |
| 202 | #214 | — | 9 | 2 | no | — |
| 203 | #214 | — | 9 | 2 | no | — |
| 204 | #214 | — | 9 | 2 | no | — |
| 205 | #214 | — | 18 | 1 | no | — |
| 206 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 207 | #204 | — | 3 | 2 | no | — |
| 208 | #204 | — | 9 | 2 | no | — |
| 209 | #204 | — | 9 | 2 | no | — |
| 210 | #204 | — | 9 | 2 | no | — |
| 211 | #204 | — | 18 | 1 | no | — |
| 212 | D3×C8 | — | 3 | 2 | no | — |
| 213 | D3×C8 | — | 9 | 2 | no | — |
| 214 | D3×C8 | — | 9 | 2 | no | — |
| 215 | D3×C8 | — | 9 | 2 | no | — |
| 216 | D3×C8 | — | 18 | 1 | no | — |
| 217 | C22×C32 | — | 6 | 1 | no | — |
| 218 | C22×C32 | — | 12 | 2 | no | — |
| 219 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 220 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 221 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 222 | C2×C3⋊D3 | — | 12 | 8 | no | — |
| 223 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 224 | C2×C3⋊D3 | — | 24 | 2 | no | — |
| 225 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 226 | C2×C3⋊D3 | — | 48 | 1 | no | — |
| 227 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 228 | C2×C3×D3 | — | 4 | 2 | no | — |
| 229 | C2×C3×D3 | — | 4 | 2 | no | — |
| 230 | C2×C3×D3 | — | 6 | 2 | no | — |
| 231 | C2×C3×D3 | — | 8 | 1 | no | — |
| 232 | C2×C3×D3 | — | 8 | 2 | no | — |
| 233 | C2×C3×D3 | — | 12 | 8 | no | — |
| 234 | C2×C3×D3 | — | 12 | 2 | no | — |
| 235 | C2×C3×D3 | — | 12 | 2 | no | — |
| 236 | C2×C3×D3 | — | 12 | 8 | no | — |
| 237 | C2×C3×D3 | — | 12 | 4 | no | — |
| 238 | C2×C3×D3 | — | 24 | 1 | no | — |
| 239 | C2×C3×D3 | — | 24 | 2 | no | — |
| 240 | C2×C3×D3 | C2×S4 | 1 | 2 | yes | — |
| 241 | D32 | — | 4 | 8 | no | — |
| 242 | D32 | — | 12 | 8 | no | — |
| 243 | D32 | — | 12 | 8 | no | — |
| 244 | D32 | — | 12 | 8 | no | — |
| 245 | D32 | — | 12 | 2 | no | — |
| 246 | D32 | — | 12 | 4 | no | — |
| 247 | D32 | — | 24 | 4 | no | — |
| 248 | D32 | — | 24 | 8 | no | — |
| 249 | D32 | GL(2,𝔽3) | 1 | 4 | yes | direct product factor |
| 250 | C32×C4 | — | 3 | 1 | no | — |
| 251 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 252 | C3×Dic3 | — | 3 | 2 | no | — |
| 253 | C22×Q8 | — | 9 | 1 | no | — |
| 254 | C22×D4 | — | 27 | 1 | no | — |
| 255 | C2×SD16 | — | 9 | 2 | no | — |
| 256 | C2×SD16 | — | 27 | 2 | no | — |
| 257 | C2×SD16 | — | 27 | 1 | no | — |
| 258 | C2×SD16 | — | 27 | 1 | no | — |
| 259 | C2×SD16 | — | 27 | 1 | no | — |
| 260 | C2×SD16 | — | 27 | 1 | no | — |
| 261 | C2×SD16 | — | 27 | 2 | no | — |
| 262 | C2×SD16 | — | 27 | 2 | no | — |
| 263 | C22×C8 | — | 27 | 1 | no | — |
| 264 | C33 | — | 4 | 1 | no | Sylow |
| 265 | C23×C3 | — | 18 | 2 | no | — |
| 266 | C23×C3 | — | 36 | 1 | no | — |
| 267 | C22×D3 | — | 3 | 2 | no | — |
| 268 | C22×D3 | — | 6 | 2 | no | — |
| 269 | C22×D3 | — | 12 | 2 | no | — |
| 270 | C22×D3 | — | 18 | 2 | no | — |
| 271 | C22×D3 | — | 18 | 2 | no | — |
| 272 | C22×D3 | — | 18 | 2 | no | — |
| 273 | C22×D3 | — | 18 | 2 | no | — |
| 274 | C22×D3 | — | 36 | 1 | no | — |
| 275 | C22×D3 | — | 36 | 1 | no | — |
| 276 | C22×D3 | — | 36 | 1 | no | — |
| 277 | C22×D3 | — | 36 | 2 | no | — |
| 278 | C22×D3 | — | 36 | 8 | no | — |
| 279 | C22×D3 | — | 36 | 8 | no | — |
| 280 | C22×D3 | — | 72 | 2 | no | — |
| 281 | C3×Q8 | — | 2 | 1 | no | — |
| 282 | C3×Q8 | — | 9 | 2 | no | — |
| 283 | C3×Q8 | C2×D32 | 1 | 2 | yes | complemented |
| 284 | C3×D4 | — | 3 | 2 | no | — |
| 285 | C3×D4 | — | 6 | 1 | no | — |
| 286 | C3×D4 | — | 9 | 2 | no | — |
| 287 | C3×D4 | — | 18 | 2 | no | — |
| 288 | C2×C3×C4 | — | 9 | 2 | no | — |
| 289 | #67 | — | 6 | 2 | no | — |
| 290 | #67 | — | 18 | 2 | no | — |
| 291 | #67 | — | 18 | 2 | no | — |
| 292 | #67 | — | 18 | 2 | no | — |
| 293 | #67 | — | 36 | 1 | no | — |
| 294 | C2×Dic3 | — | 9 | 2 | no | — |
| 295 | D12 | — | 3 | 2 | no | — |
| 296 | D12 | — | 9 | 2 | no | — |
| 297 | D12 | — | 18 | 1 | no | — |
| 298 | D12 | — | 18 | 2 | no | — |
| 299 | C4×D3 | — | 3 | 2 | no | — |
| 300 | C4×D3 | — | 9 | 2 | no | — |
| 301 | C4×D3 | — | 9 | 2 | no | — |
| 302 | C4×D3 | — | 9 | 2 | no | — |
| 303 | C4×D3 | — | 18 | 1 | no | — |
| 304 | Dic6 | — | 3 | 2 | no | — |
| 305 | Dic6 | — | 9 | 2 | no | — |
| 306 | Dic6 | — | 18 | 1 | no | — |
| 307 | Dic6 | — | 18 | 2 | no | — |
| 308 | SL(2,𝔽3) | — | 2 | 2 | no | — |
| 309 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 310 | SL(2,𝔽3) | C2×D32 | 1 | 1 | yes | characteristic, complemented |
| 311 | C3×C8 | — | 3 | 2 | no | — |
| 312 | C3×C8 | — | 6 | 1 | no | — |
| 313 | C3×C8 | — | 9 | 2 | no | — |
| 314 | C3⋊C8 | — | 3 | 2 | no | — |
| 315 | C3⋊C8 | — | 9 | 2 | no | — |
| 316 | C3⋊C8 | — | 18 | 1 | no | — |
| 317 | C2×C32 | — | 4 | 2 | no | — |
| 318 | C2×C32 | — | 8 | 1 | no | — |
| 319 | C2×C32 | — | 8 | 2 | no | — |
| 320 | C2×C32 | — | 12 | 4 | no | — |
| 321 | C2×C32 | — | 12 | 1 | no | — |
| 322 | C2×C32 | — | 16 | 1 | no | — |
| 323 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 324 | C3⋊D3 | — | 4 | 4 | no | — |
| 325 | C3⋊D3 | — | 12 | 4 | no | — |
| 326 | C3⋊D3 | — | 12 | 1 | no | — |
| 327 | C3⋊D3 | — | 24 | 2 | no | — |
| 328 | C3⋊D3 | — | 24 | 4 | no | — |
| 329 | C3⋊D3 | — | 48 | 2 | no | — |
| 330 | C3⋊D3 | C2×GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 331 | C3×D3 | — | 4 | 4 | no | — |
| 332 | C3×D3 | — | 4 | 4 | no | — |
| 333 | C3×D3 | — | 8 | 2 | no | — |
| 334 | C3×D3 | — | 8 | 4 | no | — |
| 335 | C3×D3 | — | 12 | 4 | no | — |
| 336 | C3×D3 | — | 12 | 4 | no | — |
| 337 | C3×D3 | — | 12 | 2 | no | — |
| 338 | C3×D3 | — | 24 | 2 | no | — |
| 339 | C3×D3 | — | 24 | 4 | no | — |
| 340 | C3×D3 | C2×GL(2,𝔽3) | 1 | 4 | yes | complemented |
| 341 | C24 | — | 54 | 1 | no | — |
| 342 | C2×Q8 | — | 3 | 2 | no | — |
| 343 | C2×Q8 | — | 9 | 1 | no | — |
| 344 | C2×Q8 | — | 27 | 2 | no | — |
| 345 | C2×Q8 | — | 27 | 1 | no | — |
| 346 | C2×D4 | — | 9 | 2 | no | — |
| 347 | C2×D4 | — | 27 | 2 | no | — |
| 348 | C2×D4 | — | 27 | 1 | no | — |
| 349 | C2×D4 | — | 27 | 1 | no | — |
| 350 | C2×D4 | — | 54 | 1 | no | — |
| 351 | C2×D4 | — | 54 | 2 | no | — |
| 352 | C22×C4 | — | 27 | 1 | no | — |
| 353 | SD16 | — | 3 | 1 | no | — |
| 354 | SD16 | — | 9 | 2 | no | — |
| 355 | SD16 | — | 9 | 2 | no | — |
| 356 | SD16 | — | 9 | 2 | no | — |
| 357 | SD16 | — | 27 | 1 | no | — |
| 358 | SD16 | — | 27 | 2 | no | — |
| 359 | SD16 | — | 27 | 2 | no | — |
| 360 | SD16 | — | 27 | 1 | no | — |
| 361 | SD16 | — | 27 | 2 | no | — |
| 362 | SD16 | — | 27 | 1 | no | — |
| 363 | C2×C8 | — | 9 | 2 | no | — |
| 364 | C2×C8 | — | 27 | 1 | no | — |
| 365 | C2×C8 | — | 27 | 1 | no | — |
| 366 | C2×C8 | — | 27 | 2 | no | — |
| 367 | C22×C3 | — | 3 | 2 | no | — |
| 368 | C22×C3 | — | 6 | 2 | no | — |
| 369 | C22×C3 | — | 12 | 2 | no | — |
| 370 | C22×C3 | — | 12 | 1 | no | — |
| 371 | C22×C3 | — | 18 | 2 | no | — |
| 372 | C22×C3 | — | 24 | 2 | no | — |
| 373 | C22×C3 | — | 36 | 1 | no | — |
| 374 | C22×C3 | — | 36 | 4 | no | — |
| 375 | C22×C3 | — | 36 | 4 | no | — |
| 376 | C2×D3 | — | 3 | 2 | no | — |
| 377 | C2×D3 | — | 3 | 8 | no | — |
| 378 | C2×D3 | — | 4 | 1 | no | — |
| 379 | C2×D3 | — | 6 | 1 | no | — |
| 380 | C2×D3 | — | 6 | 2 | no | — |
| 381 | C2×D3 | — | 12 | 2 | no | — |
| 382 | C2×D3 | — | 12 | 8 | no | — |
| 383 | C2×D3 | — | 12 | 4 | no | — |
| 384 | C2×D3 | — | 18 | 2 | no | — |
| 385 | C2×D3 | — | 24 | 2 | no | — |
| 386 | C2×D3 | — | 36 | 1 | no | — |
| 387 | C2×D3 | — | 36 | 1 | no | — |
| 388 | C2×D3 | — | 36 | 4 | no | — |
| 389 | C2×D3 | — | 36 | 4 | no | — |
| 390 | C2×D3 | — | 36 | 4 | no | — |
| 391 | C2×D3 | — | 36 | 4 | no | — |
| 392 | C2×D3 | — | 36 | 4 | no | — |
| 393 | C2×D3 | — | 36 | 8 | no | — |
| 394 | C2×D3 | — | 36 | 4 | no | — |
| 395 | C2×D3 | — | 72 | 2 | no | — |
| 396 | C2×D3 | — | 72 | 2 | no | — |
| 397 | C2×D3 | — | 72 | 8 | no | — |
| 398 | C2×D3 | — | 144 | 1 | no | — |
| 399 | C2×D3 | D3×S4 | 1 | 2 | yes | — |
| 400 | C3×C4 | — | 3 | 2 | no | — |
| 401 | C3×C4 | — | 6 | 1 | no | — |
| 402 | C3×C4 | — | 9 | 2 | no | — |
| 403 | Dic3 | — | 3 | 2 | no | — |
| 404 | Dic3 | — | 9 | 2 | no | — |
| 405 | Dic3 | — | 18 | 1 | no | — |
| 406 | C32 | — | 4 | 2 | no | — |
| 407 | C32 | — | 8 | 1 | no | — |
| 408 | C32 | — | 8 | 2 | no | — |
| 409 | C32 | — | 16 | 1 | no | — |
| 410 | C32 | C22×GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 411 | C23 | — | 9 | 1 | no | — |
| 412 | C23 | — | 18 | 2 | no | — |
| 413 | C23 | — | 54 | 2 | no | — |
| 414 | C23 | — | 54 | 1 | no | — |
| 415 | C23 | — | 54 | 1 | no | — |
| 416 | C23 | — | 108 | 4 | no | — |
| 417 | Q8 | — | 9 | 2 | no | — |
| 418 | Q8 | — | 27 | 1 | no | — |
| 419 | Q8 | — | 54 | 1 | no | — |
| 420 | Q8 | D33 | 1 | 1 | yes | characteristic, complemented |
| 421 | D4 | — | 3 | 1 | no | — |
| 422 | D4 | — | 9 | 2 | no | — |
| 423 | D4 | — | 18 | 2 | no | — |
| 424 | D4 | — | 27 | 1 | no | — |
| 425 | D4 | — | 54 | 2 | no | — |
| 426 | D4 | — | 54 | 1 | no | — |
| 427 | D4 | — | 54 | 1 | no | — |
| 428 | C2×C4 | — | 9 | 2 | no | — |
| 429 | C2×C4 | — | 27 | 1 | no | — |
| 430 | C2×C4 | — | 27 | 1 | no | — |
| 431 | C2×C4 | — | 27 | 2 | no | — |
| 432 | C8 | — | 3 | 1 | no | — |
| 433 | C8 | — | 9 | 2 | no | — |
| 434 | C8 | — | 27 | 1 | no | — |
| 435 | C2×C3 | — | 2 | 1 | no | — |
| 436 | C2×C3 | — | 3 | 4 | no | — |
| 437 | C2×C3 | — | 4 | 1 | no | — |
| 438 | C2×C3 | — | 8 | 2 | no | — |
| 439 | C2×C3 | — | 12 | 4 | no | — |
| 440 | C2×C3 | — | 12 | 2 | no | — |
| 441 | C2×C3 | — | 16 | 1 | no | — |
| 442 | C2×C3 | — | 24 | 4 | no | — |
| 443 | C2×C3 | — | 24 | 1 | no | — |
| 444 | C2×C3 | — | 36 | 2 | no | — |
| 445 | C2×C3 | — | 36 | 2 | no | — |
| 446 | C2×C3 | C2×D3×S4 | 1 | 2 | yes | — |
| 447 | D3 | — | 3 | 4 | no | — |
| 448 | D3 | — | 4 | 2 | no | — |
| 449 | D3 | — | 6 | 2 | no | — |
| 450 | D3 | — | 12 | 4 | no | — |
| 451 | D3 | — | 12 | 2 | no | — |
| 452 | D3 | — | 24 | 4 | no | — |
| 453 | D3 | — | 36 | 2 | no | — |
| 454 | D3 | — | 36 | 2 | no | — |
| 455 | D3 | — | 72 | 1 | no | — |
| 456 | D3 | — | 72 | 4 | no | — |
| 457 | D3 | — | 144 | 2 | no | — |
| 458 | D3 | D3×GL(2,𝔽3) | 1 | 4 | yes | direct product factor |
| 459 | C22 | — | 3 | 2 | no | — |
| 460 | C22 | — | 6 | 1 | no | — |
| 461 | C22 | — | 9 | 4 | no | — |
| 462 | C22 | — | 9 | 1 | no | — |
| 463 | C22 | — | 18 | 2 | no | — |
| 464 | C22 | — | 36 | 4 | no | — |
| 465 | C22 | — | 54 | 1 | no | — |
| 466 | C22 | — | 108 | 4 | no | — |
| 467 | C22 | — | 108 | 2 | no | — |
| 468 | C22 | — | 108 | 2 | no | — |
| 469 | C4 | — | 3 | 1 | no | — |
| 470 | C4 | — | 9 | 2 | no | — |
| 471 | C4 | — | 27 | 1 | no | — |
| 472 | C3 | — | 2 | 1 | no | — |
| 473 | C3 | — | 4 | 1 | no | — |
| 474 | C3 | — | 8 | 2 | no | — |
| 475 | C3 | — | 16 | 1 | no | — |
| 476 | C3 | C2×D3×GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 477 | C2 | — | 3 | 4 | no | — |
| 478 | C2 | — | 9 | 2 | no | — |
| 479 | C2 | — | 12 | 1 | no | — |
| 480 | C2 | — | 36 | 2 | no | — |
| 481 | C2 | — | 108 | 1 | no | — |
| 482 | C2 | D32×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C32×Q8 | C22×D3 |
| socle | C2×C32 | C22×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D32×SL(2,𝔽3) | C2 | semidirect |
| 2. #11724211 | C2 | semidirect |
| 3. D3×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 4. C3⋊D3×GL(2,𝔽3) | C2 | semidirect |
| 5. C3×D3×GL(2,𝔽3) | C2 | semidirect |
| 9. C3⋊D3×SL(2,𝔽3) | C22 | semidirect |
| 10. C3×D3×SL(2,𝔽3) | C22 | semidirect |
| 11. C3⋊(C3⋊GL(2,𝔽3)) | C22 | semidirect |
| 12. C3×C3⋊GL(2,𝔽3) | C22 | semidirect |
| 13. C32×GL(2,𝔽3) | C22 | semidirect |
| 14. D32×Q8 | D3 | semidirect |
| 24. D3×GL(2,𝔽3) | D3 | direct (N × H) |
| 40. C32×SL(2,𝔽3) | C23 | semidirect |
| 45. Q8×C3⋊D3 | C2×D3 | semidirect |
| 47. C3×D3×Q8 | C2×D3 | semidirect |
| 60. D3×SL(2,𝔽3) | C2×D3 | semidirect |
| 65. C3⋊GL(2,𝔽3) | C2×D3 | semidirect |
| 69. C3×GL(2,𝔽3) | C2×D3 | semidirect |
| 132. C32×Q8 | C22×D3 | semidirect |
| 145. C3×SL(2,𝔽3) | C22×D3 | semidirect |
| 170. D3×Q8 | D32 | semidirect |
| 189. GL(2,𝔽3) | D32 | direct (N × H) |
| 249. D32 | GL(2,𝔽3) | direct (N × H) |
| 283. C3×Q8 | C2×D32 | semidirect |
| 310. SL(2,𝔽3) | C2×D32 | semidirect |
| 330. C3⋊D3 | C2×GL(2,𝔽3) | semidirect |
| 340. C3×D3 | C2×GL(2,𝔽3) | semidirect |
| 410. C32 | C22×GL(2,𝔽3) | semidirect |
| 420. Q8 | D33 | semidirect |
| 458. D3 | D3×GL(2,𝔽3) | direct (N × H) |
| 476. C3 | C2×D3×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D32×SL(2,𝔽3) | 2 |
| #11724211 | 2 |
| D3×C3⋊GL(2,𝔽3) | 2 |
| C3⋊D3×GL(2,𝔽3) | 2 |
| C3×D3×GL(2,𝔽3) | 2 |
| C2×D3×GL(2,𝔽3) | 3 |
| D32×SD16 | 3 |
| C2×D33 | 4 |