Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
897 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 897 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×#1010⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11724470 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#64267 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×C3⋊GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11724196 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10596589 | — | 3 | 1 | no | maximal |
| 7 | #10596460 | — | 3 | 3 | no | maximal |
| 8 | #10595455 | — | 3 | 1 | no | maximal |
| 9 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #89026 | — | 4 | 1 | no | maximal |
| 11 | C3×D3×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C3⋊(C3⋊GL(2,𝔽3)) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3⋊(C3⋊#226) | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3×C3⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 15 | C3×#1010 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3×2−1+4⋊C3 | — | 3 | 1 | no | — |
| 17 | #64307 | — | 3 | 1 | no | — |
| 18 | #1010⋊C3 | D3 | 1 | 3 | yes | complemented |
| 19 | #64267 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | #64266 | — | 3 | 1 | no | — |
| 21 | #64266 | — | 3 | 3 | no | — |
| 22 | #64266 | — | 6 | 1 | no | — |
| 23 | #64266 | — | 6 | 3 | no | — |
| 24 | #64264 | — | 3 | 2 | no | — |
| 25 | C2×C3⋊GL(2,𝔽3) | — | 3 | 2 | no | — |
| 26 | C3×#986 | — | 3 | 3 | no | — |
| 27 | D3×GL(2,𝔽3) | — | 3 | 6 | no | — |
| 28 | #64201 | — | 3 | 6 | no | — |
| 29 | #64159 | — | 3 | 1 | no | — |
| 30 | C3×#949 | — | 3 | 1 | no | — |
| 31 | #63946 | — | 3 | 2 | no | — |
| 32 | D3×#215 | — | 3 | 2 | no | — |
| 33 | #63933 | — | 3 | 2 | no | — |
| 34 | #63925 | — | 3 | 2 | no | — |
| 35 | #63833 | — | 3 | 1 | no | — |
| 36 | C2×#3715 | — | 3 | 1 | no | — |
| 37 | #63815 | — | 3 | 1 | no | — |
| 38 | C2×D3×C3⋊D3 | — | 4 | 2 | no | — |
| 39 | C4×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 40 | C3×C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 41 | C32×D12 | — | 4 | 1 | no | — |
| 42 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 43 | #6290 | — | 4 | 2 | no | — |
| 44 | #5926 | — | 9 | 3 | no | — |
| 45 | #5837 | — | 9 | 1 | no | — |
| 46 | #5775 | — | 9 | 1 | no | — |
| 47 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 48 | D4×C3⋊D3 | — | 12 | 1 | no | — |
| 49 | #3828 | — | 3 | 1 | no | — |
| 50 | C3×D3×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 51 | C3×#237 | — | 3 | 2 | no | — |
| 52 | C3×#235 | — | 3 | 1 | no | — |
| 53 | C2×C3×D12 | — | 3 | 1 | no | — |
| 54 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 55 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 56 | C3×#41⋊C3 | — | 2 | 3 | no | — |
| 57 | C3×#41⋊C3 | C2×D3 | 1 | 3 | yes | complemented |
| 58 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 59 | C2×C3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 60 | D3×#67 | — | 6 | 2 | no | — |
| 61 | #3802 | — | 6 | 1 | no | — |
| 62 | #3802 | — | 12 | 3 | no | — |
| 63 | D3×D12 | — | 3 | 2 | no | — |
| 64 | #3796 | — | 6 | 1 | no | — |
| 65 | #3795 | — | 3 | 2 | no | — |
| 66 | D3×SL(2,𝔽3) | C2×D3 | 1 | 6 | yes | complemented |
| 67 | C3⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 68 | C3⋊GL(2,𝔽3) | — | 3 | 3 | no | — |
| 69 | C3⋊GL(2,𝔽3) | — | 6 | 1 | no | — |
| 70 | C3⋊GL(2,𝔽3) | — | 6 | 3 | no | — |
| 71 | C3⋊GL(2,𝔽3) | C2×D3 | 1 | 2 | yes | complemented |
| 72 | C3⋊#226 | — | 3 | 1 | no | — |
| 73 | C3⋊#226 | — | 3 | 3 | no | — |
| 74 | C3⋊#226 | — | 6 | 1 | no | — |
| 75 | C3⋊#226 | — | 6 | 3 | no | — |
| 76 | C3×GL(2,𝔽3) | — | 3 | 6 | no | — |
| 77 | #3756 | — | 3 | 1 | no | — |
| 78 | #3755 | — | 3 | 1 | no | — |
| 79 | #3754 | — | 3 | 1 | no | — |
| 80 | #3753 | — | 3 | 1 | no | — |
| 81 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 82 | C3×#215 | — | 3 | 2 | no | — |
| 83 | C3×#213 | — | 3 | 2 | no | — |
| 84 | C3×#208 | — | 3 | 1 | no | — |
| 85 | #3719 | — | 3 | 2 | no | — |
| 86 | #3717 | — | 3 | 2 | no | — |
| 87 | #3715 | — | 3 | 2 | no | — |
| 88 | #3715 | — | 3 | 2 | no | — |
| 89 | #3711 | — | 3 | 2 | no | — |
| 90 | D3×C3⋊C8 | — | 3 | 2 | no | — |
| 91 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 92 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 93 | C2×C32×D3 | — | 4 | 2 | no | — |
| 94 | D3×C3⋊D3 | — | 8 | 4 | no | — |
| 95 | C33×C4 | — | 4 | 1 | no | — |
| 96 | C3⋊(C3⋊Dic3) | — | 4 | 1 | no | — |
| 97 | C3×2−1+4 | — | 3 | 1 | no | — |
| 98 | #1010 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 99 | #1009 | — | 9 | 1 | no | — |
| 100 | #1004 | — | 9 | 1 | no | — |
| 101 | 2−1+4⋊C3 | — | 3 | 3 | no | — |
| 102 | #986 | — | 3 | 3 | no | — |
| 103 | #985 | — | 9 | 3 | no | — |
| 104 | #985 | — | 18 | 3 | no | — |
| 105 | #983 | — | 9 | 6 | no | — |
| 106 | C2×GL(2,𝔽3) | — | 9 | 6 | no | — |
| 107 | C3×AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 108 | #950 | — | 9 | 2 | no | — |
| 109 | #950 | — | 9 | 1 | no | — |
| 110 | #950 | — | 9 | 1 | no | — |
| 111 | #950 | — | 18 | 1 | no | — |
| 112 | #949 | — | 3 | 1 | no | — |
| 113 | #949 | — | 9 | 2 | no | — |
| 114 | #948 | — | 9 | 1 | no | — |
| 115 | #942 | — | 9 | 1 | no | — |
| 116 | #942 | — | 9 | 2 | no | — |
| 117 | C2×#215 | — | 9 | 2 | no | — |
| 118 | C2×#215 | — | 9 | 1 | no | — |
| 119 | C2×#214 | — | 9 | 1 | no | — |
| 120 | #915 | — | 9 | 2 | no | — |
| 121 | D3×SD16 | — | 9 | 2 | no | — |
| 122 | #912 | — | 9 | 2 | no | — |
| 123 | #911 | — | 9 | 2 | no | — |
| 124 | #909 | — | 9 | 1 | no | — |
| 125 | #907 | — | 9 | 1 | no | — |
| 126 | C22×C3⋊D3 | — | 12 | 2 | no | — |
| 127 | C22×C3×D3 | — | 6 | 1 | no | — |
| 128 | C2×D32 | — | 6 | 2 | no | — |
| 129 | C2×D32 | — | 12 | 6 | no | — |
| 130 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 131 | C32×D4 | — | 3 | 1 | no | — |
| 132 | C32×D4 | — | 12 | 1 | no | — |
| 133 | C2×C32×C4 | — | 3 | 1 | no | — |
| 134 | #638 | — | 6 | 1 | no | — |
| 135 | #638 | — | 12 | 2 | no | — |
| 136 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 137 | C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 138 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 139 | C4×C3⋊D3 | — | 12 | 1 | no | — |
| 140 | C4×C3⋊D3 | — | 12 | 3 | no | — |
| 141 | C4×C3⋊D3 | — | 24 | 1 | no | — |
| 142 | C4×C3⋊D3 | — | 24 | 3 | no | — |
| 143 | #634 | — | 3 | 1 | no | — |
| 144 | C3×#67 | — | 3 | 2 | no | — |
| 145 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 146 | C3×D12 | — | 3 | 2 | no | — |
| 147 | C3×D12 | — | 4 | 3 | no | — |
| 148 | C3×D12 | — | 6 | 1 | no | — |
| 149 | C3×D12 | — | 12 | 3 | no | — |
| 150 | C3×D12 | S4 | 1 | 1 | yes | characteristic |
| 151 | C3×C4×D3 | — | 3 | 2 | no | — |
| 152 | C3×Dic6 | — | 3 | 1 | no | — |
| 153 | C3×Dic6 | — | 3 | 2 | no | — |
| 154 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 155 | C3×SL(2,𝔽3) | — | 2 | 3 | no | — |
| 156 | C3×SL(2,𝔽3) | C22×D3 | 1 | 3 | yes | — |
| 157 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 158 | #626 | — | 6 | 2 | no | — |
| 159 | #625 | — | 6 | 2 | no | — |
| 160 | #625 | — | 12 | 6 | no | — |
| 161 | D3×Dic3 | — | 6 | 2 | no | — |
| 162 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 163 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 164 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 165 | #577 | — | 27 | 1 | no | Sylow |
| 166 | C2×C33 | — | 4 | 1 | no | — |
| 167 | C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 168 | C3×C3⋊D3 | — | 8 | 2 | no | — |
| 169 | C32×D3 | — | 8 | 2 | no | — |
| 170 | C3×#41 | — | 2 | 1 | no | — |
| 171 | C3×#41 | — | 9 | 1 | no | — |
| 172 | C3×#41 | — | 9 | 2 | no | — |
| 173 | C3×#41 | D32 | 1 | 1 | yes | characteristic, complemented |
| 174 | C3×#41 | C2×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 175 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 176 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 177 | C2×C3×D4 | — | 9 | 1 | no | — |
| 178 | C2×#67 | — | 18 | 2 | no | — |
| 179 | #239 | — | 18 | 1 | no | — |
| 180 | D3×Q8 | C2×C3⋊D3 | 1 | 2 | yes | complemented |
| 181 | #237 | — | 3 | 2 | no | — |
| 182 | #237 | — | 9 | 2 | no | — |
| 183 | #237 | — | 18 | 1 | no | — |
| 184 | D3×D4 | — | 9 | 2 | no | — |
| 185 | D3×D4 | — | 18 | 1 | no | — |
| 186 | D3×D4 | — | 18 | 2 | no | — |
| 187 | D3×D4 | — | 18 | 1 | no | — |
| 188 | D3×D4 | — | 36 | 3 | no | — |
| 189 | #235 | — | 3 | 1 | no | — |
| 190 | #235 | — | 9 | 2 | no | — |
| 191 | #235 | — | 9 | 1 | no | — |
| 192 | #235 | — | 9 | 1 | no | — |
| 193 | #235 | — | 18 | 1 | no | — |
| 194 | C2×D12 | — | 3 | 1 | no | — |
| 195 | C2×D12 | — | 9 | 2 | no | — |
| 196 | C2×Dic6 | — | 3 | 1 | no | — |
| 197 | #41⋊C3 | — | 2 | 3 | no | — |
| 198 | #41⋊C3 | D32 | 1 | 3 | yes | complemented |
| 199 | C2×SL(2,𝔽3) | — | 3 | 6 | no | — |
| 200 | GL(2,𝔽3) | — | 3 | 6 | no | — |
| 201 | GL(2,𝔽3) | — | 9 | 3 | no | — |
| 202 | GL(2,𝔽3) | — | 18 | 3 | no | — |
| 203 | #226 | — | 9 | 3 | no | — |
| 204 | #226 | — | 18 | 3 | no | — |
| 205 | C3×SD16 | — | 9 | 2 | no | — |
| 206 | C3×D8 | — | 9 | 2 | no | — |
| 207 | C3×M16 | — | 9 | 1 | no | — |
| 208 | #216 | — | 9 | 2 | no | — |
| 209 | #216 | — | 9 | 1 | no | — |
| 210 | #216 | — | 9 | 1 | no | — |
| 211 | #216 | — | 18 | 1 | no | — |
| 212 | #215 | — | 3 | 2 | no | — |
| 213 | #215 | — | 9 | 2 | no | — |
| 214 | #215 | — | 9 | 2 | no | — |
| 215 | #215 | — | 9 | 2 | no | — |
| 216 | #215 | — | 9 | 1 | no | — |
| 217 | #215 | — | 9 | 1 | no | — |
| 218 | #215 | — | 18 | 1 | no | — |
| 219 | #214 | — | 9 | 2 | no | — |
| 220 | #214 | — | 9 | 2 | no | — |
| 221 | #214 | — | 9 | 1 | no | — |
| 222 | #214 | — | 9 | 1 | no | — |
| 223 | #214 | — | 18 | 1 | no | — |
| 224 | #213 | — | 3 | 2 | no | — |
| 225 | #213 | — | 9 | 1 | no | — |
| 226 | #213 | — | 9 | 1 | no | — |
| 227 | #213 | — | 18 | 1 | no | — |
| 228 | #208 | — | 3 | 1 | no | — |
| 229 | #208 | — | 9 | 2 | no | — |
| 230 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 231 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 232 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 233 | C2×C3⋊C8 | — | 18 | 1 | no | — |
| 234 | Dic12 | — | 9 | 2 | no | — |
| 235 | #204 | — | 9 | 2 | no | — |
| 236 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 237 | D3×C8 | — | 9 | 2 | no | — |
| 238 | C22×C32 | — | 3 | 1 | no | — |
| 239 | C22×C32 | — | 12 | 2 | no | — |
| 240 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 241 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 242 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 243 | C2×C3⋊D3 | — | 12 | 3 | no | — |
| 244 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 245 | C2×C3⋊D3 | — | 24 | 3 | no | — |
| 246 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 247 | C2×C3×D3 | — | 4 | 6 | no | — |
| 248 | C2×C3×D3 | — | 6 | 2 | no | — |
| 249 | C2×C3×D3 | — | 12 | 6 | no | — |
| 250 | C2×C3×D3 | — | 24 | 1 | no | — |
| 251 | C2×C3×D3 | C2×S4 | 1 | 2 | yes | — |
| 252 | D32 | — | 24 | 2 | no | — |
| 253 | D32 | — | 24 | 12 | no | — |
| 254 | C32×C4 | — | 3 | 1 | no | — |
| 255 | C32×C4 | — | 4 | 1 | no | — |
| 256 | C32×C4 | — | 4 | 3 | no | — |
| 257 | C32×C4 | — | 8 | 3 | no | — |
| 258 | C32×C4 | — | 8 | 1 | no | — |
| 259 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 260 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 261 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 262 | C3⋊Dic3 | — | 12 | 3 | no | — |
| 263 | C3⋊Dic3 | — | 24 | 1 | no | — |
| 264 | C3⋊Dic3 | — | 24 | 3 | no | — |
| 265 | C3×Dic3 | — | 3 | 2 | no | — |
| 266 | 2−1+4 | — | 3 | 1 | no | — |
| 267 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 268 | #137 | — | 27 | 1 | no | — |
| 269 | #137 | — | 27 | 2 | no | — |
| 270 | AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 271 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 272 | #135 | — | 27 | 2 | no | — |
| 273 | #135 | — | 27 | 1 | no | — |
| 274 | C2×SD16 | — | 27 | 2 | no | — |
| 275 | C2×SD16 | — | 27 | 1 | no | — |
| 276 | #131 | — | 27 | 1 | no | — |
| 277 | C33 | — | 4 | 1 | no | Sylow |
| 278 | C23×C3 | — | 18 | 1 | no | — |
| 279 | C22×D3 | — | 6 | 1 | no | — |
| 280 | C22×D3 | — | 18 | 2 | no | — |
| 281 | C22×D3 | — | 18 | 2 | no | — |
| 282 | C22×D3 | — | 36 | 6 | no | — |
| 283 | C3×Q8 | — | 2 | 1 | no | — |
| 284 | C3×Q8 | — | 9 | 2 | no | — |
| 285 | C3×Q8 | — | 9 | 1 | no | — |
| 286 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 287 | C3×Q8 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 288 | C3×D4 | — | 3 | 1 | no | — |
| 289 | C3×D4 | — | 3 | 1 | no | — |
| 290 | C3×D4 | — | 3 | 1 | no | — |
| 291 | C3×D4 | — | 6 | 1 | no | — |
| 292 | C3×D4 | — | 9 | 2 | no | — |
| 293 | C3×D4 | — | 9 | 2 | no | — |
| 294 | C3×D4 | — | 12 | 3 | no | — |
| 295 | C3×D4 | — | 18 | 1 | no | — |
| 296 | C2×C3×C4 | — | 3 | 1 | no | — |
| 297 | C2×C3×C4 | — | 3 | 1 | no | — |
| 298 | C2×C3×C4 | — | 6 | 1 | no | — |
| 299 | C2×C3×C4 | — | 9 | 2 | no | — |
| 300 | C2×C3×C4 | — | 18 | 1 | no | — |
| 301 | #67 | — | 3 | 2 | no | — |
| 302 | #67 | — | 18 | 2 | no | — |
| 303 | #67 | — | 18 | 2 | no | — |
| 304 | #67 | — | 18 | 1 | no | — |
| 305 | #67 | — | 18 | 1 | no | — |
| 306 | #67 | — | 18 | 2 | no | — |
| 307 | #67 | — | 36 | 6 | no | — |
| 308 | #67 | — | 36 | 1 | no | — |
| 309 | C2×Dic3 | — | 6 | 1 | no | — |
| 310 | C2×Dic3 | — | 18 | 2 | no | — |
| 311 | D12 | — | 3 | 2 | no | — |
| 312 | D12 | — | 6 | 1 | no | — |
| 313 | D12 | — | 9 | 1 | no | — |
| 314 | D12 | — | 9 | 1 | no | — |
| 315 | D12 | — | 12 | 3 | no | — |
| 316 | D12 | — | 18 | 1 | no | — |
| 317 | D12 | — | 18 | 2 | no | — |
| 318 | D12 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 319 | C4×D3 | — | 3 | 2 | no | — |
| 320 | C4×D3 | — | 18 | 2 | no | — |
| 321 | C4×D3 | — | 18 | 1 | no | — |
| 322 | C4×D3 | — | 18 | 1 | no | — |
| 323 | C4×D3 | — | 36 | 1 | no | — |
| 324 | C4×D3 | — | 36 | 3 | no | — |
| 325 | C4×D3 | — | 72 | 3 | no | — |
| 326 | Dic6 | — | 3 | 1 | no | — |
| 327 | Dic6 | — | 3 | 2 | no | — |
| 328 | Dic6 | — | 9 | 1 | no | — |
| 329 | Dic6 | — | 9 | 1 | no | — |
| 330 | Dic6 | — | 18 | 1 | no | — |
| 331 | SL(2,𝔽3) | — | 2 | 3 | no | — |
| 332 | SL(2,𝔽3) | C2×D32 | 1 | 3 | yes | — |
| 333 | C3×C8 | — | 9 | 2 | no | — |
| 334 | C3⋊C8 | — | 3 | 2 | no | — |
| 335 | C3⋊C8 | — | 9 | 1 | no | — |
| 336 | C3⋊C8 | — | 9 | 1 | no | — |
| 337 | C3⋊C8 | — | 9 | 1 | no | — |
| 338 | C3⋊C8 | — | 9 | 1 | no | — |
| 339 | C3⋊C8 | — | 18 | 1 | no | — |
| 340 | C3⋊C8 | — | 18 | 1 | no | — |
| 341 | C2×C32 | — | 4 | 1 | no | — |
| 342 | C2×C32 | — | 4 | 3 | no | — |
| 343 | C2×C32 | — | 6 | 1 | no | — |
| 344 | C2×C32 | — | 8 | 3 | no | — |
| 345 | C2×C32 | — | 8 | 1 | no | — |
| 346 | C2×C32 | — | 24 | 2 | no | — |
| 347 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 348 | C3⋊D3 | — | 8 | 2 | no | — |
| 349 | C3⋊D3 | — | 12 | 2 | no | — |
| 350 | C3⋊D3 | — | 12 | 6 | no | — |
| 351 | C3⋊D3 | — | 12 | 1 | no | — |
| 352 | C3⋊D3 | — | 24 | 6 | no | — |
| 353 | C3⋊D3 | — | 24 | 2 | no | — |
| 354 | C3×D3 | — | 2 | 2 | no | — |
| 355 | C3×D3 | — | 8 | 6 | no | — |
| 356 | C3×D3 | — | 12 | 2 | no | — |
| 357 | C3×D3 | — | 24 | 6 | no | — |
| 358 | #41 | — | 9 | 1 | no | — |
| 359 | #41 | — | 9 | 2 | no | — |
| 360 | #41 | — | 27 | 1 | no | — |
| 361 | #41 | — | 27 | 2 | no | — |
| 362 | #41 | — | 54 | 1 | no | — |
| 363 | #41 | D3×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 364 | C2×Q8 | — | 3 | 2 | no | — |
| 365 | C2×Q8 | — | 9 | 1 | no | — |
| 366 | C2×D4 | — | 9 | 1 | no | — |
| 367 | C2×D4 | — | 27 | 2 | no | — |
| 368 | C2×D4 | — | 54 | 1 | no | — |
| 369 | C2×D4 | — | 54 | 2 | no | — |
| 370 | Q16 | — | 27 | 2 | no | — |
| 371 | Q16 | — | 27 | 1 | no | — |
| 372 | SD16 | — | 9 | 2 | no | — |
| 373 | SD16 | — | 27 | 2 | no | — |
| 374 | SD16 | — | 27 | 1 | no | — |
| 375 | SD16 | — | 27 | 2 | no | — |
| 376 | SD16 | — | 27 | 2 | no | — |
| 377 | SD16 | — | 27 | 1 | no | — |
| 378 | D8 | — | 9 | 2 | no | — |
| 379 | D8 | — | 27 | 1 | no | — |
| 380 | M16 | — | 9 | 1 | no | — |
| 381 | M16 | — | 27 | 2 | no | — |
| 382 | C2×C8 | — | 27 | 2 | no | — |
| 383 | C2×C8 | — | 27 | 1 | no | — |
| 384 | C22×C3 | — | 3 | 1 | no | — |
| 385 | C22×C3 | — | 3 | 2 | no | — |
| 386 | C22×C3 | — | 3 | 1 | no | — |
| 387 | C22×C3 | — | 6 | 1 | no | — |
| 388 | C22×C3 | — | 12 | 6 | no | — |
| 389 | C22×C3 | — | 18 | 2 | no | — |
| 390 | C22×C3 | — | 72 | 1 | no | — |
| 391 | C2×D3 | — | 6 | 2 | no | — |
| 392 | C2×D3 | — | 12 | 6 | no | — |
| 393 | C2×D3 | — | 18 | 1 | no | — |
| 394 | C2×D3 | — | 18 | 1 | no | — |
| 395 | C2×D3 | — | 24 | 1 | no | — |
| 396 | C2×D3 | — | 36 | 3 | no | — |
| 397 | C2×D3 | — | 36 | 1 | no | — |
| 398 | C2×D3 | — | 72 | 3 | no | — |
| 399 | C2×D3 | — | 72 | 2 | no | — |
| 400 | C2×D3 | — | 72 | 2 | no | — |
| 401 | C2×D3 | — | 72 | 12 | no | — |
| 402 | C2×D3 | C2×C3⋊S4 | 1 | 2 | yes | — |
| 403 | C3×C4 | — | 2 | 1 | no | — |
| 404 | C3×C4 | — | 3 | 1 | no | — |
| 405 | C3×C4 | — | 3 | 1 | no | — |
| 406 | C3×C4 | — | 4 | 3 | no | — |
| 407 | C3×C4 | — | 6 | 1 | no | — |
| 408 | C3×C4 | — | 8 | 3 | no | — |
| 409 | C3×C4 | — | 9 | 2 | no | — |
| 410 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 411 | C3×C4 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 412 | Dic3 | — | 3 | 2 | no | — |
| 413 | Dic3 | — | 18 | 1 | no | — |
| 414 | Dic3 | — | 18 | 1 | no | — |
| 415 | Dic3 | — | 36 | 1 | no | — |
| 416 | Dic3 | — | 36 | 3 | no | — |
| 417 | Dic3 | — | 72 | 3 | no | — |
| 418 | C32 | — | 4 | 1 | no | — |
| 419 | C32 | — | 4 | 3 | no | — |
| 420 | C32 | — | 8 | 3 | no | — |
| 421 | C32 | — | 8 | 1 | no | — |
| 422 | C32 | #5926 | 1 | 1 | yes | characteristic, complemented |
| 423 | C23 | — | 18 | 1 | no | — |
| 424 | C23 | — | 54 | 2 | no | — |
| 425 | Q8 | — | 9 | 2 | no | — |
| 426 | Q8 | — | 9 | 1 | no | — |
| 427 | Q8 | — | 27 | 1 | no | — |
| 428 | Q8 | C2×D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 429 | D4 | — | 3 | 1 | no | — |
| 430 | D4 | — | 3 | 1 | no | — |
| 431 | D4 | — | 9 | 2 | no | — |
| 432 | D4 | — | 9 | 2 | no | — |
| 433 | D4 | — | 18 | 1 | no | — |
| 434 | D4 | — | 27 | 1 | no | — |
| 435 | D4 | — | 54 | 2 | no | — |
| 436 | D4 | — | 54 | 2 | no | — |
| 437 | D4 | — | 54 | 1 | no | — |
| 438 | C2×C4 | — | 3 | 1 | no | — |
| 439 | C2×C4 | — | 9 | 2 | no | — |
| 440 | C2×C4 | — | 18 | 1 | no | — |
| 441 | C2×C4 | — | 54 | 2 | no | — |
| 442 | C2×C4 | — | 54 | 1 | no | — |
| 443 | C8 | — | 9 | 2 | no | — |
| 444 | C8 | — | 27 | 1 | no | — |
| 445 | C8 | — | 27 | 1 | no | — |
| 446 | C2×C3 | — | 2 | 1 | no | — |
| 447 | C2×C3 | — | 4 | 3 | no | — |
| 448 | C2×C3 | — | 6 | 1 | no | — |
| 449 | C2×C3 | — | 6 | 2 | no | — |
| 450 | C2×C3 | — | 6 | 1 | no | — |
| 451 | C2×C3 | — | 8 | 3 | no | — |
| 452 | C2×C3 | — | 12 | 1 | no | — |
| 453 | C2×C3 | — | 24 | 6 | no | — |
| 454 | C2×C3 | — | 36 | 2 | no | — |
| 455 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 456 | C2×C3 | C22×C3⋊S4 | 1 | 1 | yes | characteristic |
| 457 | D3 | — | 2 | 2 | no | — |
| 458 | D3 | — | 12 | 2 | no | — |
| 459 | D3 | — | 24 | 6 | no | — |
| 460 | D3 | — | 36 | 6 | no | — |
| 461 | D3 | — | 36 | 1 | no | — |
| 462 | D3 | — | 36 | 1 | no | — |
| 463 | D3 | — | 72 | 1 | no | — |
| 464 | D3 | — | 72 | 6 | no | — |
| 465 | C22 | — | 3 | 1 | no | — |
| 466 | C22 | — | 3 | 2 | no | — |
| 467 | C22 | — | 18 | 2 | no | — |
| 468 | C22 | — | 54 | 1 | no | — |
| 469 | C22 | — | 72 | 1 | no | — |
| 470 | C22 | — | 216 | 2 | no | — |
| 471 | C4 | — | 3 | 1 | no | — |
| 472 | C4 | — | 9 | 2 | no | — |
| 473 | C4 | — | 54 | 1 | no | — |
| 474 | C4 | D3×C3⋊S4 | 1 | 1 | yes | characteristic |
| 475 | C3 | — | 2 | 1 | no | — |
| 476 | C3 | — | 4 | 3 | no | — |
| 477 | C3 | — | 8 | 3 | no | — |
| 478 | C3 | #10596460 | 1 | 1 | yes | characteristic, complemented |
| 479 | C3 | #10596589 | 1 | 1 | yes | characteristic, complemented |
| 480 | C2 | — | 6 | 1 | no | — |
| 481 | C2 | — | 6 | 2 | no | — |
| 482 | C2 | — | 36 | 2 | no | — |
| 483 | C2 | — | 108 | 1 | no | — |
| 484 | C2 | C2×D3×C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C32×#41 | C2×D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C4 | D3×C3⋊S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#1010⋊C3 | C2 | semidirect |
| 2. #11724470 | C2 | semidirect |
| 3. C3×#64267 | C2 | semidirect |
| 4. D3×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 5. #11724196 | C2 | semidirect |
| 9. C32×#41⋊C3 | C22 | semidirect |
| 11. C3×D3×SL(2,𝔽3) | C22 | semidirect |
| 12. C3⋊(C3⋊GL(2,𝔽3)) | C22 | semidirect |
| 13. C3⋊(C3⋊#226) | C22 | semidirect |
| 14. C3×C3⋊GL(2,𝔽3) | C22 | semidirect |
| 15. C3×#1010 | D3 | semidirect |
| 18. #1010⋊C3 | D3 | semidirect |
| 19. #64267 | D3 | semidirect |
| 47. C32×#41 | C2×D3 | semidirect |
| 50. C3×D3×Q8 | C2×D3 | semidirect |
| 57. C3×#41⋊C3 | C2×D3 | semidirect |
| 58. C3×#41⋊C3 | C2×D3 | semidirect |
| 66. D3×SL(2,𝔽3) | C2×D3 | semidirect |
| 71. C3⋊GL(2,𝔽3) | C2×D3 | semidirect |
| 98. #1010 | C3⋊D3 | semidirect |
| 173. C3×#41 | D32 | semidirect |
| 174. C3×#41 | C2×C3⋊D3 | semidirect |
| 180. D3×Q8 | C2×C3⋊D3 | semidirect |
| 198. #41⋊C3 | D32 | semidirect |
| 363. #41 | D3×C3⋊D3 | semidirect |
| 422. C32 | #5926 | semidirect |
| 478. C3 | #10596460 | semidirect |
| 479. C3 | #10596589 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1010⋊C3 | 2 |
| #11724470 | 2 |
| C3×#64267 | 2 |
| D3×C3⋊GL(2,𝔽3) | 2 |
| #11724196 | 2 |
| #10596589 | 3 |
| #10596460 | 3 |
| #10595455 | 3 |
| #89026 | 4 |