Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
664 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 54 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 | #11724491 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#64266 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#64265 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11724207 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11724206 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10596450 | — | 3 | 1 | no | maximal |
| 7 | #10596447 | — | 3 | 1 | no | maximal |
| 8 | #10595473 | — | 3 | 1 | no | maximal |
| 9 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 10 | #89040 | — | 4 | 1 | no | maximal |
| 11 | C3⋊(C3⋊(#41⋊C3)) | C22 | 1 | 2 | yes | — |
| 12 | C3×C3⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3×C3⋊#226 | C22 | 1 | 2 | yes | complemented |
| 14 | C3×C3⋊#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #64366 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | #64297 | — | 3 | 1 | no | — |
| 17 | C3⋊#994 | — | 3 | 1 | no | — |
| 18 | C3⋊#994 | — | 3 | 1 | no | — |
| 19 | C3⋊#994 | — | 6 | 1 | no | — |
| 20 | #64266 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #64265 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C3×#985 | — | 3 | 1 | no | — |
| 23 | C3×#985 | — | 6 | 1 | no | — |
| 24 | C3×#984 | — | 3 | 1 | no | — |
| 25 | C3×#984 | — | 6 | 1 | no | — |
| 26 | #64198 | — | 3 | 2 | no | — |
| 27 | #64197 | — | 3 | 2 | no | — |
| 28 | #64196 | — | 3 | 2 | no | — |
| 29 | #64196 | — | 3 | 2 | no | — |
| 30 | C3×#951 | — | 3 | 1 | no | — |
| 31 | C3×#950 | — | 3 | 1 | no | — |
| 32 | #63950 | — | 3 | 2 | no | — |
| 33 | #63945 | — | 3 | 2 | no | — |
| 34 | #63937 | — | 3 | 2 | no | — |
| 35 | #63930 | — | 3 | 2 | no | — |
| 36 | C2×#3717 | — | 3 | 1 | no | — |
| 37 | #63825 | — | 3 | 1 | no | — |
| 38 | #63817 | — | 3 | 1 | no | — |
| 39 | C3×C4×C3⋊D3 | — | 4 | 1 | no | — |
| 40 | C3×#634 | — | 4 | 1 | no | — |
| 41 | C3×#634 | — | 4 | 1 | no | — |
| 42 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 43 | #6296 | — | 4 | 2 | no | — |
| 44 | #6294 | — | 4 | 2 | no | — |
| 45 | #5924 | — | 9 | 1 | no | — |
| 46 | #5778 | — | 9 | 1 | no | — |
| 47 | #5770 | — | 9 | 1 | no | — |
| 48 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | #3832 | C2×D3 | 1 | 2 | yes | — |
| 50 | D4×C3⋊D3 | — | 3 | 2 | no | — |
| 51 | #3828 | — | 3 | 1 | no | — |
| 52 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 53 | C3×#235 | — | 3 | 1 | no | — |
| 54 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 55 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 56 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 57 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 58 | C3×#41⋊C3 | — | 4 | 1 | no | — |
| 59 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | #3804 | — | 6 | 2 | no | — |
| 62 | #3799 | — | 6 | 1 | no | — |
| 63 | #3797 | — | 3 | 2 | no | — |
| 64 | #3797 | — | 12 | 1 | no | — |
| 65 | #3796 | — | 3 | 2 | no | — |
| 66 | D3×Dic6 | — | 6 | 1 | no | — |
| 67 | D3×Dic6 | — | 12 | 1 | no | — |
| 68 | C3⋊(#41⋊C3) | — | 3 | 2 | no | — |
| 69 | C3⋊(#41⋊C3) | — | 3 | 2 | no | — |
| 70 | C3⋊(#41⋊C3) | — | 6 | 2 | no | — |
| 71 | C3⋊GL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 72 | C3⋊#226 | C2×D3 | 1 | 2 | yes | complemented |
| 73 | C3⋊#226 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 74 | C3×GL(2,𝔽3) | — | 3 | 1 | no | — |
| 75 | C3×GL(2,𝔽3) | — | 6 | 1 | no | — |
| 76 | C3×#226 | — | 3 | 1 | no | — |
| 77 | C3×#226 | — | 3 | 2 | no | — |
| 78 | C3×#226 | — | 6 | 2 | no | — |
| 79 | C3×#226 | — | 6 | 1 | no | — |
| 80 | C3×#216 | — | 3 | 1 | no | — |
| 81 | C3×#216 | — | 3 | 2 | no | — |
| 82 | C3×#215 | — | 3 | 1 | no | — |
| 83 | C3×#214 | — | 3 | 1 | no | — |
| 84 | C3×#214 | — | 3 | 2 | no | — |
| 85 | C3×#213 | — | 3 | 1 | no | — |
| 86 | C3×#208 | — | 3 | 1 | no | — |
| 87 | C2×C3×C3⋊C8 | — | 3 | 1 | no | — |
| 88 | #3717 | — | 3 | 2 | no | — |
| 89 | #3717 | — | 3 | 2 | no | — |
| 90 | #3717 | — | 3 | 2 | no | — |
| 91 | #3714 | — | 3 | 2 | no | — |
| 92 | #3712 | — | 3 | 2 | no | — |
| 93 | #3710 | — | 3 | 2 | no | — |
| 94 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 95 | C33×C4 | — | 4 | 1 | no | — |
| 96 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 97 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 98 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 99 | #1010 | — | 9 | 1 | no | — |
| 100 | #1009 | — | 3 | 1 | no | — |
| 101 | #1009 | — | 3 | 1 | no | — |
| 102 | #1009 | — | 6 | 1 | no | — |
| 103 | #1007 | — | 9 | 1 | no | — |
| 104 | #994 | — | 9 | 1 | no | — |
| 105 | #985 | — | 3 | 1 | no | — |
| 106 | #985 | — | 6 | 1 | no | — |
| 107 | #985 | — | 9 | 2 | no | — |
| 108 | #984 | — | 3 | 1 | no | — |
| 109 | #984 | — | 6 | 1 | no | — |
| 110 | #984 | — | 9 | 2 | no | — |
| 111 | C3×#137 | — | 9 | 1 | no | — |
| 112 | C3×#135 | — | 9 | 1 | no | — |
| 113 | #951 | — | 3 | 1 | no | — |
| 114 | #950 | — | 3 | 1 | no | — |
| 115 | #949 | — | 9 | 1 | no | — |
| 116 | C2×#215 | — | 9 | 1 | no | — |
| 117 | #919 | — | 9 | 2 | no | — |
| 118 | #918 | — | 9 | 2 | no | — |
| 119 | #918 | — | 9 | 2 | no | — |
| 120 | #916 | — | 9 | 2 | no | — |
| 121 | #914 | — | 9 | 2 | no | — |
| 122 | D3×SD16 | — | 9 | 2 | no | — |
| 123 | D3×SD16 | — | 9 | 2 | no | — |
| 124 | #911 | — | 9 | 2 | no | — |
| 125 | #908 | — | 9 | 1 | no | — |
| 126 | #907 | — | 9 | 1 | no | — |
| 127 | C2×#204 | — | 9 | 1 | no | — |
| 128 | #901 | — | 9 | 1 | no | — |
| 129 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 130 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 131 | C32×D4 | — | 3 | 1 | no | — |
| 132 | C2×C32×C4 | — | 3 | 1 | no | — |
| 133 | #638 | — | 3 | 2 | no | — |
| 134 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 135 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 136 | C4×C3⋊D3 | — | 3 | 2 | no | — |
| 137 | C4×C3⋊D3 | — | 4 | 1 | no | — |
| 138 | #634 | — | 4 | 1 | no | — |
| 139 | #634 | S4 | 1 | 1 | yes | characteristic |
| 140 | C3×#67 | — | 6 | 1 | no | — |
| 141 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 142 | C3×D12 | — | 3 | 1 | no | — |
| 143 | C3×C4×D3 | — | 6 | 1 | no | — |
| 144 | C3×C4×D3 | — | 12 | 1 | no | — |
| 145 | C3×C4×D3 | — | 24 | 1 | no | — |
| 146 | C3×Dic6 | — | 3 | 2 | no | — |
| 147 | C3×Dic6 | — | 3 | 1 | no | — |
| 148 | C3×Dic6 | — | 6 | 1 | no | — |
| 149 | C3×Dic6 | — | 12 | 1 | no | — |
| 150 | C3×Dic6 | — | 12 | 1 | no | — |
| 151 | C3×Dic6 | — | 12 | 1 | no | — |
| 152 | C3×Dic6 | — | 24 | 1 | no | — |
| 153 | C3×Dic6 | — | 24 | 1 | no | — |
| 154 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 155 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 156 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 157 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 158 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 159 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 160 | #627 | — | 6 | 2 | no | — |
| 161 | #627 | — | 12 | 2 | no | — |
| 162 | #627 | — | 12 | 2 | no | — |
| 163 | #626 | — | 6 | 2 | no | — |
| 164 | #624 | — | 6 | 2 | no | — |
| 165 | #624 | — | 12 | 2 | no | — |
| 166 | D3×Dic3 | — | 6 | 2 | no | — |
| 167 | D3×Dic3 | — | 12 | 2 | no | — |
| 168 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 169 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 170 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 171 | #577 | — | 27 | 1 | no | Sylow |
| 172 | C2×C33 | — | 4 | 1 | no | — |
| 173 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 174 | C3×#41 | — | 2 | 1 | no | — |
| 175 | C3×#41 | — | 9 | 1 | no | — |
| 176 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 177 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 179 | #239 | — | 3 | 2 | no | — |
| 180 | #239 | — | 3 | 2 | no | — |
| 181 | #239 | — | 6 | 2 | no | — |
| 182 | #239 | — | 9 | 2 | no | — |
| 183 | #239 | — | 18 | 1 | no | — |
| 184 | D3×Q8 | — | 9 | 2 | no | — |
| 185 | D3×Q8 | — | 9 | 2 | no | — |
| 186 | D3×Q8 | — | 18 | 1 | no | — |
| 187 | D3×Q8 | — | 36 | 1 | no | — |
| 188 | #237 | — | 9 | 2 | no | — |
| 189 | #237 | — | 18 | 2 | no | — |
| 190 | D3×D4 | — | 9 | 2 | no | — |
| 191 | D3×D4 | — | 9 | 2 | no | — |
| 192 | D3×D4 | — | 18 | 2 | no | — |
| 193 | #235 | — | 3 | 1 | no | — |
| 194 | #235 | — | 9 | 1 | no | — |
| 195 | #235 | — | 9 | 1 | no | — |
| 196 | #235 | — | 18 | 1 | no | — |
| 197 | #235 | — | 18 | 1 | no | — |
| 198 | #235 | — | 18 | 2 | no | — |
| 199 | C2×D12 | — | 9 | 1 | no | — |
| 200 | C2×D12 | — | 9 | 1 | no | — |
| 201 | C2×D12 | — | 18 | 1 | no | — |
| 202 | C2×Dic6 | — | 3 | 1 | no | — |
| 203 | C2×Dic6 | — | 18 | 1 | no | — |
| 204 | #41⋊C3 | — | 2 | 1 | no | — |
| 205 | #41⋊C3 | — | 2 | 1 | no | — |
| 206 | #41⋊C3 | — | 4 | 1 | no | — |
| 207 | #41⋊C3 | — | 9 | 2 | no | — |
| 208 | #41⋊C3 | D32 | 1 | 1 | yes | characteristic |
| 209 | GL(2,𝔽3) | — | 3 | 1 | no | — |
| 210 | GL(2,𝔽3) | — | 6 | 1 | no | — |
| 211 | #226 | — | 3 | 2 | no | — |
| 212 | #226 | — | 3 | 1 | no | — |
| 213 | #226 | — | 6 | 2 | no | — |
| 214 | #226 | — | 6 | 1 | no | — |
| 215 | C3×Q16 | — | 9 | 1 | no | — |
| 216 | C3×Q16 | — | 9 | 2 | no | — |
| 217 | C3×SD16 | — | 9 | 1 | no | — |
| 218 | C3×SD16 | — | 9 | 2 | no | — |
| 219 | C3×SD16 | — | 9 | 1 | no | — |
| 220 | C3×D8 | — | 9 | 1 | no | — |
| 221 | C3×M16 | — | 9 | 1 | no | — |
| 222 | C2×C3×C8 | — | 9 | 1 | no | — |
| 223 | #216 | — | 3 | 2 | no | — |
| 224 | #216 | — | 3 | 1 | no | — |
| 225 | #215 | — | 3 | 1 | no | — |
| 226 | #215 | — | 9 | 2 | no | — |
| 227 | #215 | — | 9 | 2 | no | — |
| 228 | #215 | — | 9 | 2 | no | — |
| 229 | #214 | — | 3 | 2 | no | — |
| 230 | #214 | — | 3 | 1 | no | — |
| 231 | #213 | — | 3 | 1 | no | — |
| 232 | #213 | — | 9 | 2 | no | — |
| 233 | #208 | — | 3 | 1 | no | — |
| 234 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 235 | D24 | — | 9 | 2 | no | — |
| 236 | #204 | — | 9 | 2 | no | — |
| 237 | #204 | — | 9 | 2 | no | — |
| 238 | #204 | — | 9 | 2 | no | — |
| 239 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 240 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 241 | D3×C8 | — | 9 | 2 | no | — |
| 242 | D3×C8 | — | 9 | 2 | no | — |
| 243 | C22×C32 | — | 3 | 1 | no | — |
| 244 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 245 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 246 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 247 | C2×C3×D3 | — | 6 | 1 | no | — |
| 248 | C2×C3×D3 | — | 12 | 1 | no | — |
| 249 | C2×C3×D3 | — | 24 | 1 | no | — |
| 250 | C32×C4 | — | 3 | 1 | no | — |
| 251 | C32×C4 | — | 4 | 1 | no | — |
| 252 | C32×C4 | — | 4 | 1 | no | — |
| 253 | C32×C4 | — | 8 | 1 | no | — |
| 254 | C32×C4 | — | 8 | 1 | no | — |
| 255 | C32×C4 | — | 8 | 1 | no | — |
| 256 | C32×C4 | — | 16 | 1 | no | — |
| 257 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 258 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 259 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 260 | C3⋊Dic3 | C2×S4 | 1 | 2 | yes | — |
| 261 | C3×Dic3 | — | 6 | 2 | no | — |
| 262 | C3×Dic3 | — | 6 | 1 | no | — |
| 263 | C3×Dic3 | — | 12 | 2 | no | — |
| 264 | C3×Dic3 | — | 12 | 2 | no | — |
| 265 | C3×Dic3 | — | 12 | 2 | no | — |
| 266 | C3×Dic3 | — | 12 | 1 | no | — |
| 267 | C3×Dic3 | — | 24 | 2 | no | — |
| 268 | C3×Dic3 | — | 24 | 2 | no | — |
| 269 | C3×Dic3 | — | 24 | 1 | no | — |
| 270 | 2−1+4 | — | 27 | 1 | no | — |
| 271 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 272 | #137 | — | 9 | 1 | no | — |
| 273 | #137 | — | 27 | 2 | no | — |
| 274 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 275 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 276 | #135 | — | 9 | 1 | no | — |
| 277 | #135 | — | 27 | 2 | no | — |
| 278 | C2×SD16 | — | 27 | 1 | no | — |
| 279 | C2×SD16 | — | 27 | 2 | no | — |
| 280 | #131 | — | 27 | 1 | no | — |
| 281 | C33 | — | 4 | 1 | no | Sylow |
| 282 | C22×D3 | — | 18 | 1 | no | — |
| 283 | C22×D3 | — | 18 | 1 | no | — |
| 284 | C22×D3 | — | 18 | 2 | no | — |
| 285 | C3×Q8 | — | 2 | 1 | no | — |
| 286 | C3×Q8 | — | 9 | 1 | no | — |
| 287 | C3×Q8 | — | 9 | 2 | no | — |
| 288 | C3×Q8 | — | 18 | 1 | no | — |
| 289 | C3×Q8 | — | 36 | 1 | no | — |
| 290 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 291 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 292 | C3×D4 | — | 3 | 1 | no | — |
| 293 | C3×D4 | — | 3 | 1 | no | — |
| 294 | C3×D4 | — | 6 | 1 | no | — |
| 295 | C3×D4 | — | 9 | 1 | no | — |
| 296 | C3×D4 | — | 18 | 1 | no | — |
| 297 | C2×C3×C4 | — | 3 | 1 | no | — |
| 298 | C2×C3×C4 | — | 3 | 1 | no | — |
| 299 | C2×C3×C4 | — | 6 | 1 | no | — |
| 300 | C2×C3×C4 | — | 18 | 1 | no | — |
| 301 | C2×C3×C4 | — | 18 | 1 | no | — |
| 302 | #67 | — | 6 | 1 | no | — |
| 303 | #67 | — | 9 | 2 | no | — |
| 304 | #67 | — | 9 | 2 | no | — |
| 305 | #67 | — | 18 | 2 | no | — |
| 306 | #67 | — | 18 | 2 | no | — |
| 307 | C2×Dic3 | — | 6 | 1 | no | — |
| 308 | C2×Dic3 | — | 18 | 2 | no | — |
| 309 | D12 | — | 3 | 1 | no | — |
| 310 | D12 | — | 9 | 2 | no | — |
| 311 | D12 | — | 9 | 2 | no | — |
| 312 | D12 | — | 9 | 1 | no | — |
| 313 | D12 | — | 9 | 1 | no | — |
| 314 | D12 | — | 18 | 1 | no | — |
| 315 | D12 | — | 18 | 2 | no | — |
| 316 | D12 | — | 18 | 2 | no | — |
| 317 | C4×D3 | — | 6 | 1 | no | — |
| 318 | C4×D3 | — | 9 | 2 | no | — |
| 319 | C4×D3 | — | 9 | 2 | no | — |
| 320 | C4×D3 | — | 12 | 1 | no | — |
| 321 | C4×D3 | — | 18 | 2 | no | — |
| 322 | C4×D3 | — | 18 | 2 | no | — |
| 323 | C4×D3 | — | 18 | 2 | no | — |
| 324 | C4×D3 | — | 18 | 2 | no | — |
| 325 | C4×D3 | — | 24 | 1 | no | — |
| 326 | C4×D3 | — | 36 | 2 | no | — |
| 327 | Dic6 | — | 3 | 1 | no | — |
| 328 | Dic6 | — | 3 | 1 | no | — |
| 329 | Dic6 | — | 3 | 2 | no | — |
| 330 | Dic6 | — | 3 | 1 | no | — |
| 331 | Dic6 | — | 6 | 1 | no | — |
| 332 | Dic6 | — | 6 | 1 | no | — |
| 333 | Dic6 | — | 12 | 1 | no | — |
| 334 | Dic6 | — | 18 | 2 | no | — |
| 335 | Dic6 | — | 18 | 2 | no | — |
| 336 | Dic6 | — | 24 | 1 | no | — |
| 337 | Dic6 | — | 36 | 2 | no | — |
| 338 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 339 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 340 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 341 | SL(2,𝔽3) | C2×D32 | 1 | 1 | yes | characteristic |
| 342 | C3×C8 | — | 9 | 1 | no | — |
| 343 | C3×C8 | — | 9 | 2 | no | — |
| 344 | C3×C8 | — | 9 | 1 | no | — |
| 345 | C3⋊C8 | — | 3 | 1 | no | — |
| 346 | C3⋊C8 | — | 3 | 2 | no | — |
| 347 | C3⋊C8 | — | 3 | 1 | no | — |
| 348 | C2×C32 | — | 4 | 1 | no | — |
| 349 | C2×C32 | — | 4 | 1 | no | — |
| 350 | C2×C32 | — | 6 | 1 | no | — |
| 351 | C2×C32 | — | 8 | 1 | no | — |
| 352 | C2×C32 | — | 8 | 1 | no | — |
| 353 | C2×C32 | — | 8 | 1 | no | — |
| 354 | C2×C32 | — | 16 | 1 | no | — |
| 355 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 356 | C3⋊D3 | — | 4 | 2 | no | — |
| 357 | C3⋊D3 | — | 6 | 2 | no | — |
| 358 | C3×D3 | — | 12 | 2 | no | — |
| 359 | C3×D3 | — | 12 | 1 | no | — |
| 360 | C3×D3 | — | 24 | 2 | no | — |
| 361 | #41 | — | 9 | 1 | no | — |
| 362 | #41 | — | 9 | 2 | no | — |
| 363 | #41 | — | 27 | 1 | no | — |
| 364 | #41 | — | 27 | 2 | no | — |
| 365 | #41 | — | 54 | 2 | no | — |
| 366 | #41 | — | 54 | 1 | no | — |
| 367 | #41 | #1113 | 1 | 1 | yes | characteristic |
| 368 | C2×Q8 | — | 9 | 1 | no | — |
| 369 | C2×Q8 | — | 27 | 2 | no | — |
| 370 | C2×Q8 | — | 54 | 1 | no | — |
| 371 | C2×D4 | — | 27 | 1 | no | — |
| 372 | C2×D4 | — | 27 | 2 | no | — |
| 373 | Q16 | — | 9 | 1 | no | — |
| 374 | Q16 | — | 9 | 2 | no | — |
| 375 | SD16 | — | 9 | 1 | no | — |
| 376 | SD16 | — | 9 | 2 | no | — |
| 377 | SD16 | — | 9 | 1 | no | — |
| 378 | SD16 | — | 27 | 2 | no | — |
| 379 | SD16 | — | 27 | 2 | no | — |
| 380 | SD16 | — | 27 | 2 | no | — |
| 381 | D8 | — | 9 | 1 | no | — |
| 382 | D8 | — | 27 | 2 | no | — |
| 383 | M16 | — | 9 | 1 | no | — |
| 384 | M16 | — | 27 | 2 | no | — |
| 385 | C2×C8 | — | 9 | 1 | no | — |
| 386 | C2×C8 | — | 27 | 2 | no | — |
| 387 | C22×C3 | — | 3 | 1 | no | — |
| 388 | C22×C3 | — | 3 | 1 | no | — |
| 389 | C22×C3 | — | 6 | 1 | no | — |
| 390 | C22×C3 | — | 18 | 1 | no | — |
| 391 | C2×D3 | — | 6 | 1 | no | — |
| 392 | C2×D3 | — | 9 | 2 | no | — |
| 393 | C2×D3 | — | 9 | 2 | no | — |
| 394 | C2×D3 | — | 12 | 1 | no | — |
| 395 | C2×D3 | — | 18 | 2 | no | — |
| 396 | C2×D3 | — | 24 | 1 | no | — |
| 397 | C2×D3 | — | 72 | 1 | no | — |
| 398 | C2×D3 | — | 72 | 1 | no | — |
| 399 | C2×D3 | — | 72 | 2 | no | — |
| 400 | C3×C4 | — | 2 | 1 | no | — |
| 401 | C3×C4 | — | 3 | 1 | no | — |
| 402 | C3×C4 | — | 3 | 1 | no | — |
| 403 | C3×C4 | — | 4 | 1 | no | — |
| 404 | C3×C4 | — | 6 | 1 | no | — |
| 405 | C3×C4 | — | 8 | 1 | no | — |
| 406 | C3×C4 | — | 8 | 1 | no | — |
| 407 | C3×C4 | — | 16 | 1 | no | — |
| 408 | C3×C4 | — | 18 | 1 | no | — |
| 409 | C3×C4 | — | 18 | 2 | no | — |
| 410 | C3×C4 | — | 36 | 2 | no | — |
| 411 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 412 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 413 | Dic3 | — | 3 | 2 | no | — |
| 414 | Dic3 | — | 3 | 2 | no | — |
| 415 | Dic3 | — | 6 | 2 | no | — |
| 416 | Dic3 | — | 6 | 2 | no | — |
| 417 | Dic3 | — | 6 | 1 | no | — |
| 418 | Dic3 | — | 12 | 2 | no | — |
| 419 | Dic3 | — | 12 | 1 | no | — |
| 420 | Dic3 | — | 24 | 2 | no | — |
| 421 | Dic3 | — | 24 | 1 | no | — |
| 422 | C32 | — | 4 | 1 | no | — |
| 423 | C32 | — | 4 | 1 | no | — |
| 424 | C32 | — | 8 | 1 | no | — |
| 425 | C32 | — | 8 | 1 | no | — |
| 426 | C32 | — | 8 | 1 | no | — |
| 427 | C32 | — | 16 | 1 | no | — |
| 428 | C32 | #5924 | 1 | 1 | yes | characteristic, complemented |
| 429 | C23 | — | 54 | 1 | no | — |
| 430 | Q8 | — | 9 | 2 | no | — |
| 431 | Q8 | — | 9 | 1 | no | — |
| 432 | Q8 | — | 9 | 1 | no | — |
| 433 | Q8 | — | 18 | 1 | no | — |
| 434 | Q8 | — | 54 | 2 | no | — |
| 435 | Q8 | C2×#1113 | 1 | 1 | yes | characteristic |
| 436 | D4 | — | 3 | 1 | no | — |
| 437 | D4 | — | 9 | 1 | no | — |
| 438 | D4 | — | 18 | 1 | no | — |
| 439 | D4 | — | 27 | 2 | no | — |
| 440 | D4 | — | 27 | 2 | no | — |
| 441 | D4 | — | 27 | 1 | no | — |
| 442 | D4 | — | 54 | 2 | no | — |
| 443 | C2×C4 | — | 3 | 1 | no | — |
| 444 | C2×C4 | — | 18 | 1 | no | — |
| 445 | C2×C4 | — | 18 | 1 | no | — |
| 446 | C2×C4 | — | 27 | 2 | no | — |
| 447 | C2×C4 | — | 54 | 2 | no | — |
| 448 | C2×C4 | — | 54 | 2 | no | — |
| 449 | C8 | — | 9 | 1 | no | — |
| 450 | C8 | — | 9 | 1 | no | — |
| 451 | C8 | — | 9 | 2 | no | — |
| 452 | C2×C3 | — | 2 | 1 | no | — |
| 453 | C2×C3 | — | 4 | 1 | no | — |
| 454 | C2×C3 | — | 6 | 1 | no | — |
| 455 | C2×C3 | — | 6 | 1 | no | — |
| 456 | C2×C3 | — | 8 | 1 | no | — |
| 457 | C2×C3 | — | 8 | 1 | no | — |
| 458 | C2×C3 | — | 12 | 1 | no | — |
| 459 | C2×C3 | — | 16 | 1 | no | — |
| 460 | C2×C3 | — | 36 | 1 | no | — |
| 461 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 462 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 463 | D3 | — | 12 | 2 | no | — |
| 464 | D3 | — | 12 | 1 | no | — |
| 465 | D3 | — | 18 | 2 | no | — |
| 466 | D3 | — | 18 | 2 | no | — |
| 467 | D3 | — | 24 | 2 | no | — |
| 468 | D3 | — | 36 | 2 | no | — |
| 469 | C22 | — | 3 | 1 | no | — |
| 470 | C22 | — | 18 | 1 | no | — |
| 471 | C22 | — | 27 | 2 | no | — |
| 472 | C22 | — | 216 | 1 | no | — |
| 473 | C4 | — | 3 | 1 | no | — |
| 474 | C4 | — | 9 | 2 | no | — |
| 475 | C4 | — | 18 | 1 | no | — |
| 476 | C4 | — | 18 | 2 | no | — |
| 477 | C4 | #89101 | 1 | 1 | yes | characteristic |
| 478 | C3 | — | 2 | 1 | no | — |
| 479 | C3 | — | 4 | 1 | no | — |
| 480 | C3 | — | 8 | 1 | no | — |
| 481 | C3 | — | 8 | 1 | no | — |
| 482 | C3 | — | 16 | 1 | no | — |
| 483 | C3 | #10596447 | 1 | 1 | yes | characteristic, complemented |
| 484 | C3 | #10596450 | 1 | 1 | yes | characteristic, complemented |
| 485 | C2 | — | 6 | 1 | no | — |
| 486 | C2 | — | 36 | 1 | no | — |
| 487 | C2 | — | 54 | 2 | no | — |
| 488 | C2 | C2×#89101 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#89101 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C32×#41 | C2×D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C4 | #89101 |
| 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. #11724491 | C2 | semidirect |
| 2. C3×#64266 | C2 | semidirect |
| 3. C3×#64265 | C2 | semidirect |
| 4. #11724207 | C2 | semidirect |
| 5. #11724206 | C2 | semidirect |
| 12. C3×C3⋊GL(2,𝔽3) | C22 | semidirect |
| 13. C3×C3⋊#226 | C22 | semidirect |
| 14. C3×C3⋊#226 | C22 | semidirect |
| 15. #64366 | D3 | semidirect |
| 20. #64266 | D3 | semidirect |
| 21. #64265 | D3 | semidirect |
| 71. C3⋊GL(2,𝔽3) | C2×D3 | semidirect |
| 72. C3⋊#226 | C2×D3 | semidirect |
| 73. C3⋊#226 | C2×D3 | semidirect |
| 428. C32 | #5924 | semidirect |
| 483. C3 | #10596447 | semidirect |
| 484. C3 | #10596450 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724491 | 2 |
| C3×#64266 | 2 |
| C3×#64265 | 2 |
| #11724207 | 2 |
| #11724206 | 2 |
| #10596450 | 3 |
| #10596447 | 3 |
| #10595473 | 3 |
| #89040 | 4 |