Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
384 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 46 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×C3⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×C3⋊(#131⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10595449 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5835⋊C3 | — | 3 | 1 | no | maximal |
| 6 | #5752⋊C3 | — | 3 | 1 | no | maximal |
| 7 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3⋊(C3⋊(#41⋊C3)) | — | 2 | 1 | no | — |
| 9 | C3×C3⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 10 | C3×#3719 | — | 4 | 1 | no | maximal |
| 11 | #64366 | C2×C3 | 1 | 1 | yes | characteristic |
| 12 | C3×#1009 | C2×C3 | 1 | 1 | yes | characteristic |
| 13 | C3×#994 | — | 3 | 1 | no | — |
| 14 | C3⋊#994 | — | 2 | 1 | no | — |
| 15 | C3⋊#994 | — | 3 | 1 | no | — |
| 16 | C3⋊#994 | — | 3 | 1 | no | — |
| 17 | C3⋊#994 | — | 6 | 1 | no | — |
| 18 | C3⋊#994 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C3×#948 | C2×C3 | 1 | 1 | yes | characteristic |
| 20 | C3×#131⋊C3 | — | 3 | 1 | no | — |
| 21 | #63830 | — | 3 | 1 | no | — |
| 22 | #63829 | — | 3 | 1 | no | — |
| 23 | #63825 | — | 3 | 1 | no | — |
| 24 | C2×#3714 | — | 3 | 1 | no | — |
| 25 | C3⋊(#131⋊C3) | — | 2 | 1 | no | — |
| 26 | C3⋊(#131⋊C3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C3×#634 | — | 4 | 1 | no | — |
| 28 | C32×Dic6 | — | 4 | 1 | no | — |
| 29 | C32×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 30 | C32×C3⋊C8 | — | 4 | 1 | no | — |
| 31 | #5835 | — | 3 | 1 | no | — |
| 32 | #5752 | — | 3 | 1 | no | — |
| 33 | #576⋊C3 | — | 9 | 1 | no | — |
| 34 | C32×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 35 | #3832 | — | 2 | 1 | no | — |
| 36 | D4×C3⋊D3 | — | 6 | 1 | no | — |
| 37 | #3828 | — | 3 | 1 | no | — |
| 38 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 39 | C3×#239 | — | 2 | 1 | no | — |
| 40 | C3×D3×D4 | — | 6 | 1 | no | — |
| 41 | C3×#235 | — | 3 | 1 | no | — |
| 42 | C2×C3×D12 | — | 3 | 1 | no | — |
| 43 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 44 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 45 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 46 | C3×#41⋊C3 | — | 4 | 1 | no | — |
| 47 | C3×#41⋊C3 | — | 6 | 1 | no | — |
| 48 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 50 | C3⋊(#41⋊C3) | — | 2 | 2 | no | — |
| 51 | C3⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 52 | C3⋊(#41⋊C3) | — | 6 | 2 | no | — |
| 53 | C3⋊(#41⋊C3) | — | 6 | 1 | no | — |
| 54 | C3⋊(#41⋊C3) | — | 6 | 1 | no | — |
| 55 | C3×#216 | — | 12 | 1 | no | — |
| 56 | C3×#208 | — | 3 | 1 | no | — |
| 57 | C2×C3×C3⋊C8 | — | 3 | 1 | no | — |
| 58 | C3×Dic12 | — | 12 | 1 | no | — |
| 59 | #3719 | A4 | 1 | 1 | yes | characteristic |
| 60 | #3717 | — | 3 | 1 | no | — |
| 61 | #3716 | — | 3 | 1 | no | — |
| 62 | #3714 | — | 3 | 2 | no | — |
| 63 | #3714 | — | 3 | 1 | no | — |
| 64 | C33×C4 | — | 4 | 1 | no | — |
| 65 | C3×C3⋊Dic3 | — | 8 | 1 | no | — |
| 66 | C32×Dic3 | — | 8 | 1 | no | — |
| 67 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 68 | #1009 | — | 3 | 1 | no | — |
| 69 | #1009 | — | 3 | 1 | no | — |
| 70 | #1009 | — | 6 | 1 | no | — |
| 71 | #1009 | C3×D3 | 1 | 1 | yes | characteristic |
| 72 | #994 | — | 3 | 1 | no | — |
| 73 | #994 | — | 6 | 1 | no | — |
| 74 | #994 | — | 9 | 1 | no | — |
| 75 | C3×#131 | — | 3 | 1 | no | — |
| 76 | #950 | — | 9 | 1 | no | — |
| 77 | #949 | — | 9 | 1 | no | — |
| 78 | #948 | C3×D3 | 1 | 1 | yes | characteristic |
| 79 | #932 | — | 9 | 1 | no | — |
| 80 | C2×#213 | — | 9 | 1 | no | — |
| 81 | #908 | — | 9 | 1 | no | — |
| 82 | #908 | — | 9 | 1 | no | — |
| 83 | #904 | — | 9 | 1 | no | — |
| 84 | C2×D24 | — | 9 | 1 | no | — |
| 85 | #131⋊C3 | — | 3 | 1 | no | — |
| 86 | #131⋊C3 | — | 6 | 1 | no | — |
| 87 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 88 | C22×C3×D3 | — | 6 | 1 | no | — |
| 89 | C32×Q8 | — | 12 | 1 | no | — |
| 90 | C32×Q8 | C3×D4 | 1 | 1 | yes | characteristic |
| 91 | C32×D4 | — | 3 | 1 | no | — |
| 92 | C2×C32×C4 | — | 3 | 1 | no | — |
| 93 | #638 | — | 6 | 1 | no | — |
| 94 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 95 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 96 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 97 | #634 | C2×A4 | 1 | 1 | yes | characteristic |
| 98 | C3×#67 | — | 6 | 1 | no | — |
| 99 | C3×D12 | — | 3 | 1 | no | — |
| 100 | C3×D12 | — | 6 | 1 | no | — |
| 101 | C3×C4×D3 | — | 6 | 1 | no | — |
| 102 | C3×Dic6 | — | 4 | 1 | no | — |
| 103 | C3×Dic6 | — | 8 | 1 | no | — |
| 104 | C3×Dic6 | — | 12 | 1 | no | — |
| 105 | C3×Dic6 | — | 12 | 1 | no | — |
| 106 | C3×Dic6 | — | 24 | 1 | no | — |
| 107 | C3×Dic6 | C2×A4 | 1 | 1 | yes | characteristic |
| 108 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 109 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 110 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 111 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 112 | C3×SL(2,𝔽3) | D12 | 1 | 1 | yes | characteristic |
| 113 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 114 | C32×C8 | — | 12 | 1 | no | — |
| 115 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 116 | C3×C3⋊C8 | — | 4 | 1 | no | — |
| 117 | C3×C3⋊C8 | — | 8 | 1 | no | — |
| 118 | C3×C3⋊C8 | C2×A4 | 1 | 1 | yes | characteristic |
| 119 | #576 | — | 9 | 1 | no | Sylow |
| 120 | C2×C33 | — | 4 | 1 | no | — |
| 121 | C3×#41 | — | 2 | 1 | no | — |
| 122 | C3×#41 | — | 6 | 1 | no | — |
| 123 | C3×#41 | — | 9 | 1 | no | — |
| 124 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 125 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×D4 | — | 9 | 1 | no | — |
| 127 | C2×C3×D4 | — | 18 | 1 | no | — |
| 128 | #239 | — | 2 | 1 | no | — |
| 129 | #239 | — | 6 | 2 | no | — |
| 130 | #239 | — | 6 | 1 | no | — |
| 131 | #239 | — | 6 | 1 | no | — |
| 132 | D3×D4 | — | 6 | 1 | no | — |
| 133 | D3×D4 | — | 18 | 1 | no | — |
| 134 | D3×D4 | — | 18 | 1 | no | — |
| 135 | D3×D4 | — | 18 | 2 | no | — |
| 136 | #235 | — | 3 | 1 | no | — |
| 137 | #235 | — | 9 | 1 | no | — |
| 138 | #235 | — | 9 | 1 | no | — |
| 139 | #235 | — | 18 | 1 | no | — |
| 140 | C2×D12 | — | 3 | 1 | no | — |
| 141 | C2×D12 | — | 9 | 1 | no | — |
| 142 | C2×D12 | — | 9 | 1 | no | — |
| 143 | C2×D12 | — | 18 | 1 | no | — |
| 144 | #41⋊C3 | — | 2 | 1 | no | — |
| 145 | #41⋊C3 | — | 2 | 1 | no | — |
| 146 | #41⋊C3 | — | 4 | 1 | no | — |
| 147 | #41⋊C3 | — | 6 | 2 | no | — |
| 148 | #41⋊C3 | — | 6 | 1 | no | — |
| 149 | #41⋊C3 | — | 18 | 1 | no | — |
| 150 | #41⋊C3 | D32 | 1 | 1 | yes | characteristic |
| 151 | C3×Q16 | — | 36 | 1 | no | — |
| 152 | C3×M16 | — | 9 | 1 | no | — |
| 153 | C2×C3×C8 | — | 9 | 1 | no | — |
| 154 | #216 | — | 3 | 1 | no | — |
| 155 | #215 | — | 9 | 1 | no | — |
| 156 | #214 | — | 9 | 1 | no | — |
| 157 | #213 | — | 9 | 2 | no | — |
| 158 | #213 | — | 9 | 1 | no | — |
| 159 | #208 | — | 3 | 1 | no | — |
| 160 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 161 | Dic12 | — | 3 | 1 | no | — |
| 162 | D24 | — | 9 | 1 | no | — |
| 163 | D24 | — | 9 | 2 | no | — |
| 164 | #204 | — | 9 | 1 | no | — |
| 165 | #204 | — | 9 | 1 | no | — |
| 166 | C22×C32 | — | 3 | 1 | no | — |
| 167 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 168 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 169 | C2×C3×D3 | — | 6 | 1 | no | — |
| 170 | C2×C3×D3 | — | 24 | 1 | no | — |
| 171 | C32×C4 | — | 3 | 1 | no | — |
| 172 | C32×C4 | — | 4 | 1 | no | — |
| 173 | C32×C4 | — | 4 | 1 | no | — |
| 174 | C32×C4 | — | 8 | 1 | no | — |
| 175 | C32×C4 | — | 8 | 1 | no | — |
| 176 | C32×C4 | — | 8 | 1 | no | — |
| 177 | C32×C4 | — | 16 | 1 | no | — |
| 178 | C32×C4 | — | 24 | 1 | no | — |
| 179 | C32×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 180 | C3⋊Dic3 | — | 2 | 1 | no | — |
| 181 | C3×Dic3 | — | 2 | 1 | no | — |
| 182 | C3×Dic3 | — | 8 | 1 | no | — |
| 183 | C3×Dic3 | — | 8 | 2 | no | — |
| 184 | C3×Dic3 | — | 24 | 1 | no | — |
| 185 | C3×Dic3 | — | 24 | 1 | no | — |
| 186 | C3×Dic3 | — | 24 | 2 | no | — |
| 187 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 188 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 189 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 190 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 191 | #135 | — | 27 | 1 | no | — |
| 192 | C2×D8 | — | 27 | 1 | no | — |
| 193 | #131 | — | 3 | 1 | no | — |
| 194 | C33 | — | 4 | 1 | no | Sylow |
| 195 | C23×C3 | — | 18 | 1 | no | — |
| 196 | C22×D3 | — | 6 | 1 | no | — |
| 197 | C22×D3 | — | 18 | 2 | no | — |
| 198 | C22×D3 | — | 18 | 1 | no | — |
| 199 | C22×D3 | — | 18 | 1 | no | — |
| 200 | C3×Q8 | — | 2 | 1 | no | — |
| 201 | C3×Q8 | — | 3 | 1 | no | — |
| 202 | C3×Q8 | — | 12 | 1 | no | — |
| 203 | C3×Q8 | — | 24 | 1 | no | — |
| 204 | C3×Q8 | — | 36 | 1 | no | — |
| 205 | C3×Q8 | C3×D12 | 1 | 1 | yes | characteristic |
| 206 | C3×Q8 | C3×#67 | 1 | 1 | yes | characteristic |
| 207 | C3×D4 | — | 3 | 1 | no | — |
| 208 | C3×D4 | — | 3 | 1 | no | — |
| 209 | C3×D4 | — | 6 | 1 | no | — |
| 210 | C3×D4 | — | 9 | 1 | no | — |
| 211 | C3×D4 | — | 18 | 1 | no | — |
| 212 | C3×D4 | — | 18 | 1 | no | — |
| 213 | C2×C3×C4 | — | 3 | 1 | no | — |
| 214 | C2×C3×C4 | — | 3 | 1 | no | — |
| 215 | C2×C3×C4 | — | 6 | 1 | no | — |
| 216 | C2×C3×C4 | — | 18 | 1 | no | — |
| 217 | #67 | — | 6 | 1 | no | — |
| 218 | #67 | — | 18 | 1 | no | — |
| 219 | #67 | — | 18 | 1 | no | — |
| 220 | #67 | — | 18 | 2 | no | — |
| 221 | D12 | — | 3 | 1 | no | — |
| 222 | D12 | — | 6 | 1 | no | — |
| 223 | D12 | — | 9 | 1 | no | — |
| 224 | D12 | — | 9 | 1 | no | — |
| 225 | D12 | — | 18 | 1 | no | — |
| 226 | D12 | — | 18 | 1 | no | — |
| 227 | D12 | — | 18 | 1 | no | — |
| 228 | D12 | — | 18 | 2 | no | — |
| 229 | C4×D3 | — | 6 | 1 | no | — |
| 230 | C4×D3 | — | 18 | 1 | no | — |
| 231 | C4×D3 | — | 18 | 1 | no | — |
| 232 | C4×D3 | — | 18 | 2 | no | — |
| 233 | Dic6 | — | 3 | 1 | no | — |
| 234 | Dic6 | — | 3 | 1 | no | — |
| 235 | Dic6 | — | 6 | 1 | no | — |
| 236 | Dic6 | D3×A4 | 1 | 1 | yes | characteristic |
| 237 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 238 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 239 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 240 | SL(2,𝔽3) | #626 | 1 | 1 | yes | characteristic |
| 241 | C3×C8 | — | 3 | 1 | no | — |
| 242 | C3×C8 | — | 9 | 1 | no | — |
| 243 | C3×C8 | — | 12 | 1 | no | — |
| 244 | C3×C8 | — | 24 | 1 | no | — |
| 245 | C3⋊C8 | — | 3 | 1 | no | — |
| 246 | C3⋊C8 | D3×A4 | 1 | 1 | yes | characteristic |
| 247 | C2×C32 | — | 4 | 1 | no | — |
| 248 | C2×C32 | — | 4 | 1 | no | — |
| 249 | C2×C32 | — | 6 | 1 | no | — |
| 250 | C2×C32 | — | 8 | 1 | no | — |
| 251 | C2×C32 | — | 8 | 1 | no | — |
| 252 | C2×C32 | — | 8 | 1 | no | — |
| 253 | C2×C32 | — | 16 | 1 | no | — |
| 254 | C2×C32 | D4×A4 | 1 | 1 | yes | characteristic |
| 255 | C3⋊D3 | — | 12 | 1 | no | — |
| 256 | C3×D3 | — | 12 | 1 | no | — |
| 257 | #41 | — | 6 | 1 | no | — |
| 258 | #41 | — | 9 | 1 | no | — |
| 259 | #41 | — | 18 | 1 | no | — |
| 260 | #41 | — | 27 | 1 | no | — |
| 261 | #41 | C3×D32 | 1 | 1 | yes | characteristic |
| 262 | C2×D4 | — | 9 | 1 | no | — |
| 263 | C2×D4 | — | 18 | 1 | no | — |
| 264 | C2×D4 | — | 27 | 1 | no | — |
| 265 | C2×D4 | — | 54 | 1 | no | — |
| 266 | Q16 | — | 9 | 1 | no | — |
| 267 | SD16 | — | 27 | 1 | no | — |
| 268 | SD16 | — | 27 | 1 | no | — |
| 269 | D8 | — | 27 | 1 | no | — |
| 270 | D8 | — | 27 | 2 | no | — |
| 271 | M16 | — | 9 | 1 | no | — |
| 272 | C2×C8 | — | 9 | 1 | no | — |
| 273 | C22×C3 | — | 3 | 1 | no | — |
| 274 | C22×C3 | — | 3 | 1 | no | — |
| 275 | C22×C3 | — | 6 | 1 | no | — |
| 276 | C22×C3 | — | 18 | 1 | no | — |
| 277 | C22×C3 | — | 72 | 1 | no | — |
| 278 | C2×D3 | — | 6 | 1 | no | — |
| 279 | C2×D3 | — | 18 | 2 | no | — |
| 280 | C2×D3 | — | 18 | 1 | no | — |
| 281 | C2×D3 | — | 18 | 1 | no | — |
| 282 | C2×D3 | — | 24 | 1 | no | — |
| 283 | C2×D3 | — | 72 | 2 | no | — |
| 284 | C2×D3 | — | 72 | 1 | no | — |
| 285 | C2×D3 | — | 72 | 1 | no | — |
| 286 | C3×C4 | — | 2 | 1 | no | — |
| 287 | C3×C4 | — | 3 | 1 | no | — |
| 288 | C3×C4 | — | 3 | 1 | no | — |
| 289 | C3×C4 | — | 4 | 1 | no | — |
| 290 | C3×C4 | — | 6 | 1 | no | — |
| 291 | C3×C4 | — | 6 | 1 | no | — |
| 292 | C3×C4 | — | 8 | 1 | no | — |
| 293 | C3×C4 | — | 8 | 1 | no | — |
| 294 | C3×C4 | — | 16 | 1 | no | — |
| 295 | C3×C4 | — | 24 | 1 | no | — |
| 296 | C3×C4 | — | 24 | 2 | no | — |
| 297 | C3×C4 | — | 72 | 1 | no | — |
| 298 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 299 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 300 | Dic3 | — | 2 | 1 | no | — |
| 301 | Dic3 | — | 6 | 2 | no | — |
| 302 | Dic3 | — | 6 | 1 | no | — |
| 303 | Dic3 | — | 6 | 1 | no | — |
| 304 | C32 | — | 4 | 1 | no | — |
| 305 | C32 | — | 4 | 1 | no | — |
| 306 | C32 | — | 8 | 1 | no | — |
| 307 | C32 | — | 8 | 1 | no | — |
| 308 | C32 | — | 8 | 1 | no | — |
| 309 | C32 | — | 16 | 1 | no | — |
| 310 | C32 | #576⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 311 | C23 | — | 18 | 1 | no | — |
| 312 | C23 | — | 54 | 1 | no | — |
| 313 | Q8 | — | 3 | 1 | no | — |
| 314 | Q8 | — | 9 | 1 | no | — |
| 315 | Q8 | C3×#626 | 1 | 1 | yes | characteristic |
| 316 | D4 | — | 3 | 1 | no | — |
| 317 | D4 | — | 9 | 1 | no | — |
| 318 | D4 | — | 18 | 1 | no | — |
| 319 | D4 | — | 18 | 1 | no | — |
| 320 | D4 | — | 27 | 1 | no | — |
| 321 | D4 | — | 54 | 1 | no | — |
| 322 | D4 | — | 54 | 1 | no | — |
| 323 | C2×C4 | — | 3 | 1 | no | — |
| 324 | C2×C4 | — | 18 | 1 | no | — |
| 325 | C2×C4 | — | 54 | 1 | no | — |
| 326 | C8 | — | 3 | 1 | no | — |
| 327 | C8 | — | 9 | 1 | no | — |
| 328 | C2×C3 | — | 2 | 1 | no | — |
| 329 | C2×C3 | — | 4 | 1 | no | — |
| 330 | C2×C3 | — | 6 | 1 | no | — |
| 331 | C2×C3 | — | 6 | 1 | no | — |
| 332 | C2×C3 | — | 8 | 1 | no | — |
| 333 | C2×C3 | — | 8 | 1 | no | — |
| 334 | C2×C3 | — | 12 | 1 | no | — |
| 335 | C2×C3 | — | 16 | 1 | no | — |
| 336 | C2×C3 | — | 36 | 1 | no | — |
| 337 | C2×C3 | A4×D12 | 1 | 1 | yes | characteristic |
| 338 | C2×C3 | A4×#67 | 1 | 1 | yes | characteristic |
| 339 | D3 | — | 12 | 1 | no | — |
| 340 | D3 | — | 36 | 2 | no | — |
| 341 | D3 | — | 36 | 1 | no | — |
| 342 | D3 | — | 36 | 1 | no | — |
| 343 | C22 | — | 3 | 1 | no | — |
| 344 | C22 | — | 18 | 1 | no | — |
| 345 | C22 | — | 54 | 1 | no | — |
| 346 | C22 | — | 72 | 1 | no | — |
| 347 | C22 | — | 216 | 1 | no | — |
| 348 | C4 | — | 3 | 1 | no | — |
| 349 | C4 | — | 6 | 1 | no | — |
| 350 | C4 | — | 18 | 1 | no | — |
| 351 | C4 | D32×A4 | 1 | 1 | yes | characteristic |
| 352 | C3 | — | 2 | 1 | no | — |
| 353 | C3 | — | 4 | 1 | no | — |
| 354 | C3 | — | 8 | 1 | no | — |
| 355 | C3 | — | 8 | 1 | no | — |
| 356 | C3 | — | 16 | 1 | no | — |
| 357 | C3 | #5752⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 358 | C3 | #5835⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 359 | C2 | — | 6 | 1 | no | — |
| 360 | C2 | — | 36 | 1 | no | — |
| 361 | C2 | — | 108 | 1 | no | — |
| 362 | C2 | A4×#626 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×#626 |
| commutator, Fitting, cosocle | C32×#41 | C22×C3 |
| Frattini, hypercenter | C4 | D32×A4 |
| socle | C2×C32 | D4×A4 |
| hypocenter | C32×Q8 | C3×D4 |
| LCS | G ▷ C32×#41 ▷ C32×Q8 | |
| 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×C3⋊#994 | C2 | semidirect |
| 3. C3×C3⋊(#131⋊C3) | C2 | semidirect |
| 4. #10595449 | C3 | semidirect |
| 18. C3⋊#994 | D3 | semidirect |
| 26. C3⋊(#131⋊C3) | D3 | semidirect |
| 310. C32 | #576⋊C3 | semidirect |
| 357. C3 | #5752⋊C3 | semidirect |
| 358. C3 | #5835⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724491 | 2 |
| C3×C3⋊#994 | 2 |
| C3×C3⋊(#131⋊C3) | 2 |
| #10595449 | 3 |
| #5835⋊C3 | 3 |
| #5752⋊C3 | 3 |
| C3×#3719 | 4 |