Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
444 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 85 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 | #3584 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2148 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #2143 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #2117 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1815 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #585 | — | 2 | 1 | no | — |
| 9 | #578 | — | 2 | 1 | no | — |
| 10 | #578 | C4 | 1 | 2 | yes | complemented |
| 11 | #577 | — | 2 | 2 | no | — |
| 12 | #577 | — | 2 | 2 | no | — |
| 13 | #576 | — | 2 | 1 | no | — |
| 14 | #576 | C4 | 1 | 2 | yes | complemented |
| 15 | #575 | — | 2 | 2 | no | — |
| 16 | #575 | — | 2 | 1 | no | — |
| 17 | C2×#135 | — | 2 | 2 | no | — |
| 18 | C2×#135 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#135 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#131 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×M32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×M32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×#108 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×#106 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | #417 | — | 2 | 1 | no | — |
| 26 | #362 | — | 2 | 1 | no | — |
| 27 | #361 | — | 2 | 1 | no | — |
| 28 | #360 | — | 2 | 2 | no | — |
| 29 | #360 | — | 2 | 2 | no | — |
| 30 | #359 | — | 2 | 2 | no | — |
| 31 | #350 | — | 2 | 2 | no | — |
| 32 | #343 | — | 2 | 1 | no | — |
| 33 | 2−1+4 | — | 2 | 2 | no | — |
| 34 | 2−1+4 | — | 2 | 2 | no | — |
| 35 | 2−1+4 | — | 4 | 1 | no | — |
| 36 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 37 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 38 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 39 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 40 | C2×#41 | — | 2 | 4 | no | — |
| 41 | C2×#41 | — | 2 | 1 | no | — |
| 42 | C2×#41 | — | 2 | 1 | no | — |
| 43 | C2×#41 | — | 4 | 2 | no | — |
| 44 | C2×#41 | — | 4 | 2 | no | — |
| 45 | C2×#41 | D4 | 1 | 1 | yes | characteristic |
| 46 | #137 | — | 2 | 2 | no | — |
| 47 | #137 | — | 2 | 4 | no | — |
| 48 | #137 | — | 2 | 4 | no | — |
| 49 | #137 | — | 2 | 2 | no | — |
| 50 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 51 | AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 52 | AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 53 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 54 | #135 | — | 2 | 4 | no | — |
| 55 | #135 | — | 2 | 4 | no | — |
| 56 | #135 | — | 2 | 2 | no | — |
| 57 | #135 | — | 2 | 2 | no | — |
| 58 | #135 | — | 2 | 1 | no | — |
| 59 | #135 | — | 2 | 1 | no | — |
| 60 | #135 | — | 2 | 1 | no | — |
| 61 | #135 | — | 2 | 1 | no | — |
| 62 | #135 | C2×C4 | 1 | 2 | yes | — |
| 63 | #135 | D4 | 1 | 2 | yes | — |
| 64 | #135 | D4 | 1 | 2 | yes | — |
| 65 | #135 | D4 | 1 | 2 | yes | — |
| 66 | C2×Q16 | — | 2 | 2 | no | — |
| 67 | C2×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×Q16 | D4 | 1 | 1 | yes | characteristic |
| 69 | C2×SD16 | — | 2 | 4 | no | — |
| 70 | C2×SD16 | — | 2 | 1 | no | — |
| 71 | C2×SD16 | — | 2 | 1 | no | — |
| 72 | C2×D8 | — | 2 | 2 | no | — |
| 73 | C2×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×D8 | D4 | 1 | 1 | yes | characteristic |
| 75 | #131 | — | 2 | 1 | no | — |
| 76 | #131 | — | 2 | 1 | no | — |
| 77 | #131 | C2×C4 | 1 | 2 | yes | — |
| 78 | #131 | D4 | 1 | 2 | yes | — |
| 79 | C2×M16 | — | 2 | 1 | no | — |
| 80 | C2×M16 | — | 2 | 1 | no | — |
| 81 | C2×M16 | D4 | 1 | 1 | yes | characteristic |
| 82 | C22×C8 | — | 2 | 1 | no | — |
| 83 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 84 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 85 | M32 | — | 2 | 1 | no | — |
| 86 | M32 | — | 2 | 1 | no | — |
| 87 | M32 | — | 2 | 1 | no | — |
| 88 | M32 | — | 2 | 1 | no | — |
| 89 | C2×C16 | — | 2 | 1 | no | — |
| 90 | C2×C16 | — | 2 | 1 | no | — |
| 91 | #108 | — | 2 | 1 | no | — |
| 92 | #108 | — | 2 | 1 | no | — |
| 93 | #106 | — | 2 | 1 | no | — |
| 94 | #106 | — | 2 | 1 | no | — |
| 95 | #103 | — | 4 | 1 | no | — |
| 96 | #102 | — | 4 | 1 | no | — |
| 97 | #41 | — | 2 | 1 | no | — |
| 98 | #41 | — | 2 | 1 | no | — |
| 99 | #41 | — | 2 | 4 | no | — |
| 100 | #41 | — | 2 | 4 | no | — |
| 101 | #41 | — | 2 | 2 | no | — |
| 102 | #41 | — | 2 | 2 | no | — |
| 103 | #41 | — | 2 | 2 | no | — |
| 104 | #41 | — | 2 | 2 | no | — |
| 105 | #41 | — | 4 | 4 | no | — |
| 106 | #41 | — | 4 | 4 | no | — |
| 107 | #41 | — | 4 | 4 | no | — |
| 108 | #41 | — | 4 | 4 | no | — |
| 109 | #41 | — | 4 | 2 | no | — |
| 110 | #41 | — | 4 | 2 | no | — |
| 111 | #41 | — | 4 | 2 | no | — |
| 112 | #41 | — | 4 | 2 | no | — |
| 113 | #41 | — | 4 | 1 | no | — |
| 114 | #41 | — | 4 | 1 | no | — |
| 115 | #41 | — | 4 | 1 | no | — |
| 116 | #41 | — | 4 | 1 | no | — |
| 117 | #41 | C22⋊C4 | 1 | 2 | yes | — |
| 118 | #41 | SD16 | 1 | 2 | yes | — |
| 119 | C2×Q8 | — | 2 | 4 | no | — |
| 120 | C2×Q8 | — | 2 | 1 | no | — |
| 121 | C2×Q8 | — | 2 | 1 | no | — |
| 122 | C2×Q8 | — | 4 | 2 | no | — |
| 123 | C2×Q8 | — | 4 | 2 | no | — |
| 124 | C2×Q8 | D8 | 1 | 1 | yes | characteristic |
| 125 | C2×D4 | — | 2 | 1 | no | — |
| 126 | C2×D4 | — | 2 | 4 | no | — |
| 127 | C2×D4 | — | 2 | 1 | no | — |
| 128 | C2×D4 | — | 2 | 1 | no | — |
| 129 | C2×D4 | — | 4 | 4 | no | — |
| 130 | C2×D4 | — | 4 | 4 | no | — |
| 131 | C2×D4 | — | 4 | 2 | no | — |
| 132 | C2×D4 | — | 4 | 4 | no | — |
| 133 | C2×D4 | — | 4 | 1 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | — | 4 | 2 | no | — |
| 136 | C2×D4 | D8 | 1 | 1 | yes | characteristic |
| 137 | C22×C4 | — | 2 | 1 | no | — |
| 138 | C22×C4 | — | 4 | 4 | no | — |
| 139 | C22×C4 | — | 4 | 1 | no | — |
| 140 | C22×C4 | — | 4 | 1 | no | — |
| 141 | C22×C4 | — | 4 | 1 | no | — |
| 142 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 143 | Q16 | — | 2 | 4 | no | — |
| 144 | Q16 | — | 2 | 1 | no | — |
| 145 | Q16 | — | 2 | 1 | no | — |
| 146 | Q16 | C22⋊C4 | 1 | 2 | yes | — |
| 147 | Q16 | C22⋊C4 | 1 | 2 | yes | — |
| 148 | SD16 | — | 2 | 4 | no | — |
| 149 | SD16 | — | 2 | 4 | no | — |
| 150 | SD16 | — | 2 | 2 | no | — |
| 151 | SD16 | — | 2 | 2 | no | — |
| 152 | SD16 | — | 2 | 2 | no | — |
| 153 | SD16 | — | 2 | 2 | no | — |
| 154 | D8 | — | 2 | 4 | no | — |
| 155 | D8 | — | 2 | 1 | no | — |
| 156 | D8 | — | 2 | 1 | no | — |
| 157 | D8 | C22⋊C4 | 1 | 2 | yes | — |
| 158 | D8 | C22⋊C4 | 1 | 2 | yes | — |
| 159 | M16 | — | 2 | 1 | no | — |
| 160 | M16 | — | 2 | 2 | no | — |
| 161 | M16 | — | 2 | 2 | no | — |
| 162 | M16 | — | 4 | 1 | no | — |
| 163 | M16 | — | 4 | 1 | no | — |
| 164 | M16 | C22⋊C4 | 1 | 2 | yes | — |
| 165 | C2×C8 | — | 2 | 1 | no | — |
| 166 | C2×C8 | — | 2 | 2 | no | — |
| 167 | C2×C8 | — | 2 | 2 | no | — |
| 168 | C2×C8 | — | 4 | 1 | no | — |
| 169 | C2×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 175 | C4⋊C4 | — | 4 | 1 | no | — |
| 176 | C4⋊C4 | — | 4 | 1 | no | — |
| 177 | C16 | — | 4 | 1 | no | — |
| 178 | C16 | — | 4 | 1 | no | — |
| 179 | C23 | — | 2 | 1 | no | — |
| 180 | C23 | — | 4 | 4 | no | — |
| 181 | C23 | — | 4 | 1 | no | — |
| 182 | C23 | — | 4 | 1 | no | — |
| 183 | C23 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 184 | Q8 | — | 2 | 1 | no | — |
| 185 | Q8 | — | 2 | 2 | no | — |
| 186 | Q8 | — | 2 | 2 | no | — |
| 187 | Q8 | — | 4 | 4 | no | — |
| 188 | Q8 | — | 4 | 1 | no | — |
| 189 | Q8 | — | 4 | 1 | no | — |
| 190 | Q8 | #102 | 1 | 2 | yes | — |
| 191 | D4 | — | 2 | 2 | no | — |
| 192 | D4 | — | 2 | 2 | no | — |
| 193 | D4 | — | 2 | 2 | no | — |
| 194 | D4 | — | 2 | 2 | no | — |
| 195 | D4 | — | 2 | 1 | no | — |
| 196 | D4 | — | 4 | 4 | no | — |
| 197 | D4 | — | 4 | 4 | no | — |
| 198 | D4 | — | 4 | 4 | no | — |
| 199 | D4 | — | 4 | 2 | no | — |
| 200 | D4 | — | 4 | 2 | no | — |
| 201 | D4 | — | 4 | 2 | no | — |
| 202 | D4 | — | 4 | 2 | no | — |
| 203 | D4 | — | 4 | 1 | no | — |
| 204 | D4 | — | 4 | 1 | no | — |
| 205 | D4 | #102 | 1 | 2 | yes | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 2 | no | — |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 4 | no | — |
| 210 | C2×C4 | — | 4 | 4 | no | — |
| 211 | C2×C4 | — | 4 | 4 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 4 | 2 | no | — |
| 215 | C2×C4 | — | 4 | 2 | no | — |
| 216 | C2×C4 | — | 4 | 2 | no | — |
| 217 | C2×C4 | — | 4 | 2 | no | — |
| 218 | C2×C4 | — | 8 | 1 | no | — |
| 219 | C2×C4 | — | 16 | 1 | no | — |
| 220 | C2×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 226 | C8 | — | 2 | 2 | no | — |
| 227 | C8 | — | 8 | 1 | no | — |
| 228 | C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 229 | C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 230 | C8 | #120 | 1 | 1 | yes | characteristic |
| 231 | C8 | #120 | 1 | 1 | yes | characteristic |
| 232 | C22 | — | 2 | 2 | no | — |
| 233 | C22 | — | 4 | 1 | no | — |
| 234 | C22 | — | 4 | 2 | no | — |
| 235 | C22 | — | 4 | 2 | no | — |
| 236 | C22 | — | 8 | 1 | no | — |
| 237 | C22 | — | 16 | 1 | no | — |
| 238 | C22 | — | 16 | 1 | no | — |
| 239 | C22 | — | 16 | 4 | no | — |
| 240 | C22 | #418 | 1 | 1 | yes | characteristic |
| 241 | C22 | #447 | 1 | 1 | yes | characteristic |
| 242 | C22 | #450 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 2 | no | — |
| 244 | C4 | — | 4 | 2 | no | — |
| 245 | C4 | — | 4 | 2 | no | — |
| 246 | C4 | — | 16 | 1 | no | — |
| 247 | C4 | #379 | 1 | 1 | yes | characteristic |
| 248 | C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 249 | C4 | #447 | 1 | 1 | yes | characteristic |
| 250 | C4 | #450 | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 2 | 1 | no | — |
| 252 | C2 | — | 2 | 1 | no | — |
| 253 | C2 | — | 2 | 1 | no | — |
| 254 | C2 | — | 4 | 2 | no | — |
| 255 | C2 | — | 8 | 2 | no | — |
| 256 | C2 | — | 8 | 2 | no | — |
| 257 | C2 | #1786 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2148 | C2 | semidirect |
| 3. C2×#360 | C2 | semidirect |
| 4. C2×#360 | C2 | semidirect |
| 5. #2143 | C2 | semidirect |
| 6. #2117 | C2 | semidirect |
| 7. #1815 | C2 | semidirect |
| 10. #578 | C4 | semidirect |
| 14. #576 | C4 | semidirect |
| 21. C2×M32 | C22 | semidirect |
| 22. C2×M32 | C22 | semidirect |
| 23. C2×#108 | C22 | semidirect |
| 24. C2×#106 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3584 | 2 |
| #2148 | 2 |
| C2×#360 | 2 |
| #2143 | 2 |
| #2117 | 2 |
| #1815 | 2 |