Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
696 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 111 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 | C2×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×B(4,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#455 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2199 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 5 | C2×#354 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×AΣL(1,𝔽16) | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#137 | — | 2 | 1 | no | — |
| 9 | H(3,𝔽4) | — | 2 | 2 | no | — |
| 10 | H(3,𝔽4) | — | 2 | 4 | no | — |
| 11 | H(3,𝔽4) | C22 | 1 | 4 | yes | complemented |
| 12 | C2×#124 | — | 2 | 1 | no | — |
| 13 | C2×#124 | — | 2 | 2 | no | — |
| 14 | C2×#124 | — | 2 | 1 | no | — |
| 15 | C2×#124 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#120 | — | 2 | 2 | no | — |
| 17 | C2×#120 | — | 2 | 2 | no | — |
| 18 | C2×#120 | C22 | 1 | 2 | yes | — |
| 19 | B(4,𝔽2) | — | 2 | 4 | no | — |
| 20 | B(4,𝔽2) | C22 | 1 | 8 | yes | complemented |
| 21 | #455 | — | 2 | 2 | no | — |
| 22 | #455 | C22 | 1 | 4 | yes | complemented |
| 23 | C2×#104 | — | 2 | 1 | no | — |
| 24 | C2×#100 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#99 | — | 2 | 2 | no | — |
| 26 | C2×#99 | C22 | 1 | 2 | yes | — |
| 27 | #354 | C22 | 1 | 4 | yes | complemented |
| 28 | AΣL(1,𝔽16) | C22 | 1 | 8 | yes | complemented |
| 29 | C25 | D4 | 1 | 2 | yes | complemented |
| 30 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 31 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 32 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 33 | H(5,𝔽2) | C23 | 1 | 2 | yes | — |
| 34 | C2×#41 | — | 2 | 1 | no | — |
| 35 | C2×#41 | — | 4 | 1 | no | — |
| 36 | C22×Q8 | — | 2 | 1 | no | — |
| 37 | C22×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 38 | C22×D4 | — | 2 | 2 | no | — |
| 39 | C22×D4 | — | 2 | 1 | no | — |
| 40 | C22×D4 | — | 2 | 1 | no | — |
| 41 | C22×D4 | — | 2 | 2 | no | — |
| 42 | C22×D4 | — | 4 | 1 | no | — |
| 43 | C22×D4 | — | 4 | 2 | no | — |
| 44 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 45 | #137 | — | 2 | 4 | no | — |
| 46 | #137 | — | 4 | 2 | no | — |
| 47 | C2×Q16 | — | 4 | 1 | no | — |
| 48 | C2×SD16 | — | 4 | 1 | no | — |
| 49 | C2×M16 | — | 2 | 1 | no | — |
| 50 | #124 | — | 2 | 8 | no | — |
| 51 | #124 | — | 2 | 4 | no | — |
| 52 | #124 | — | 2 | 4 | no | — |
| 53 | #124 | — | 2 | 4 | no | — |
| 54 | #124 | — | 2 | 2 | no | — |
| 55 | #124 | — | 2 | 8 | no | — |
| 56 | #124 | — | 2 | 4 | no | — |
| 57 | #124 | D4 | 1 | 2 | yes | complemented |
| 58 | #124 | C23 | 1 | 2 | yes | — |
| 59 | #120 | — | 2 | 8 | no | — |
| 60 | #120 | — | 2 | 8 | no | — |
| 61 | #120 | — | 2 | 2 | no | — |
| 62 | #120 | — | 2 | 2 | no | — |
| 63 | #120 | — | 2 | 8 | no | — |
| 64 | #120 | — | 4 | 4 | no | — |
| 65 | #120 | D4 | 1 | 4 | yes | complemented |
| 66 | #120 | C23 | 1 | 4 | yes | — |
| 67 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 68 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 69 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 70 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 71 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 72 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 73 | C2×C22⋊C4 | D4 | 1 | 2 | yes | complemented |
| 74 | C2×C42 | — | 2 | 1 | no | — |
| 75 | C2×C42 | D4 | 1 | 1 | yes | characteristic, complemented |
| 76 | #104 | — | 2 | 4 | no | — |
| 77 | #100 | — | 2 | 1 | no | — |
| 78 | #100 | C23 | 1 | 2 | yes | — |
| 79 | #99 | — | 2 | 2 | no | — |
| 80 | #99 | — | 2 | 8 | no | — |
| 81 | #99 | C23 | 1 | 4 | yes | — |
| 82 | C24 | — | 2 | 2 | no | — |
| 83 | C24 | — | 2 | 2 | no | — |
| 84 | C24 | — | 2 | 2 | no | — |
| 85 | C24 | — | 4 | 2 | no | — |
| 86 | C24 | — | 4 | 2 | no | — |
| 87 | C24 | — | 8 | 4 | no | — |
| 88 | C24 | C2×D4 | 1 | 2 | yes | — |
| 89 | C24 | C2×D4 | 1 | 4 | yes | complemented |
| 90 | #41 | — | 2 | 2 | no | — |
| 91 | #41 | — | 2 | 2 | no | — |
| 92 | #41 | — | 4 | 2 | no | — |
| 93 | #41 | — | 4 | 1 | no | — |
| 94 | #41 | — | 4 | 4 | no | — |
| 95 | #41 | — | 4 | 2 | no | — |
| 96 | #41 | — | 4 | 1 | no | — |
| 97 | C2×Q8 | — | 2 | 4 | no | — |
| 98 | C2×Q8 | — | 2 | 1 | no | — |
| 99 | C2×Q8 | — | 2 | 1 | no | — |
| 100 | C2×Q8 | — | 4 | 1 | no | — |
| 101 | C2×Q8 | — | 4 | 1 | no | — |
| 102 | C2×Q8 | — | 8 | 2 | no | — |
| 103 | C2×Q8 | C2×D4 | 1 | 2 | yes | complemented |
| 104 | C2×D4 | — | 2 | 1 | no | — |
| 105 | C2×D4 | — | 2 | 4 | no | — |
| 106 | C2×D4 | — | 2 | 4 | no | — |
| 107 | C2×D4 | — | 2 | 4 | no | — |
| 108 | C2×D4 | — | 2 | 4 | no | — |
| 109 | C2×D4 | — | 2 | 4 | no | — |
| 110 | C2×D4 | — | 2 | 4 | no | — |
| 111 | C2×D4 | — | 4 | 1 | no | — |
| 112 | C2×D4 | — | 4 | 1 | no | — |
| 113 | C2×D4 | — | 4 | 2 | no | — |
| 114 | C2×D4 | — | 4 | 2 | no | — |
| 115 | C2×D4 | — | 4 | 2 | no | — |
| 116 | C2×D4 | — | 4 | 2 | no | — |
| 117 | C2×D4 | — | 4 | 4 | no | — |
| 118 | C2×D4 | — | 4 | 2 | no | — |
| 119 | C2×D4 | — | 4 | 1 | no | — |
| 120 | C2×D4 | — | 4 | 8 | no | — |
| 121 | C2×D4 | — | 8 | 4 | no | — |
| 122 | C2×D4 | — | 8 | 2 | no | — |
| 123 | C2×D4 | — | 8 | 1 | no | — |
| 124 | C2×D4 | — | 8 | 1 | no | — |
| 125 | C2×D4 | — | 8 | 8 | no | — |
| 126 | C2×D4 | C2×D4 | 1 | 2 | yes | — |
| 127 | C2×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×D4 | C24 | 1 | 1 | yes | characteristic |
| 129 | C22×C4 | — | 2 | 2 | no | — |
| 130 | C22×C4 | — | 2 | 1 | no | — |
| 131 | C22×C4 | — | 2 | 2 | no | — |
| 132 | C22×C4 | — | 2 | 1 | no | — |
| 133 | C22×C4 | — | 4 | 1 | no | — |
| 134 | C22×C4 | — | 4 | 2 | no | — |
| 135 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | Q16 | — | 4 | 4 | no | — |
| 137 | SD16 | — | 4 | 4 | no | — |
| 138 | M16 | — | 2 | 2 | no | — |
| 139 | M16 | — | 4 | 1 | no | — |
| 140 | C2×C8 | — | 4 | 1 | no | — |
| 141 | C22⋊C4 | — | 2 | 8 | no | — |
| 142 | C22⋊C4 | — | 2 | 8 | no | — |
| 143 | C22⋊C4 | — | 2 | 4 | no | — |
| 144 | C22⋊C4 | — | 2 | 2 | no | — |
| 145 | C22⋊C4 | — | 2 | 4 | no | — |
| 146 | C22⋊C4 | — | 2 | 4 | no | — |
| 147 | C22⋊C4 | — | 2 | 4 | no | — |
| 148 | C22⋊C4 | — | 2 | 4 | no | — |
| 149 | C22⋊C4 | — | 4 | 8 | no | — |
| 150 | C22⋊C4 | — | 4 | 2 | no | — |
| 151 | C22⋊C4 | C2×D4 | 1 | 4 | yes | complemented |
| 152 | C42 | — | 2 | 4 | no | — |
| 153 | C42 | — | 2 | 1 | no | — |
| 154 | C42 | C2×D4 | 1 | 2 | yes | complemented |
| 155 | C23 | — | 2 | 2 | no | — |
| 156 | C23 | — | 2 | 2 | no | — |
| 157 | C23 | — | 2 | 4 | no | — |
| 158 | C23 | — | 2 | 4 | no | — |
| 159 | C23 | — | 4 | 4 | no | — |
| 160 | C23 | — | 4 | 4 | no | — |
| 161 | C23 | — | 4 | 4 | no | — |
| 162 | C23 | — | 4 | 4 | no | — |
| 163 | C23 | — | 4 | 2 | no | — |
| 164 | C23 | — | 4 | 8 | no | — |
| 165 | C23 | — | 4 | 2 | no | — |
| 166 | C23 | — | 4 | 1 | no | — |
| 167 | C23 | — | 4 | 2 | no | — |
| 168 | C23 | — | 8 | 2 | no | — |
| 169 | C23 | — | 8 | 4 | no | — |
| 170 | C23 | — | 8 | 4 | no | — |
| 171 | C23 | — | 8 | 4 | no | — |
| 172 | C23 | — | 8 | 8 | no | — |
| 173 | C23 | — | 8 | 4 | no | — |
| 174 | C23 | #120 | 1 | 2 | yes | — |
| 175 | C23 | #120 | 1 | 1 | yes | characteristic |
| 176 | C23 | C22×D4 | 1 | 2 | yes | — |
| 177 | Q8 | — | 2 | 2 | no | — |
| 178 | Q8 | — | 4 | 2 | no | — |
| 179 | Q8 | — | 4 | 1 | no | — |
| 180 | Q8 | — | 8 | 1 | no | — |
| 181 | Q8 | — | 8 | 4 | no | — |
| 182 | D4 | — | 2 | 2 | no | — |
| 183 | D4 | — | 4 | 1 | no | — |
| 184 | D4 | — | 4 | 2 | no | — |
| 185 | D4 | — | 4 | 1 | no | — |
| 186 | D4 | — | 4 | 1 | no | — |
| 187 | D4 | — | 4 | 4 | no | — |
| 188 | D4 | — | 4 | 4 | no | — |
| 189 | D4 | — | 4 | 2 | no | — |
| 190 | D4 | — | 4 | 2 | no | — |
| 191 | D4 | — | 8 | 4 | no | — |
| 192 | D4 | — | 8 | 4 | no | — |
| 193 | D4 | — | 8 | 4 | no | — |
| 194 | D4 | — | 8 | 4 | no | — |
| 195 | D4 | — | 8 | 16 | no | — |
| 196 | C2×C4 | — | 2 | 1 | no | — |
| 197 | C2×C4 | — | 2 | 1 | no | — |
| 198 | C2×C4 | — | 2 | 4 | no | — |
| 199 | C2×C4 | — | 2 | 2 | no | — |
| 200 | C2×C4 | — | 2 | 4 | no | — |
| 201 | C2×C4 | — | 2 | 2 | no | — |
| 202 | C2×C4 | — | 4 | 2 | no | — |
| 203 | C2×C4 | — | 4 | 2 | no | — |
| 204 | C2×C4 | — | 4 | 2 | no | — |
| 205 | C2×C4 | — | 4 | 4 | no | — |
| 206 | C2×C4 | — | 4 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 1 | no | — |
| 208 | C2×C4 | — | 4 | 1 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 4 | 4 | no | — |
| 211 | C2×C4 | — | 8 | 4 | no | — |
| 212 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 4 | 2 | no | — |
| 215 | C22 | — | 2 | 2 | no | — |
| 216 | C22 | — | 2 | 2 | no | — |
| 217 | C22 | — | 2 | 2 | no | — |
| 218 | C22 | — | 4 | 2 | no | — |
| 219 | C22 | — | 4 | 2 | no | — |
| 220 | C22 | — | 4 | 4 | no | — |
| 221 | C22 | — | 4 | 4 | no | — |
| 222 | C22 | — | 4 | 4 | no | — |
| 223 | C22 | — | 4 | 8 | no | — |
| 224 | C22 | — | 4 | 2 | no | — |
| 225 | C22 | — | 4 | 4 | no | — |
| 226 | C22 | — | 8 | 2 | no | — |
| 227 | C22 | — | 8 | 4 | no | — |
| 228 | C22 | — | 8 | 4 | no | — |
| 229 | C22 | — | 8 | 4 | no | — |
| 230 | C22 | — | 8 | 8 | no | — |
| 231 | C22 | — | 8 | 4 | no | — |
| 232 | C22 | — | 8 | 2 | no | — |
| 233 | C22 | — | 16 | 2 | no | — |
| 234 | C22 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 235 | C22 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 236 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 237 | C4 | — | 2 | 1 | no | — |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 4 | 4 | no | — |
| 240 | C4 | — | 4 | 2 | no | — |
| 241 | C4 | — | 4 | 2 | no | — |
| 242 | C4 | — | 4 | 2 | no | — |
| 243 | C4 | — | 8 | 4 | no | — |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 2 | 1 | no | — |
| 246 | C2 | — | 4 | 2 | no | — |
| 247 | C2 | — | 4 | 2 | no | — |
| 248 | C2 | — | 8 | 4 | no | — |
| 249 | C2 | — | 8 | 2 | no | — |
| 250 | C2 | #2199 | 1 | 2 | yes | direct product factor |
| 251 | C2 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽4) | C2 | semidirect |
| 2. C2×B(4,𝔽2) | C2 | semidirect |
| 3. C2×#455 | C2 | semidirect |
| 4. #2199 | C2 | direct (N × H) |
| 5. C2×#354 | C2 | semidirect |
| 6. C2×AΣL(1,𝔽16) | C2 | semidirect |
| 11. H(3,𝔽4) | C22 | semidirect |
| 20. B(4,𝔽2) | C22 | semidirect |
| 22. #455 | C22 | semidirect |
| 27. #354 | C22 | semidirect |
| 28. AΣL(1,𝔽16) | C22 | semidirect |
| 29. C25 | D4 | semidirect |
| 37. C22×Q8 | D4 | semidirect |
| 57. #124 | D4 | semidirect |
| 65. #120 | D4 | semidirect |
| 73. C2×C22⋊C4 | D4 | semidirect |
| 75. C2×C42 | D4 | semidirect |
| 89. C24 | C2×D4 | semidirect |
| 103. C2×Q8 | C2×D4 | semidirect |
| 151. C22⋊C4 | C2×D4 | semidirect |
| 154. C42 | C2×D4 | semidirect |
| 250. C2 | #2199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4) | 2 |
| C2×B(4,𝔽2) | 2 |
| C2×#455 | 2 |
| #2199 | 2 |
| C2×#354 | 2 |
| C2×AΣL(1,𝔽16) | 2 |