Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
210 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #3288 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2202 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2199 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #1655 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #1406 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #1404 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1403 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #577 | — | 2 | 1 | no | — |
| 9 | #576 | — | 2 | 1 | no | — |
| 10 | H(3,𝔽4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #496 | — | 2 | 1 | no | — |
| 12 | #496 | C22 | 1 | 1 | yes | characteristic |
| 13 | #472 | — | 2 | 1 | no | — |
| 14 | #471 | — | 2 | 1 | no | — |
| 15 | B(4,𝔽2) | — | 2 | 1 | no | — |
| 16 | #455 | C22 | 1 | 1 | yes | characteristic |
| 17 | #454 | — | 2 | 1 | no | — |
| 18 | #454 | C22 | 1 | 1 | yes | characteristic |
| 19 | #453 | — | 2 | 1 | no | — |
| 20 | #444 | — | 2 | 1 | no | — |
| 21 | #355 | — | 2 | 1 | no | — |
| 22 | #355 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #354 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | AΣL(1,𝔽16) | — | 2 | 1 | no | — |
| 25 | #329 | — | 2 | 1 | no | — |
| 26 | #329 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | 2−1+4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 28 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 29 | H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 30 | #137 | — | 2 | 1 | no | — |
| 31 | #137 | — | 4 | 1 | no | — |
| 32 | AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 33 | AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 34 | AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 35 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 36 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 37 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 38 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 39 | #135 | — | 2 | 1 | no | — |
| 40 | #135 | — | 4 | 1 | no | — |
| 41 | #135 | — | 4 | 1 | no | — |
| 42 | C2×SD16 | — | 2 | 1 | no | — |
| 43 | C2×SD16 | — | 4 | 1 | no | — |
| 44 | C2×SD16 | — | 4 | 1 | no | — |
| 45 | C2×D8 | — | 2 | 1 | no | — |
| 46 | C2×D8 | — | 4 | 1 | no | — |
| 47 | C2×D8 | — | 4 | 1 | no | — |
| 48 | #131 | — | 2 | 1 | no | — |
| 49 | #131 | — | 2 | 1 | no | — |
| 50 | #127 | D4 | 1 | 1 | yes | characteristic, complemented |
| 51 | #124 | — | 2 | 1 | no | — |
| 52 | #124 | — | 2 | 1 | no | — |
| 53 | #124 | — | 4 | 1 | no | — |
| 54 | #124 | C23 | 1 | 1 | yes | characteristic |
| 55 | #120 | — | 2 | 1 | no | — |
| 56 | #120 | — | 4 | 1 | no | — |
| 57 | #120 | — | 4 | 1 | no | — |
| 58 | #108 | — | 2 | 1 | no | — |
| 59 | #104 | — | 2 | 1 | no | — |
| 60 | #104 | — | 2 | 1 | no | — |
| 61 | #104 | — | 4 | 1 | no | — |
| 62 | #101 | — | 2 | 1 | no | — |
| 63 | #101 | D4 | 1 | 1 | yes | characteristic, complemented |
| 64 | #100 | — | 2 | 1 | no | — |
| 65 | #100 | D4 | 1 | 1 | yes | characteristic |
| 66 | #99 | — | 2 | 1 | no | — |
| 67 | #99 | — | 4 | 1 | no | — |
| 68 | #97 | — | 2 | 1 | no | — |
| 69 | #97 | D4 | 1 | 1 | yes | characteristic, complemented |
| 70 | C24 | — | 2 | 1 | no | — |
| 71 | #41 | — | 2 | 1 | no | — |
| 72 | #41 | — | 2 | 1 | no | — |
| 73 | #41 | — | 4 | 1 | no | — |
| 74 | #41 | — | 4 | 1 | no | — |
| 75 | #41 | — | 4 | 1 | no | — |
| 76 | #41 | — | 4 | 1 | no | — |
| 77 | #41 | — | 4 | 1 | no | — |
| 78 | C2×Q8 | — | 2 | 1 | no | — |
| 79 | C2×Q8 | — | 4 | 1 | no | — |
| 80 | C2×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×D4 | — | 2 | 1 | no | — |
| 82 | C2×D4 | — | 2 | 1 | no | — |
| 83 | C2×D4 | — | 2 | 1 | no | — |
| 84 | C2×D4 | — | 4 | 1 | no | — |
| 85 | C2×D4 | — | 4 | 1 | no | — |
| 86 | C2×D4 | — | 4 | 1 | no | — |
| 87 | C2×D4 | — | 4 | 1 | no | — |
| 88 | C2×D4 | — | 4 | 1 | no | — |
| 89 | C2×D4 | — | 4 | 1 | no | — |
| 90 | C2×D4 | — | 8 | 1 | no | — |
| 91 | C2×D4 | — | 8 | 1 | no | — |
| 92 | C2×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | Q16 | — | 2 | 1 | no | — |
| 94 | Q16 | — | 4 | 1 | no | — |
| 95 | SD16 | — | 2 | 1 | no | — |
| 96 | SD16 | — | 2 | 1 | no | — |
| 97 | SD16 | — | 4 | 1 | no | — |
| 98 | SD16 | — | 4 | 1 | no | — |
| 99 | SD16 | — | 4 | 1 | no | — |
| 100 | SD16 | — | 4 | 1 | no | — |
| 101 | SD16 | — | 4 | 1 | no | — |
| 102 | SD16 | — | 4 | 1 | no | — |
| 103 | SD16 | — | 8 | 2 | no | — |
| 104 | SD16 | — | 8 | 1 | no | — |
| 105 | D8 | — | 2 | 1 | no | — |
| 106 | D8 | — | 4 | 1 | no | — |
| 107 | D8 | — | 4 | 1 | no | — |
| 108 | D8 | — | 4 | 1 | no | — |
| 109 | D8 | — | 4 | 1 | no | — |
| 110 | D8 | — | 4 | 1 | no | — |
| 111 | D8 | — | 8 | 2 | no | — |
| 112 | D8 | — | 8 | 1 | no | — |
| 113 | M16 | — | 2 | 1 | no | — |
| 114 | M16 | — | 2 | 1 | no | — |
| 115 | M16 | — | 4 | 1 | no | — |
| 116 | M16 | — | 4 | 1 | no | — |
| 117 | M16 | — | 4 | 1 | no | — |
| 118 | C2×C8 | — | 2 | 1 | no | — |
| 119 | C2×C8 | — | 4 | 1 | no | — |
| 120 | C2×C8 | — | 4 | 1 | no | — |
| 121 | C2×C8 | — | 4 | 1 | no | — |
| 122 | C22⋊C4 | — | 2 | 1 | no | — |
| 123 | C22⋊C4 | — | 4 | 1 | no | — |
| 124 | C22⋊C4 | — | 4 | 1 | no | — |
| 125 | C22⋊C4 | — | 4 | 1 | no | — |
| 126 | C22⋊C4 | — | 4 | 1 | no | — |
| 127 | C22⋊C4 | — | 4 | 1 | no | — |
| 128 | C22⋊C4 | — | 8 | 1 | no | — |
| 129 | C42 | — | 2 | 1 | no | — |
| 130 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 131 | C23 | — | 2 | 1 | no | — |
| 132 | C23 | — | 4 | 1 | no | — |
| 133 | C23 | — | 4 | 1 | no | — |
| 134 | C23 | — | 4 | 1 | no | — |
| 135 | C23 | — | 8 | 1 | no | — |
| 136 | C23 | — | 8 | 1 | no | — |
| 137 | C23 | — | 16 | 1 | no | — |
| 138 | Q8 | — | 2 | 1 | no | — |
| 139 | Q8 | — | 2 | 1 | no | — |
| 140 | Q8 | — | 4 | 1 | no | — |
| 141 | Q8 | — | 4 | 1 | no | — |
| 142 | Q8 | — | 8 | 1 | no | — |
| 143 | D4 | — | 2 | 1 | no | — |
| 144 | D4 | — | 2 | 1 | no | — |
| 145 | D4 | — | 4 | 1 | no | — |
| 146 | D4 | — | 4 | 1 | no | — |
| 147 | D4 | — | 4 | 1 | no | — |
| 148 | D4 | — | 4 | 1 | no | — |
| 149 | D4 | — | 4 | 1 | no | — |
| 150 | D4 | — | 4 | 1 | no | — |
| 151 | D4 | — | 8 | 1 | no | — |
| 152 | D4 | — | 8 | 1 | no | — |
| 153 | D4 | — | 8 | 1 | no | — |
| 154 | D4 | — | 8 | 1 | no | — |
| 155 | D4 | — | 8 | 1 | no | — |
| 156 | D4 | — | 8 | 1 | no | — |
| 157 | D4 | — | 8 | 1 | no | — |
| 158 | D4 | — | 16 | 1 | no | — |
| 159 | D4 | — | 16 | 2 | no | — |
| 160 | C2×C4 | — | 2 | 1 | no | — |
| 161 | C2×C4 | — | 2 | 1 | no | — |
| 162 | C2×C4 | — | 4 | 1 | no | — |
| 163 | C2×C4 | — | 4 | 1 | no | — |
| 164 | C2×C4 | — | 4 | 1 | no | — |
| 165 | C2×C4 | — | 4 | 1 | no | — |
| 166 | C2×C4 | — | 8 | 1 | no | — |
| 167 | C2×C4 | — | 8 | 1 | no | — |
| 168 | C2×C4 | — | 8 | 1 | no | — |
| 169 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 170 | C8 | — | 2 | 1 | no | — |
| 171 | C8 | — | 2 | 1 | no | — |
| 172 | C8 | — | 4 | 1 | no | — |
| 173 | C8 | — | 4 | 1 | no | — |
| 174 | C8 | — | 4 | 2 | no | — |
| 175 | C8 | — | 8 | 1 | no | — |
| 176 | C22 | — | 4 | 1 | no | — |
| 177 | C22 | — | 4 | 1 | no | — |
| 178 | C22 | — | 4 | 1 | no | — |
| 179 | C22 | — | 8 | 1 | no | — |
| 180 | C22 | — | 8 | 1 | no | — |
| 181 | C22 | — | 8 | 1 | no | — |
| 182 | C22 | — | 8 | 2 | no | — |
| 183 | C22 | — | 8 | 1 | no | — |
| 184 | C22 | — | 8 | 1 | no | — |
| 185 | C22 | — | 16 | 1 | no | — |
| 186 | C22 | — | 16 | 1 | no | — |
| 187 | C22 | — | 16 | 1 | no | — |
| 188 | C22 | — | 32 | 1 | no | — |
| 189 | C22 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 190 | C4 | — | 2 | 1 | no | — |
| 191 | C4 | — | 4 | 1 | no | — |
| 192 | C4 | — | 4 | 1 | no | — |
| 193 | C4 | — | 4 | 1 | no | — |
| 194 | C4 | — | 4 | 1 | no | — |
| 195 | C4 | — | 8 | 1 | no | — |
| 196 | C4 | — | 16 | 1 | no | — |
| 197 | C2 | — | 2 | 1 | no | — |
| 198 | C2 | — | 8 | 1 | no | — |
| 199 | C2 | — | 8 | 1 | no | — |
| 200 | C2 | — | 8 | 1 | no | — |
| 201 | C2 | — | 16 | 1 | no | — |
| 202 | C2 | — | 16 | 1 | no | — |
| 203 | C2 | #2196 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #124 | C23 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3288 | C2 | semidirect |
| 2. #2202 | C2 | semidirect |
| 3. #2199 | C2 | semidirect |
| 4. #1655 | C2 | semidirect |
| 5. #1406 | C2 | semidirect |
| 6. #1404 | C2 | semidirect |
| 7. #1403 | C2 | semidirect |
| 10. H(3,𝔽4) | C22 | semidirect |
| 22. #355 | C22 | semidirect |
| 23. #354 | C22 | semidirect |
| 26. #329 | C22 | semidirect |
| 27. 2−1+4 | D4 | semidirect |
| 50. #127 | D4 | semidirect |
| 63. #101 | D4 | semidirect |
| 69. #97 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3288 | 2 |
| #2202 | 2 |
| #2199 | 2 |
| #1655 | 2 |
| #1406 | 2 |
| #1404 | 2 |
| #1403 | 2 |