Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
446 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 167 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×#497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2950 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #2949 | C2 | 1 | 2 | yes | maximal |
| 4 | #1979 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #1977 | C2 | 1 | 2 | yes | maximal |
| 6 | #1888 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #1886 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #1885 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #1773 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C22×Q16 | C22 | 1 | 2 | yes | — |
| 11 | C22×SD16 | C22 | 1 | 2 | yes | — |
| 12 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#124 | C22 | 1 | 1 | yes | characteristic |
| 14 | #519 | C22 | 1 | 2 | yes | — |
| 15 | #519 | C22 | 1 | 1 | yes | characteristic |
| 16 | #518 | C22 | 1 | 1 | yes | characteristic |
| 17 | #497 | C4 | 1 | 8 | yes | complemented |
| 18 | #441 | — | 2 | 4 | no | — |
| 19 | #440 | — | 2 | 4 | no | — |
| 20 | C4×Q16 | — | 2 | 4 | no | — |
| 21 | C4×SD16 | — | 2 | 4 | no | — |
| 22 | C2×#108 | C22 | 1 | 2 | yes | — |
| 23 | #427 | C22 | 1 | 2 | yes | — |
| 24 | #419 | C22 | 1 | 2 | yes | — |
| 25 | #419 | C22 | 1 | 2 | yes | — |
| 26 | #419 | C22 | 1 | 2 | yes | — |
| 27 | #418 | C22 | 1 | 2 | yes | — |
| 28 | C2×#101 | C22 | 1 | 2 | yes | — |
| 29 | C2×#101 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×#100 | C22 | 1 | 1 | yes | characteristic |
| 31 | #405 | C22 | 1 | 2 | yes | — |
| 32 | #405 | C22 | 1 | 2 | yes | — |
| 33 | C2×#97 | C22 | 1 | 1 | yes | characteristic |
| 34 | #339 | C22 | 1 | 2 | yes | — |
| 35 | #339 | C22 | 1 | 2 | yes | — |
| 36 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 37 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 38 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×Q16 | — | 2 | 2 | no | — |
| 40 | C2×Q16 | — | 2 | 2 | no | — |
| 41 | C2×Q16 | — | 2 | 4 | no | — |
| 42 | C2×Q16 | C2×C4 | 1 | 8 | yes | complemented |
| 43 | C2×SD16 | — | 2 | 2 | no | — |
| 44 | C2×SD16 | — | 2 | 2 | no | — |
| 45 | C2×SD16 | — | 2 | 4 | no | — |
| 46 | C2×SD16 | C2×C4 | 1 | 8 | yes | — |
| 47 | C2×M16 | C23 | 1 | 2 | yes | — |
| 48 | C2×M16 | C23 | 1 | 2 | yes | — |
| 49 | C22×C8 | C23 | 1 | 2 | yes | — |
| 50 | C4⋊Q8 | — | 4 | 1 | no | — |
| 51 | C4⋊Q8 | C2×C4 | 1 | 4 | yes | complemented |
| 52 | #124 | — | 4 | 1 | no | — |
| 53 | #124 | C2×C4 | 1 | 4 | yes | — |
| 54 | C4×Q8 | — | 2 | 4 | no | — |
| 55 | C4×Q8 | — | 2 | 4 | no | — |
| 56 | C4×Q8 | — | 2 | 4 | no | — |
| 57 | C4×D4 | — | 2 | 4 | no | — |
| 58 | #117 | — | 4 | 2 | no | — |
| 59 | #117 | — | 4 | 1 | no | — |
| 60 | #117 | C23 | 1 | 2 | yes | — |
| 61 | #117 | C23 | 1 | 2 | yes | — |
| 62 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 63 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 64 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 65 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 66 | #108 | — | 2 | 4 | no | — |
| 67 | #107 | D4 | 1 | 4 | yes | — |
| 68 | #106 | D4 | 1 | 4 | yes | — |
| 69 | #103 | — | 2 | 4 | no | — |
| 70 | #103 | — | 2 | 4 | no | — |
| 71 | #103 | — | 2 | 4 | no | — |
| 72 | #102 | — | 2 | 4 | no | — |
| 73 | #101 | — | 4 | 2 | no | — |
| 74 | #101 | — | 4 | 1 | no | — |
| 75 | #100 | — | 4 | 1 | no | — |
| 76 | #97 | — | 2 | 2 | no | — |
| 77 | #97 | — | 2 | 2 | no | — |
| 78 | #97 | C2×C4 | 1 | 4 | yes | complemented |
| 79 | C4×C8 | — | 2 | 2 | no | — |
| 80 | C4×C8 | — | 2 | 2 | no | — |
| 81 | C24 | — | 2 | 1 | no | — |
| 82 | C2×Q8 | — | 2 | 1 | no | — |
| 83 | C2×Q8 | — | 2 | 2 | no | — |
| 84 | C2×Q8 | — | 2 | 2 | no | — |
| 85 | C2×Q8 | — | 2 | 2 | no | — |
| 86 | C2×Q8 | — | 2 | 2 | no | — |
| 87 | C2×Q8 | — | 2 | 2 | no | — |
| 88 | C2×Q8 | — | 2 | 2 | no | — |
| 89 | C2×Q8 | — | 2 | 2 | no | — |
| 90 | C2×Q8 | C22×C4 | 1 | 2 | yes | — |
| 91 | C2×Q8 | C22×C4 | 1 | 4 | yes | — |
| 92 | C2×D4 | — | 2 | 2 | no | — |
| 93 | C2×D4 | — | 2 | 2 | no | — |
| 94 | C2×D4 | — | 2 | 1 | no | — |
| 95 | C2×D4 | C22×C4 | 1 | 2 | yes | — |
| 96 | C22×C4 | — | 2 | 1 | no | — |
| 97 | C22×C4 | — | 2 | 1 | no | — |
| 98 | C22×C4 | — | 2 | 2 | no | — |
| 99 | C22×C4 | — | 4 | 2 | no | — |
| 100 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 101 | Q16 | — | 2 | 8 | no | — |
| 102 | Q16 | — | 2 | 8 | no | — |
| 103 | SD16 | — | 2 | 8 | no | — |
| 104 | SD16 | — | 2 | 8 | no | — |
| 105 | M16 | — | 2 | 2 | no | — |
| 106 | M16 | — | 2 | 2 | no | — |
| 107 | M16 | — | 2 | 2 | no | — |
| 108 | M16 | — | 2 | 2 | no | — |
| 109 | C2×C8 | — | 2 | 2 | no | — |
| 110 | C2×C8 | — | 2 | 2 | no | — |
| 111 | C2×C8 | C22×C4 | 1 | 4 | yes | — |
| 112 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 113 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 114 | C2×C8 | #41 | 1 | 4 | yes | — |
| 115 | C4⋊C4 | — | 4 | 4 | no | — |
| 116 | C4⋊C4 | — | 4 | 2 | no | — |
| 117 | C4⋊C4 | — | 4 | 4 | no | — |
| 118 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 119 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 120 | C22⋊C4 | — | 2 | 2 | no | — |
| 121 | C22⋊C4 | — | 2 | 2 | no | — |
| 122 | C22⋊C4 | — | 4 | 4 | no | — |
| 123 | C22⋊C4 | — | 4 | 2 | no | — |
| 124 | C42 | — | 2 | 2 | no | — |
| 125 | C42 | — | 2 | 2 | no | — |
| 126 | C42 | — | 2 | 1 | no | — |
| 127 | C42 | — | 4 | 2 | no | — |
| 128 | C42 | — | 4 | 4 | no | — |
| 129 | C42 | C22×C4 | 1 | 2 | yes | — |
| 130 | C23 | — | 2 | 2 | no | — |
| 131 | C23 | — | 4 | 1 | no | — |
| 132 | C23 | — | 4 | 1 | no | — |
| 133 | C23 | — | 8 | 2 | no | — |
| 134 | C23 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 135 | Q8 | — | 2 | 4 | no | — |
| 136 | Q8 | — | 2 | 4 | no | — |
| 137 | Q8 | — | 2 | 4 | no | — |
| 138 | Q8 | — | 4 | 2 | no | — |
| 139 | Q8 | — | 4 | 2 | no | — |
| 140 | Q8 | — | 4 | 2 | no | — |
| 141 | D4 | — | 2 | 4 | no | — |
| 142 | D4 | — | 4 | 2 | no | — |
| 143 | C2×C4 | — | 2 | 2 | no | — |
| 144 | C2×C4 | — | 2 | 2 | no | — |
| 145 | C2×C4 | — | 2 | 4 | no | — |
| 146 | C2×C4 | — | 2 | 4 | no | — |
| 147 | C2×C4 | — | 2 | 4 | no | — |
| 148 | C2×C4 | — | 4 | 2 | no | — |
| 149 | C2×C4 | — | 4 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 2 | no | — |
| 151 | C2×C4 | — | 4 | 1 | no | — |
| 152 | C2×C4 | — | 4 | 2 | no | — |
| 153 | C2×C4 | — | 16 | 2 | no | — |
| 154 | C2×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 156 | C2×C4 | C2×#41 | 1 | 2 | yes | — |
| 157 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 158 | C8 | — | 4 | 2 | no | — |
| 159 | C8 | — | 4 | 2 | no | — |
| 160 | C8 | C4×D4 | 1 | 4 | yes | — |
| 161 | C8 | C4×D4 | 1 | 4 | yes | — |
| 162 | C22 | — | 2 | 2 | no | — |
| 163 | C22 | — | 4 | 2 | no | — |
| 164 | C22 | — | 8 | 2 | no | — |
| 165 | C22 | — | 8 | 2 | no | — |
| 166 | C22 | — | 16 | 1 | no | — |
| 167 | C22 | #518 | 1 | 1 | yes | characteristic |
| 168 | C22 | D42 | 1 | 1 | yes | characteristic |
| 169 | C22 | #559 | 1 | 1 | yes | characteristic |
| 170 | C4 | — | 4 | 2 | no | — |
| 171 | C4 | — | 4 | 2 | no | — |
| 172 | C4 | — | 4 | 4 | no | — |
| 173 | C4 | — | 4 | 4 | no | — |
| 174 | C4 | — | 4 | 4 | no | — |
| 175 | C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 176 | C4 | #550 | 1 | 2 | yes | — |
| 177 | C2 | — | 2 | 1 | no | — |
| 178 | C2 | — | 2 | 1 | no | — |
| 179 | C2 | — | 8 | 2 | no | — |
| 180 | C2 | #2366 | 1 | 1 | yes | characteristic |
| 181 | C2 | #3289 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2950 | C2 | semidirect |
| 4. #1979 | C2 | semidirect |
| 7. #1886 | C2 | semidirect |
| 9. #1773 | C2 | semidirect |
| 17. #497 | C4 | semidirect |
| 42. C2×Q16 | C2×C4 | semidirect |
| 51. C4⋊Q8 | C2×C4 | semidirect |
| 78. #97 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#497 | 2 |
| #2950 | 2 |
| #2949 | 2 |
| #1979 | 2 |
| #1977 | 2 |
| #1888 | 2 |
| #1886 | 2 |
| #1885 | 2 |
| #1773 | 2 |