Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
311 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 143 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 | #3164 | C2 | 1 | 2 | yes | maximal |
| 2 | #3163 | C2 | 1 | 2 | yes | maximal |
| 3 | #2061 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #2061 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #2060 | C2 | 1 | 2 | yes | maximal |
| 6 | #1952 | C2 | 1 | 2 | yes | maximal |
| 7 | #1951 | C2 | 1 | 2 | yes | maximal |
| 8 | #1847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #1773 | C2 | 1 | 2 | yes | maximal |
| 10 | #544 | C22 | 1 | 2 | yes | — |
| 11 | #544 | C22 | 1 | 1 | yes | characteristic |
| 12 | #544 | C22 | 1 | 1 | yes | characteristic |
| 13 | #501 | — | 2 | 2 | no | — |
| 14 | #501 | — | 2 | 2 | no | — |
| 15 | #501 | — | 2 | 2 | no | — |
| 16 | #501 | — | 2 | 2 | no | — |
| 17 | #500 | — | 2 | 2 | no | — |
| 18 | #499 | — | 2 | 2 | no | — |
| 19 | #499 | — | 2 | 2 | no | — |
| 20 | #498 | — | 2 | 2 | no | — |
| 21 | C2×#108 | C22 | 1 | 2 | yes | — |
| 22 | #428 | C22 | 1 | 2 | yes | — |
| 23 | #428 | C22 | 1 | 2 | yes | — |
| 24 | #428 | C22 | 1 | 2 | yes | — |
| 25 | #428 | C22 | 1 | 2 | yes | — |
| 26 | #427 | C22 | 1 | 2 | yes | — |
| 27 | C2×#107 | C22 | 1 | 2 | yes | — |
| 28 | C2×#106 | C22 | 1 | 2 | yes | — |
| 29 | #405 | C22 | 1 | 2 | yes | — |
| 30 | #405 | C22 | 1 | 2 | yes | — |
| 31 | C2×#97 | C22 | 1 | 1 | yes | characteristic |
| 32 | #384 | C22 | 1 | 1 | yes | characteristic |
| 33 | #382 | C22 | 1 | 1 | yes | characteristic |
| 34 | #343 | C22 | 1 | 2 | yes | — |
| 35 | #343 | C22 | 1 | 1 | yes | characteristic |
| 36 | #343 | C22 | 1 | 1 | yes | characteristic |
| 37 | #339 | C22 | 1 | 2 | yes | — |
| 38 | #339 | C22 | 1 | 2 | yes | — |
| 39 | C2×M16 | C23 | 1 | 2 | yes | — |
| 40 | C2×M16 | C23 | 1 | 2 | yes | — |
| 41 | C22×C8 | C23 | 1 | 2 | yes | — |
| 42 | C4⋊Q8 | — | 2 | 2 | no | — |
| 43 | C4⋊Q8 | — | 2 | 2 | no | — |
| 44 | C4⋊Q8 | — | 2 | 2 | no | — |
| 45 | C4⋊Q8 | — | 2 | 2 | no | — |
| 46 | #125 | — | 2 | 2 | no | — |
| 47 | #125 | — | 2 | 2 | no | — |
| 48 | #125 | — | 2 | 2 | no | — |
| 49 | #125 | — | 2 | 2 | no | — |
| 50 | C22⋊Q8 | — | 4 | 1 | no | — |
| 51 | C22⋊Q8 | — | 4 | 1 | no | — |
| 52 | C22⋊Q8 | — | 4 | 2 | no | — |
| 53 | #117 | C23 | 1 | 2 | yes | — |
| 54 | #117 | C23 | 1 | 2 | yes | — |
| 55 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 56 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 58 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 59 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 60 | #108 | Q8 | 1 | 8 | yes | complemented |
| 61 | #107 | — | 2 | 2 | no | — |
| 62 | #107 | — | 2 | 2 | no | — |
| 63 | #107 | — | 2 | 2 | no | — |
| 64 | #107 | — | 2 | 2 | no | — |
| 65 | #107 | — | 2 | 2 | no | — |
| 66 | #107 | — | 2 | 2 | no | — |
| 67 | #107 | D4 | 1 | 2 | yes | — |
| 68 | #107 | D4 | 1 | 2 | yes | — |
| 69 | #106 | — | 2 | 2 | no | — |
| 70 | #106 | — | 2 | 2 | no | — |
| 71 | #106 | — | 2 | 2 | no | — |
| 72 | #106 | — | 2 | 2 | no | — |
| 73 | #106 | — | 2 | 2 | no | — |
| 74 | #106 | — | 2 | 2 | no | — |
| 75 | #106 | D4 | 1 | 2 | yes | — |
| 76 | #106 | D4 | 1 | 2 | yes | — |
| 77 | #97 | — | 2 | 2 | no | — |
| 78 | #97 | — | 2 | 2 | no | — |
| 79 | #97 | Q8 | 1 | 4 | yes | complemented |
| 80 | C4×C8 | — | 2 | 2 | no | — |
| 81 | C4×C8 | — | 2 | 2 | no | — |
| 82 | #95 | — | 4 | 1 | no | — |
| 83 | C2×Q8 | — | 4 | 1 | no | — |
| 84 | C2×Q8 | — | 4 | 2 | no | — |
| 85 | C2×Q8 | — | 4 | 1 | no | — |
| 86 | C22×C4 | — | 2 | 1 | no | — |
| 87 | C22×C4 | — | 2 | 1 | no | — |
| 88 | C22×C4 | — | 2 | 1 | no | — |
| 89 | C22×C4 | — | 2 | 2 | no | — |
| 90 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 91 | M16 | — | 2 | 2 | no | — |
| 92 | M16 | — | 2 | 2 | no | — |
| 93 | M16 | C2×Q8 | 1 | 4 | yes | — |
| 94 | M16 | C2×Q8 | 1 | 4 | yes | — |
| 95 | C2×C8 | — | 2 | 2 | no | — |
| 96 | C2×C8 | — | 2 | 2 | no | — |
| 97 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 99 | C2×C8 | C2×Q8 | 1 | 4 | yes | — |
| 100 | C2×C8 | C2×Q8 | 1 | 4 | yes | — |
| 101 | C4⋊C4 | — | 2 | 2 | no | — |
| 102 | C4⋊C4 | — | 2 | 2 | no | — |
| 103 | C4⋊C4 | — | 2 | 2 | no | — |
| 104 | C4⋊C4 | — | 2 | 2 | no | — |
| 105 | C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C4⋊C4 | — | 4 | 2 | no | — |
| 107 | C4⋊C4 | — | 4 | 1 | no | — |
| 108 | C4⋊C4 | — | 4 | 2 | no | — |
| 109 | C4⋊C4 | — | 4 | 2 | no | — |
| 110 | C4⋊C4 | — | 4 | 2 | no | — |
| 111 | C4⋊C4 | — | 4 | 2 | no | — |
| 112 | C4⋊C4 | — | 8 | 2 | no | — |
| 113 | C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 114 | C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 115 | C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 116 | C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 117 | C22⋊C4 | — | 2 | 2 | no | — |
| 118 | C22⋊C4 | — | 2 | 2 | no | — |
| 119 | C42 | — | 2 | 2 | no | — |
| 120 | C42 | — | 2 | 2 | no | — |
| 121 | C42 | — | 2 | 1 | no | — |
| 122 | C42 | C2×Q8 | 1 | 2 | yes | — |
| 123 | C23 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 124 | Q8 | — | 16 | 2 | no | — |
| 125 | Q8 | — | 16 | 1 | no | — |
| 126 | Q8 | — | 16 | 1 | no | — |
| 127 | C2×C4 | — | 2 | 2 | no | — |
| 128 | C2×C4 | — | 2 | 2 | no | — |
| 129 | C2×C4 | — | 2 | 2 | no | — |
| 130 | C2×C4 | — | 2 | 2 | no | — |
| 131 | C2×C4 | — | 2 | 2 | no | — |
| 132 | C2×C4 | — | 4 | 1 | no | — |
| 133 | C2×C4 | — | 4 | 1 | no | — |
| 134 | C2×C4 | — | 4 | 2 | no | — |
| 135 | C2×C4 | — | 4 | 1 | no | — |
| 136 | C2×C4 | — | 4 | 1 | no | — |
| 137 | C2×C4 | — | 8 | 2 | no | — |
| 138 | C2×C4 | — | 8 | 1 | no | — |
| 139 | C2×C4 | — | 8 | 1 | no | — |
| 140 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 141 | C2×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 142 | C2×C4 | C22×Q8 | 1 | 2 | yes | — |
| 143 | C2×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 144 | C8 | — | 2 | 4 | no | — |
| 145 | C8 | — | 2 | 4 | no | — |
| 146 | C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 147 | C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 148 | C22 | — | 2 | 1 | no | — |
| 149 | C22 | — | 2 | 1 | no | — |
| 150 | C22 | #544 | 1 | 1 | yes | characteristic |
| 151 | C22 | D42 | 1 | 1 | yes | characteristic |
| 152 | C22 | Q82 | 1 | 1 | yes | characteristic |
| 153 | C4 | — | 4 | 2 | no | — |
| 154 | C4 | — | 4 | 2 | no | — |
| 155 | C4 | — | 4 | 2 | no | — |
| 156 | C4 | — | 4 | 2 | no | — |
| 157 | C4 | — | 4 | 2 | no | — |
| 158 | C4 | — | 8 | 2 | no | — |
| 159 | C4 | — | 8 | 1 | no | — |
| 160 | C4 | — | 8 | 1 | no | — |
| 161 | C4 | C2×C4⋊Q8 | 1 | 2 | yes | — |
| 162 | C4 | D4×Q8 | 1 | 2 | yes | — |
| 163 | C2 | — | 2 | 1 | no | — |
| 164 | C2 | — | 2 | 1 | no | — |
| 165 | C2 | #2551 | 1 | 1 | yes | characteristic |
| 166 | C2 | #3288 | 1 | 1 | yes | characteristic |
| 167 | C2 | #3289 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×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 |
|---|---|---|
| 60. #108 | Q8 | semidirect |
| 79. #97 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3164 | 2 |
| #3163 | 2 |
| #2061 | 2 |
| #2060 | 2 |
| #1952 | 2 |
| #1951 | 2 |
| #1847 | 2 |
| #1773 | 2 |