Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
841 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 841 conjugacy classes of subgroups — too many to draw. Only its 160 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 | #23474 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11285 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11115 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #11026 | C2 | 1 | 3 | yes | maximal |
| 6 | C22×#135 | C22 | 1 | 1 | yes | characteristic |
| 7 | #3458 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2911 | C22 | 1 | 3 | yes | — |
| 9 | C2×#421 | C22 | 1 | 6 | yes | — |
| 10 | C2×#410 | C22 | 1 | 6 | yes | — |
| 11 | #2032 | — | 2 | 8 | no | — |
| 12 | #1742 | C22 | 1 | 3 | yes | — |
| 13 | #1740 | C22 | 1 | 6 | yes | — |
| 14 | #1735 | C22 | 1 | 3 | yes | — |
| 15 | C2×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 16 | #1729 | C22 | 1 | 3 | yes | — |
| 17 | C22×#41 | C23 | 1 | 2 | yes | — |
| 18 | C2×#135 | — | 4 | 3 | no | — |
| 19 | C2×#135 | — | 4 | 3 | no | — |
| 20 | C22×Q16 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 21 | C22×SD16 | Q8 | 1 | 2 | yes | complemented |
| 22 | C22×D8 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 23 | C22×M16 | C23 | 1 | 3 | yes | — |
| 24 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 25 | #544 | — | 4 | 3 | no | — |
| 26 | #543 | D4 | 1 | 4 | yes | — |
| 27 | #533 | — | 4 | 3 | no | — |
| 28 | C2×#117 | C23 | 1 | 6 | yes | — |
| 29 | C2×#117 | C23 | 1 | 3 | yes | — |
| 30 | #428 | — | 4 | 6 | no | — |
| 31 | #427 | — | 4 | 3 | no | — |
| 32 | #421 | — | 4 | 6 | no | — |
| 33 | #421 | — | 4 | 6 | no | — |
| 34 | C2×#104 | — | 2 | 12 | no | — |
| 35 | #410 | — | 4 | 6 | no | — |
| 36 | #410 | — | 4 | 6 | no | — |
| 37 | C2×#99 | — | 2 | 12 | no | — |
| 38 | #343 | — | 4 | 3 | no | — |
| 39 | #342 | — | 2 | 12 | no | — |
| 40 | #340 | — | 4 | 3 | no | — |
| 41 | H(3,ℤ/4) | — | 2 | 6 | no | — |
| 42 | #336 | — | 4 | 3 | no | — |
| 43 | C2×#41 | — | 2 | 6 | no | — |
| 44 | C2×#41 | — | 2 | 6 | no | — |
| 45 | C2×#41 | — | 2 | 6 | no | — |
| 46 | C2×#41 | — | 2 | 6 | no | — |
| 47 | C22×Q8 | C2×Q8 | 1 | 2 | yes | — |
| 48 | C22×D4 | — | 2 | 2 | no | — |
| 49 | C22×D4 | C2×Q8 | 1 | 2 | yes | — |
| 50 | C23×C4 | — | 2 | 2 | no | — |
| 51 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 52 | #135 | — | 4 | 12 | no | — |
| 53 | #135 | — | 4 | 4 | no | — |
| 54 | C2×Q16 | — | 4 | 3 | no | — |
| 55 | C2×SD16 | — | 4 | 6 | no | — |
| 56 | C2×D8 | — | 4 | 3 | no | — |
| 57 | C2×M16 | — | 2 | 3 | no | — |
| 58 | C2×M16 | — | 2 | 6 | no | — |
| 59 | C2×M16 | — | 2 | 6 | no | — |
| 60 | C2×M16 | #41 | 1 | 6 | yes | — |
| 61 | C22×C8 | — | 2 | 3 | no | — |
| 62 | C22×C8 | — | 2 | 3 | no | — |
| 63 | C22×C8 | — | 2 | 3 | no | — |
| 64 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 66 | C4⋊Q8 | — | 4 | 6 | no | — |
| 67 | #125 | — | 4 | 6 | no | — |
| 68 | C22⋊Q8 | — | 4 | 12 | no | — |
| 69 | C4×Q8 | — | 4 | 4 | no | — |
| 70 | #117 | — | 2 | 6 | no | — |
| 71 | #117 | — | 2 | 6 | no | — |
| 72 | #117 | — | 2 | 6 | no | — |
| 73 | #117 | — | 2 | 6 | no | — |
| 74 | #117 | — | 2 | 6 | no | — |
| 75 | #117 | — | 2 | 6 | no | — |
| 76 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 78 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 79 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 80 | C2×C42 | — | 2 | 3 | no | — |
| 81 | C2×C42 | — | 2 | 6 | no | — |
| 82 | #107 | — | 4 | 6 | no | — |
| 83 | #106 | — | 4 | 6 | no | — |
| 84 | #104 | — | 4 | 24 | no | — |
| 85 | #99 | — | 4 | 24 | no | — |
| 86 | C24 | — | 2 | 2 | no | — |
| 87 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 88 | #41 | — | 4 | 6 | no | — |
| 89 | #41 | — | 4 | 6 | no | — |
| 90 | #41 | — | 4 | 6 | no | — |
| 91 | #41 | — | 4 | 6 | no | — |
| 92 | #41 | — | 4 | 6 | no | — |
| 93 | #41 | — | 4 | 2 | no | — |
| 94 | C2×Q8 | — | 2 | 6 | no | — |
| 95 | C2×Q8 | — | 2 | 6 | no | — |
| 96 | C2×Q8 | — | 4 | 4 | no | — |
| 97 | C2×D4 | — | 2 | 6 | no | — |
| 98 | C2×D4 | — | 2 | 6 | no | — |
| 99 | C2×D4 | — | 4 | 6 | no | — |
| 100 | C2×D4 | — | 4 | 6 | no | — |
| 101 | C22×C4 | — | 2 | 2 | no | — |
| 102 | C22×C4 | — | 2 | 6 | no | — |
| 103 | C22×C4 | — | 4 | 6 | no | — |
| 104 | C22×C4 | — | 4 | 6 | no | — |
| 105 | C22×C4 | — | 4 | 6 | no | — |
| 106 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 107 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 108 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 109 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 110 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 111 | C22×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 112 | Q16 | — | 4 | 4 | no | — |
| 113 | SD16 | — | 4 | 8 | no | — |
| 114 | D8 | — | 4 | 4 | no | — |
| 115 | M16 | — | 2 | 12 | no | — |
| 116 | M16 | — | 4 | 6 | no | — |
| 117 | C2×C8 | — | 2 | 6 | no | — |
| 118 | C2×C8 | — | 2 | 3 | no | — |
| 119 | C2×C8 | — | 2 | 3 | no | — |
| 120 | C2×C8 | — | 4 | 3 | no | — |
| 121 | C2×C8 | — | 4 | 3 | no | — |
| 122 | C2×C8 | — | 4 | 3 | no | — |
| 123 | C2×C8 | — | 4 | 1 | no | — |
| 124 | C4⋊C4 | — | 4 | 12 | no | — |
| 125 | C4⋊C4 | — | 4 | 6 | no | — |
| 126 | C4⋊C4 | — | 4 | 6 | no | — |
| 127 | C4⋊C4 | — | 4 | 6 | no | — |
| 128 | C22⋊C4 | — | 2 | 12 | no | — |
| 129 | C22⋊C4 | — | 4 | 6 | no | — |
| 130 | C22⋊C4 | — | 4 | 6 | no | — |
| 131 | C42 | — | 4 | 3 | no | — |
| 132 | C42 | — | 4 | 3 | no | — |
| 133 | C42 | — | 4 | 6 | no | — |
| 134 | C42 | — | 4 | 6 | no | — |
| 135 | C23 | — | 4 | 6 | no | — |
| 136 | C23 | — | 8 | 1 | no | — |
| 137 | C23 | — | 16 | 2 | no | — |
| 138 | C23 | #534 | 1 | 3 | yes | — |
| 139 | C23 | D4×Q8 | 1 | 1 | yes | characteristic |
| 140 | C23 | #550 | 1 | 3 | yes | — |
| 141 | Q8 | — | 4 | 2 | no | — |
| 142 | Q8 | — | 4 | 6 | no | — |
| 143 | Q8 | — | 16 | 4 | no | — |
| 144 | D4 | — | 4 | 2 | no | — |
| 145 | D4 | — | 4 | 6 | no | — |
| 146 | D4 | — | 8 | 6 | no | — |
| 147 | D4 | — | 8 | 2 | no | — |
| 148 | C2×C4 | — | 4 | 6 | no | — |
| 149 | C2×C4 | — | 4 | 6 | no | — |
| 150 | C2×C4 | — | 8 | 6 | no | — |
| 151 | C2×C4 | — | 8 | 2 | no | — |
| 152 | C2×C4 | — | 8 | 6 | no | — |
| 153 | C2×C4 | — | 8 | 6 | no | — |
| 154 | C2×C4 | — | 16 | 6 | no | — |
| 155 | C2×C4 | #538 | 1 | 6 | yes | — |
| 156 | C2×C4 | #539 | 1 | 3 | yes | — |
| 157 | C2×C4 | #543 | 1 | 3 | yes | — |
| 158 | C2×C4 | #554 | 1 | 3 | yes | — |
| 159 | C2×C4 | #555 | 1 | 3 | yes | — |
| 160 | C2×C4 | #560 | 1 | 3 | yes | — |
| 161 | C2×C4 | H(3,𝔽4) | 1 | 3 | yes | — |
| 162 | C2×C4 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 163 | C2×C4 | #562 | 1 | 3 | yes | — |
| 164 | C8 | — | 4 | 6 | no | — |
| 165 | C8 | — | 4 | 1 | no | — |
| 166 | C8 | — | 4 | 1 | no | — |
| 167 | C22 | — | 4 | 3 | no | — |
| 168 | C22 | — | 4 | 3 | no | — |
| 169 | C22 | — | 4 | 1 | no | — |
| 170 | C22 | — | 8 | 2 | no | — |
| 171 | C22 | — | 16 | 6 | no | — |
| 172 | C22 | #2732 | 1 | 3 | yes | — |
| 173 | C22 | #2733 | 1 | 3 | yes | — |
| 174 | C22 | #2812 | 1 | 1 | yes | characteristic |
| 175 | C4 | — | 8 | 6 | no | — |
| 176 | C4 | — | 8 | 2 | no | — |
| 177 | C4 | — | 16 | 6 | no | — |
| 178 | C4 | #2784 | 1 | 3 | yes | — |
| 179 | C4 | #2785 | 1 | 1 | yes | characteristic |
| 180 | C4 | #2824 | 1 | 3 | yes | — |
| 181 | C4 | #2840 | 1 | 1 | yes | characteristic |
| 182 | C2 | — | 2 | 1 | no | — |
| 183 | C2 | — | 2 | 3 | no | — |
| 184 | C2 | — | 2 | 3 | no | — |
| 185 | C2 | — | 16 | 2 | no | — |
| 186 | C2 | #17216 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2785 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17216 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23474 | C2 | semidirect |
| 2. #11391 | C2 | semidirect |
| 3. #11285 | C2 | semidirect |
| 4. #11115 | C2 | semidirect |
| 20. C22×Q16 | Q8 | semidirect |
| 21. C22×SD16 | Q8 | semidirect |
| 22. C22×D8 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23474 | 2 |
| #11391 | 2 |
| #11285 | 2 |
| #11115 | 2 |
| #11026 | 2 |