Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1057 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1057 conjugacy classes of subgroups — too many to draw. Only its 242 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 | #34639 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20773 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13856 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11226 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #11075 | C2 | 1 | 6 | yes | maximal |
| 6 | C22×#104 | C22 | 1 | 4 | yes | — |
| 7 | #2645 | C4 | 1 | 4 | yes | complemented |
| 8 | #2644 | C4 | 1 | 4 | yes | complemented |
| 9 | #2643 | — | 2 | 4 | no | — |
| 10 | #2401 | — | 2 | 4 | no | — |
| 11 | #2297 | C22 | 1 | 3 | yes | — |
| 12 | #2296 | C22 | 1 | 3 | yes | — |
| 13 | C2×C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 14 | C2×#376 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1840 | — | 2 | 4 | no | — |
| 16 | C2×#339 | C22 | 1 | 12 | yes | — |
| 17 | #1728 | C22 | 1 | 6 | yes | — |
| 18 | #1444 | — | 2 | 4 | no | — |
| 19 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 20 | C22×M16 | C23 | 1 | 4 | yes | — |
| 21 | C2×#117 | C23 | 1 | 3 | yes | — |
| 22 | C22×C4⋊C4 | — | 2 | 6 | no | — |
| 23 | C22×C42 | C23 | 1 | 4 | yes | — |
| 24 | C22×C42 | C23 | 1 | 3 | yes | — |
| 25 | C2×#104 | — | 4 | 12 | no | — |
| 26 | #401 | — | 2 | 4 | no | — |
| 27 | #400 | — | 2 | 12 | no | — |
| 28 | #397 | — | 2 | 12 | no | — |
| 29 | #396 | — | 2 | 4 | no | — |
| 30 | #391 | C2×C4 | 1 | 6 | yes | — |
| 31 | #390 | C2×C4 | 1 | 6 | yes | — |
| 32 | #389 | C2×C4 | 1 | 12 | yes | — |
| 33 | #386 | — | 2 | 6 | no | — |
| 34 | #385 | — | 2 | 6 | no | — |
| 35 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 36 | C4×C4⋊C4 | — | 4 | 12 | no | — |
| 37 | C4×C22⋊C4 | — | 2 | 6 | no | — |
| 38 | #376 | C2×C4 | 1 | 4 | yes | — |
| 39 | #376 | D4 | 1 | 4 | yes | — |
| 40 | C2×#95 | — | 2 | 6 | no | — |
| 41 | C2×#95 | D4 | 1 | 4 | yes | — |
| 42 | #339 | — | 4 | 12 | no | — |
| 43 | #336 | — | 2 | 12 | no | — |
| 44 | C2×#41 | — | 2 | 6 | no | — |
| 45 | C2×#41 | — | 2 | 6 | no | — |
| 46 | C22×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 47 | C22×D4 | — | 2 | 1 | no | — |
| 48 | C22×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 49 | C23×C4 | — | 2 | 4 | no | — |
| 50 | C23×C4 | — | 2 | 1 | no | — |
| 51 | C23×C4 | C2×D4 | 1 | 6 | yes | — |
| 52 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 53 | C2×M16 | — | 2 | 12 | no | — |
| 54 | C2×M16 | — | 2 | 12 | no | — |
| 55 | C22×C8 | — | 2 | 4 | no | — |
| 56 | #117 | — | 2 | 6 | no | — |
| 57 | #117 | — | 2 | 6 | no | — |
| 58 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 59 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 60 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 61 | C2×C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 62 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 63 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 64 | C2×C42 | — | 2 | 6 | no | — |
| 65 | C2×C42 | — | 2 | 6 | no | — |
| 66 | C2×C42 | — | 2 | 3 | no | — |
| 67 | C2×C42 | — | 2 | 12 | no | — |
| 68 | C2×C42 | — | 2 | 12 | no | — |
| 69 | C2×C42 | C22×C4 | 1 | 6 | yes | — |
| 70 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 71 | #104 | — | 4 | 16 | no | — |
| 72 | #95 | — | 2 | 12 | no | — |
| 73 | #95 | — | 2 | 12 | no | — |
| 74 | #95 | — | 4 | 12 | no | — |
| 75 | #95 | C22⋊C4 | 1 | 4 | yes | — |
| 76 | #95 | C22⋊C4 | 1 | 12 | yes | — |
| 77 | C24 | — | 2 | 1 | no | — |
| 78 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 79 | #41 | — | 4 | 4 | no | — |
| 80 | #41 | — | 4 | 12 | no | — |
| 81 | C2×Q8 | — | 2 | 6 | no | — |
| 82 | C2×D4 | — | 2 | 6 | no | — |
| 83 | C2×D4 | — | 4 | 3 | no | — |
| 84 | C2×D4 | — | 4 | 3 | no | — |
| 85 | C22×C4 | — | 2 | 6 | no | — |
| 86 | C22×C4 | — | 2 | 6 | no | — |
| 87 | C22×C4 | — | 2 | 8 | no | — |
| 88 | C22×C4 | — | 2 | 1 | no | — |
| 89 | C22×C4 | — | 2 | 6 | no | — |
| 90 | C22×C4 | — | 4 | 12 | no | — |
| 91 | C22×C4 | — | 4 | 12 | no | — |
| 92 | C22×C4 | — | 4 | 12 | no | — |
| 93 | C22×C4 | — | 4 | 6 | no | — |
| 94 | C22×C4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 95 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 96 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 97 | C22×C4 | H(5,𝔽2) | 1 | 6 | yes | — |
| 98 | C22×C4 | 2−1+4 | 1 | 6 | yes | — |
| 99 | M16 | — | 4 | 4 | no | — |
| 100 | M16 | — | 4 | 12 | no | — |
| 101 | C2×C8 | — | 4 | 12 | no | — |
| 102 | C4⋊C4 | — | 2 | 12 | no | — |
| 103 | C4⋊C4 | — | 4 | 48 | no | — |
| 104 | C4⋊C4 | — | 8 | 12 | no | — |
| 105 | C22⋊C4 | — | 4 | 3 | no | — |
| 106 | C22⋊C4 | — | 4 | 3 | no | — |
| 107 | C22⋊C4 | — | 8 | 12 | no | — |
| 108 | C42 | — | 2 | 12 | no | — |
| 109 | C42 | — | 2 | 12 | no | — |
| 110 | C42 | — | 4 | 6 | no | — |
| 111 | C42 | — | 4 | 4 | no | — |
| 112 | C42 | — | 4 | 12 | no | — |
| 113 | C23 | — | 2 | 4 | no | — |
| 114 | C23 | — | 4 | 3 | no | — |
| 115 | C23 | — | 4 | 4 | no | — |
| 116 | C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C23 | #535 | 1 | 3 | yes | — |
| 118 | C23 | #536 | 1 | 3 | yes | — |
| 119 | Q8 | — | 4 | 4 | no | — |
| 120 | D4 | — | 4 | 4 | no | — |
| 121 | D4 | — | 8 | 1 | no | — |
| 122 | D4 | — | 8 | 3 | no | — |
| 123 | C2×C4 | — | 2 | 12 | no | — |
| 124 | C2×C4 | — | 2 | 12 | no | — |
| 125 | C2×C4 | — | 2 | 12 | no | — |
| 126 | C2×C4 | — | 4 | 12 | no | — |
| 127 | C2×C4 | — | 4 | 12 | no | — |
| 128 | C2×C4 | — | 4 | 24 | no | — |
| 129 | C2×C4 | — | 4 | 6 | no | — |
| 130 | C2×C4 | — | 4 | 3 | no | — |
| 131 | C2×C4 | — | 4 | 12 | no | — |
| 132 | C2×C4 | — | 8 | 12 | no | — |
| 133 | C2×C4 | — | 8 | 4 | no | — |
| 134 | C2×C4 | — | 8 | 4 | no | — |
| 135 | C2×C4 | #520 | 1 | 6 | yes | — |
| 136 | C2×C4 | #537 | 1 | 6 | yes | — |
| 137 | C2×C4 | #560 | 1 | 4 | yes | — |
| 138 | C2×C4 | #563 | 1 | 12 | yes | — |
| 139 | C8 | — | 8 | 4 | no | — |
| 140 | C22 | — | 2 | 6 | no | — |
| 141 | C22 | — | 2 | 6 | no | — |
| 142 | C22 | — | 8 | 1 | no | — |
| 143 | C22 | — | 8 | 6 | no | — |
| 144 | C22 | #2311 | 1 | 3 | yes | — |
| 145 | C22 | #2818 | 1 | 1 | yes | characteristic |
| 146 | C22 | #2821 | 1 | 3 | yes | — |
| 147 | C22 | #2901 | 1 | 4 | yes | — |
| 148 | C4 | — | 4 | 12 | no | — |
| 149 | C4 | — | 8 | 8 | no | — |
| 150 | C4 | — | 8 | 1 | no | — |
| 151 | C4 | — | 8 | 6 | no | — |
| 152 | C4 | #2383 | 1 | 4 | yes | — |
| 153 | C4 | #2823 | 1 | 4 | yes | — |
| 154 | C2 | — | 2 | 1 | no | — |
| 155 | C2 | — | 2 | 3 | no | — |
| 156 | C2 | — | 16 | 1 | no | — |
| 157 | C2 | #14665 | 1 | 1 | yes | characteristic |
| 158 | C2 | #21901 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20773 | C2 | semidirect |
| 3. #13856 | C2 | semidirect |
| 4. #11226 | C2 | semidirect |
| 7. #2645 | C4 | semidirect |
| 8. #2644 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #34639 | 2 |
| #20773 | 2 |
| #13856 | 2 |
| #11226 | 2 |
| #11075 | 2 |