Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
662 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 202 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 | #23214 | C2 | 1 | 4 | yes | maximal |
| 2 | C2×#1710 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #9736 | C2 | 1 | 4 | yes | maximal |
| 4 | #9058 | C2 | 1 | 4 | yes | maximal |
| 5 | C8×#106 | C2 | 1 | 2 | yes | maximal |
| 6 | C2×#500 | C22 | 1 | 2 | yes | — |
| 7 | C2×#487 | C22 | 1 | 4 | yes | — |
| 8 | #2365 | C22 | 1 | 4 | yes | — |
| 9 | #1988 | C22 | 1 | 4 | yes | — |
| 10 | #1981 | C22 | 1 | 4 | yes | — |
| 11 | #1933 | — | 2 | 4 | no | — |
| 12 | #1927 | — | 2 | 4 | no | — |
| 13 | #1865 | — | 2 | 4 | no | — |
| 14 | C4×#106 | C22 | 1 | 2 | yes | — |
| 15 | C8×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 16 | C4×#103 | — | 2 | 4 | no | — |
| 17 | #1710 | C4 | 1 | 8 | yes | complemented |
| 18 | C2×#334 | C22 | 1 | 2 | yes | — |
| 19 | C2×#332 | C22 | 1 | 4 | yes | — |
| 20 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1279 | C22 | 1 | 4 | yes | — |
| 22 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 23 | C2×C4×Q8 | — | 2 | 4 | no | — |
| 24 | #500 | — | 4 | 2 | no | — |
| 25 | #500 | C2×C4 | 1 | 8 | yes | — |
| 26 | #487 | — | 2 | 8 | no | — |
| 27 | #487 | C2×C4 | 1 | 16 | yes | — |
| 28 | C2×#107 | — | 4 | 2 | no | — |
| 29 | C2×#106 | D4 | 1 | 4 | yes | — |
| 30 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 31 | C2×#103 | — | 2 | 4 | no | — |
| 32 | C2×#103 | — | 2 | 4 | no | — |
| 33 | C2×#103 | — | 2 | 4 | no | — |
| 34 | C2×#103 | — | 2 | 4 | no | — |
| 35 | C2×C4×C8 | — | 2 | 4 | no | — |
| 36 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 38 | #391 | — | 2 | 4 | no | — |
| 39 | #389 | — | 2 | 4 | no | — |
| 40 | #387 | — | 4 | 4 | no | — |
| 41 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 42 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 43 | #340 | — | 2 | 4 | no | — |
| 44 | #340 | — | 2 | 4 | no | — |
| 45 | #340 | D4 | 1 | 2 | yes | — |
| 46 | #340 | D4 | 1 | 2 | yes | — |
| 47 | #336 | — | 2 | 4 | no | — |
| 48 | #334 | — | 2 | 4 | no | — |
| 49 | #332 | — | 4 | 4 | no | — |
| 50 | C82 | C2×C4 | 1 | 4 | yes | — |
| 51 | C22×Q8 | — | 2 | 4 | no | — |
| 52 | C22×C8 | — | 2 | 1 | no | — |
| 53 | C22×C8 | — | 2 | 4 | no | — |
| 54 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 55 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 56 | C4⋊Q8 | — | 2 | 4 | no | — |
| 57 | C4⋊Q8 | — | 2 | 4 | no | — |
| 58 | C4⋊Q8 | — | 2 | 4 | no | — |
| 59 | C4⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 60 | C4×Q8 | — | 4 | 8 | no | — |
| 61 | C4×Q8 | — | 8 | 4 | no | — |
| 62 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 63 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 64 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 65 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 66 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 67 | C2×C42 | — | 2 | 4 | no | — |
| 68 | C2×C42 | — | 4 | 4 | no | — |
| 69 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 70 | #107 | — | 4 | 8 | no | — |
| 71 | #106 | — | 2 | 8 | no | — |
| 72 | C4⋊C8 | — | 2 | 4 | no | — |
| 73 | C4⋊C8 | — | 2 | 4 | no | — |
| 74 | #103 | — | 2 | 16 | no | — |
| 75 | #103 | — | 2 | 16 | no | — |
| 76 | #103 | — | 4 | 8 | no | — |
| 77 | #103 | — | 8 | 4 | no | — |
| 78 | C4×C8 | — | 2 | 1 | no | — |
| 79 | C4×C8 | — | 8 | 4 | no | — |
| 80 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 81 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 82 | C4×C8 | #41 | 1 | 4 | yes | — |
| 83 | #95 | — | 2 | 4 | no | — |
| 84 | #95 | — | 4 | 4 | no | — |
| 85 | #95 | — | 4 | 4 | no | — |
| 86 | C2×Q8 | — | 2 | 8 | no | — |
| 87 | C2×Q8 | — | 4 | 4 | no | — |
| 88 | C2×Q8 | — | 4 | 4 | no | — |
| 89 | C2×Q8 | — | 4 | 4 | no | — |
| 90 | C2×Q8 | — | 4 | 4 | no | — |
| 91 | C2×Q8 | — | 4 | 4 | no | — |
| 92 | C22×C4 | — | 2 | 4 | no | — |
| 93 | C22×C4 | — | 2 | 4 | no | — |
| 94 | C22×C4 | — | 4 | 4 | no | — |
| 95 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 96 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 97 | C2×C8 | — | 2 | 4 | no | — |
| 98 | C2×C8 | — | 2 | 2 | no | — |
| 99 | C2×C8 | — | 2 | 4 | no | — |
| 100 | C2×C8 | — | 2 | 4 | no | — |
| 101 | C2×C8 | — | 4 | 4 | no | — |
| 102 | C2×C8 | — | 4 | 4 | no | — |
| 103 | C2×C8 | — | 4 | 4 | no | — |
| 104 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 105 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 106 | C4⋊C4 | — | 2 | 4 | no | — |
| 107 | C4⋊C4 | — | 2 | 8 | no | — |
| 108 | C4⋊C4 | — | 2 | 8 | no | — |
| 109 | C4⋊C4 | — | 2 | 4 | no | — |
| 110 | C4⋊C4 | — | 2 | 4 | no | — |
| 111 | C4⋊C4 | — | 4 | 4 | no | — |
| 112 | C4⋊C4 | — | 4 | 4 | no | — |
| 113 | C4⋊C4 | — | 8 | 8 | no | — |
| 114 | C4⋊C4 | #135 | 1 | 8 | yes | — |
| 115 | C42 | — | 4 | 4 | no | — |
| 116 | C42 | — | 4 | 4 | no | — |
| 117 | C42 | — | 8 | 8 | no | — |
| 118 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 119 | C42 | C2×#41 | 1 | 2 | yes | — |
| 120 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 121 | C23 | D42 | 1 | 1 | yes | characteristic |
| 122 | Q8 | — | 4 | 8 | no | — |
| 123 | Q8 | — | 4 | 8 | no | — |
| 124 | Q8 | — | 8 | 4 | no | — |
| 125 | Q8 | — | 8 | 4 | no | — |
| 126 | C2×C4 | — | 2 | 2 | no | — |
| 127 | C2×C4 | — | 2 | 8 | no | — |
| 128 | C2×C4 | — | 4 | 4 | no | — |
| 129 | C2×C4 | — | 4 | 4 | no | — |
| 130 | C2×C4 | — | 4 | 8 | no | — |
| 131 | C2×C4 | — | 4 | 4 | no | — |
| 132 | C2×C4 | — | 4 | 4 | no | — |
| 133 | C2×C4 | — | 4 | 4 | no | — |
| 134 | C2×C4 | — | 8 | 4 | no | — |
| 135 | C2×C4 | — | 8 | 4 | no | — |
| 136 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 137 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 138 | C2×C4 | #550 | 1 | 2 | yes | — |
| 139 | C2×C4 | #559 | 1 | 1 | yes | characteristic |
| 140 | C2×C4 | C2×#135 | 1 | 4 | yes | — |
| 141 | C2×C4 | #577 | 1 | 4 | yes | — |
| 142 | C8 | — | 2 | 4 | no | — |
| 143 | C8 | — | 2 | 4 | no | — |
| 144 | C8 | — | 2 | 4 | no | — |
| 145 | C8 | — | 8 | 4 | no | — |
| 146 | C22 | #2366 | 1 | 1 | yes | characteristic |
| 147 | C22 | #3282 | 1 | 4 | yes | — |
| 148 | C22 | #3293 | 1 | 2 | yes | — |
| 149 | C4 | — | 2 | 1 | no | — |
| 150 | C4 | — | 2 | 1 | no | — |
| 151 | C4 | — | 4 | 8 | no | — |
| 152 | C4 | — | 8 | 4 | no | — |
| 153 | C4 | — | 8 | 8 | no | — |
| 154 | C4 | C4×#135 | 1 | 2 | yes | — |
| 155 | C4 | #2952 | 1 | 2 | yes | — |
| 156 | C4 | #3394 | 1 | 4 | yes | — |
| 157 | C2 | #19998 | 1 | 4 | yes | — |
| 158 | C2 | #23183 | 1 | 2 | yes | — |
| 159 | C2 | #23200 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. #1710 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23214 | 2 |
| C2×#1710 | 2 |
| #9736 | 2 |
| #9058 | 2 |
| C8×#106 | 2 |