Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
307 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 84 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 | #28824 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13186 | C2 | 1 | 2 | yes | maximal |
| 3 | #8813 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7268 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #7179 | C2 | 1 | 2 | yes | maximal |
| 6 | #3163 | — | 2 | 2 | no | — |
| 7 | C2×#365 | — | 2 | 2 | no | — |
| 8 | C2×#364 | C22 | 1 | 2 | yes | — |
| 9 | #2060 | — | 2 | 2 | no | — |
| 10 | #1951 | — | 2 | 2 | no | — |
| 11 | #1844 | — | 2 | 2 | no | — |
| 12 | #1844 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1844 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1772 | C22 | 1 | 2 | yes | — |
| 15 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1386 | — | 2 | 2 | no | — |
| 17 | #544 | — | 2 | 2 | no | — |
| 18 | C2×M32 | — | 2 | 2 | no | — |
| 19 | #501 | — | 4 | 4 | no | — |
| 20 | #499 | — | 4 | 2 | no | — |
| 21 | #498 | — | 4 | 2 | no | — |
| 22 | C2×#108 | D4 | 1 | 2 | yes | — |
| 23 | #428 | — | 2 | 4 | no | — |
| 24 | #427 | D4 | 1 | 2 | yes | — |
| 25 | C2×#106 | — | 2 | 2 | no | — |
| 26 | C2×#106 | — | 2 | 2 | no | — |
| 27 | C2×#106 | — | 4 | 4 | no | — |
| 28 | #405 | — | 2 | 2 | no | — |
| 29 | C2×C4×C8 | — | 2 | 1 | no | — |
| 30 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 31 | #384 | — | 2 | 1 | no | — |
| 32 | #384 | D4 | 1 | 2 | yes | — |
| 33 | #365 | — | 4 | 4 | no | — |
| 34 | #364 | — | 2 | 2 | no | — |
| 35 | #364 | Q8 | 1 | 4 | yes | complemented |
| 36 | #343 | — | 2 | 2 | no | — |
| 37 | #339 | — | 2 | 2 | no | — |
| 38 | C82 | — | 2 | 1 | no | — |
| 39 | C82 | Q8 | 1 | 2 | yes | complemented |
| 40 | C2×M16 | — | 2 | 2 | no | — |
| 41 | C22×C8 | — | 2 | 2 | no | — |
| 42 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 43 | C4⋊Q8 | — | 4 | 4 | no | — |
| 44 | #125 | — | 4 | 4 | no | — |
| 45 | C22⋊Q8 | — | 8 | 2 | no | — |
| 46 | #117 | — | 2 | 2 | no | — |
| 47 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 48 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 49 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 50 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 51 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | M32 | — | 2 | 4 | no | — |
| 53 | M32 | — | 4 | 2 | no | — |
| 54 | C2×C16 | — | 4 | 2 | no | — |
| 55 | #108 | — | 2 | 2 | no | — |
| 56 | #108 | SD16 | 1 | 4 | yes | — |
| 57 | #107 | — | 4 | 4 | no | — |
| 58 | #107 | SD16 | 1 | 4 | yes | — |
| 59 | #106 | — | 2 | 2 | no | — |
| 60 | #106 | — | 4 | 4 | no | — |
| 61 | #106 | — | 4 | 2 | no | — |
| 62 | #106 | — | 4 | 2 | no | — |
| 63 | #106 | — | 4 | 2 | no | — |
| 64 | #106 | — | 4 | 2 | no | — |
| 65 | #106 | — | 8 | 8 | no | — |
| 66 | #97 | — | 4 | 2 | no | — |
| 67 | C4×C8 | — | 2 | 2 | no | — |
| 68 | C4×C8 | — | 2 | 2 | no | — |
| 69 | C4×C8 | — | 4 | 2 | no | — |
| 70 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 71 | C4×C8 | C2×Q8 | 1 | 2 | yes | — |
| 72 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 73 | C2×Q8 | — | 8 | 2 | no | — |
| 74 | C22×C4 | — | 2 | 1 | no | — |
| 75 | C22×C4 | — | 4 | 2 | no | — |
| 76 | C22×C4 | — | 8 | 1 | no | — |
| 77 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 78 | M16 | — | 2 | 4 | no | — |
| 79 | M16 | — | 4 | 2 | no | — |
| 80 | C2×C8 | — | 2 | 4 | no | — |
| 81 | C2×C8 | — | 2 | 8 | no | — |
| 82 | C2×C8 | — | 4 | 2 | no | — |
| 83 | C2×C8 | C22⋊Q8 | 1 | 2 | yes | — |
| 84 | C2×C8 | C4⋊Q8 | 1 | 2 | yes | — |
| 85 | C2×C8 | C2×SD16 | 1 | 2 | yes | — |
| 86 | C2×C8 | C2×SD16 | 1 | 2 | yes | — |
| 87 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 88 | C2×C8 | #137 | 1 | 2 | yes | — |
| 89 | C4⋊C4 | — | 2 | 4 | no | — |
| 90 | C4⋊C4 | — | 4 | 4 | no | — |
| 91 | C4⋊C4 | — | 8 | 4 | no | — |
| 92 | C4⋊C4 | — | 8 | 4 | no | — |
| 93 | C4⋊C4 | — | 8 | 2 | no | — |
| 94 | C4⋊C4 | — | 8 | 4 | no | — |
| 95 | C4⋊C4 | — | 8 | 2 | no | — |
| 96 | C22⋊C4 | — | 4 | 2 | no | — |
| 97 | C42 | — | 2 | 1 | no | — |
| 98 | C42 | — | 4 | 2 | no | — |
| 99 | C42 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 100 | C42 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 101 | C16 | — | 4 | 4 | no | — |
| 102 | C23 | #453 | 1 | 1 | yes | characteristic |
| 103 | Q8 | — | 32 | 2 | no | — |
| 104 | C2×C4 | — | 2 | 4 | no | — |
| 105 | C2×C4 | — | 2 | 1 | no | — |
| 106 | C2×C4 | — | 2 | 1 | no | — |
| 107 | C2×C4 | — | 4 | 4 | no | — |
| 108 | C2×C4 | — | 8 | 2 | no | — |
| 109 | C2×C4 | — | 16 | 2 | no | — |
| 110 | C2×C4 | — | 16 | 2 | no | — |
| 111 | C2×C4 | — | 16 | 1 | no | — |
| 112 | C2×C4 | #395 | 1 | 1 | yes | characteristic |
| 113 | C2×C4 | #448 | 1 | 2 | yes | — |
| 114 | C2×C4 | #450 | 1 | 2 | yes | — |
| 115 | C2×C4 | #456 | 1 | 1 | yes | characteristic |
| 116 | C8 | — | 2 | 8 | no | — |
| 117 | C8 | — | 4 | 4 | no | — |
| 118 | C8 | #475 | 1 | 2 | yes | — |
| 119 | C8 | #476 | 1 | 2 | yes | — |
| 120 | C8 | #498 | 1 | 2 | yes | — |
| 121 | C8 | #501 | 1 | 2 | yes | — |
| 122 | C22 | — | 2 | 2 | no | — |
| 123 | C22 | #1623 | 1 | 1 | yes | characteristic |
| 124 | C22 | #1629 | 1 | 1 | yes | characteristic |
| 125 | C22 | #2060 | 1 | 1 | yes | characteristic |
| 126 | C4 | — | 2 | 2 | no | — |
| 127 | C4 | — | 2 | 2 | no | — |
| 128 | C4 | — | 8 | 4 | no | — |
| 129 | C4 | — | 16 | 2 | no | — |
| 130 | C4 | — | 32 | 1 | no | — |
| 131 | C4 | #1651 | 1 | 2 | yes | — |
| 132 | C4 | #2058 | 1 | 2 | yes | — |
| 133 | C2 | — | 2 | 1 | no | — |
| 134 | C2 | — | 2 | 1 | no | — |
| 135 | C2 | #10574 | 1 | 1 | yes | characteristic |
| 136 | C2 | #12364 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #453 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 35. #364 | Q8 | semidirect |
| 39. C82 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28824 | 2 |
| #13186 | 2 |
| #8813 | 2 |
| #7268 | 2 |
| #7179 | 2 |