Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
295 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #20830 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12605 | C2 | 1 | 2 | yes | maximal |
| 3 | #7267 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #2920 | C4 | 1 | 2 | yes | — |
| 5 | C22×#97 | C22 | 1 | 1 | yes | characteristic |
| 6 | #2112 | — | 2 | 2 | no | — |
| 7 | C2×#347 | C22 | 1 | 4 | yes | — |
| 8 | #1849 | C4 | 1 | 1 | yes | characteristic |
| 9 | #1849 | C4 | 1 | 1 | yes | characteristic |
| 10 | #1838 | C22 | 1 | 2 | yes | — |
| 11 | #1810 | — | 2 | 2 | no | — |
| 12 | #1778 | — | 2 | 2 | no | — |
| 13 | #1375 | — | 2 | 2 | no | — |
| 14 | C23×C8 | D4 | 1 | 2 | yes | — |
| 15 | C22×C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C2×M32 | — | 2 | 4 | no | — |
| 17 | #432 | — | 4 | 2 | no | — |
| 18 | #431 | — | 4 | 2 | no | — |
| 19 | C2×#108 | — | 2 | 2 | no | — |
| 20 | C2×#108 | — | 2 | 2 | no | — |
| 21 | #423 | C2×C4 | 1 | 2 | yes | — |
| 22 | #423 | C2×C4 | 1 | 2 | yes | — |
| 23 | #407 | — | 2 | 2 | no | — |
| 24 | C4×M16 | — | 2 | 2 | no | — |
| 25 | C2×#97 | — | 2 | 4 | no | — |
| 26 | C2×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×#97 | D4 | 1 | 2 | yes | — |
| 28 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 29 | C22⋊C16 | — | 2 | 8 | no | — |
| 30 | #347 | — | 2 | 8 | no | — |
| 31 | #341 | — | 2 | 4 | no | — |
| 32 | C23×C4 | — | 2 | 1 | no | — |
| 33 | C23×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 34 | C2×M16 | — | 2 | 2 | no | — |
| 35 | C2×M16 | — | 2 | 2 | no | — |
| 36 | C22×C8 | — | 2 | 4 | no | — |
| 37 | C22×C8 | — | 2 | 4 | no | — |
| 38 | C22×C8 | — | 4 | 2 | no | — |
| 39 | C22×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 40 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 41 | C2×C42 | — | 2 | 1 | no | — |
| 42 | C2×C42 | — | 2 | 2 | no | — |
| 43 | C2×C42 | — | 2 | 2 | no | — |
| 44 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 46 | M32 | — | 4 | 4 | no | — |
| 47 | M32 | — | 4 | 4 | no | — |
| 48 | C2×C16 | — | 2 | 8 | no | — |
| 49 | #108 | — | 4 | 4 | no | — |
| 50 | #108 | — | 4 | 4 | no | — |
| 51 | C4⋊C8 | — | 4 | 2 | no | — |
| 52 | C4⋊C8 | — | 4 | 2 | no | — |
| 53 | C22⋊C8 | — | 4 | 4 | no | — |
| 54 | #97 | — | 2 | 4 | no | — |
| 55 | #97 | — | 2 | 2 | no | — |
| 56 | #97 | — | 4 | 2 | no | — |
| 57 | #97 | #41 | 1 | 4 | yes | — |
| 58 | C4×C8 | — | 4 | 2 | no | — |
| 59 | C24 | #101 | 1 | 1 | yes | characteristic |
| 60 | C22×C4 | — | 2 | 2 | no | — |
| 61 | C22×C4 | — | 2 | 2 | no | — |
| 62 | C22×C4 | — | 2 | 1 | no | — |
| 63 | C22×C4 | — | 2 | 1 | no | — |
| 64 | C22×C4 | — | 2 | 1 | no | — |
| 65 | C22×C4 | — | 2 | 1 | no | — |
| 66 | C22×C4 | — | 2 | 2 | no | — |
| 67 | C22×C4 | — | 2 | 2 | no | — |
| 68 | C22×C4 | — | 8 | 2 | no | — |
| 69 | C22×C4 | #101 | 1 | 1 | yes | characteristic |
| 70 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | M16 | — | 4 | 4 | no | — |
| 72 | M16 | — | 4 | 4 | no | — |
| 73 | C2×C8 | — | 2 | 2 | no | — |
| 74 | C2×C8 | — | 2 | 2 | no | — |
| 75 | C2×C8 | — | 2 | 2 | no | — |
| 76 | C2×C8 | — | 2 | 2 | no | — |
| 77 | C2×C8 | — | 4 | 8 | no | — |
| 78 | C2×C8 | — | 4 | 2 | no | — |
| 79 | C2×C8 | — | 4 | 2 | no | — |
| 80 | C2×C8 | #104 | 1 | 4 | yes | — |
| 81 | C2×C8 | #123 | 1 | 4 | yes | — |
| 82 | C42 | — | 2 | 4 | no | — |
| 83 | C42 | — | 2 | 1 | no | — |
| 84 | C42 | — | 2 | 1 | no | — |
| 85 | C42 | #100 | 1 | 2 | yes | — |
| 86 | C42 | #117 | 1 | 2 | yes | — |
| 87 | C16 | — | 4 | 8 | no | — |
| 88 | C23 | — | 2 | 1 | no | — |
| 89 | C23 | — | 2 | 2 | no | — |
| 90 | C23 | — | 4 | 2 | no | — |
| 91 | C23 | C2×#101 | 1 | 1 | yes | characteristic |
| 92 | C2×C4 | — | 2 | 1 | no | — |
| 93 | C2×C4 | — | 2 | 2 | no | — |
| 94 | C2×C4 | — | 2 | 2 | no | — |
| 95 | C2×C4 | — | 2 | 2 | no | — |
| 96 | C2×C4 | — | 4 | 4 | no | — |
| 97 | C2×C4 | — | 4 | 1 | no | — |
| 98 | C2×C4 | — | 4 | 2 | no | — |
| 99 | C2×C4 | — | 4 | 1 | no | — |
| 100 | C2×C4 | — | 8 | 1 | no | — |
| 101 | C2×C4 | — | 8 | 1 | no | — |
| 102 | C2×C4 | — | 8 | 2 | no | — |
| 103 | C2×C4 | #381 | 1 | 1 | yes | characteristic |
| 104 | C2×C4 | C2×#100 | 1 | 1 | yes | characteristic |
| 105 | C2×C4 | C2×#104 | 1 | 2 | yes | — |
| 106 | C2×C4 | #421 | 1 | 2 | yes | — |
| 107 | C8 | — | 2 | 4 | no | — |
| 108 | C8 | — | 2 | 4 | no | — |
| 109 | C8 | — | 8 | 2 | no | — |
| 110 | C8 | — | 8 | 2 | no | — |
| 111 | C22 | — | 2 | 2 | no | — |
| 112 | C22 | — | 2 | 4 | no | — |
| 113 | C22 | — | 4 | 1 | no | — |
| 114 | C22 | — | 4 | 1 | no | — |
| 115 | C22 | — | 4 | 2 | no | — |
| 116 | C22 | — | 4 | 1 | no | — |
| 117 | C22 | #1531 | 1 | 2 | yes | — |
| 118 | C22 | #1821 | 1 | 1 | yes | characteristic |
| 119 | C4 | — | 4 | 2 | no | — |
| 120 | C4 | — | 4 | 2 | no | — |
| 121 | C4 | — | 4 | 1 | no | — |
| 122 | C4 | #1525 | 1 | 1 | yes | characteristic |
| 123 | C4 | #1536 | 1 | 1 | yes | characteristic |
| 124 | C4 | #1826 | 1 | 2 | yes | — |
| 125 | C2 | — | 2 | 1 | no | — |
| 126 | C2 | — | 2 | 2 | no | — |
| 127 | C2 | — | 4 | 2 | no | — |
| 128 | C2 | #8721 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1525 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #8721 |
| prime radical | C24 | #101 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7267 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20830 | 2 |
| #12605 | 2 |
| #7267 | 2 |