Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1181 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 1181 conjugacy classes of subgroups — too many to draw. Only its 230 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 | #33175 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×#1712 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#1455 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #9837 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #8813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#493 | C22 | 1 | 6 | yes | — |
| 7 | C2×#469 | — | 2 | 12 | no | — |
| 8 | C2×C4×M16 | C22 | 1 | 3 | yes | — |
| 9 | #2856 | C22 | 1 | 4 | yes | — |
| 10 | #1972 | — | 2 | 12 | no | — |
| 11 | #1957 | C22 | 1 | 12 | yes | — |
| 12 | #1844 | C22 | 1 | 6 | yes | — |
| 13 | #1712 | — | 2 | 4 | no | — |
| 14 | #1455 | — | 2 | 4 | no | — |
| 15 | C2×#322 | C22 | 1 | 3 | yes | — |
| 16 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×D8 | — | 2 | 12 | no | — |
| 18 | C22×M16 | D4 | 1 | 6 | yes | — |
| 19 | C2×#127 | C23 | 1 | 4 | yes | — |
| 20 | C2×#121 | — | 2 | 12 | no | — |
| 21 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 22 | #493 | — | 2 | 12 | no | — |
| 23 | #493 | — | 2 | 12 | no | — |
| 24 | #469 | — | 4 | 48 | no | — |
| 25 | C2×#106 | — | 2 | 12 | no | — |
| 26 | C2×#102 | — | 2 | 24 | no | — |
| 27 | C4×M16 | — | 2 | 6 | no | — |
| 28 | C4×M16 | — | 2 | 6 | no | — |
| 29 | C2×#97 | D4 | 1 | 6 | yes | — |
| 30 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 31 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 32 | #390 | — | 2 | 12 | no | — |
| 33 | #384 | C23 | 1 | 4 | yes | — |
| 34 | #322 | — | 2 | 6 | no | — |
| 35 | C82 | D4 | 1 | 4 | yes | complemented |
| 36 | C22×D4 | — | 2 | 12 | no | — |
| 37 | C22×D4 | — | 4 | 4 | no | — |
| 38 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 39 | C2×D8 | — | 4 | 24 | no | — |
| 40 | C2×D8 | — | 4 | 24 | no | — |
| 41 | C2×D8 | — | 4 | 24 | no | — |
| 42 | C2×M16 | — | 2 | 12 | no | — |
| 43 | C2×M16 | — | 2 | 24 | no | — |
| 44 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 45 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 46 | #127 | — | 2 | 4 | no | — |
| 47 | #127 | — | 2 | 12 | no | — |
| 48 | #121 | — | 4 | 48 | no | — |
| 49 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 50 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 51 | C2×C42 | — | 2 | 1 | no | — |
| 52 | C2×C42 | — | 2 | 3 | no | — |
| 53 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 54 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 55 | #106 | — | 4 | 24 | no | — |
| 56 | #102 | — | 4 | 48 | no | — |
| 57 | #97 | — | 2 | 12 | no | — |
| 58 | C4×C8 | — | 2 | 3 | no | — |
| 59 | C4×C8 | — | 2 | 3 | no | — |
| 60 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 61 | C4×C8 | #41 | 1 | 6 | yes | — |
| 62 | C24 | — | 4 | 4 | no | — |
| 63 | C24 | #127 | 1 | 1 | yes | characteristic |
| 64 | C2×D4 | — | 4 | 12 | no | — |
| 65 | C2×D4 | — | 4 | 12 | no | — |
| 66 | C2×D4 | — | 4 | 24 | no | — |
| 67 | C2×D4 | — | 4 | 24 | no | — |
| 68 | C2×D4 | — | 8 | 24 | no | — |
| 69 | C22×C4 | — | 2 | 3 | no | — |
| 70 | C22×C4 | — | 2 | 3 | no | — |
| 71 | C22×C4 | — | 2 | 12 | no | — |
| 72 | C22×C4 | — | 4 | 4 | no | — |
| 73 | C22×C4 | #127 | 1 | 3 | yes | — |
| 74 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 75 | D8 | — | 8 | 48 | no | — |
| 76 | M16 | — | 4 | 24 | no | — |
| 77 | C2×C8 | — | 2 | 6 | no | — |
| 78 | C2×C8 | — | 2 | 12 | no | — |
| 79 | C2×C8 | #121 | 1 | 12 | yes | — |
| 80 | C2×C8 | AGL(1,ℤ/8) | 1 | 24 | yes | — |
| 81 | C4⋊C4 | — | 4 | 24 | no | — |
| 82 | C22⋊C4 | — | 8 | 24 | no | — |
| 83 | C42 | — | 2 | 3 | no | — |
| 84 | C42 | — | 2 | 3 | no | — |
| 85 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 86 | C42 | C2×#41 | 1 | 3 | yes | — |
| 87 | C23 | — | 2 | 1 | no | — |
| 88 | C23 | — | 2 | 6 | no | — |
| 89 | C23 | — | 8 | 4 | no | — |
| 90 | C23 | — | 8 | 12 | no | — |
| 91 | C23 | — | 8 | 12 | no | — |
| 92 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 93 | D4 | — | 8 | 24 | no | — |
| 94 | D4 | — | 8 | 24 | no | — |
| 95 | D4 | — | 16 | 16 | no | — |
| 96 | C2×C4 | — | 2 | 3 | no | — |
| 97 | C2×C4 | — | 2 | 6 | no | — |
| 98 | C2×C4 | — | 4 | 6 | no | — |
| 99 | C2×C4 | — | 4 | 6 | no | — |
| 100 | C2×C4 | — | 8 | 12 | no | — |
| 101 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 102 | C2×C4 | #532 | 1 | 3 | yes | — |
| 103 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 12 | yes | — |
| 104 | C8 | — | 4 | 6 | no | — |
| 105 | C8 | #469 | 1 | 24 | yes | complemented |
| 106 | C22 | — | 4 | 4 | no | — |
| 107 | C22 | — | 4 | 3 | no | — |
| 108 | C22 | — | 16 | 4 | no | — |
| 109 | C22 | — | 16 | 12 | no | — |
| 110 | C22 | — | 16 | 12 | no | — |
| 111 | C22 | #2856 | 1 | 1 | yes | characteristic |
| 112 | C22 | #3151 | 1 | 6 | yes | — |
| 113 | C4 | — | 4 | 3 | no | — |
| 114 | C4 | — | 16 | 4 | no | — |
| 115 | C4 | C2×#469 | 1 | 6 | yes | — |
| 116 | C4 | #3184 | 1 | 6 | yes | — |
| 117 | C2 | — | 8 | 1 | no | — |
| 118 | C2 | — | 32 | 4 | no | — |
| 119 | C2 | #19745 | 1 | 4 | yes | — |
| 120 | C2 | #33175 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33175 | C2 | semidirect |
| 2. C2×#1712 | C2 | semidirect |
| 3. C2×#1455 | C2 | semidirect |
| 4. #9837 | C2 | semidirect |
| 5. #8813 | C2 | semidirect |
| 35. C82 | D4 | semidirect |
| 105. C8 | #469 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33175 | 2 |
| C2×#1712 | 2 |
| C2×#1455 | 2 |
| #9837 | 2 |
| #8813 | 2 |