Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
858 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 858 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 | C2×#2845 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1836 | C2 | 1 | 3 | yes | maximal |
| 3 | #10586 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×#1295 | C2 | 1 | 3 | yes | maximal |
| 5 | C22×#97 | C22 | 1 | 3 | yes | — |
| 6 | #2845 | — | 2 | 6 | no | — |
| 7 | #2845 | C22 | 1 | 4 | yes | — |
| 8 | C2×#398 | — | 2 | 6 | no | — |
| 9 | C2×#384 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#382 | C22 | 1 | 3 | yes | — |
| 11 | #1836 | — | 2 | 6 | no | — |
| 12 | #1836 | C22 | 1 | 12 | yes | — |
| 13 | C2×#340 | — | 2 | 6 | no | — |
| 14 | #1295 | C22 | 1 | 12 | yes | — |
| 15 | C23×C8 | — | 2 | 3 | no | — |
| 16 | C22×C4⋊C4 | — | 2 | 6 | no | — |
| 17 | C22×C4⋊C4 | — | 2 | 3 | no | — |
| 18 | C22×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 19 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 20 | C2×#97 | — | 2 | 3 | no | — |
| 21 | C2×#97 | — | 2 | 6 | no | — |
| 22 | C2×#97 | Q8 | 1 | 12 | yes | — |
| 23 | C2×#97 | C23 | 1 | 6 | yes | — |
| 24 | #398 | — | 2 | 24 | no | — |
| 25 | #398 | — | 2 | 24 | no | — |
| 26 | #384 | — | 2 | 3 | no | — |
| 27 | #384 | C23 | 1 | 2 | yes | — |
| 28 | #382 | — | 2 | 6 | no | — |
| 29 | #382 | D4 | 1 | 6 | yes | — |
| 30 | #382 | C23 | 1 | 6 | yes | — |
| 31 | C2×#95 | — | 2 | 2 | no | — |
| 32 | C2×#95 | — | 2 | 6 | no | — |
| 33 | #340 | — | 2 | 24 | no | — |
| 34 | C23×C4 | — | 2 | 6 | no | — |
| 35 | C23×C4 | — | 4 | 1 | no | — |
| 36 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 37 | C22×C8 | — | 2 | 12 | no | — |
| 38 | C22×C8 | — | 2 | 6 | no | — |
| 39 | C22×C8 | — | 4 | 6 | no | — |
| 40 | C22×C8 | — | 4 | 6 | no | — |
| 41 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 42 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 43 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 44 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 45 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 46 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 47 | C2×C4⋊C4 | — | 8 | 12 | no | — |
| 48 | C2×C4⋊C4 | C2×D4 | 1 | 12 | yes | — |
| 49 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 50 | C2×C42 | C2×Q8 | 1 | 6 | yes | — |
| 51 | C2×C42 | #41 | 1 | 2 | yes | — |
| 52 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 53 | #97 | — | 2 | 12 | no | — |
| 54 | #97 | — | 2 | 6 | no | — |
| 55 | #97 | C2×Q8 | 1 | 12 | yes | — |
| 56 | #95 | — | 2 | 8 | no | — |
| 57 | #95 | — | 2 | 12 | no | — |
| 58 | #95 | — | 2 | 12 | no | — |
| 59 | C24 | #120 | 1 | 1 | yes | characteristic |
| 60 | C22×C4 | — | 2 | 6 | no | — |
| 61 | C22×C4 | — | 2 | 6 | no | — |
| 62 | C22×C4 | — | 2 | 12 | no | — |
| 63 | C22×C4 | — | 4 | 12 | no | — |
| 64 | C22×C4 | — | 4 | 12 | no | — |
| 65 | C22×C4 | — | 4 | 12 | no | — |
| 66 | C22×C4 | — | 4 | 2 | no | — |
| 67 | C22×C4 | — | 8 | 6 | no | — |
| 68 | C22×C4 | #120 | 1 | 3 | yes | — |
| 69 | C22×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 70 | C22×C4 | C4⋊Q8 | 1 | 6 | yes | — |
| 71 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 72 | C2×C8 | — | 2 | 12 | no | — |
| 73 | C2×C8 | — | 4 | 24 | no | — |
| 74 | C2×C8 | — | 4 | 6 | no | — |
| 75 | C2×C8 | — | 4 | 6 | no | — |
| 76 | C4⋊C4 | — | 4 | 12 | no | — |
| 77 | C4⋊C4 | — | 4 | 12 | no | — |
| 78 | C4⋊C4 | — | 8 | 6 | no | — |
| 79 | C4⋊C4 | — | 8 | 6 | no | — |
| 80 | C4⋊C4 | — | 8 | 24 | no | — |
| 81 | C42 | C22⋊Q8 | 1 | 6 | yes | — |
| 82 | C42 | C4⋊Q8 | 1 | 6 | yes | — |
| 83 | C42 | C22×Q8 | 1 | 3 | yes | — |
| 84 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 85 | C23 | #395 | 1 | 2 | yes | — |
| 86 | C23 | #454 | 1 | 6 | yes | — |
| 87 | C23 | #455 | 1 | 6 | yes | — |
| 88 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 89 | C2×C4 | — | 2 | 12 | no | — |
| 90 | C2×C4 | — | 2 | 12 | no | — |
| 91 | C2×C4 | — | 2 | 3 | no | — |
| 92 | C2×C4 | — | 2 | 3 | no | — |
| 93 | C2×C4 | — | 2 | 3 | no | — |
| 94 | C2×C4 | — | 2 | 3 | no | — |
| 95 | C2×C4 | — | 4 | 12 | no | — |
| 96 | C2×C4 | — | 4 | 12 | no | — |
| 97 | C2×C4 | — | 4 | 12 | no | — |
| 98 | C2×C4 | — | 8 | 24 | no | — |
| 99 | C2×C4 | — | 8 | 6 | no | — |
| 100 | C2×C4 | — | 16 | 4 | no | — |
| 101 | C2×C4 | #395 | 1 | 6 | yes | — |
| 102 | C2×C4 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 103 | C2×C4 | C2×C4⋊Q8 | 1 | 3 | yes | — |
| 104 | C8 | — | 4 | 12 | no | — |
| 105 | C22 | #1658 | 1 | 4 | yes | — |
| 106 | C22 | #1662 | 1 | 12 | yes | — |
| 107 | C22 | #2061 | 1 | 12 | yes | — |
| 108 | C22 | C2×#395 | 1 | 1 | yes | characteristic |
| 109 | C22 | C2×#455 | 1 | 3 | yes | — |
| 110 | C22 | C2×#454 | 1 | 3 | yes | — |
| 111 | C4 | — | 2 | 3 | no | — |
| 112 | C4 | — | 2 | 3 | no | — |
| 113 | C4 | — | 2 | 6 | no | — |
| 114 | C4 | — | 8 | 12 | no | — |
| 115 | C4 | — | 16 | 2 | no | — |
| 116 | C2 | #10586 | 1 | 8 | yes | direct product factor |
| 117 | C2 | C2×#1662 | 1 | 3 | yes | — |
| 118 | C2 | C2×#1658 | 1 | 1 | yes | characteristic |
| 119 | C2 | C2×#2061 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10586 | C2 | direct (N × H) |
| 116. C2 | #10586 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2845 | 2 |
| C2×#1836 | 2 |
| #10586 | 2 |
| C2×#1295 | 2 |