Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
170 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #398 | C2 | 1 | 12 | yes | maximal |
| 2 | #382 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 4 | C2×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 5 | C2×C42 | C22 | 1 | 1 | yes | characteristic |
| 6 | #95 | C22 | 1 | 12 | yes | — |
| 7 | #95 | C22 | 1 | 4 | yes | — |
| 8 | C22×C4 | C23 | 1 | 12 | yes | — |
| 9 | C22×C4 | C23 | 1 | 3 | yes | — |
| 10 | C4⋊C4 | — | 4 | 6 | no | — |
| 11 | C4⋊C4 | — | 4 | 12 | no | — |
| 12 | C42 | Q8 | 1 | 4 | yes | — |
| 13 | C23 | C24 | 1 | 1 | yes | characteristic |
| 14 | C2×C4 | — | 2 | 3 | no | — |
| 15 | C2×C4 | — | 2 | 3 | no | — |
| 16 | C2×C4 | — | 2 | 12 | no | — |
| 17 | C2×C4 | — | 2 | 12 | no | — |
| 18 | C2×C4 | — | 2 | 12 | no | — |
| 19 | C2×C4 | C2×Q8 | 1 | 6 | yes | — |
| 20 | C22 | C22×Q8 | 1 | 1 | yes | characteristic |
| 21 | C22 | H(5,𝔽2) | 1 | 3 | yes | — |
| 22 | C22 | 2−1+4 | 1 | 3 | yes | — |
| 23 | C4 | — | 2 | 6 | no | — |
| 24 | C4 | — | 4 | 12 | no | — |
| 25 | C2 | #544 | 1 | 3 | yes | — |
| 26 | C2 | #562 | 1 | 3 | yes | — |
| 27 | C2 | #564 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #398 | 2 |
| #382 | 2 |