Subgroups — C₂²×Q₃₂ #3410

Order 128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆	C₂	1	2	yes	maximal
2	C₂×Q₃₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈	—	2	2	no	—
5	C₂×Q₁₆	—	2	6	no	—
6	C₂×Q₁₆	C₂²	1	12	yes	—
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	Q₃₂	C₂²	1	16	yes	direct product factor
9	C₂×C₁₆	C₂²	1	6	yes	—
10	C₂×Q₈	—	2	12	no	—
11	C₂×Q₈	—	4	6	no	—
12	C₂²×C₄	—	4	2	no	—
13	C₂²×C₄	D₄	1	1	yes	characteristic
14	Q₁₆	—	2	12	no	—
15	Q₁₆	C₂³	1	8	yes	—
16	C₂×C₈	D₄	1	3	yes	—
17	C₂×C₈	C₂³	1	3	yes	—
18	C₁₆	C₂³	1	4	yes	—
19	C₂³	D₈	1	1	yes	characteristic
20	Q₈	—	2	8	no	—
21	Q₈	—	4	12	no	—
22	C₂×C₄	—	4	12	no	—
23	C₂×C₄	D₈	1	3	yes	—
24	C₂×C₄	C₂×D₄	1	3	yes	—
25	C₈	C₂×D₄	1	3	yes	—
26	C₈	C₂⁴	1	1	yes	characteristic
27	C₂²	Q₃₂	1	4	yes	direct product factor
28	C₂²	C₂×D₈	1	3	yes	—
29	C₄	—	4	8	no	—
30	C₄	C₂×D₈	1	3	yes	—
31	C₄	C₂²×D₄	1	1	yes	characteristic
32	C₂	C₂×Q₃₂	1	6	yes	direct product factor
33	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	D₈
commutator, Frattini, cosocle	C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.