Subgroups — C₂×#543 #3456

Order 128.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#543	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
3	C₂²×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂⁵	C₂²	1	1	yes	characteristic
5	C₂²×Q₈	C₂²	1	4	yes	complemented
6	C₂³×C₄	C₂²	1	6	yes	—
7	C₂²⋊Q₈	C₂²	1	96	yes	complemented
8	C₂×C₄⋊C₄	C₂²	1	12	yes	complemented
9	C₂×C₂²⋊C₄	C₂²	1	12	yes	—
10	C₂×C₂²⋊C₄	C₂²	1	24	yes	—
11	C₂⁴	—	4	4	no	—
12	C₂⁴	Q₈	1	8	yes	complemented
13	C₂⁴	C₂³	1	3	yes	—
14	C₂⁴	C₂³	1	4	yes	—
15	C₂×Q₈	—	4	8	no	—
16	C₂×Q₈	C₂³	1	16	yes	complemented
17	C₂²×C₄	—	4	12	no	—
18	C₂²×C₄	C₂³	1	12	yes	—
19	C₂²×C₄	C₂³	1	24	yes	—
20	C₄⋊C₄	C₂³	1	48	yes	complemented
21	C₂²⋊C₄	C₂³	1	48	yes	—
22	C₂³	—	2	12	no	—
23	C₂³	—	4	16	no	—
24	C₂³	—	4	8	no	—
25	C₂³	C₂×Q₈	1	12	yes	—
26	C₂³	C₂×Q₈	1	16	yes	complemented
27	C₂³	C₂⁴	1	6	yes	—
28	C₂³	C₂⁴	1	1	yes	characteristic
29	Q₈	—	4	16	no	—
30	C₂×C₄	—	2	24	no	—
31	C₂×C₄	—	4	24	no	—
32	C₂×C₄	C₂⁴	1	24	yes	—
33	C₂²	—	2	24	no	—
34	C₂²	—	2	12	no	—
35	C₂²	—	4	16	no	—
36	C₂²	C₂²×Q₈	1	12	yes	—
37	C₂²	C₂²×Q₈	1	2	yes	—
38	C₂²	H(5,𝔽₂)	1	4	yes	—
39	C₂²	C₂⁵	1	1	yes	characteristic
40	C₄	—	2	24	no	—
41	C₂	—	2	12	no	—
42	C₂	#543	1	4	yes	direct product factor
43	C₂	C₂³×Q₈	1	1	yes	characteristic
44	C₂	C₂×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂⁵	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ 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.