Subgroups — C₂×#123 #524

Order 64.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#123	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁴	C₂²	1	1	yes	characteristic
8	C₂⁴	C₂²	1	1	yes	characteristic
9	C₂×D₄	—	2	4	no	—
10	C₂×D₄	C₂²	1	4	yes	—
11	C₂²×C₄	—	2	4	no	—
12	C₂²×C₄	C₂²	1	2	yes	—
13	C₂²×C₄	C₂²	1	2	yes	—
14	C₂²×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂²	1	4	yes	complemented
16	C₄⋊C₄	C₂²	1	8	yes	complemented
17	C₂²⋊C₄	C₂²	1	4	yes	complemented
18	C₂²⋊C₄	C₂²	1	8	yes	complemented
19	C₂³	—	2	4	no	—
20	C₂³	—	2	2	no	—
21	C₂³	—	2	4	no	—
22	C₂³	D₄	1	4	yes	—
23	C₂³	C₂³	1	2	yes	—
24	C₂³	C₂³	1	2	yes	—
25	C₂³	C₂³	1	1	yes	characteristic
26	D₄	—	2	8	no	—
27	C₂×C₄	—	2	4	no	—
28	C₂×C₄	—	2	8	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	4	no	—
31	C₂×C₄	C₂³	1	4	yes	—
32	C₂×C₄	C₂³	1	4	yes	—
33	C₂×C₄	C₂³	1	2	yes	—
34	C₂²	—	2	8	no	—
35	C₂²	—	2	4	no	—
36	C₂²	—	2	2	no	—
37	C₂²	—	2	4	no	—
38	C₂²	—	4	4	no	—
39	C₂²	C₂×D₄	1	4	yes	—
40	C₂²	C₂×D₄	1	2	yes	—
41	C₂²	#41	1	4	yes	—
42	C₂²	C₂⁴	1	1	yes	characteristic
43	C₄	—	2	4	no	—
44	C₄	—	2	2	no	—
45	C₄	—	2	4	no	—
46	C₂	—	2	4	no	—
47	C₂	—	4	2	no	—
48	C₂	#123	1	4	yes	direct product factor
49	C₂	C₂²×D₄	1	1	yes	characteristic
50	C₂	C₂×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴
prime radical	C₂²×D₄	C₂
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.