Subgroups — #539

Order 64.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#126	C₂	1	2	yes	maximal, complemented
2	#125	C₂	1	2	yes	maximal, complemented
3	#123	C₂	1	2	yes	maximal, complemented
4	#123	C₂	1	2	yes	maximal, complemented
5	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	#121	C₂	1	1	yes	maximal, characteristic
8	C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈	C₂²	1	1	yes	characteristic
11	C₂×D₄	C₂²	1	2	yes	—
12	C₂×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂²	1	1	yes	characteristic
16	C₂²×C₄	C₂²	1	2	yes	—
17	C₄⋊C₄	C₂²	1	2	yes	—
18	C₄⋊C₄	C₂²	1	2	yes	—
19	C₄⋊C₄	C₂²	1	2	yes	—
20	C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊C₄	C₂²	1	2	yes	—
22	C₄⋊C₄	C₂²	1	2	yes	—
23	C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₄⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂²⋊C₄	C₂²	1	2	yes	—
30	C₄²	C₂²	1	2	yes	—
31	C₂³	C₂³	1	2	yes	—
32	C₂³	C₂³	1	1	yes	characteristic
33	Q₈	—	4	1	no	—
34	D₄	—	2	4	no	—
35	D₄	—	4	1	no	—
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	C₂³	1	2	yes	—
41	C₂×C₄	C₂³	1	2	yes	—
42	C₂×C₄	C₂³	1	2	yes	—
43	C₂×C₄	C₂³	1	2	yes	—
44	C₂×C₄	C₂³	1	2	yes	—
45	C₂×C₄	C₂³	1	1	yes	characteristic
46	C₂×C₄	C₂³	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	2	no	—
50	C₂²	—	2	2	no	—
51	C₂²	—	2	2	no	—
52	C₂²	#41	1	2	yes	—
53	C₂²	C₂⁴	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	2	2	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	1	no	—
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₂	—	2	2	no	—
62	C₂	—	4	2	no	—
63	C₂	C₂×#41	1	1	yes	characteristic
64	C₂	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂	2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
prime radical	#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #126	C₂	semidirect
2. #125	C₂	semidirect
3. #123	C₂	semidirect
4. #123	C₂	semidirect
5. C₂²⋊Q₈	C₂	semidirect
6. C₂²⋊Q₈	C₂	semidirect
8. C₄×D₄	C₂	semidirect
9. C₂×C₄⋊C₄	C₂	semidirect

Maximal subgroups

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

subgroup	index
#126	2
#125	2
#123	2
C₂²⋊Q₈	2
#121	2
C₄×D₄	2
C₂×C₄⋊C₄	2