Subgroups — #538

Order 64.

125 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	#124	C₂	1	2	yes	maximal, complemented
3	#123	C₂	1	2	yes	maximal, complemented
4	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#121	C₂	1	2	yes	maximal, complemented
6	#121	C₂	1	1	yes	maximal, characteristic, complemented
7	#120	C₂	1	2	yes	maximal, complemented
8	C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴	C₂²	1	1	yes	characteristic
11	C₂×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄	C₂²	1	2	yes	—
13	C₂×D₄	C₂²	1	2	yes	—
14	C₂×D₄	C₂²	1	2	yes	—
15	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄	C₂²	1	2	yes	complemented
19	C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂²⋊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	complemented
29	C₂²⋊C₄	C₂²	1	2	yes	complemented
30	C₄²	C₂²	1	2	yes	complemented
31	C₂³	—	2	2	no	—
32	C₂³	—	4	1	no	—
33	C₂³	—	4	1	no	—
34	C₂³	C₂³	1	2	yes	—
35	C₂³	C₂³	1	2	yes	—
36	C₂³	C₂³	1	1	yes	characteristic
37	Q₈	—	4	1	no	—
38	D₄	—	2	4	no	—
39	D₄	—	4	2	no	—
40	D₄	—	4	2	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	C₂³	1	2	yes	—
46	C₂×C₄	C₂³	1	2	yes	—
47	C₂×C₄	C₂³	1	2	yes	—
48	C₂×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₄	C₂³	1	2	yes	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	—	2	2	no	—
54	C₂²	—	2	2	no	—
55	C₂²	—	2	2	no	—
56	C₂²	—	2	2	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	2	no	—
59	C₂²	—	4	4	no	—
60	C₂²	#41	1	2	yes	—
61	C₂²	C₂⁴	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	2	2	no	—
66	C₄	—	2	1	no	—
67	C₄	—	2	2	no	—
68	C₂	—	2	2	no	—
69	C₂	—	4	2	no	—
70	C₂	—	4	2	no	—
71	C₂	C₂×#41	1	1	yes	characteristic
72	C₂	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂	H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
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. #124	C₂	semidirect
3. #123	C₂	semidirect
4. C₂²⋊Q₈	C₂	semidirect
5. #121	C₂	semidirect
6. #121	C₂	semidirect
7. #120	C₂	semidirect
8. C₄×D₄	C₂	semidirect
9. C₂×C₂²⋊C₄	C₂	semidirect
11. C₂×Q₈	C₂²	semidirect
15. C₂×D₄	C₂²	semidirect
18. C₂²×C₄	C₂²	semidirect
19. C₄⋊C₄	C₂²	semidirect
20. C₄⋊C₄	C₂²	semidirect
21. C₄⋊C₄	C₂²	semidirect
22. C₄⋊C₄	C₂²	semidirect
28. C₂²⋊C₄	C₂²	semidirect
29. C₂²⋊C₄	C₂²	semidirect
30. C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#126	2
#124	2
#123	2
C₂²⋊Q₈	2
#121	2
#120	2
C₄×D₄	2
C₂×C₂²⋊C₄	2