Subgroups — #546

Order 64.

167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#123	C₂	1	2	yes	maximal, complemented
5	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#121	C₂	1	1	yes	maximal, characteristic, complemented
7	#121	C₂	1	2	yes	maximal, complemented
8	#120	C₂	1	2	yes	maximal, complemented
9	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂⁴	C₂²	1	2	yes	—
13	#41	—	2	4	no	—
14	C₂×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄	—	2	4	no	—
16	C₂×D₄	C₂²	1	1	yes	characteristic
17	C₂×D₄	C₂²	1	2	yes	—
18	C₂×D₄	C₂²	1	1	yes	characteristic
19	C₂×D₄	C₂²	1	1	yes	characteristic
20	C₂×D₄	C₂²	1	2	yes	complemented
21	C₂×D₄	C₂²	1	2	yes	complemented
22	C₂²×C₄	C₂²	1	1	yes	characteristic
23	C₂²×C₄	C₂²	1	2	yes	complemented
24	C₂²×C₄	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄	C₂²	1	2	yes	complemented
26	C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂²⋊C₄	C₂²	1	2	yes	complemented
32	C₂²⋊C₄	C₂²	1	2	yes	complemented
33	C₂²⋊C₄	C₂²	1	2	yes	complemented
34	C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	C₂²	1	2	yes	complemented
36	C₄²	C₂²	1	1	yes	characteristic, complemented
37	C₂³	—	2	4	no	—
38	C₂³	—	2	4	no	—
39	C₂³	—	4	2	no	—
40	C₂³	C₂³	1	2	yes	—
41	C₂³	C₂³	1	1	yes	characteristic
42	C₂³	C₂³	1	2	yes	—
43	C₂³	C₂³	1	1	yes	characteristic
44	Q₈	—	2	2	no	—
45	D₄	—	2	4	no	—
46	D₄	—	2	4	no	—
47	D₄	—	2	2	no	—
48	D₄	—	2	2	no	—
49	D₄	—	4	2	no	—
50	D₄	D₄	1	4	yes	complemented
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	C₂³	1	1	yes	characteristic
56	C₂×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₄	C₂³	1	2	yes	—
58	C₂×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₄	C₂³	1	2	yes	—
60	C₂×C₄	C₂³	1	1	yes	characteristic
61	C₂×C₄	C₂³	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	8	no	—
73	C₂²	C₂×D₄	1	4	yes	—
74	C₂²	#41	1	2	yes	complemented
75	C₂²	C₂⁴	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	C₂×D₄	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	—	2	4	no	—
85	C₂	—	4	1	no	—
86	C₂	—	4	2	no	—
87	C₂	C₂²×D₄	1	1	yes	characteristic
88	C₂	C₂×#41	1	1	yes	characteristic
89	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. C₂×#41	C₂	semidirect
2. C₂²×D₄	C₂	semidirect
3. #124	C₂	semidirect
4. #123	C₂	semidirect
5. C₂²⋊Q₈	C₂	semidirect
6. #121	C₂	semidirect
7. #121	C₂	semidirect
8. #120	C₂	semidirect
9. C₄×D₄	C₂	semidirect
10. C₄×D₄	C₂	semidirect
11. C₂×C₂²⋊C₄	C₂	semidirect
14. C₂×Q₈	C₂²	semidirect
20. C₂×D₄	C₂²	semidirect
21. C₂×D₄	C₂²	semidirect
23. C₂²×C₄	C₂²	semidirect
24. C₂²×C₄	C₂²	semidirect
25. C₂²×C₄	C₂²	semidirect
26. C₄⋊C₄	C₂²	semidirect
27. C₄⋊C₄	C₂²	semidirect
28. C₄⋊C₄	C₂²	semidirect
31. C₂²⋊C₄	C₂²	semidirect
32. C₂²⋊C₄	C₂²	semidirect
33. C₂²⋊C₄	C₂²	semidirect
34. C₂²⋊C₄	C₂²	semidirect
35. C₂²⋊C₄	C₂²	semidirect
36. C₄²	C₂²	semidirect
50. D₄	D₄	semidirect
74. C₂²	#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41	2
C₂²×D₄	2
#124	2
#123	2
C₂²⋊Q₈	2
#121	2
#120	2
C₄×D₄	2
C₂×C₂²⋊C₄	2