Subgroups — #542

Order 64.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#126	C₂	1	2	yes	maximal, complemented
3	#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#124	C₂	1	2	yes	maximal, complemented
5	#123	C₂	1	2	yes	maximal, complemented
6	C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#121	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈	C₂²	1	1	yes	characteristic
13	C₂×Q₈	C₂²	1	2	yes	—
14	C₂×D₄	C₂²	1	1	yes	characteristic
15	C₂×D₄	C₂²	1	2	yes	—
16	C₂²×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄	C₂²	1	2	yes	—
18	C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₄⋊C₄	C₂²	1	2	yes	—
23	C₄⋊C₄	C₂²	1	2	yes	—
24	C₄⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂²⋊C₄	C₂²	1	2	yes	—
32	C₄²	C₂²	1	1	yes	characteristic
33	C₄²	C₂²	1	1	yes	characteristic
34	C₄²	C₂²	1	2	yes	—
35	C₂³	C₂³	1	1	yes	characteristic
36	C₂³	C₂³	1	2	yes	—
37	Q₈	—	2	2	no	—
38	Q₈	—	4	2	no	—
39	D₄	—	2	2	no	—
40	D₄	—	4	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₂³	1	1	yes	characteristic
44	C₂×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₄	C₂³	1	2	yes	—
46	C₂×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₄	C₂³	1	2	yes	—
49	C₂×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₄	C₂³	1	2	yes	—
51	C₂×C₄	C₂³	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	2	no	—
54	C₂²	—	2	2	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	2	2	no	—
58	C₂²	C₂⁴	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	—	2	1	no	—
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	#41	1	2	yes	—
68	C₂	—	4	1	no	—
69	C₂	—	4	2	no	—
70	C₂	C₂×#41	1	1	yes	characteristic
71	C₂	H(5,𝔽₂)	1	1	yes	characteristic
72	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. C₄⋊Q₈	C₂	semidirect
2. #126	C₂	semidirect
3. #124	C₂	semidirect
4. #124	C₂	semidirect
5. #123	C₂	semidirect
6. C₂²⋊Q₈	C₂	semidirect
7. C₂²⋊Q₈	C₂	semidirect
9. C₄×Q₈	C₂	semidirect
10. C₄×D₄	C₂	semidirect
11. #117	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈	2
#126	2
#124	2
#123	2
C₂²⋊Q₈	2
#121	2
C₄×Q₈	2
C₄×D₄	2
#117	2