Subgroups — #388

Order 64.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	#95	C₂	1	1	yes	maximal, characteristic, complemented
7	#95	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴	C₂²	1	1	yes	characteristic
9	C₂²×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄	C₂²	1	1	yes	characteristic
15	C₄⋊C₄	—	2	2	no	—
16	C₂²⋊C₄	—	2	2	no	—
17	C₂²⋊C₄	—	2	2	no	—
18	C₂²⋊C₄	C₄	1	4	yes	complemented
19	C₄²	—	2	2	no	—
20	C₂³	—	2	1	no	—
21	C₂³	—	2	1	no	—
22	C₂³	—	2	1	no	—
23	C₂³	—	2	1	no	—
24	C₂³	—	2	1	no	—
25	C₂³	—	2	1	no	—
26	C₂³	C₂×C₄	1	2	yes	—
27	C₂³	C₂³	1	1	yes	characteristic
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄	C₂×C₄	1	2	yes	—
44	C₂×C₄	D₄	1	2	yes	—
45	C₂×C₄	D₄	1	2	yes	—
46	C₂²	—	2	2	no	—
47	C₂²	—	2	2	no	—
48	C₂²	—	2	2	no	—
49	C₂²	—	4	1	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	1	no	—
52	C₂²	—	4	1	no	—
53	C₂²	C₂²×C₄	1	1	yes	characteristic
54	C₂²	C₂×D₄	1	1	yes	characteristic
55	C₂²	C₂×D₄	1	1	yes	characteristic
56	C₂²	#41	1	1	yes	characteristic
57	C₂²	#41	1	1	yes	characteristic
58	C₂²	#41	1	1	yes	characteristic
59	C₂²	#41	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	2	2	no	—
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	—	4	1	no	—
65	C₄	—	4	1	no	—
66	C₂	—	4	2	no	—
67	C₂	#117	1	1	yes	characteristic
68	C₂	C₄×D₄	1	1	yes	characteristic
69	C₂	C₄×D₄	1	1	yes	characteristic
70	C₂	#121	1	1	yes	characteristic
71	C₂	#123	1	1	yes	characteristic
72	C₂	#124	1	1	yes	characteristic
73	C₂	#126	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂³
commutator	C₂²	C₂²×C₄
prime radical	C₂⁴	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊C₄	C₂	semidirect
5. C₂×C₄²	C₂	semidirect
6. #95	C₂	semidirect
7. #95	C₂	semidirect
18. C₂²⋊C₄	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄	2
C₂×C₂²⋊C₄	2
C₂×C₄²	2
#95	2