Subgroups — #5152

Order 192.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₃	C₂	1	2	yes	maximal, complemented
3	#832	C₂	1	2	yes	maximal, complemented
4	#817	C₂	1	1	yes	maximal, characteristic, complemented
5	#815	C₂	1	1	yes	maximal, characteristic, complemented
6	#442	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	2	yes	—
8	C₂²×Dic₃	—	2	2	no	—
9	C₃×D₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	#217	—	2	2	no	—
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×D₈	D₃	1	1	yes	characteristic
16	C₄×D₄	—	3	2	no	—
17	#106	—	3	1	no	—
18	#102	—	3	2	no	—
19	#97	—	3	1	no	—
20	C₂³×C₃	—	2	2	no	—
21	C₃×D₄	—	2	2	no	—
22	C₃×D₄	C₂×C₄	1	4	yes	complemented
23	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×Dic₃	—	2	2	no	—
25	C₂×Dic₃	—	4	4	no	—
26	C₂×Dic₃	D₄	1	2	yes	—
27	C₃×C₈	C₂×C₄	1	2	yes	complemented
28	C₃⋊C₈	—	2	1	no	—
29	C₂×D₄	C₂×D₃	1	2	yes	—
30	C₂²×C₄	—	6	2	no	—
31	D₈	Dic₃	1	4	yes	complemented
32	C₂×C₈	—	3	1	no	—
33	C₂×C₈	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₄	—	3	2	no	—
35	C₂²⋊C₄	—	6	2	no	—
36	C₄²	—	3	1	no	—
37	C₂²×C₃	—	2	4	no	—
38	C₂²×C₃	—	4	2	no	—
39	C₂²×C₃	—	4	2	no	—
40	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
41	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄	#41	1	1	yes	characteristic
43	Dic₃	—	2	2	no	—
44	Dic₃	—	4	2	no	—
45	C₂³	—	2	2	no	—
46	D₄	—	2	2	no	—
47	D₄	C₂×Dic₃	1	4	yes	complemented
48	C₂×C₄	—	3	2	no	—
49	C₂×C₄	—	6	2	no	—
50	C₂×C₄	—	12	4	no	—
51	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₈	—	6	1	no	—
53	C₈	C₂×Dic₃	1	2	yes	complemented
54	C₂×C₃	—	4	4	no	—
55	C₂×C₃	C₄×D₄	1	1	yes	characteristic
56	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²	—	2	4	no	—
58	C₂²	—	4	2	no	—
59	C₂²	—	4	2	no	—
60	C₂²	D₃×D₄	1	1	yes	characteristic
61	C₄	—	6	2	no	—
62	C₄	—	12	2	no	—
63	C₄	#237	1	1	yes	characteristic
64	C₄	C₂²×Dic₃	1	1	yes	characteristic
65	C₃	#442	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	4	4	no	—
67	C₂	#911	1	2	yes	—
68	C₂	D₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#442
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₃	C₂	semidirect
3. #832	C₂	semidirect
4. #817	C₂	semidirect
5. #815	C₂	semidirect
9. C₃×D₈	C₄	semidirect
22. C₃×D₄	C₂×C₄	semidirect
27. C₃×C₈	C₂×C₄	semidirect
31. D₈	Dic₃	semidirect
47. D₄	C₂×Dic₃	semidirect
53. C₈	C₂×Dic₃	semidirect
65. C₃	#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₈	2
D₄×Dic₃	2
#832	2
#817	2
#815	2
#442	3