Subgroups — #4539

Order 192.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#123	C₂	1	1	yes	maximal, characteristic
2	#834	C₂	1	1	yes	maximal, characteristic
3	#833	C₂	1	1	yes	maximal, characteristic
4	#352	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₄	1	1	yes	characteristic, complemented
7	C₃×C₄⋊C₄	—	2	1	no	—
8	C₃×C₂²⋊C₄	—	2	1	no	—
9	C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
10	#217	—	2	1	no	—
11	#208	—	2	1	no	—
12	#123	D₃	1	1	yes	characteristic
13	#100	—	3	1	no	—
14	#99	—	3	1	no	—
15	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₃	D₄	1	1	yes	characteristic
17	C₃×D₄	—	4	1	no	—
18	C₂×C₃×C₄	—	2	1	no	—
19	C₂×C₃×C₄	—	2	1	no	—
20	C₂×C₃×C₄	—	4	1	no	—
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂×Dic₃	—	4	1	no	—
23	C₃⋊C₈	—	4	1	no	—
24	C₂×D₄	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₄	Dic₃	1	1	yes	characteristic, complemented
26	M₁₆	—	6	1	no	—
27	C₄⋊C₄	—	2	1	no	—
28	C₂²⋊C₄	—	2	1	no	—
29	C₂²⋊C₄	—	6	1	no	—
30	C₂²⋊C₄	Dic₃	1	1	yes	characteristic, complemented
31	C₂²×C₃	—	2	1	no	—
32	C₂²×C₃	—	2	1	no	—
33	C₂²×C₃	—	4	1	no	—
34	C₂²×C₃	—	4	1	no	—
35	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₃×C₄	—	2	1	no	—
37	C₃×C₄	—	4	1	no	—
38	C₃×C₄	—	4	1	no	—
39	Dic₃	—	8	1	no	—
40	C₂³	C₂×Dic₃	1	1	yes	characteristic
41	C₂³	#67	1	1	yes	characteristic
42	D₄	—	4	1	no	—
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	4	1	no	—
46	C₂×C₄	—	12	1	no	—
47	C₂×C₄	#67	1	1	yes	characteristic
48	C₈	—	12	1	no	—
49	C₂×C₃	—	2	1	no	—
50	C₂×C₃	—	4	1	no	—
51	C₂×C₃	—	4	1	no	—
52	C₂×C₃	#99	1	1	yes	characteristic
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	#217	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	4	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	24	1	no	—
62	C₃	#352	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#834	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#834
commutator	C₂³×C₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₃
Fitting	C₃×#123	C₂
socle	C₂×C₃	#99
prime radical, cosocle	C₂×C₃×D₄	C₂²
hypercenter	#123	D₃
hypocenter	C₃	#352
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #123

Complements

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

N	H ≅ G/N	type
6. C₂²×C₃×C₄	C₄	semidirect
9. C₃×C₂²⋊C₄	C₄	semidirect
25. C₂²×C₄	Dic₃	semidirect
30. C₂²⋊C₄	Dic₃	semidirect
62. C₃	#352	semidirect

Maximal subgroups

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

subgroup	index
C₃×#123	2
#834	2
#833	2
#352	3