Subgroups — #5197

Order 192.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#135	C₂	1	1	yes	maximal, characteristic
2	#948	C₂	1	2	yes	maximal, complemented
3	#837	C₂	1	2	yes	maximal, complemented
4	#819	C₂	1	1	yes	maximal, characteristic, complemented
5	#815	C₂	1	1	yes	maximal, characteristic, complemented
6	#444	—	3	1	no	maximal, Sylow
7	C₃×#41	C₂²	1	2	yes	—
8	C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
9	C₃×SD₁₆	C₄	1	2	yes	complemented
10	C₃×D₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₄×Dic₃	C₂²	1	1	yes	characteristic
13	#208	—	2	2	no	—
14	#208	C₂²	1	2	yes	—
15	C₂×C₃⋊C₈	—	2	2	no	—
16	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	#135	D₃	1	1	yes	characteristic
18	#131	—	3	2	no	—
19	#108	—	3	1	no	—
20	#104	—	3	2	no	—
21	#97	—	3	1	no	—
22	C₃×Q₈	C₂×C₄	1	2	yes	—
23	C₃×D₄	—	2	2	no	—
24	C₃×D₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	—	2	2	no	—
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃⋊C₈	—	2	2	no	—
31	C₃⋊C₈	D₄	1	2	yes	—
32	#41	C₂×D₃	1	2	yes	—
33	Q₁₆	Dic₃	1	1	yes	characteristic, complemented
34	SD₁₆	Dic₃	1	2	yes	complemented
35	D₈	Dic₃	1	1	yes	characteristic, complemented
36	M₁₆	—	3	2	no	—
37	M₁₆	—	6	2	no	—
38	C₂×C₈	—	3	1	no	—
39	C₂×C₈	—	6	2	no	—
40	C₂×C₈	C₂×D₃	1	1	yes	characteristic
41	C₄²	—	3	1	no	—
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	#41	1	1	yes	characteristic
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	Dic₃	—	4	1	no	—
48	Q₈	C₂×Dic₃	1	2	yes	—
49	D₄	—	2	2	no	—
50	D₄	C₂×Dic₃	1	2	yes	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	6	1	no	—
53	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
54	C₈	—	3	2	no	—
55	C₈	—	6	2	no	—
56	C₈	C₂×Dic₃	1	1	yes	characteristic
57	C₈	C₂×Dic₃	1	1	yes	characteristic
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	4	2	no	—
60	C₂×C₃	C₄×D₄	1	1	yes	characteristic
61	C₂²	—	2	2	no	—
62	C₂²	#237	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	12	1	no	—
65	C₄	D₃×D₄	1	1	yes	characteristic
66	C₄	C₂²×Dic₃	1	1	yes	characteristic
67	C₃	#444	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₃×#135	C₂
socle	C₂×C₃	C₄×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#135	D₃
hypocenter	C₃	#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #948	C₂	semidirect
3. #837	C₂	semidirect
4. #819	C₂	semidirect
5. #815	C₂	semidirect
8. C₃×Q₁₆	C₄	semidirect
9. C₃×SD₁₆	C₄	semidirect
10. C₃×D₈	C₄	semidirect
33. Q₁₆	Dic₃	semidirect
34. SD₁₆	Dic₃	semidirect
35. D₈	Dic₃	semidirect
67. C₃	#444	semidirect

Maximal subgroups

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

subgroup	index
C₃×#135	2
#948	2
#837	2
#819	2
#815	2
#444	3