Subgroups — #4671

Order 192.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₃)⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	#392	—	3	1	no	maximal, Sylow
5	C₂³×D₃	C₂²	1	3	yes	—
6	C₂²×C₃×C₄	C₂²	1	3	yes	—
7	C₂²×Dic₃	C₂²	1	1	yes	characteristic
8	C₂×D₁₂	—	2	12	no	—
9	C₂×D₁₂	—	2	6	no	—
10	(C₂×D₃)⋊C₄	—	2	6	no	—
11	C₂²×D₄	—	3	3	no	—
12	C₂×C₂²⋊C₄	—	3	3	no	—
13	#95	D₃	1	1	yes	characteristic, complemented
14	C₂³×C₃	C₂³	1	1	yes	characteristic
15	C₂²×D₃	—	2	3	no	—
16	C₂²×D₃	—	2	6	no	—
17	C₂²×D₃	—	2	6	no	—
18	C₂²×D₃	—	2	3	no	—
19	C₂²×D₃	D₄	1	6	yes	complemented
20	C₂×C₃×C₄	—	2	6	no	—
21	C₂×C₃×C₄	D₄	1	6	yes	complemented
22	C₂×Dic₃	—	2	3	no	—
23	D₁₂	—	2	24	no	—
24	C₂⁴	—	3	3	no	—
25	C₂×D₄	—	6	12	no	—
26	C₂×D₄	—	6	6	no	—
27	C₂²×C₄	—	3	1	no	—
28	C₂²×C₄	C₂×D₃	1	3	yes	—
29	C₂²⋊C₄	—	6	6	no	—
30	C₂²×C₃	C₂×D₄	1	3	yes	—
31	C₂²×C₃	C₂×D₄	1	3	yes	—
32	C₂²×C₃	#41	1	1	yes	characteristic
33	C₂×D₃	—	2	12	no	—
34	C₂×D₃	—	2	6	no	—
35	C₂×D₃	—	4	3	no	—
36	C₂×D₃	—	4	6	no	—
37	C₂×D₃	—	4	3	no	—
38	C₃×C₄	—	2	6	no	—
39	Dic₃	—	4	1	no	—
40	C₂³	—	3	6	no	—
41	C₂³	—	6	3	no	—
42	C₂³	—	6	6	no	—
43	C₂³	—	6	6	no	—
44	C₂³	—	6	3	no	—
45	C₂³	C₂²×D₃	1	1	yes	characteristic
46	D₄	—	6	24	no	—
47	C₂×C₄	—	2	6	no	—
48	C₂×C₄	—	6	3	no	—
49	C₂×C₄	D₁₂	1	6	yes	complemented
50	C₂×C₃	#120	1	3	yes	—
51	C₂×C₃	#121	1	3	yes	—
52	C₂×C₃	#127	1	1	yes	characteristic
53	D₃	—	4	6	no	—
54	C₂²	—	6	12	no	—
55	C₂²	—	6	6	no	—
56	C₂²	—	12	3	no	—
57	C₂²	—	12	3	no	—
58	C₂²	—	12	6	no	—
59	C₂²	C₂×D₁₂	1	3	yes	—
60	C₂²	D₃×D₄	1	3	yes	—
61	C₂²	#239	1	1	yes	characteristic
62	C₄	—	2	6	no	—
63	C₄	—	12	1	no	—
64	C₃	#392	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	12	6	no	—
66	C₂	#874	1	1	yes	characteristic
67	C₂	#882	1	3	yes	—
68	C₂	#895	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator, socle, cosocle	C₂³×C₃	C₂³
Fitting	C₃×#95	C₂
hypercenter	#95	D₃
hypocenter	C₃	#392
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂	C₂	semidirect
2. C₂×(C₂×D₃)⋊C₄	C₂	semidirect
3. C₃×#95	C₂	semidirect
13. #95	D₃	semidirect
19. C₂²×D₃	D₄	semidirect
21. C₂×C₃×C₄	D₄	semidirect
49. C₂×C₄	D₁₂	semidirect
64. C₃	#392	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂	2
C₂×(C₂×D₃)⋊C₄	2
C₃×#95	2
#392	3