Subgroups — #4711

Order 192.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₂	C₂	1	2	yes	maximal, complemented
3	C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	C₂	1	2	yes	maximal, complemented
5	#817	C₂	1	1	yes	maximal, characteristic, complemented
6	#442	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₂²	1	2	yes	—
8	C₂×C₄×D₃	—	2	2	no	—
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₄⋊Dic₃	C₂²	1	2	yes	—
14	D₂₄	C₄	1	4	yes	complemented
15	C₂×D₈	—	3	1	no	—
16	C₄×D₄	—	3	2	no	—
17	#106	—	3	1	no	—
18	#102	—	3	2	no	—
19	#97	D₃	1	1	yes	characteristic, complemented
20	C₂²×D₃	—	2	2	no	—
21	C₂×C₃×C₄	D₄	1	2	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	—	2	2	no	—
24	D₁₂	—	2	2	no	—
25	D₁₂	C₂×C₄	1	4	yes	complemented
26	C₄×D₃	—	4	4	no	—
27	C₃×C₈	—	2	1	no	—
28	C₃×C₈	C₂×C₄	1	2	yes	complemented
29	C₂×D₄	—	3	2	no	—
30	C₂²×C₄	—	6	2	no	—
31	D₈	—	3	4	no	—
32	C₂×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₈	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₄	—	3	2	no	—
35	C₂²⋊C₄	—	6	2	no	—
36	C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
38	C₂×D₃	—	2	4	no	—
39	C₂×D₃	—	4	2	no	—
40	C₂×D₃	—	4	2	no	—
41	C₃×C₄	—	2	2	no	—
42	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₄	#41	1	1	yes	characteristic
44	Dic₃	—	4	2	no	—
45	C₂³	—	6	2	no	—
46	D₄	—	3	4	no	—
47	D₄	—	6	2	no	—
48	C₂×C₄	—	6	2	no	—
49	C₂×C₄	—	12	4	no	—
50	C₂×C₄	D₁₂	1	2	yes	—
51	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	C₄×D₃	1	2	yes	complemented
54	C₂×C₃	C₄×D₄	1	1	yes	characteristic
55	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
56	D₃	—	4	4	no	—
57	C₂²	—	6	4	no	—
58	C₂²	—	12	2	no	—
59	C₂²	—	12	2	no	—
60	C₂²	C₂×D₁₂	1	1	yes	characteristic
61	C₄	—	2	2	no	—
62	C₄	—	12	2	no	—
63	C₄	C₂×C₄×D₃	1	1	yes	characteristic
64	C₄	#235	1	1	yes	characteristic
65	C₃	#442	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	12	4	no	—
67	C₂	C₄×D₁₂	1	1	yes	characteristic
68	C₂	#908	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#97	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#97	D₃
hypocenter	C₃	#442
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₂	C₂	semidirect
3. C₃×#97	C₂	semidirect
4. #821	C₂	semidirect
5. #817	C₂	semidirect
14. D₂₄	C₄	semidirect
19. #97	D₃	semidirect
25. D₁₂	C₂×C₄	semidirect
28. C₃×C₈	C₂×C₄	semidirect
53. C₈	C₄×D₃	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×D₂₄	2
C₄×D₁₂	2
C₃×#97	2
#821	2
#817	2
#442	3