Subgroups — #4712

Order 192.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄	C₂	1	2	yes	maximal, complemented
2	C₂×#204	C₂	1	2	yes	maximal, complemented
3	#875	C₂	1	1	yes	maximal, characteristic, complemented
4	#874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	#496	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	—	2	2	no	—
8	C₂×D₁₂	C₂²	1	2	yes	—
9	C₂×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×Dic₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	2	yes	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	—	2	2	no	—
14	D₂₄	—	2	4	no	—
15	#204	—	2	4	no	—
16	C₂×SD₁₆	—	3	2	no	—
17	C₂×D₈	—	3	2	no	—
18	#127	—	3	1	no	—
19	#124	—	3	1	no	—
20	#97	D₃	1	1	yes	characteristic, complemented
21	C₂²×D₃	—	2	1	no	—
22	C₂²×D₃	—	2	2	no	—
23	C₂×C₃×C₄	D₄	1	2	yes	—
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	—	2	1	no	—
26	D₁₂	—	2	2	no	—
27	D₁₂	—	2	2	no	—
28	D₁₂	—	2	2	no	—
29	D₁₂	—	4	4	no	—
30	Dic₆	—	2	2	no	—
31	C₃×C₈	D₄	1	4	yes	complemented
32	C₂×Q₈	—	3	1	no	—
33	C₂×D₄	—	3	1	no	—
34	C₂×D₄	—	3	2	no	—
35	C₂×D₄	—	6	2	no	—
36	SD₁₆	—	6	4	no	—
37	D₈	—	6	4	no	—
38	C₂×C₈	C₂×D₃	1	2	yes	—
39	C₂²⋊C₄	—	6	2	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
42	C₂×D₃	—	4	2	no	—
43	C₂×D₃	—	4	1	no	—
44	C₂×D₃	—	4	4	no	—
45	C₂×D₃	—	4	2	no	—
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	C₂×D₄	1	2	yes	—
48	Dic₃	—	4	1	no	—
49	C₂³	—	6	1	no	—
50	C₂³	—	6	2	no	—
51	Q₈	—	6	2	no	—
52	D₄	—	6	2	no	—
53	D₄	—	6	2	no	—
54	D₄	—	6	2	no	—
55	D₄	—	12	4	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	D₁₂	1	2	yes	—
58	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₈	D₁₂	1	4	yes	complemented
60	C₂×C₃	#127	1	1	yes	characteristic
61	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
62	D₃	—	8	2	no	—
63	D₃	—	8	1	no	—
64	C₂²	—	12	1	no	—
65	C₂²	—	12	2	no	—
66	C₂²	—	12	4	no	—
67	C₂²	—	12	2	no	—
68	C₂²	C₂×D₁₂	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	12	1	no	—
71	C₄	C₂×D₁₂	1	2	yes	—
72	C₃	#496	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	24	2	no	—
74	C₂	—	24	1	no	—
75	C₂	#874	1	1	yes	characteristic
76	C₂	#908	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#97	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	#97	D₃
hypocenter	C₃	#496
LCS	G ▷ C₂×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
1. C₂×D₂₄	C₂	semidirect
2. C₂×#204	C₂	semidirect
3. #875	C₂	semidirect
4. #874	C₂	semidirect
5. C₃×#97	C₂	semidirect
20. #97	D₃	semidirect
31. C₃×C₈	D₄	semidirect
59. C₈	D₁₂	semidirect
72. C₃	#496	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
C₂×#204	2
#875	2
#874	2
C₃×#97	2
#496	3