Subgroups — #4751

Order 192.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#239	C₂	1	1	yes	maximal, characteristic
2	D₃×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#836	C₂	1	1	yes	maximal, characteristic, complemented
5	#823	C₂	1	2	yes	maximal, complemented
6	#413	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	#239	—	2	2	no	—
9	#239	—	2	2	no	—
10	C₂×D₁₂	C₄	1	2	yes	—
11	C₂×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃	C₄	1	2	yes	—
13	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
14	C₃×M₁₆	C₂²	1	2	yes	—
15	#208	C₂²	1	2	yes	—
16	ΓL(1,𝔽₂₅)	—	2	2	no	—
17	D₃×C₈	—	2	2	no	—
18	C₂×#41	—	3	1	no	—
19	C₂×M₁₆	—	3	2	no	—
20	#101	—	3	1	no	—
21	#101	D₃	1	1	yes	characteristic, complemented
22	#100	—	3	2	no	—
23	C₂²×D₃	C₂×C₄	1	2	yes	—
24	C₂²×D₃	C₂×C₄	1	1	yes	characteristic
25	C₃×Q₈	—	2	2	no	—
26	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
29	D₁₂	—	2	4	no	—
30	D₁₂	—	2	2	no	—
31	C₄×D₃	—	2	2	no	—
32	C₄×D₃	—	2	2	no	—
33	C₄×D₃	D₄	1	2	yes	—
34	C₄×D₃	D₄	1	2	yes	—
35	C₃×C₈	—	2	2	no	—
36	C₃⋊C₈	—	2	2	no	—
37	#41	—	6	2	no	—
38	#41	—	6	2	no	—
39	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
40	C₂×D₄	—	3	2	no	—
41	C₂×D₄	—	3	1	no	—
42	C₂²×C₄	—	3	1	no	—
43	C₂²×C₄	—	3	2	no	—
44	M₁₆	—	3	2	no	—
45	M₁₆	—	6	2	no	—
46	M₁₆	C₂×D₃	1	2	yes	—
47	C₂×C₈	—	6	2	no	—
48	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×D₃	—	2	2	no	—
50	C₂×D₃	—	4	1	no	—
51	C₂×D₃	—	4	2	no	—
52	C₂×D₃	C₂²⋊C₄	1	2	yes	—
53	C₃×C₄	—	2	2	no	—
54	C₃×C₄	C₂×D₄	1	2	yes	—
55	Dic₃	C₂²⋊C₄	1	1	yes	characteristic
56	Dic₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³	—	3	1	no	—
58	C₂³	—	3	2	no	—
59	Q₈	—	2	2	no	—
60	D₄	—	6	4	no	—
61	D₄	—	6	2	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	3	2	no	—
64	C₂×C₄	—	3	2	no	—
65	C₂×C₄	—	6	2	no	—
66	C₂×C₄	—	6	2	no	—
67	C₂×C₄	C₄×D₃	1	2	yes	—
68	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	6	2	no	—
71	C₂×C₃	—	2	1	no	—
72	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	D₃	—	2	2	no	—
74	D₃	—	4	2	no	—
75	C₂²	—	3	2	no	—
76	C₂²	—	6	2	no	—
77	C₂²	—	12	1	no	—
78	C₂²	—	12	2	no	—
79	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	3	1	no	—
82	C₄	—	3	1	no	—
83	C₄	D₃×D₄	1	2	yes	—
84	C₃	#413	1	1	yes	characteristic, complemented, Sylow
85	C₂	—	2	1	no	—
86	C₂	—	6	2	no	—
87	C₂	—	12	2	no	—
88	C₂	D₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#101	C₂
socle	C₂×C₃	C₂×C₂²⋊C₄
prime radical	C₂×#239	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#101	D₃
hypocenter	C₃	#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. D₃×M₁₆	C₂	semidirect
3. C₃×#101	C₂	semidirect
4. #836	C₂	semidirect
5. #823	C₂	semidirect
21. #101	D₃	semidirect
84. C₃	#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#239	2
D₃×M₁₆	2
C₃×#101	2
#836	2
#823	2
#413	3