Subgroups — #5126

Order 192.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊C₈	C₂	1	4	yes	maximal, complemented
5	#407	—	3	1	no	maximal, Sylow
6	C₂³×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃	—	2	2	no	—
10	C₂×C₄×D₃	—	2	2	no	—
11	C₂×C₄×D₃	C₄	1	2	yes	—
12	C₂×C₄×D₃	C₂²	1	2	yes	—
13	C₃×M₁₆	—	2	2	no	—
14	C₂×C₃×C₈	C₂²	1	2	yes	complemented
15	#208	—	2	2	no	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	complemented
17	C₂³×C₄	—	3	1	no	—
18	C₂×M₁₆	—	3	1	no	—
19	C₂×M₁₆	D₃	1	1	yes	characteristic, complemented
20	C₂²⋊C₈	—	3	4	no	—
21	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₃	—	2	2	no	—
23	C₂²×D₃	—	2	4	no	—
24	C₂²×D₃	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	D₄	1	2	yes	—
27	C₂×C₃×C₄	D₄	1	2	yes	—
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	C₂×C₄	1	2	yes	—
31	C₄×D₃	—	2	4	no	—
32	C₄×D₃	—	2	2	no	—
33	C₄×D₃	—	2	2	no	—
34	C₃×C₈	—	2	2	no	—
35	C₃⋊C₈	—	2	2	no	—
36	C₂⁴	—	3	1	no	—
37	C₂²×C₄	—	3	1	no	—
38	C₂²×C₄	—	3	2	no	—
39	C₂²×C₄	—	3	2	no	—
40	C₂²×C₄	—	6	2	no	—
41	C₂²×C₄	—	6	2	no	—
42	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	6	2	no	—
45	C₂×C₈	—	3	2	no	—
46	C₂×C₈	C₂×D₃	1	2	yes	complemented
47	C₂²×C₃	—	2	2	no	—
48	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×D₃	—	2	8	no	—
51	C₂×D₃	—	2	2	no	—
52	C₂×D₃	—	2	2	no	—
53	C₂×D₃	M₁₆	1	4	yes	complemented
54	C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	Dic₃	—	2	2	no	—
58	C₂³	—	3	2	no	—
59	C₂³	—	6	4	no	—
60	C₂³	—	6	2	no	—
61	C₂³	C₄×D₃	1	1	yes	characteristic
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	6	2	no	—
64	C₂×C₄	—	6	4	no	—
65	C₂×C₄	—	6	2	no	—
66	C₂×C₄	—	6	2	no	—
67	C₂×C₄	C₄×D₃	1	1	yes	characteristic
68	C₂×C₄	D₁₂	1	2	yes	—
69	C₂×C₄	#67	1	2	yes	—
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	6	2	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃	C₂×M₁₆	1	2	yes	—
76	D₃	—	2	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	—	3	4	no	—
79	C₂²	—	6	8	no	—
80	C₂²	—	6	2	no	—
81	C₂²	—	6	2	no	—
82	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
83	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
84	C₄	—	6	2	no	—
85	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
86	C₄	C₂×D₁₂	1	1	yes	characteristic
87	C₄	C₂×#67	1	1	yes	characteristic
88	C₃	#407	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	2	2	no	—
90	C₂	—	6	4	no	—
91	C₂	D₃×M₁₆	1	2	yes	—
92	C₂	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₂×C₃×M₁₆	C₂
prime radical	C₂³×D₃	C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#407
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×M₁₆	C₂	semidirect
3. C₂×#208	C₂	semidirect
4. (C₂×D₃)⋊C₈	C₂	semidirect
14. C₂×C₃×C₈	C₂²	semidirect
16. C₂×C₃⋊C₈	C₂²	semidirect
19. C₂×M₁₆	D₃	semidirect
46. C₂×C₈	C₂×D₃	semidirect
53. C₂×D₃	M₁₆	semidirect
88. C₃	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃	2
C₂×C₃×M₁₆	2
C₂×#208	2
(C₂×D₃)⋊C₈	2
#407	3