Subgroups — #5040

Order 192.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#121	C₂	1	1	yes	maximal, characteristic
3	C₂×#214	C₂	1	2	yes	maximal, complemented
4	#830	C₂	1	1	yes	maximal, characteristic, complemented
5	#810	C₂	1	2	yes	maximal, complemented
6	#448	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	—	2	1	no	—
8	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	—	2	1	no	—
11	C₂×Dic₆	—	2	1	no	—
12	C₂×Dic₆	—	2	4	no	—
13	C₂×Dic₆	C₂²	1	2	yes	—
14	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊C₄	—	2	1	no	—
16	#214	—	2	4	no	—
17	C₂×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×Q₈	—	3	1	no	—
19	C₂×SD₁₆	—	3	2	no	—
20	#121	D₃	1	1	yes	characteristic
21	#103	—	3	2	no	—
22	C₂²⋊C₈	—	3	1	no	—
23	C₂³×C₃	—	2	1	no	—
24	C₂³×C₃	D₄	1	1	yes	characteristic
25	C₃×D₄	—	2	2	no	—
26	C₃×D₄	—	4	2	no	—
27	C₂×C₃×C₄	—	2	2	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂×Dic₃	—	2	4	no	—
32	C₂×Dic₃	—	2	2	no	—
33	Dic₆	—	2	4	no	—
34	Dic₆	—	2	2	no	—
35	Dic₆	D₄	1	4	yes	complemented
36	C₃⋊C₈	—	2	2	no	—
37	C₂×Q₈	—	3	2	no	—
38	C₂×Q₈	—	6	4	no	—
39	C₂×Q₈	—	6	1	no	—
40	C₂×D₄	—	2	1	no	—
41	C₂×D₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₄	—	6	1	no	—
43	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
44	SD₁₆	—	6	4	no	—
45	C₂×C₈	—	3	2	no	—
46	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²⋊C₄	—	2	1	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	—	4	1	no	—
51	C₂²×C₃	—	4	1	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂²×C₃	SD₁₆	1	2	yes	complemented
54	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	C₂×D₄	1	2	yes	—
58	Dic₃	—	2	4	no	—
59	C₂³	—	2	1	no	—
60	C₂³	#67	1	1	yes	characteristic
61	Q₈	—	3	4	no	—
62	Q₈	—	6	4	no	—
63	Q₈	—	6	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	4	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	6	4	no	—
69	C₂×C₄	—	6	2	no	—
70	C₂×C₄	#67	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₈	—	6	2	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	—	8	1	no	—
75	C₂×C₃	#120	1	1	yes	characteristic
76	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₃	#137	1	1	yes	characteristic
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	#214	1	2	yes	complemented
84	C₂²	C₂×#67	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	6	4	no	—
88	C₄	D₃×D₄	1	2	yes	—
89	C₃	#448	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	—	8	1	no	—
92	C₂	C₂×#214	1	1	yes	characteristic
93	C₂	#937	1	1	yes	characteristic
94	C₂	#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₃×#121	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	#121	D₃
hypocenter	C₃	#448
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₆	C₂	semidirect
3. C₂×#214	C₂	semidirect
4. #830	C₂	semidirect
5. #810	C₂	semidirect
35. Dic₆	D₄	semidirect
53. C₂²×C₃	SD₁₆	semidirect
83. C₂²	#214	semidirect
89. C₃	#448	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆	2
C₃×#121	2
C₂×#214	2
#830	2
#810	2
#448	3