Subgroups — #5182

Order 192.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	#944	C₂	1	1	yes	maximal, characteristic
4	C₂×#216	C₂	1	1	yes	maximal, characteristic
5	#835	C₂	1	1	yes	maximal, characteristic
6	#816	C₂	1	1	yes	maximal, characteristic
7	Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#462	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₆	C₂²	1	1	yes	characteristic
12	C₃×Q₁₆	—	2	2	no	—
13	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
14	#216	—	2	2	no	—
15	C₄⋊Dic₃	—	2	1	no	—
16	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	Dic₃⋊C₄	—	2	1	no	—
18	Dic₃⋊C₄	—	2	1	no	—
19	C₄×Dic₃	—	2	1	no	—
20	C₄×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆	—	3	1	no	—
23	C₂×Q₁₆	D₃	1	1	yes	characteristic
24	C₄⋊Q₈	—	3	1	no	—
25	C₄×Q₈	—	3	1	no	—
26	C₄⋊C₈	—	3	1	no	—
27	#103	—	3	1	no	—
28	#103	—	3	1	no	—
29	C₃×Q₈	—	2	1	no	—
30	C₃×Q₈	—	2	1	no	—
31	C₃×Q₈	—	2	1	no	—
32	C₃×Q₈	D₄	1	2	yes	—
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂×Dic₃	—	2	1	no	—
37	C₂×Dic₃	—	2	1	no	—
38	C₂×Dic₃	D₄	1	1	yes	characteristic
39	C₂×Dic₃	D₄	1	1	yes	characteristic
40	Dic₆	—	2	1	no	—
41	Dic₆	—	2	1	no	—
42	C₃×C₈	—	2	1	no	—
43	C₃⋊C₈	—	2	1	no	—
44	C₂×Q₈	—	3	1	no	—
45	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
46	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
47	Q₁₆	—	2	2	no	—
48	Q₁₆	—	6	2	no	—
49	C₂×C₈	—	3	1	no	—
50	C₂×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄⋊C₄	—	3	1	no	—
52	C₄⋊C₄	—	6	1	no	—
53	C₄⋊C₄	—	6	1	no	—
54	C₄⋊C₄	—	6	1	no	—
55	C₄²	—	3	1	no	—
56	C₄²	—	6	1	no	—
57	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	—	4	1	no	—
60	C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₃×C₄	#41	1	1	yes	characteristic
62	Dic₃	—	2	1	no	—
63	Dic₃	—	4	1	no	—
64	Dic₃	—	4	1	no	—
65	Dic₃	Q₁₆	1	2	yes	complemented
66	Q₈	—	2	1	no	—
67	Q₈	—	2	1	no	—
68	Q₈	—	2	1	no	—
69	Q₈	—	6	1	no	—
70	Q₈	—	6	1	no	—
71	Q₈	#67	1	2	yes	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	3	1	no	—
75	C₂×C₄	—	3	1	no	—
76	C₂×C₄	—	6	1	no	—
77	C₂×C₄	—	6	1	no	—
78	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	6	1	no	—
81	C₂×C₃	#121	1	1	yes	characteristic
82	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₃	#137	1	1	yes	characteristic
84	C₂²	D₃×D₄	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	3	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	6	1	no	—
89	C₄	—	12	1	no	—
90	C₄	—	12	1	no	—
91	C₄	#237	1	1	yes	characteristic
92	C₄	C₂×#67	1	1	yes	characteristic
93	C₃	#462	1	1	yes	characteristic, complemented, Sylow
94	C₂	D₃×Q₁₆	1	1	yes	characteristic
95	C₂	#918	1	1	yes	characteristic
96	C₂	#939	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×Q₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	C₂×Q₁₆	D₃
hypocenter	C₃	#462
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
65. Dic₃	Q₁₆	semidirect
93. C₃	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₁₆	2
Q₈×Dic₃	2
#944	2
C₂×#216	2
#835	2
#816	2
Dic₃⋊C₈	2
#462	3