Subgroups — #5118

Order 192.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#208	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₈	C₂	1	4	yes	maximal, complemented
5	#423	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃	C₄	1	2	yes	—
8	C₃×M₁₆	—	2	2	no	—
9	C₂×C₃×C₈	C₂²	1	2	yes	—
10	C₄×Dic₃	C₄	1	1	yes	characteristic
11	C₄×Dic₃	C₄	1	1	yes	characteristic
12	C₄×Dic₃	C₂²	1	2	yes	—
13	#208	—	2	2	no	—
14	C₂×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×M₁₆	—	3	1	no	—
16	C₂×M₁₆	D₃	1	1	yes	characteristic
17	C₂×C₄²	—	3	1	no	—
18	C₄⋊C₈	—	3	4	no	—
19	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	D₄	1	2	yes	—
22	C₂×C₃×C₄	Q₈	1	2	yes	—
23	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×Dic₃	—	2	4	no	—
25	C₂×Dic₃	C₂×C₄	1	2	yes	—
26	C₂×Dic₃	C₂×C₄	1	2	yes	—
27	C₃×C₈	—	2	2	no	—
28	C₃⋊C₈	—	2	2	no	—
29	C₂²×C₄	—	3	2	no	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	M₁₆	—	6	2	no	—
33	C₂×C₈	—	3	2	no	—
34	C₂×C₈	C₂×D₃	1	2	yes	—
35	C₄²	—	3	1	no	—
36	C₄²	—	3	1	no	—
37	C₄²	—	3	2	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
41	C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
44	Dic₃	—	2	2	no	—
45	Dic₃	M₁₆	1	4	yes	complemented
46	C₂³	C₄×D₃	1	1	yes	characteristic
47	C₂×C₄	—	3	2	no	—
48	C₂×C₄	—	3	2	no	—
49	C₂×C₄	—	6	4	no	—
50	C₂×C₄	Dic₆	1	2	yes	—
51	C₂×C₄	C₄×D₃	1	1	yes	characteristic
52	C₂×C₄	#67	1	2	yes	—
53	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	6	2	no	—
56	C₂×C₃	—	2	2	no	—
57	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃	C₂×M₁₆	1	2	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	Dic₃⋊C₄	1	2	yes	—
61	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
62	C₄	—	3	4	no	—
63	C₄	—	6	2	no	—
64	C₄	Dic₃⋊C₄	1	2	yes	—
65	C₄	C₂×Dic₆	1	1	yes	characteristic
66	C₄	C₂×#67	1	1	yes	characteristic
67	C₃	#423	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	2	no	—
69	C₂	D₃×M₁₆	1	2	yes	—
70	C₂	C₂×Dic₃⋊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×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#423
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
4. Dic₃⋊C₈	C₂	semidirect
45. Dic₃	M₁₆	semidirect
67. C₃	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₁₆	2
C₂×C₄×Dic₃	2
C₂×#208	2
Dic₃⋊C₈	2
#423	3