Subgroups — #5184

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	2	yes	maximal
3	#835	C₂	1	2	yes	maximal
4	#817	C₂	1	1	yes	maximal, characteristic
5	#815	C₂	1	1	yes	maximal, characteristic
6	#441	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₃×Q₁₆	C₄	1	4	yes	complemented
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃	—	2	2	no	—
11	C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₄×Dic₃	—	2	2	no	—
13	C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	D₃	1	1	yes	characteristic
16	C₄×Q₈	—	3	2	no	—
17	#106	—	3	1	no	—
18	#103	—	3	2	no	—
19	#97	—	3	1	no	—
20	C₃×Q₈	—	2	2	no	—
21	C₃×Q₈	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×Dic₃	—	2	2	no	—
25	C₂×Dic₃	D₄	1	2	yes	—
26	C₃×C₈	C₂×C₄	1	2	yes	—
27	C₃⋊C₈	—	2	1	no	—
28	C₂×Q₈	C₂×D₃	1	2	yes	—
29	Q₁₆	Dic₃	1	4	yes	complemented
30	C₂×C₈	—	3	1	no	—
31	C₂×C₈	C₂×D₃	1	1	yes	characteristic
32	C₄⋊C₄	—	3	2	no	—
33	C₄⋊C₄	—	6	2	no	—
34	C₄²	—	3	1	no	—
35	C₄²	—	6	2	no	—
36	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
37	C₃×C₄	—	2	4	no	—
38	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₃×C₄	#41	1	1	yes	characteristic
40	Dic₃	—	2	2	no	—
41	Dic₃	—	4	2	no	—
42	Q₈	—	2	2	no	—
43	Q₈	C₂×Dic₃	1	4	yes	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	3	2	no	—
46	C₂×C₄	—	6	2	no	—
47	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
48	C₈	—	6	1	no	—
49	C₈	C₂×Dic₃	1	2	yes	—
50	C₂×C₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₃	#137	1	2	yes	—
52	C₂²	D₃×D₄	1	1	yes	characteristic
53	C₄	—	2	4	no	—
54	C₄	—	6	2	no	—
55	C₄	—	12	2	no	—
56	C₄	#237	1	1	yes	characteristic
57	C₄	C₂²×Dic₃	1	1	yes	characteristic
58	C₃	#441	1	1	yes	characteristic, complemented, Sylow
59	C₂	#918	1	2	yes	—
60	C₂	D₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×Q₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×Q₁₆	D₃
hypocenter	C₃	#441
LCS	G ▷ 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
8. C₃×Q₁₆	C₄	semidirect
29. Q₁₆	Dic₃	semidirect
58. C₃	#441	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
#835	2
#817	2
#815	2
#441	3