Subgroups — #5119

Order 192.

114 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	#926	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₈	C₂	1	4	yes	maximal, complemented
5	#424	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×C₈	C₂²	1	2	yes	—
9	#217	C₄	1	2	yes	—
10	C₄⋊Dic₃	C₄	1	2	yes	—
11	C₄×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃⋊C₈	—	2	2	no	—
13	C₂×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×M₁₆	D₃	1	1	yes	characteristic
15	C₂²×C₈	—	3	1	no	—
16	#117	—	3	1	no	—
17	C₄⋊C₈	—	3	4	no	—
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	D₄	1	2	yes	—
21	C₂×C₃×C₄	Q₈	1	2	yes	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	C₂×C₄	1	4	yes	—
24	C₃×C₈	—	2	2	no	—
25	C₃⋊C₈	—	2	2	no	—
26	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
27	M₁₆	—	2	2	no	—
28	C₂×C₈	—	3	2	no	—
29	C₂×C₈	—	6	2	no	—
30	C₂×C₈	C₂×D₃	1	2	yes	—
31	C₄⋊C₄	—	3	2	no	—
32	C₂²⋊C₄	—	3	2	no	—
33	C₄²	—	3	2	no	—
34	C₂²×C₃	—	2	2	no	—
35	C₂²×C₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
37	C₃×C₄	C₄⋊C₄	1	2	yes	—
38	C₃×C₄	C₂×D₄	1	1	yes	characteristic
39	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
40	Dic₃	—	2	4	no	—
41	C₂³	C₄×D₃	1	1	yes	characteristic
42	C₂×C₄	—	3	4	no	—
43	C₂×C₄	Dic₆	1	2	yes	—
44	C₂×C₄	C₄×D₃	1	1	yes	characteristic
45	C₂×C₄	#67	1	2	yes	—
46	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
47	C₈	—	2	2	no	—
48	C₈	—	6	2	no	—
49	C₂×C₃	—	2	2	no	—
50	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃	#131	1	2	yes	—
52	C₂²	—	2	2	no	—
53	C₂²	Dic₃⋊C₄	1	2	yes	—
54	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
55	C₄	—	6	4	no	—
56	C₄	Dic₃⋊C₄	1	2	yes	—
57	C₄	C₂×Dic₆	1	1	yes	characteristic
58	C₄	C₂×#67	1	1	yes	characteristic
59	C₃	#424	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	2	no	—
61	C₂	#907	1	2	yes	—
62	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	#424
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
59. C₃	#424	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
#926	2
C₂²×C₃⋊C₈	2
Dic₃⋊C₈	2
#424	3