Subgroups — #5106

Order 192.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	2	yes	maximal
3	#818	C₂	1	2	yes	maximal, complemented
4	#817	C₂	1	2	yes	maximal, complemented
5	#427	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₈	C₄	1	4	yes	complemented
8	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	#217	—	2	2	no	—
11	C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₄×Dic₃	—	2	2	no	—
13	C₂²×C₈	D₃	1	1	yes	characteristic
14	#117	—	3	2	no	—
15	#107	—	3	2	no	—
16	#106	—	3	2	no	—
17	C₂³×C₃	D₄	1	1	yes	characteristic
18	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄	Q₈	1	2	yes	—
21	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×Dic₃	—	2	4	no	—
23	C₃×C₈	C₂×C₄	1	2	yes	—
24	C₃×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₂×C₈	Dic₃	1	4	yes	complemented
27	C₂×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₈	C₂×D₃	1	1	yes	characteristic
29	C₄⋊C₄	—	3	4	no	—
30	C₂²⋊C₄	—	6	2	no	—
31	C₄²	—	6	2	no	—
32	C₂²×C₃	—	2	2	no	—
33	C₂²×C₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
35	C₃×C₄	C₄⋊C₄	1	2	yes	—
36	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
38	Dic₃	—	4	4	no	—
39	C₂³	D₁₂	1	1	yes	characteristic
40	C₂×C₄	—	6	4	no	—
41	C₂×C₄	Dic₆	1	2	yes	—
42	C₂×C₄	D₁₂	1	1	yes	characteristic
43	C₂×C₄	C₂×Dic₃	1	2	yes	—
44	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
45	C₈	C₂×Dic₃	1	2	yes	—
46	C₈	C₂×Dic₃	1	2	yes	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃	#135	1	2	yes	—
50	C₂²	—	2	2	no	—
51	C₂²	C₄⋊Dic₃	1	2	yes	—
52	C₂²	C₂×D₁₂	1	1	yes	characteristic
53	C₄	—	12	4	no	—
54	C₄	C₄⋊Dic₃	1	2	yes	—
55	C₄	C₂×Dic₆	1	1	yes	characteristic
56	C₄	C₂²×Dic₃	1	1	yes	characteristic
57	C₃	#427	1	1	yes	characteristic, complemented, Sylow
58	C₂	—	2	2	no	—
59	C₂	#904	1	2	yes	—
60	C₂	C₂×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#427
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #818	C₂	semidirect
4. #817	C₂	semidirect
7. C₂×C₃×C₈	C₄	semidirect
26. C₂×C₈	Dic₃	semidirect
57. C₃	#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈	2
#926	2
#818	2
#817	2
#427	3