Subgroups — #4700

Order 192.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#904	C₂	1	1	yes	maximal, characteristic
2	#901	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#819	C₂	1	1	yes	maximal, characteristic, complemented
5	#805	C₂	1	2	yes	maximal, complemented
6	#443	—	3	1	no	maximal, Sylow
7	#235	C₂²	1	2	yes	—
8	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	#208	C₂²	1	2	yes	—
12	Dic₁₂	C₄	1	1	yes	characteristic, complemented
13	D₂₄	C₄	1	1	yes	characteristic, complemented
14	#204	C₄	1	2	yes	complemented
15	ΓL(1,𝔽₂₅)	—	2	2	no	—
16	D₃×C₈	—	2	2	no	—
17	#135	—	3	1	no	—
18	#131	—	3	2	no	—
19	#108	—	3	1	no	—
20	#104	—	3	2	no	—
21	C₄×C₈	D₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
24	#67	—	2	2	no	—
25	D₁₂	C₂×C₄	1	2	yes	—
26	C₄×D₃	—	2	2	no	—
27	Dic₆	C₂×C₄	1	2	yes	—
28	C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₈	D₄	1	1	yes	characteristic
31	C₃×C₈	D₄	1	1	yes	characteristic
32	C₃⋊C₈	—	2	2	no	—
33	#41	—	3	2	no	—
34	Q₁₆	—	3	1	no	—
35	SD₁₆	—	3	2	no	—
36	D₈	—	3	1	no	—
37	M₁₆	—	3	2	no	—
38	M₁₆	—	6	2	no	—
39	C₂×C₈	—	6	2	no	—
40	C₂×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₈	C₂×D₃	1	1	yes	characteristic
42	C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃	#41	1	1	yes	characteristic
44	C₂×D₃	—	2	2	no	—
45	C₃×C₄	—	2	2	no	—
46	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	Dic₃	—	2	2	no	—
49	Q₈	—	3	2	no	—
50	D₄	—	3	2	no	—
51	D₄	—	6	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
55	C₈	—	6	2	no	—
56	C₈	C₄×D₃	1	1	yes	characteristic
57	C₈	C₄×D₃	1	1	yes	characteristic
58	C₈	D₁₂	1	1	yes	characteristic
59	C₈	D₁₂	1	1	yes	characteristic
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	C₄×D₄	1	1	yes	characteristic
62	D₃	—	4	2	no	—
63	C₂²	—	6	2	no	—
64	C₂²	#235	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	6	2	no	—
67	C₄	C₂×C₄×D₃	1	1	yes	characteristic
68	C₄	C₂×D₁₂	1	1	yes	characteristic
69	C₃	#443	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	1	no	—
71	C₂	—	12	2	no	—
72	C₂	C₄×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×C₈	C₂
socle	C₂×C₃	C₄×D₄
prime radical	#904	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄×C₈	D₃
hypocenter	C₃	#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #901	C₂	semidirect
3. C₃×C₄×C₈	C₂	semidirect
4. #819	C₂	semidirect
5. #805	C₂	semidirect
12. Dic₁₂	C₄	semidirect
13. D₂₄	C₄	semidirect
14. #204	C₄	semidirect
21. C₄×C₈	D₃	semidirect
69. C₃	#443	semidirect

Maximal subgroups

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

subgroup	index
#904	2
#901	2
C₃×C₄×C₈	2
#819	2
#805	2
#443	3