Subgroups — #5120

Order 192.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#926	C₂	1	1	yes	maximal, characteristic
4	#816	C₂	1	4	yes	maximal, complemented
5	#419	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃	—	2	1	no	—
8	C₂×Dic₆	—	2	1	no	—
9	C₂×Dic₆	—	2	2	no	—
10	C₂×Dic₆	C₄	1	4	yes	complemented
11	C₂×Dic₆	C₂²	1	2	yes	—
12	C₃×M₁₆	—	2	2	no	—
13	C₂×C₃×C₈	C₂²	1	2	yes	—
14	#217	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₄×Dic₃	—	2	1	no	—
17	C₂²×Q₈	—	3	1	no	—
18	C₂×M₁₆	D₃	1	1	yes	characteristic
19	#117	—	3	1	no	—
20	#103	—	3	4	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	D₄	1	2	yes	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	—	2	4	no	—
27	C₂×Dic₃	—	2	2	no	—
28	C₂×Dic₃	—	2	2	no	—
29	Dic₆	—	2	4	no	—
30	Dic₆	—	2	2	no	—
31	Dic₆	C₂×C₄	1	4	yes	—
32	C₃×C₈	—	2	2	no	—
33	C₂×Q₈	—	3	4	no	—
34	C₂×Q₈	—	3	2	no	—
35	C₂×Q₈	—	6	1	no	—
36	C₂×Q₈	—	6	2	no	—
37	C₂²×C₄	—	6	1	no	—
38	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
39	M₁₆	—	2	2	no	—
40	C₂×C₈	C₂×D₃	1	2	yes	—
41	C₄⋊C₄	—	3	2	no	—
42	C₂²⋊C₄	—	6	1	no	—
43	C₄²	—	6	1	no	—
44	C₂²×C₃	—	2	2	no	—
45	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄	C₂²⋊C₄	1	2	yes	—
48	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	Dic₃	—	2	4	no	—
51	Dic₃	—	4	2	no	—
52	C₂³	D₁₂	1	1	yes	characteristic
53	Q₈	—	3	4	no	—
54	Q₈	—	6	4	no	—
55	Q₈	—	6	2	no	—
56	C₂×C₄	—	6	4	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	C₄×D₃	1	2	yes	—
60	C₂×C₄	D₁₂	1	1	yes	characteristic
61	C₂×C₄	#67	1	2	yes	—
62	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃	#137	1	2	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
69	C₂²	C₂×D₁₂	1	1	yes	characteristic
70	C₄	—	6	4	no	—
71	C₄	—	12	2	no	—
72	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
73	C₄	C₂×C₄×D₃	1	1	yes	characteristic
74	C₄	C₂×#67	1	1	yes	characteristic
75	C₃	#419	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	2	no	—
77	C₂	#909	1	2	yes	—
78	C₂	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#419
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #816	C₂	semidirect
10. C₂×Dic₆	C₄	semidirect
75. C₃	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆	2
C₂×C₃×M₁₆	2
#926	2
#816	2
#419	3