Subgroups — #5131

Order 192.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#926	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	C₂	1	4	yes	maximal, complemented
5	#418	—	3	1	no	maximal, Sylow
6	C₂³×D₃	—	2	1	no	—
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×D₁₂	—	2	1	no	—
9	C₂×D₁₂	—	2	2	no	—
10	C₂×D₁₂	C₄	1	4	yes	complemented
11	C₂×D₁₂	C₂²	1	2	yes	—
12	C₃×M₁₆	—	2	2	no	—
13	C₂×C₃×C₈	C₂²	1	2	yes	complemented
14	#217	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	2	yes	complemented
16	C₄×Dic₃	—	2	1	no	—
17	C₂²×D₄	—	3	1	no	—
18	C₂×M₁₆	D₃	1	1	yes	characteristic, complemented
19	#117	—	3	1	no	—
20	#102	—	3	4	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂²×D₃	—	2	4	no	—
23	C₂²×D₃	—	2	2	no	—
24	C₂²×D₃	—	4	4	no	—
25	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	D₄	1	2	yes	—
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×Dic₃	—	2	2	no	—
30	D₁₂	—	2	4	no	—
31	D₁₂	—	2	2	no	—
32	D₁₂	C₂×C₄	1	4	yes	—
33	C₃×C₈	—	2	2	no	—
34	C₂⁴	—	6	1	no	—
35	C₂×D₄	—	3	4	no	—
36	C₂×D₄	—	3	2	no	—
37	C₂×D₄	—	6	1	no	—
38	C₂×D₄	—	6	2	no	—
39	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
40	M₁₆	—	2	2	no	—
41	C₂×C₈	C₂×D₃	1	2	yes	complemented
42	C₄⋊C₄	—	3	2	no	—
43	C₂²⋊C₄	—	6	1	no	—
44	C₄²	—	6	1	no	—
45	C₂²×C₃	—	2	2	no	—
46	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂×D₃	—	2	4	no	—
49	C₂×D₃	—	4	8	no	—
50	C₂×D₃	—	4	4	no	—
51	C₃×C₄	C₂²⋊C₄	1	2	yes	—
52	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	Dic₃	—	4	2	no	—
55	C₂³	—	6	4	no	—
56	C₂³	—	6	2	no	—
57	C₂³	—	12	4	no	—
58	C₂³	D₁₂	1	1	yes	characteristic
59	D₄	—	3	4	no	—
60	D₄	—	6	4	no	—
61	D₄	—	6	2	no	—
62	C₂×C₄	—	6	2	no	—
63	C₂×C₄	C₄×D₃	1	2	yes	—
64	C₂×C₄	D₁₂	1	1	yes	characteristic
65	C₂×C₄	#67	1	2	yes	—
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
71	D₃	—	4	4	no	—
72	C₂²	—	2	2	no	—
73	C₂²	—	6	4	no	—
74	C₂²	—	12	8	no	—
75	C₂²	—	12	4	no	—
76	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
77	C₂²	C₂×D₁₂	1	1	yes	characteristic
78	C₄	—	12	2	no	—
79	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
80	C₄	C₂×C₄×D₃	1	1	yes	characteristic
81	C₄	C₂×#67	1	1	yes	characteristic
82	C₃	#418	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	2	2	no	—
84	C₂	—	12	4	no	—
85	C₂	#908	1	2	yes	—
86	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	C₂²×D₁₂	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#418
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
2. C₂×C₃×M₁₆	C₂	semidirect
3. #926	C₂	semidirect
4. #821	C₂	semidirect
10. C₂×D₁₂	C₄	semidirect
13. C₂×C₃×C₈	C₂²	semidirect
15. C₄⋊Dic₃	C₂²	semidirect
18. C₂×M₁₆	D₃	semidirect
41. C₂×C₈	C₂×D₃	semidirect
82. C₃	#418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂	2
C₂×C₃×M₁₆	2
#926	2
#821	2
#418	3