Subgroups — #5122

Order 192.

130 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	#815	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₃	C₂	1	2	yes	maximal, complemented
6	#405	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₃×M₁₆	C₄	1	4	yes	complemented
9	C₂×C₃×C₈	C₂²	1	2	yes	—
10	#217	C₄	1	2	yes	—
11	C₄⋊Dic₃	C₄	1	2	yes	—
12	C₄×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×M₁₆	D₃	1	1	yes	characteristic
17	C₂²×C₈	—	3	1	no	—
18	#117	—	3	1	no	—
19	#97	—	3	2	no	—
20	C₄×C₈	—	3	2	no	—
21	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂×Dic₃	C₂×C₄	1	4	yes	—
27	C₃×C₈	C₂×C₄	1	4	yes	—
28	C₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
31	M₁₆	Dic₃	1	4	yes	complemented
32	C₂×C₈	—	3	4	no	—
33	C₂×C₈	—	3	1	no	—
34	C₂×C₈	—	3	1	no	—
35	C₂×C₈	C₂×D₃	1	2	yes	—
36	C₄⋊C₄	—	3	2	no	—
37	C₂²⋊C₄	—	3	2	no	—
38	C₄²	—	3	2	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	C₄²	1	2	yes	—
41	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄	C₄²	1	2	yes	—
43	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	Dic₃	—	2	4	no	—
46	C₂³	C₄×D₃	1	1	yes	characteristic
47	C₂×C₄	—	3	4	no	—
48	C₂×C₄	C₄×D₃	1	1	yes	characteristic
49	C₂×C₄	C₄×D₃	1	2	yes	—
50	C₂×C₄	C₂×Dic₃	1	2	yes	—
51	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₈	—	3	2	no	—
53	C₈	—	3	2	no	—
54	C₈	C₂×Dic₃	1	4	yes	—
55	C₂×C₃	—	2	2	no	—
56	C₂×C₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₃	#131	1	2	yes	—
58	C₂²	—	2	2	no	—
59	C₂²	C₄×Dic₃	1	2	yes	—
60	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
61	C₄	—	6	4	no	—
62	C₄	C₄×Dic₃	1	2	yes	—
63	C₄	C₂×C₄×D₃	1	1	yes	characteristic
64	C₄	C₂²×Dic₃	1	1	yes	characteristic
65	C₃	#405	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	2	2	no	—
67	C₂	#907	1	2	yes	—
68	C₂	C₂×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#405
LCS	G ▷ C₂×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. #815	C₂	semidirect
5. C₈×Dic₃	C₂	semidirect
8. C₃×M₁₆	C₄	semidirect
13. C₂×C₃⋊C₈	C₄	semidirect
31. M₁₆	Dic₃	semidirect
65. C₃	#405	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
#815	2
C₈×Dic₃	2
#405	3