Subgroups — #4719

Order 192.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#830	C₂	1	1	yes	maximal, characteristic
4	#815	C₂	1	2	yes	maximal, complemented
5	Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
6	#432	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	2	yes	—
11	C₄×Dic₃	C₄	1	1	yes	characteristic
12	C₄×Dic₃	C₄	1	1	yes	characteristic
13	C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄²	—	3	1	no	—
17	C₄⋊C₈	—	3	2	no	—
18	C₂²⋊C₈	—	3	1	no	—
19	C₂²⋊C₈	D₃	1	1	yes	characteristic
20	#97	—	3	2	no	—
21	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	—	2	2	no	—
27	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₃×C₈	—	2	2	no	—
32	C₃⋊C₈	—	2	2	no	—
33	C₂²×C₄	—	3	1	no	—
34	C₂²×C₄	—	3	1	no	—
35	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂×C₈	—	3	2	no	—
37	C₂×C₈	C₂×D₃	1	2	yes	—
38	C₄²	—	3	1	no	—
39	C₄²	—	3	1	no	—
40	C₄²	—	3	1	no	—
41	C₄²	—	3	1	no	—
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	—	2	1	no	—
44	C₂²×C₃	M₁₆	1	2	yes	—
45	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	#41	1	2	yes	—
48	Dic₃	—	2	1	no	—
49	Dic₃	—	2	1	no	—
50	Dic₃	—	2	1	no	—
51	Dic₃	M₁₆	1	2	yes	—
52	C₂³	C₄×D₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	3	1	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	3	1	no	—
57	C₂×C₄	—	3	1	no	—
58	C₂×C₄	—	6	2	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	C₄×D₃	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	6	2	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	#117	1	1	yes	characteristic
66	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	ΓL(1,𝔽₂₅)	1	2	yes	—
71	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	3	2	no	—
74	C₄	—	6	1	no	—
75	C₄	—	6	1	no	—
76	C₄	—	6	1	no	—
77	C₄	#237	1	2	yes	—
78	C₃	#432	1	1	yes	characteristic, complemented, Sylow
79	C₂	—	2	2	no	—
80	C₂	#877	1	1	yes	characteristic
81	C₂	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
82	C₂	D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₂²⋊C₈	C₂
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#432
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. #815	C₂	semidirect
5. Dic₃⋊C₈	C₂	semidirect
78. C₃	#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃	2
C₃×C₂²⋊C₈	2
#830	2
#815	2
Dic₃⋊C₈	2
#432	3