Subgroups — #4925

Order 192.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	#830	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#431	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×C₄²	C₄	1	1	yes	characteristic
10	C₃×C₄²	C₄	1	1	yes	characteristic
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄²	D₃	1	1	yes	characteristic
15	C₄⋊C₈	—	3	2	no	—
16	C₂²⋊C₈	—	3	2	no	—
17	C₄×C₈	—	3	2	no	—
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄	—	2	2	no	—
20	C₂×C₃×C₄	—	2	2	no	—
21	C₂×C₃×C₄	C₈	1	4	yes	complemented
22	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₃⋊C₈	—	2	4	no	—
29	C₂²×C₄	Dic₃	1	1	yes	characteristic
30	C₂²×C₄	Dic₃	1	1	yes	characteristic
31	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₂×C₈	—	3	4	no	—
33	C₄²	Dic₃	1	1	yes	characteristic
34	C₄²	Dic₃	1	1	yes	characteristic
35	C₄²	C₂×D₃	1	1	yes	characteristic
36	C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃	—	2	1	no	—
38	C₂²×C₃	—	2	1	no	—
39	C₂²×C₃	C₂×C₈	1	2	yes	—
40	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	C₂×C₈	1	2	yes	—
44	C₃×C₄	C₂×C₈	1	2	yes	—
45	C₃×C₄	M₁₆	1	2	yes	—
46	C₃×C₄	#41	1	2	yes	—
47	C₂³	C₂×Dic₃	1	1	yes	characteristic
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	C₃⋊C₈	1	4	yes	complemented
51	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
53	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₈	—	6	4	no	—
58	C₂×C₃	—	2	2	no	—
59	C₂×C₃	#117	1	1	yes	characteristic
60	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	C₂×C₃⋊C₈	1	2	yes	—
65	C₂²	C₂²×Dic₃	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2	1	no	—
68	C₄	C₂×C₃⋊C₈	1	2	yes	—
69	C₄	C₂×C₃⋊C₈	1	2	yes	—
70	C₄	#208	1	2	yes	—
71	C₄	#235	1	2	yes	—
72	C₃	#431	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	2	2	no	—
74	C₂	C₂²×C₃⋊C₈	1	1	yes	characteristic
75	C₂	C₂×#208	1	1	yes	characteristic
76	C₂	#926	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃
commutator	C₂×C₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄²	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	#431
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₄×C₃⋊C₈	C₂	semidirect
21. C₂×C₃×C₄	C₈	semidirect
50. C₂×C₄	C₃⋊C₈	semidirect
72. C₃	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²	2
#830	2
C₄⋊(C₃⋊C₈)	2
C₄×C₃⋊C₈	2
#431	3