Subgroups — #4526

Order 192.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#217	C₂	1	2	yes	maximal
3	#342	—	3	1	no	maximal, Sylow
4	C₂⁴×C₃	C₂²	1	1	yes	characteristic
5	C₂²×C₃×C₄	C₄	1	2	yes	complemented
6	C₂²×Dic₃	—	2	2	no	—
7	C₃×C₂²⋊C₄	—	2	2	no	—
8	#217	—	2	2	no	—
9	#217	C₄	1	4	yes	complemented
10	C₂×C₂²⋊C₄	—	3	2	no	—
11	C₂×C₂²⋊C₄	D₃	1	1	yes	characteristic
12	C₂³×C₃	—	4	2	no	—
13	C₂³×C₃	C₂×C₄	1	2	yes	—
14	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₃	D₄	1	1	yes	characteristic
16	C₂³×C₃	D₄	1	2	yes	—
17	C₂³×C₃	Q₈	1	1	yes	characteristic
18	C₂×C₃×C₄	—	2	2	no	—
19	C₂×C₃×C₄	—	4	2	no	—
20	C₂×Dic₃	—	2	4	no	—
21	C₂×Dic₃	—	4	4	no	—
22	C₂⁴	C₂×D₃	1	1	yes	characteristic
23	C₂²×C₄	—	6	2	no	—
24	C₂²×C₄	Dic₃	1	2	yes	complemented
25	C₂²⋊C₄	—	2	2	no	—
26	C₂²⋊C₄	—	3	4	no	—
27	C₂²⋊C₄	—	6	2	no	—
28	C₂²×C₃	—	2	4	no	—
29	C₂²×C₃	—	2	2	no	—
30	C₂²×C₃	—	4	2	no	—
31	C₂²×C₃	—	4	1	no	—
32	C₂²×C₃	—	4	1	no	—
33	C₂²×C₃	C₄²	1	1	yes	characteristic
34	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
38	C₃×C₄	—	4	2	no	—
39	Dic₃	—	4	4	no	—
40	C₂³	—	4	2	no	—
41	C₂³	Dic₆	1	1	yes	characteristic
42	C₂³	C₄×D₃	1	2	yes	—
43	C₂³	D₁₂	1	1	yes	characteristic
44	C₂³	C₂×Dic₃	1	1	yes	characteristic
45	C₂³	#67	1	2	yes	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	6	4	no	—
49	C₂×C₄	—	12	4	no	—
50	C₂×C₃	—	2	2	no	—
51	C₂×C₃	—	2	1	no	—
52	C₂×C₃	—	2	2	no	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	#95	1	1	yes	characteristic
55	C₂×C₃	#99	1	2	yes	—
56	C₂²	—	2	2	no	—
57	C₂²	—	2	4	no	—
58	C₂²	—	4	1	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	1	no	—
61	C₂²	C₄×Dic₃	1	1	yes	characteristic
62	C₂²	Dic₃⋊C₄	1	2	yes	—
63	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
64	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
65	C₂²	#217	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	12	4	no	—
68	C₃	#342	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	2	no	—
72	C₂	—	2	1	no	—
73	C₂	C₃⋊#95	1	1	yes	characteristic
74	C₂	#834	1	2	yes	—

	subgroup	G/H
center	C₂²	#217
commutator	C₂²×C₃	C₄²
Frattini	C₂⁴	C₂×D₃
Fitting	C₂×C₃×C₂²⋊C₄	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₃
hypocenter	C₃	#342
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
5. C₂²×C₃×C₄	C₄	semidirect
9. #217	C₄	semidirect
24. C₂²×C₄	Dic₃	semidirect
68. C₃	#342	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₂²⋊C₄	2
C₂×#217	2
#342	3