Subgroups — #5211

Order 192.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#217	C₂	1	2	yes	maximal
3	C₃⋊#95	C₂	1	4	yes	maximal, complemented
4	#381	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₄	—	2	4	no	—
7	C₂²×C₃×C₄	C₄	1	4	yes	complemented
8	C₂²×C₃×C₄	C₂²	1	2	yes	—
9	C₂²×Dic₃	C₂²	1	4	yes	—
10	#217	—	2	4	no	—
11	C₂³×C₄	D₃	1	1	yes	characteristic
12	C₂×C₂²⋊C₄	—	3	2	no	—
13	#95	—	3	4	no	—
14	C₂³×C₃	—	2	4	no	—
15	C₂³×C₃	C₂×C₄	1	2	yes	—
16	C₂³×C₃	D₄	1	4	yes	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄	—	2	8	no	—
19	C₂×C₃×C₄	—	2	4	no	—
20	C₂×C₃×C₄	C₂×C₄	1	4	yes	—
21	C₂×Dic₃	—	2	4	no	—
22	C₂×Dic₃	—	2	8	no	—
23	C₂⁴	C₂×D₃	1	1	yes	characteristic
24	C₂²×C₄	—	2	4	no	—
25	C₂²×C₄	—	3	4	no	—
26	C₂²×C₄	Dic₃	1	4	yes	complemented
27	C₂²×C₄	C₂×D₃	1	2	yes	—
28	C₂²⋊C₄	—	6	4	no	—
29	C₂²×C₃	—	2	4	no	—
30	C₂²×C₃	—	2	8	no	—
31	C₂²×C₃	C₂²⋊C₄	1	4	yes	—
32	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₃	C₂×D₄	1	2	yes	—
34	C₂²×C₃	#41	1	4	yes	—
35	C₃×C₄	—	2	4	no	—
36	Dic₃	—	4	4	no	—
37	C₂³	—	2	4	no	—
38	C₂³	C₂×Dic₃	1	2	yes	—
39	C₂³	#67	1	4	yes	—
40	C₂³	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₄	—	2	8	no	—
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	—	6	8	no	—
44	C₂×C₄	—	6	4	no	—
45	C₂×C₄	C₂×Dic₃	1	4	yes	—
46	C₂×C₃	—	2	4	no	—
47	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃	#117	1	2	yes	—
49	C₂×C₃	#123	1	4	yes	—
50	C₂²	—	2	4	no	—
51	C₂²	—	2	8	no	—
52	C₂²	#217	1	4	yes	—
53	C₂²	#235	1	4	yes	—
54	C₂²	C₂²×Dic₃	1	1	yes	characteristic
55	C₂²	C₂×#67	1	2	yes	—
56	C₄	—	2	4	no	—
57	C₄	—	12	4	no	—
58	C₃	#381	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	2	4	no	—
60	C₂	#926	1	2	yes	—
61	C₂	#929	1	4	yes	—
62	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄
Fitting	C₂³×C₃×C₄	C₂
socle, cosocle	C₂³×C₃	C₂³
prime radical	C₂⁴×C₃	C₂²
hypercenter	C₂³×C₄	D₃
hypocenter	C₃	#381
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₃⋊#95	C₂	semidirect
7. C₂²×C₃×C₄	C₄	semidirect
26. C₂²×C₄	Dic₃	semidirect
58. C₃	#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂×#217	2
C₃⋊#95	2
#381	3