Subgroups — #4981

Order 192.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃⋊C₄	C₂	1	2	yes	maximal
4	C₃⋊#95	C₂	1	2	yes	maximal
5	C₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	#398	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	C₂²	1	2	yes	—
11	C₂²×Dic₃	C₂²	1	2	yes	—
12	C₃×C₄⋊C₄	—	2	2	no	—
13	C₄⋊Dic₃	—	2	2	no	—
14	Dic₃⋊C₄	—	2	4	no	—
15	C₂×C₄⋊C₄	—	3	2	no	—
16	C₂×C₄⋊C₄	—	3	1	no	—
17	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
18	#95	—	3	1	no	—
19	#95	—	3	2	no	—
20	C₂³×C₃	C₂³	1	1	yes	characteristic
21	C₂×C₃×C₄	—	2	1	no	—
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	—	2	2	no	—
26	C₂×C₃×C₄	D₄	1	2	yes	—
27	C₂×C₃×C₄	D₄	1	2	yes	—
28	C₂×Dic₃	—	2	2	no	—
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	—	2	2	no	—
31	C₂×Dic₃	—	2	2	no	—
32	C₂×Dic₃	—	2	2	no	—
33	C₂×Dic₃	Q₈	1	4	yes	—
34	C₂²×C₄	—	3	2	no	—
35	C₂²×C₄	—	3	2	no	—
36	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	6	4	no	—
41	C₄⋊C₄	—	6	2	no	—
42	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₃	#41	1	1	yes	characteristic
46	C₂²×C₃	#41	1	2	yes	—
47	C₃×C₄	—	2	2	no	—
48	C₃×C₄	—	2	2	no	—
49	C₃×C₄	—	4	1	no	—
50	Dic₃	—	2	4	no	—
51	Dic₃	—	4	2	no	—
52	C₂³	C₂²×D₃	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	3	4	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	—	6	2	no	—
61	C₂×C₄	—	6	2	no	—
62	C₂×C₄	—	6	2	no	—
63	C₂×C₄	—	6	2	no	—
64	C₂×C₄	D₁₂	1	2	yes	—
65	C₂×C₄	#67	1	2	yes	—
66	C₂×C₃	#121	1	1	yes	characteristic
67	C₂×C₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₃	#123	1	1	yes	characteristic
69	C₂×C₃	#125	1	2	yes	—
70	C₂×C₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²	C₂×D₁₂	1	1	yes	characteristic
72	C₂²	#235	1	1	yes	characteristic
73	C₂²	#237	1	2	yes	—
74	C₂²	D₃×Q₈	1	2	yes	—
75	C₂²	C₂×#67	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	4	1	no	—
79	C₄	—	6	4	no	—
80	C₄	—	12	2	no	—
81	C₃	#398	1	1	yes	characteristic, complemented, Sylow
82	C₂	#889	1	2	yes	—
83	C₂	#897	1	2	yes	—
84	C₂	#930	1	1	yes	characteristic
85	C₂	#935	1	1	yes	characteristic
86	C₂	#944	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂³
Fitting	C₂×C₃×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
81. C₃	#398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
C₂×C₄⋊Dic₃	2
C₂×Dic₃⋊C₄	2
C₃⋊#95	2
#398	3