Subgroups — #4986

Order 192.

134 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	C₃⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₃⋊#95	C₂	1	2	yes	maximal
5	C₃⋊#95	C₂	1	2	yes	maximal
6	#400	—	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	C₂×C₄⋊C₄	—	3	1	no	—
15	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
16	#95	—	3	1	no	—
17	#95	—	3	2	no	—
18	#95	—	3	2	no	—
19	C₂³×C₃	C₂³	1	1	yes	characteristic
20	C₂×C₃×C₄	—	2	1	no	—
21	C₂×C₃×C₄	—	2	1	no	—
22	C₂×C₃×C₄	—	2	2	no	—
23	C₂×C₃×C₄	—	2	1	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	D₄	1	2	yes	—
26	C₂×C₃×C₄	Q₈	1	2	yes	—
27	C₂×Dic₃	—	2	2	no	—
28	C₂×Dic₃	—	2	2	no	—
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	—	2	4	no	—
31	C₂×Dic₃	—	2	2	no	—
32	C₂²×C₄	—	3	2	no	—
33	C₂²×C₄	—	3	2	no	—
34	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	—	6	2	no	—
39	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₃	#41	1	1	yes	characteristic
42	C₂²×C₃	#41	1	2	yes	—
43	C₂²×C₃	#41	1	2	yes	—
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	—	2	2	no	—
46	C₃×C₄	—	4	1	no	—
47	Dic₃	—	4	2	no	—
48	Dic₃	—	4	2	no	—
49	C₂³	C₂²×D₃	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	6	2	no	—
56	C₂×C₄	—	6	2	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	—	6	4	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	Dic₆	1	2	yes	—
61	C₂×C₄	#67	1	2	yes	—
62	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₃	#123	1	1	yes	characteristic
64	C₂×C₃	#124	1	1	yes	characteristic
65	C₂×C₃	#125	1	2	yes	—
66	C₂×C₃	#126	1	2	yes	—
67	C₂²	C₂×Dic₆	1	1	yes	characteristic
68	C₂²	#235	1	1	yes	characteristic
69	C₂²	#237	1	2	yes	—
70	C₂²	#239	1	2	yes	—
71	C₂²	C₂×#67	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	2	2	no	—
74	C₄	—	4	1	no	—
75	C₄	—	12	2	no	—
76	C₄	—	12	2	no	—
77	C₃	#400	1	1	yes	characteristic, complemented, Sylow
78	C₂	#890	1	2	yes	—
79	C₂	#898	1	2	yes	—
80	C₂	#924	1	1	yes	characteristic
81	C₂	#935	1	1	yes	characteristic
82	C₂	#947	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	#400
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
77. C₃	#400	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
C₃⋊#95	2
#400	3