Subgroups — #4529

Order 192.

114 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	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#340	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	—	2	1	no	—
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃	—	2	1	no	—
8	C₃×C₄⋊C₄	—	2	1	no	—
9	C₃×C₄⋊C₄	C₄	1	2	yes	complemented
10	C₄⋊Dic₃	—	2	1	no	—
11	C₄⋊Dic₃	C₄	1	2	yes	complemented
12	C₂×C₃⋊C₈	—	2	1	no	—
13	C₂×C₃⋊C₈	—	2	1	no	—
14	C₂×C₃⋊C₈	C₄	1	2	yes	complemented
15	C₂²×C₈	—	3	1	no	—
16	C₂×C₄⋊C₄	—	3	1	no	—
17	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
18	C₂³×C₃	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄	—	2	2	no	—
20	C₂×C₃×C₄	—	4	1	no	—
21	C₂×C₃×C₄	—	4	1	no	—
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	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
28	C₂×Dic₃	—	2	2	no	—
29	C₂×Dic₃	—	4	1	no	—
30	C₂×Dic₃	—	4	1	no	—
31	C₃⋊C₈	—	2	2	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	—	6	1	no	—
34	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂×C₈	—	3	2	no	—
36	C₂×C₈	—	6	1	no	—
37	C₂×C₈	—	6	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	3	2	no	—
40	C₄⋊C₄	—	6	1	no	—
41	C₄⋊C₄	Dic₃	1	2	yes	complemented
42	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₃	D₈	1	1	yes	characteristic
46	C₂²×C₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₃	SD₁₆	1	1	yes	characteristic
48	C₂²×C₃	Q₁₆	1	1	yes	characteristic
49	C₃×C₄	—	4	2	no	—
50	C₃×C₄	C₄²	1	1	yes	characteristic
51	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
54	Dic₃	—	4	2	no	—
55	C₂³	#67	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	—	12	1	no	—
61	C₂×C₄	—	12	1	no	—
62	C₂×C₄	Dic₆	1	1	yes	characteristic
63	C₂×C₄	C₄×D₃	1	1	yes	characteristic
64	C₂×C₄	C₄×D₃	1	1	yes	characteristic
65	C₂×C₄	D₁₂	1	1	yes	characteristic
66	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₄	#67	1	1	yes	characteristic
68	C₈	—	6	2	no	—
69	C₂×C₃	#95	1	1	yes	characteristic
70	C₂×C₃	#102	1	1	yes	characteristic
71	C₂×C₃	#102	1	1	yes	characteristic
72	C₂×C₃	#103	1	1	yes	characteristic
73	C₂×C₃	#103	1	1	yes	characteristic
74	C₂×C₃	#106	1	1	yes	characteristic
75	C₂×C₃	#107	1	1	yes	characteristic
76	C₂²	Dic₃⋊C₄	1	1	yes	characteristic
77	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
78	C₂²	#213	1	1	yes	characteristic
79	C₂²	#214	1	1	yes	characteristic
80	C₂²	#215	1	1	yes	characteristic
81	C₂²	#216	1	1	yes	characteristic
82	C₂²	#217	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	—	12	2	no	—
85	C₄	C₄×Dic₃	1	1	yes	characteristic
86	C₄	Dic₃⋊C₄	1	1	yes	characteristic
87	C₄	C₄⋊Dic₃	1	1	yes	characteristic
88	C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
89	C₃	#340	1	1	yes	characteristic, complemented, Sylow
90	C₂	#807	1	1	yes	characteristic
91	C₂	#808	1	1	yes	characteristic
92	C₂	#809	1	1	yes	characteristic
93	C₂	#810	1	1	yes	characteristic
94	C₂	C₃⋊#95	1	1	yes	characteristic
95	C₂	#832	1	1	yes	characteristic
96	C₂	#835	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#67
commutator	C₃×C₄	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#340
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₃×C₄⋊C₄	C₄	semidirect
11. C₄⋊Dic₃	C₄	semidirect
14. C₂×C₃⋊C₈	C₄	semidirect
41. C₄⋊C₄	Dic₃	semidirect
89. C₃	#340	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
C₂²×C₃⋊C₈	2
#340	3