Subgroups — #5019

Order 192.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#214	C₂	1	2	yes	maximal
3	#869	C₂	1	1	yes	maximal, characteristic, complemented
4	#810	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#461	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄	—	2	1	no	—
9	C₂×Dic₆	C₂²	1	2	yes	—
10	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₃×C₂²⋊C₄	—	2	1	no	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	#214	—	2	4	no	—
14	C₄⋊Dic₃	—	2	1	no	—
15	C₄⋊Dic₃	—	2	1	no	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×SD₁₆	—	3	2	no	—
18	C₄⋊Q₈	—	3	1	no	—
19	C₄×D₄	D₃	1	1	yes	characteristic
20	C₄⋊C₈	—	3	1	no	—
21	#103	—	3	2	no	—
22	C₂³×C₃	—	2	1	no	—
23	C₃×D₄	—	2	1	no	—
24	C₃×D₄	D₄	1	2	yes	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	—	4	2	no	—
27	C₂×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×Dic₃	—	2	2	no	—
31	Dic₆	—	2	2	no	—
32	Dic₆	—	2	2	no	—
33	C₃⋊C₈	—	2	2	no	—
34	C₂×Q₈	—	3	2	no	—
35	C₂×D₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₄	—	2	1	no	—
37	SD₁₆	—	6	4	no	—
38	C₂×C₈	—	3	2	no	—
39	C₄⋊C₄	—	6	1	no	—
40	C₄⋊C₄	—	6	1	no	—
41	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²⋊C₄	—	2	1	no	—
43	C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃	—	2	2	no	—
45	C₂²×C₃	—	4	1	no	—
46	C₂²×C₃	—	4	1	no	—
47	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	—	4	1	no	—
50	C₃×C₄	SD₁₆	1	2	yes	complemented
51	C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄	#41	1	1	yes	characteristic
53	Dic₃	—	4	2	no	—
54	C₂³	—	2	1	no	—
55	Q₈	—	6	2	no	—
56	Q₈	—	6	2	no	—
57	D₄	—	2	1	no	—
58	D₄	D₁₂	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	6	2	no	—
62	C₂×C₄	#67	1	1	yes	characteristic
63	C₂×C₄	#67	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₈	—	6	2	no	—
66	C₂×C₃	—	4	2	no	—
67	C₂×C₃	#121	1	1	yes	characteristic
68	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₃	#137	1	1	yes	characteristic
70	C₂²	—	2	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	C₂×#67	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	12	2	no	—
77	C₄	#214	1	2	yes	complemented
78	C₄	C₂×D₁₂	1	1	yes	characteristic
79	C₄	#235	1	1	yes	characteristic
80	C₃	#461	1	1	yes	characteristic, complemented, Sylow
81	C₂	—	4	2	no	—
82	C₂	#930	1	1	yes	characteristic
83	C₂	C₂×#214	1	1	yes	characteristic
84	C₂	#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×D₄	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×C₃×D₄	C₂²
hypercenter	C₄×D₄	D₃
hypocenter	C₃	#461
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
3. #869	C₂	semidirect
4. #810	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
50. C₃×C₄	SD₁₆	semidirect
77. C₄	#214	semidirect
80. C₃	#461	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄	2
C₂×#214	2
#869	2
#810	2
C₄⋊(C₃⋊C₈)	2
#461	3