Subgroups — #5228

Order 192.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#216	C₂	1	2	yes	maximal, complemented
3	#924	C₂	1	1	yes	maximal, characteristic
4	#835	C₂	1	2	yes	maximal, complemented
5	#830	C₂	1	1	yes	maximal, characteristic
6	#451	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	—	2	4	no	—
8	C₂×C₃×Q₈	—	2	1	no	—
9	C₂×C₃×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	—	2	1	no	—
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂×Dic₆	C₂²	1	1	yes	characteristic
13	#217	—	2	1	no	—
14	#216	—	2	4	no	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	Dic₃⋊C₄	—	2	1	no	—
17	C₂×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×Q₈	D₃	1	1	yes	characteristic
19	C₂×Q₁₆	—	3	2	no	—
20	C₂²⋊Q₈	—	3	1	no	—
21	#103	—	3	2	no	—
22	C₂²⋊C₈	—	3	1	no	—
23	C₂³×C₃	D₄	1	1	yes	characteristic
24	C₃×Q₈	—	2	4	no	—
25	C₃×Q₈	—	2	2	no	—
26	C₃×Q₈	D₄	1	4	yes	—
27	C₂×C₃×C₄	—	2	4	no	—
28	C₂×C₃×C₄	—	2	2	no	—
29	C₂×C₃×C₄	—	2	2	no	—
30	C₂×C₃×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	—	2	1	no	—
33	C₂×Dic₃	—	2	1	no	—
34	Dic₆	—	2	2	no	—
35	C₃⋊C₈	—	2	2	no	—
36	C₂×Q₈	—	2	4	no	—
37	C₂×Q₈	—	2	1	no	—
38	C₂×Q₈	—	3	1	no	—
39	C₂×Q₈	C₂×D₃	1	2	yes	—
40	C₂²×C₄	—	2	1	no	—
41	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
42	Q₁₆	—	6	4	no	—
43	C₂×C₈	—	3	2	no	—
44	C₄⋊C₄	—	3	1	no	—
45	C₄⋊C₄	—	6	1	no	—
46	C₂²⋊C₄	—	6	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	Q₁₆	1	2	yes	complemented
50	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	—	2	1	no	—
53	C₃×C₄	C₂×D₄	1	2	yes	—
54	Dic₃	—	4	1	no	—
55	Dic₃	—	4	1	no	—
56	C₂³	#67	1	1	yes	characteristic
57	Q₈	—	2	4	no	—
58	Q₈	—	2	2	no	—
59	Q₈	—	6	2	no	—
60	Q₈	#67	1	4	yes	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	6	1	no	—
65	C₂×C₄	—	6	1	no	—
66	C₂×C₄	#67	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₈	—	6	2	no	—
69	C₂×C₃	—	2	2	no	—
70	C₂×C₃	#120	1	1	yes	characteristic
71	C₂×C₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₃	#137	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	#216	1	2	yes	complemented
76	C₂²	C₂×#67	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	1	no	—
79	C₄	—	12	1	no	—
80	C₄	—	12	1	no	—
81	C₄	C₂×#67	1	2	yes	—
82	C₃	#451	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	2	2	no	—
84	C₂	#942	1	1	yes	characteristic
85	C₂	C₂×#216	1	1	yes	characteristic
86	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂³×C₃	D₄
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	#451
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#216	C₂	semidirect
4. #835	C₂	semidirect
49. C₂²×C₃	Q₁₆	semidirect
75. C₂²	#216	semidirect
82. C₃	#451	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂×#216	2
#924	2
#835	2
#830	2
#451	3