Subgroups — #4843

Order 192.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄	C₂	1	2	yes	maximal, complemented
2	#874	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#821	C₂	1	2	yes	maximal, complemented
6	#459	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	—	2	1	no	—
8	C₂×D₁₂	—	2	1	no	—
9	C₂×D₁₂	C₂²	1	2	yes	—
10	C₂×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃	—	2	1	no	—
12	C₂×C₃×C₈	C₂²	1	2	yes	—
13	C₃×C₄²	C₂²	1	1	yes	characteristic
14	(C₂×D₃)⋊C₄	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	D₂₄	—	2	4	no	—
17	C₂×D₈	—	3	2	no	—
18	#127	—	3	1	no	—
19	C₄×D₄	—	3	1	no	—
20	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
21	#102	—	3	2	no	—
22	C₂²×D₃	—	2	1	no	—
23	C₂²×D₃	—	2	2	no	—
24	C₂×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	D₁₂	—	2	2	no	—
29	D₁₂	—	2	1	no	—
30	D₁₂	—	2	2	no	—
31	D₁₂	—	4	2	no	—
32	D₁₂	—	4	2	no	—
33	D₁₂	D₄	1	2	yes	complemented
34	C₄×D₃	—	4	2	no	—
35	C₃×C₈	—	2	2	no	—
36	C₂×D₄	—	3	1	no	—
37	C₂×D₄	—	3	2	no	—
38	C₂×D₄	—	6	1	no	—
39	C₂×D₄	—	6	1	no	—
40	C₂²×C₄	—	6	1	no	—
41	D₈	—	6	4	no	—
42	C₂×C₈	C₂×D₃	1	2	yes	—
43	C₄⋊C₄	—	3	1	no	—
44	C₂²⋊C₄	—	6	1	no	—
45	C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×D₃	—	2	2	no	—
48	C₂×D₃	—	4	1	no	—
49	C₂×D₃	—	4	1	no	—
50	C₂×D₃	—	4	2	no	—
51	C₂×D₃	—	4	2	no	—
52	C₂×D₃	—	4	2	no	—
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	D₈	1	2	yes	complemented
55	C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₃×C₄	#41	1	1	yes	characteristic
57	Dic₃	—	4	1	no	—
58	C₂³	—	6	1	no	—
59	C₂³	—	6	2	no	—
60	D₄	—	3	2	no	—
61	D₄	—	6	2	no	—
62	D₄	—	6	1	no	—
63	D₄	—	6	2	no	—
64	D₄	—	12	2	no	—
65	D₄	—	12	2	no	—
66	C₂×C₄	—	6	1	no	—
67	C₂×C₄	—	12	2	no	—
68	C₂×C₄	D₁₂	1	1	yes	characteristic
69	C₂×C₄	D₁₂	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₂×C₃	#121	1	1	yes	characteristic
73	C₂×C₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	D₃	—	4	2	no	—
76	D₃	—	8	2	no	—
77	C₂²	—	6	2	no	—
78	C₂²	—	12	1	no	—
79	C₂²	—	12	1	no	—
80	C₂²	—	12	2	no	—
81	C₂²	—	12	2	no	—
82	C₂²	—	12	2	no	—
83	C₂²	C₂×D₁₂	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	12	1	no	—
86	C₄	D₂₄	1	2	yes	complemented
87	C₄	D₃×D₄	1	1	yes	characteristic
88	C₄	#239	1	1	yes	characteristic
89	C₃	#459	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	12	2	no	—
91	C₂	—	24	2	no	—
92	C₂	#895	1	1	yes	characteristic
93	C₂	C₂×D₂₄	1	1	yes	characteristic
94	C₂	#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#459
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₄	C₂	semidirect
2. #874	C₂	semidirect
3. C₄×D₁₂	C₂	semidirect
4. C₃×C₄⋊C₈	C₂	semidirect
5. #821	C₂	semidirect
20. C₄⋊C₈	D₃	semidirect
33. D₁₂	D₄	semidirect
54. C₃×C₄	D₈	semidirect
86. C₄	D₂₄	semidirect
89. C₃	#459	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₄	2
#874	2
C₄×D₁₂	2
C₃×C₄⋊C₈	2
#821	2
#459	3