Subgroups — #5058

Order 192.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#215	C₂	1	2	yes	maximal, complemented
3	C₂×#213	C₂	1	2	yes	maximal, complemented
4	#874	C₂	1	1	yes	maximal, characteristic, complemented
5	#803	C₂	1	1	yes	maximal, characteristic, complemented
6	#496	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
9	C₂×D₁₂	—	2	2	no	—
10	C₂×D₁₂	C₂²	1	2	yes	—
11	C₃×C₂²⋊C₄	—	2	2	no	—
12	C₃×C₄²	C₂²	1	1	yes	characteristic
13	#215	—	2	4	no	—
14	#213	—	2	4	no	—
15	C₂×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×SD₁₆	—	3	2	no	—
17	C₂×D₈	—	3	2	no	—
18	#127	—	3	1	no	—
19	#124	D₃	1	1	yes	characteristic, complemented
20	#97	—	3	1	no	—
21	C₂³×C₃	—	2	1	no	—
22	C₂²×D₃	—	2	2	no	—
23	C₃×Q₈	—	2	2	no	—
24	C₃×D₄	—	2	2	no	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	D₄	1	2	yes	—
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	D₁₂	—	2	2	no	—
29	D₁₂	—	2	2	no	—
30	D₁₂	—	4	4	no	—
31	C₃⋊C₈	D₄	1	4	yes	complemented
32	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
33	C₂×D₄	—	3	2	no	—
34	C₂×D₄	—	6	2	no	—
35	C₂×D₄	C₂×D₃	1	1	yes	characteristic
36	SD₁₆	—	6	4	no	—
37	D₈	—	6	4	no	—
38	C₂×C₈	—	3	2	no	—
39	C₂²⋊C₄	—	2	2	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	—	4	1	no	—
42	C₂²×C₃	—	4	2	no	—
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₂×D₃	—	4	2	no	—
45	C₂×D₃	—	4	4	no	—
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	—	4	1	no	—
48	C₃×C₄	C₂×D₄	1	2	yes	—
49	C₂³	—	2	1	no	—
50	C₂³	—	6	2	no	—
51	Q₈	—	2	2	no	—
52	D₄	—	2	2	no	—
53	D₄	—	6	2	no	—
54	D₄	—	6	2	no	—
55	D₄	—	12	4	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	#67	1	2	yes	—
58	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₈	—	3	4	no	—
60	C₂×C₃	—	8	1	no	—
61	C₂×C₃	#127	1	1	yes	characteristic
62	C₂×C₃	AGL(1,ℤ/8)	1	2	yes	—
63	D₃	—	8	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	12	2	no	—
67	C₂²	—	12	4	no	—
68	C₂²	C₂×#67	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	4	1	no	—
71	C₄	D₃×D₄	1	2	yes	—
72	C₃	#496	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	8	1	no	—
74	C₂	—	24	2	no	—
75	C₂	#940	1	1	yes	characteristic
76	C₂	#949	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#124	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	#124	D₃
hypocenter	C₃	#496
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₃×#124	C₂	semidirect
2. C₂×#215	C₂	semidirect
3. C₂×#213	C₂	semidirect
4. #874	C₂	semidirect
5. #803	C₂	semidirect
19. #124	D₃	semidirect
31. C₃⋊C₈	D₄	semidirect
72. C₃	#496	semidirect

Maximal subgroups

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

subgroup	index
C₃×#124	2
C₂×#215	2
C₂×#213	2
#874	2
#803	2
#496	3