Subgroups — #4872

Order 192.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic
2	#893	C₂	1	2	yes	maximal, complemented
3	C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#809	C₂	1	2	yes	maximal, complemented
6	C₄×D₈	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₂²	1	2	yes	—
8	C₂×C₄×D₃	—	2	2	no	—
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	(C₂×D₃)⋊C₄	—	2	2	no	—
12	C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	D₂₄	C₄	1	4	yes	complemented
15	C₂×D₈	—	3	1	no	—
16	C₄×D₄	—	3	2	no	—
17	#107	D₃	1	1	yes	characteristic, complemented
18	#102	—	3	2	no	—
19	C₄×C₈	—	3	1	no	—
20	C₂²×D₃	—	2	2	no	—
21	C₂×C₃×C₄	—	2	2	no	—
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	D₄	1	1	yes	characteristic
24	C₂×Dic₃	D₄	1	1	yes	characteristic
25	D₁₂	—	2	2	no	—
26	D₁₂	C₂×C₄	1	4	yes	complemented
27	C₄×D₃	—	4	4	no	—
28	C₃×C₈	C₂×C₄	1	2	yes	complemented
29	C₃⋊C₈	—	2	1	no	—
30	C₂×D₄	—	3	2	no	—
31	C₂²×C₄	—	6	2	no	—
32	D₈	—	3	4	no	—
33	C₂×C₈	—	3	1	no	—
34	C₂×C₈	C₂×D₃	1	1	yes	characteristic
35	C₄⋊C₄	C₂×D₃	1	2	yes	—
36	C₂²⋊C₄	—	6	2	no	—
37	C₄²	—	3	1	no	—
38	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
39	C₂×D₃	—	2	4	no	—
40	C₂×D₃	—	4	2	no	—
41	C₂×D₃	—	4	2	no	—
42	C₃×C₄	—	4	2	no	—
43	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₃×C₄	#41	1	1	yes	characteristic
45	Dic₃	—	2	1	no	—
46	Dic₃	D₈	1	2	yes	complemented
47	C₂³	—	6	2	no	—
48	D₄	—	3	4	no	—
49	D₄	—	6	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	3	1	no	—
52	C₂×C₄	—	3	1	no	—
53	C₂×C₄	—	12	4	no	—
54	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
55	C₈	—	6	1	no	—
56	C₈	C₄×D₃	1	2	yes	complemented
57	C₂×C₃	C₄×D₄	1	1	yes	characteristic
58	C₂×C₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₃	#135	1	1	yes	characteristic
60	D₃	—	4	4	no	—
61	C₂²	—	6	4	no	—
62	C₂²	—	12	2	no	—
63	C₂²	—	12	2	no	—
64	C₂²	D₃×D₄	1	1	yes	characteristic
65	C₄	—	3	2	no	—
66	C₄	—	4	2	no	—
67	C₄	—	6	1	no	—
68	C₄	C₂×C₄×D₃	1	1	yes	characteristic
69	C₄	#239	1	1	yes	characteristic
70	C₃	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	12	4	no	—
72	C₂	#893	1	1	yes	characteristic
73	C₂	D₃×D₈	1	1	yes	characteristic
74	C₂	#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#107	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#107	D₃
hypocenter	C₃	C₄×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #893	C₂	semidirect
3. C₃×#107	C₂	semidirect
4. C₈×Dic₃	C₂	semidirect
5. #809	C₂	semidirect
14. D₂₄	C₄	semidirect
17. #107	D₃	semidirect
26. D₁₂	C₂×C₄	semidirect
28. C₃×C₈	C₂×C₄	semidirect
46. Dic₃	D₈	semidirect
56. C₈	C₄×D₃	semidirect
70. C₃	C₄×D₈	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
#893	2
C₃×#107	2
C₈×Dic₃	2
#809	2
C₄×D₈	3