Subgroups — #4846

Order 192.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
2	#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	C₂	1	2	yes	maximal
5	#818	C₂	1	2	yes	maximal, complemented
6	#474	—	3	1	no	maximal, Sylow
7	C₂×D₁₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃	—	2	1	no	—
9	C₂×Dic₆	—	2	1	no	—
10	C₂×C₃×C₈	C₂²	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	(C₂×D₃)⋊C₄	—	2	1	no	—
13	C₄⋊Dic₃	—	2	1	no	—
14	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₄⋊Q₈	—	3	1	no	—
17	C₄×D₄	—	3	1	no	—
18	#107	—	3	2	no	—
19	C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
20	#102	—	3	2	no	—
21	C₂²×D₃	—	2	1	no	—
22	C₂×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	—	2	1	no	—
27	D₁₂	—	2	1	no	—
28	D₁₂	Q₈	1	2	yes	complemented
29	C₄×D₃	—	4	2	no	—
30	Dic₆	—	4	2	no	—
31	C₃×C₈	—	2	2	no	—
32	C₂×Q₈	—	6	1	no	—
33	C₂×D₄	—	3	1	no	—
34	C₂²×C₄	—	6	1	no	—
35	C₂×C₈	C₂×D₃	1	2	yes	—
36	C₄⋊C₄	—	3	2	no	—
37	C₄⋊C₄	—	3	1	no	—
38	C₄⋊C₄	—	6	1	no	—
39	C₂²⋊C₄	—	6	1	no	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
42	C₂×D₃	—	2	2	no	—
43	C₂×D₃	—	4	1	no	—
44	C₂×D₃	—	4	1	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	D₈	1	2	yes	—
47	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₃×C₄	#41	1	1	yes	characteristic
49	Dic₃	—	4	2	no	—
50	Dic₃	—	4	1	no	—
51	C₂³	—	6	1	no	—
52	Q₈	—	12	2	no	—
53	D₄	—	3	2	no	—
54	D₄	—	6	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	—	6	2	no	—
57	C₂×C₄	—	12	2	no	—
58	C₂×C₄	D₁₂	1	1	yes	characteristic
59	C₂×C₄	D₁₂	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₃	#137	1	1	yes	characteristic
65	D₃	—	4	2	no	—
66	C₂²	—	6	2	no	—
67	C₂²	—	12	1	no	—
68	C₂²	—	12	1	no	—
69	C₂²	C₂×D₁₂	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	12	1	no	—
72	C₄	—	12	2	no	—
73	C₄	D₂₄	1	2	yes	—
74	C₄	#237	1	1	yes	characteristic
75	C₄	D₃×Q₈	1	1	yes	characteristic
76	C₃	#474	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	12	2	no	—
78	C₂	#897	1	1	yes	characteristic
79	C₂	C₂×D₂₄	1	1	yes	characteristic
80	C₂	#909	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
prime radical	C₂×D₁₂	C₂²
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#474
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
2. #869	C₂	semidirect
3. C₃×C₄⋊C₈	C₂	semidirect
5. #818	C₂	semidirect
19. C₄⋊C₈	D₃	semidirect
28. D₁₂	Q₈	semidirect
76. C₃	#474	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂	2
#869	2
C₃×C₄⋊C₈	2
#821	2
#818	2
#474	3