Subgroups — #5115

Order 192.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#930	C₂	1	2	yes	maximal, complemented
3	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	C₂	1	2	yes	maximal, complemented
5	#818	C₂	1	1	yes	maximal, characteristic, complemented
6	#466	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×#67	—	2	2	no	—
9	C₂×D₁₂	C₂²	1	2	yes	—
10	C₂×C₃×C₈	—	2	2	no	—
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₄⋊Dic₃	C₂²	1	2	yes	—
15	D₂₄	—	2	2	no	—
16	C₂×D₈	—	3	1	no	—
17	C₂²×C₈	D₃	1	1	yes	characteristic, complemented
18	#121	—	3	2	no	—
19	#107	—	3	1	no	—
20	#102	—	3	2	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂²×D₃	—	2	2	no	—
23	C₂×C₃×C₄	—	2	1	no	—
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
27	#67	—	4	4	no	—
28	C₂×Dic₃	—	2	2	no	—
29	D₁₂	—	2	2	no	—
30	D₁₂	—	2	2	no	—
31	C₃×C₈	—	2	1	no	—
32	C₃×C₈	D₄	1	2	yes	complemented
33	C₂×D₄	—	3	2	no	—
34	C₂×D₄	—	6	2	no	—
35	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
36	D₈	—	6	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₈	C₂×D₃	1	1	yes	characteristic
40	C₄⋊C₄	—	3	2	no	—
41	C₂²⋊C₄	—	6	2	no	—
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	—	2	1	no	—
44	C₂²×C₃	D₈	1	2	yes	complemented
45	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
46	C₂×D₃	—	4	2	no	—
47	C₂×D₃	—	4	2	no	—
48	C₂×D₃	—	4	2	no	—
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₃×C₄	#41	1	1	yes	characteristic
52	Dic₃	—	4	2	no	—
53	C₂³	—	6	2	no	—
54	C₂³	D₁₂	1	1	yes	characteristic
55	D₄	—	6	2	no	—
56	D₄	—	6	2	no	—
57	D₄	—	12	4	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	6	2	no	—
61	C₂×C₄	D₁₂	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	#67	1	2	yes	complemented
65	C₂×C₃	—	2	2	no	—
66	C₂×C₃	#121	1	1	yes	characteristic
67	C₂×C₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₃	#135	1	1	yes	characteristic
69	D₃	—	8	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	—	12	2	no	—
73	C₂²	—	12	2	no	—
74	C₂²	—	12	2	no	—
75	C₂²	D₂₄	1	2	yes	complemented
76	C₂²	C₂×D₁₂	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	12	2	no	—
79	C₄	#235	1	1	yes	characteristic
80	C₄	C₂×#67	1	1	yes	characteristic
81	C₃	#466	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	2	2	no	—
83	C₂	—	24	2	no	—
84	C₂	C₂×D₂₄	1	1	yes	characteristic
85	C₂	#904	1	1	yes	characteristic
86	C₂	#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#466
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₈	C₂	semidirect
2. #930	C₂	semidirect
3. C₂×D₂₄	C₂	semidirect
4. #821	C₂	semidirect
5. #818	C₂	semidirect
17. C₂²×C₈	D₃	semidirect
32. C₃×C₈	D₄	semidirect
44. C₂²×C₃	D₈	semidirect
64. C₈	#67	semidirect
75. C₂²	D₂₄	semidirect
81. C₃	#466	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈	2
#930	2
C₂×D₂₄	2
#821	2
#818	2
#466	3