Subgroups — #5227

Order 192.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#215	C₂	1	2	yes	maximal, complemented
3	#930	C₂	1	1	yes	maximal, characteristic
4	#835	C₂	1	2	yes	maximal, complemented
5	#830	C₂	1	1	yes	maximal, characteristic, complemented
6	#448	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	—	2	4	no	—
8	C₂×C₃×Q₈	—	2	1	no	—
9	C₂×C₃×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	—	2	1	no	—
11	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂×#67	—	2	1	no	—
13	C₂×D₁₂	C₂²	1	1	yes	characteristic
14	#215	—	2	4	no	—
15	(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×Q₈	D₃	1	1	yes	characteristic, complemented
19	C₂×SD₁₆	—	3	2	no	—
20	#121	—	3	1	no	—
21	#103	—	3	2	no	—
22	C₂²⋊C₈	—	3	1	no	—
23	C₂³×C₃	D₄	1	1	yes	characteristic
24	C₂²×D₃	—	2	1	no	—
25	C₃×Q₈	—	2	4	no	—
26	C₃×Q₈	—	2	2	no	—
27	C₃×Q₈	D₄	1	4	yes	complemented
28	C₂×C₃×C₄	—	2	4	no	—
29	C₂×C₃×C₄	—	2	2	no	—
30	C₂×C₃×C₄	—	2	2	no	—
31	C₂×C₃×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	#67	—	4	2	no	—
34	C₂×Dic₃	—	2	1	no	—
35	D₁₂	—	2	2	no	—
36	C₃⋊C₈	—	2	2	no	—
37	C₂×Q₈	—	2	4	no	—
38	C₂×Q₈	—	2	1	no	—
39	C₂×Q₈	C₂×D₃	1	2	yes	—
40	C₂×D₄	—	3	1	no	—
41	C₂×D₄	—	6	1	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
44	SD₁₆	—	6	4	no	—
45	C₂×C₈	—	3	2	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₂²⋊C₄	—	6	1	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	SD₁₆	1	2	yes	complemented
51	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
52	C₂×D₃	—	4	1	no	—
53	C₂×D₃	—	4	1	no	—
54	C₂×D₃	—	4	1	no	—
55	C₃×C₄	—	2	4	no	—
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	C₂×D₄	1	2	yes	—
58	Dic₃	—	4	1	no	—
59	C₂³	—	6	1	no	—
60	C₂³	#67	1	1	yes	characteristic
61	Q₈	—	2	4	no	—
62	Q₈	—	2	2	no	—
63	Q₈	#67	1	4	yes	complemented
64	D₄	—	6	2	no	—
65	D₄	—	12	2	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	#67	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₈	—	6	2	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	#120	1	1	yes	characteristic
75	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₃	#137	1	1	yes	characteristic
77	D₃	—	8	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	12	1	no	—
81	C₂²	—	12	1	no	—
82	C₂²	—	12	1	no	—
83	C₂²	#215	1	2	yes	complemented
84	C₂²	C₂×#67	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	2	1	no	—
87	C₄	—	12	1	no	—
88	C₄	C₂×#67	1	2	yes	—
89	C₃	#448	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	—	24	1	no	—
92	C₂	C₂×#215	1	1	yes	characteristic
93	C₂	#942	1	1	yes	characteristic
94	C₂	#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	#930	C₂
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	#448
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
2. C₂×#215	C₂	semidirect
4. #835	C₂	semidirect
5. #830	C₂	semidirect
18. C₂²×Q₈	D₃	semidirect
27. C₃×Q₈	D₄	semidirect
50. C₂²×C₃	SD₁₆	semidirect
63. Q₈	#67	semidirect
83. C₂²	#215	semidirect
89. C₃	#448	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂×#215	2
#930	2
#835	2
#830	2
#448	3