Subgroups — #5493

Order 192.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#930	C₂	1	4	yes	maximal, complemented
4	#886	C₂	1	4	yes	maximal, complemented
5	#882	C₂	1	4	yes	maximal, complemented
6	#534	—	3	1	no	maximal, Sylow
7	C₂⁴×C₃	C₂²	1	1	yes	characteristic
8	C₂³×D₃	C₂²	1	2	yes	complemented
9	C₂²×C₃×C₄	C₂²	1	2	yes	complemented
10	C₂×#67	—	2	8	no	—
11	C₂×#67	C₂²	1	4	yes	—
12	C₂×#67	C₂²	1	4	yes	complemented
13	C₂²×Dic₃	C₂²	1	2	yes	complemented
14	C₂×D₁₂	C₂²	1	4	yes	complemented
15	C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
16	(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
17	C₄⋊Dic₃	C₂²	1	4	yes	complemented
18	C₂²×D₄	—	3	2	no	—
19	#123	—	3	4	no	—
20	#121	—	3	4	no	—
21	#120	—	3	4	no	—
22	C₂×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
23	C₂³×C₃	—	4	2	no	—
24	C₂³×C₃	D₄	1	4	yes	complemented
25	C₂³×C₃	C₂³	1	2	yes	—
26	C₂³×C₃	C₂³	1	1	yes	characteristic
27	C₂²×D₃	—	2	4	no	—
28	C₂²×D₃	—	4	4	no	—
29	C₂²×D₃	C₂³	1	4	yes	—
30	C₂×C₃×C₄	—	4	2	no	—
31	C₂×C₃×C₄	C₂³	1	4	yes	—
32	#67	—	2	8	no	—
33	#67	—	2	8	no	—
34	C₂×Dic₃	—	2	4	no	—
35	C₂×Dic₃	C₂³	1	4	yes	—
36	D₁₂	—	4	4	no	—
37	C₂⁴	—	3	2	no	—
38	C₂⁴	C₂×D₃	1	1	yes	characteristic
39	C₂×D₄	—	3	4	no	—
40	C₂×D₄	—	3	4	no	—
41	C₂×D₄	—	3	4	no	—
42	C₂×D₄	—	6	8	no	—
43	C₂²×C₄	—	3	2	no	—
44	C₂²×C₄	C₂×D₃	1	2	yes	complemented
45	C₄⋊C₄	—	3	4	no	—
46	C₂²⋊C₄	—	3	8	no	—
47	C₂²⋊C₄	C₂×D₃	1	4	yes	complemented
48	C₂²×C₃	—	2	2	no	—
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	4	4	no	—
51	C₂²×C₃	C₂×D₄	1	4	yes	—
52	C₂²×C₃	C₂×D₄	1	2	yes	—
53	C₂²×C₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₃	—	2	4	no	—
55	C₂×D₃	—	2	8	no	—
56	C₂×D₃	—	4	8	no	—
57	C₃×C₄	—	2	4	no	—
58	Dic₃	—	2	4	no	—
59	C₂³	—	3	4	no	—
60	C₂³	—	4	2	no	—
61	C₂³	—	6	4	no	—
62	C₂³	—	12	4	no	—
63	C₂³	D₁₂	1	4	yes	complemented
64	C₂³	C₂²×D₃	1	2	yes	—
65	C₂³	C₂²×D₃	1	1	yes	characteristic
66	D₄	—	6	8	no	—
67	D₄	—	6	8	no	—
68	D₄	—	12	4	no	—
69	C₂×C₄	—	3	4	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	6	4	no	—
72	C₂×C₄	C₂²×D₃	1	4	yes	—
73	C₂×C₃	—	2	4	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
77	D₃	—	4	4	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	2	4	no	—
80	C₂²	—	4	4	no	—
81	C₂²	—	6	4	no	—
82	C₂²	—	6	8	no	—
83	C₂²	—	12	8	no	—
84	C₂²	C₂×D₁₂	1	4	yes	—
85	C₂²	C₂×D₁₂	1	2	yes	—
86	C₂²	C₂³×D₃	1	1	yes	characteristic
87	C₄	—	2	4	no	—
88	C₄	—	6	4	no	—
89	C₃	#534	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	12	4	no	—
93	C₂	C₂²×D₁₂	1	1	yes	characteristic
94	C₂	#1004	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₂×C₃×C₂²⋊C₄	C₂
hypercenter	C₂×C₂²⋊C₄	D₃
hypocenter	C₃	#534
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#67	C₂	semidirect
2. C₂×C₃×C₂²⋊C₄	C₂	semidirect
3. #930	C₂	semidirect
4. #886	C₂	semidirect
5. #882	C₂	semidirect
8. C₂³×D₃	C₂²	semidirect
9. C₂²×C₃×C₄	C₂²	semidirect
12. C₂×#67	C₂²	semidirect
13. C₂²×Dic₃	C₂²	semidirect
14. C₂×D₁₂	C₂²	semidirect
15. C₃×C₂²⋊C₄	C₂²	semidirect
16. (C₂×D₃)⋊C₄	C₂²	semidirect
17. C₄⋊Dic₃	C₂²	semidirect
22. C₂×C₂²⋊C₄	D₃	semidirect
24. C₂³×C₃	D₄	semidirect
44. C₂²×C₄	C₂×D₃	semidirect
47. C₂²⋊C₄	C₂×D₃	semidirect
63. C₂³	D₁₂	semidirect
89. C₃	#534	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67	2
C₂×C₃×C₂²⋊C₄	2
#930	2
#886	2
#882	2
#534	3