Subgroups — #5798

Order 192.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#217	C₂	1	4	yes	maximal, complemented
3	D₄×Dic₃	C₂	1	8	yes	maximal, complemented
4	#926	C₂	1	2	yes	maximal, complemented
5	#518	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃	C₂²	1	2	yes	—
7	C₂×C₃×D₄	C₄	1	8	yes	complemented
8	C₂×C₃×D₄	C₂²	1	4	yes	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	C₂²	1	8	yes	complemented
11	#217	C₂²	1	4	yes	—
12	#217	C₂²	1	8	yes	complemented
13	C₄⋊Dic₃	C₂²	1	4	yes	complemented
14	C₄×Dic₃	C₂²	1	4	yes	complemented
15	C₂²×D₄	D₃	1	1	yes	characteristic
16	C₄×D₄	—	3	8	no	—
17	#117	—	3	2	no	—
18	C₂×C₂²⋊C₄	—	3	4	no	—
19	C₂³×C₃	—	4	4	no	—
20	C₂³×C₃	C₂×C₄	1	8	yes	complemented
21	C₂³×C₃	C₂³	1	1	yes	characteristic
22	C₂³×C₃	C₂³	1	4	yes	—
23	C₃×D₄	C₂×C₄	1	16	yes	complemented
24	C₂×C₃×C₄	C₂×C₄	1	4	yes	complemented
25	C₂×C₃×C₄	C₂³	1	2	yes	—
26	C₂×Dic₃	—	4	8	no	—
27	C₂×Dic₃	C₂³	1	8	yes	—
28	C₂⁴	C₂×D₃	1	2	yes	—
29	C₂×D₄	Dic₃	1	8	yes	complemented
30	C₂×D₄	C₂×D₃	1	4	yes	—
31	C₂²×C₄	—	3	8	no	—
32	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₄⋊C₄	—	3	4	no	—
34	C₂²⋊C₄	—	3	4	no	—
35	C₂²⋊C₄	—	3	8	no	—
36	C₄²	—	3	4	no	—
37	C₂²×C₃	—	2	8	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	4	8	no	—
40	C₂²×C₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₃	C₂²×C₄	1	8	yes	—
42	C₂²×C₃	C₂⁴	1	1	yes	characteristic
43	C₃×C₄	C₂²×C₄	1	4	yes	—
44	Dic₃	—	2	8	no	—
45	C₂³	—	4	4	no	—
46	C₂³	C₂×Dic₃	1	8	yes	complemented
47	C₂³	C₂²×D₃	1	1	yes	characteristic
48	C₂³	C₂²×D₃	1	4	yes	—
49	D₄	C₂×Dic₃	1	16	yes	complemented
50	C₂×C₄	—	3	8	no	—
51	C₂×C₄	—	12	8	no	—
52	C₂×C₄	C₂×Dic₃	1	4	yes	complemented
53	C₂×C₄	C₂²×D₃	1	2	yes	—
54	C₂×C₃	—	2	2	no	—
55	C₂×C₃	—	2	8	no	—
56	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
57	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
58	C₂²	—	2	8	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	8	no	—
61	C₂²	C₂²×Dic₃	1	2	yes	—
62	C₂²	C₂²×Dic₃	1	8	yes	—
63	C₂²	C₂³×D₃	1	1	yes	characteristic
64	C₄	—	6	8	no	—
65	C₄	C₂²×Dic₃	1	4	yes	—
66	C₃	#518	1	1	yes	characteristic, complemented, Sylow
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	#1004	1	2	yes	—
70	C₂	C₂³×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting, prime radical	C₂²×C₃×D₄	C₂
socle, cosocle	C₂²×C₃	C₂⁴
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	#518
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#217	C₂	semidirect
3. D₄×Dic₃	C₂	semidirect
4. #926	C₂	semidirect
7. C₂×C₃×D₄	C₄	semidirect
10. C₂²×Dic₃	C₂²	semidirect
12. #217	C₂²	semidirect
13. C₄⋊Dic₃	C₂²	semidirect
14. C₄×Dic₃	C₂²	semidirect
20. C₂³×C₃	C₂×C₄	semidirect
23. C₃×D₄	C₂×C₄	semidirect
24. C₂×C₃×C₄	C₂×C₄	semidirect
29. C₂×D₄	Dic₃	semidirect
46. C₂³	C₂×Dic₃	semidirect
49. D₄	C₂×Dic₃	semidirect
52. C₂×C₄	C₂×Dic₃	semidirect
66. C₃	#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄	2
C₂×#217	2
D₄×Dic₃	2
#926	2
#518	3