Subgroups — #5521

Order 192.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#926	C₂	1	1	yes	maximal, characteristic, complemented
4	#893	C₂	1	4	yes	maximal, complemented
5	D₃×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₄×D₁₂	C₂	1	4	yes	maximal, complemented
7	#518	—	3	1	no	maximal, Sylow
8	C₂³×D₃	C₂²	1	2	yes	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×D₁₂	C₄	1	8	yes	complemented
11	C₂×D₁₂	C₂²	1	2	yes	—
12	C₂×D₁₂	C₂²	1	2	yes	—
13	C₂×C₄×D₃	C₂²	1	8	yes	complemented
14	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
15	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²	C₂²	1	2	yes	complemented
17	#217	C₂²	1	2	yes	—
18	(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
19	C₄⋊Dic₃	C₂²	1	2	yes	complemented
20	C₄×Dic₃	C₂²	1	2	yes	complemented
21	C₂²×D₄	—	3	1	no	—
22	C₄×D₄	—	3	4	no	—
23	C₄×D₄	—	3	4	no	—
24	#117	—	3	1	no	—
25	#117	D₃	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊C₄	—	3	4	no	—
27	C₂³×C₃	C₂³	1	1	yes	characteristic
28	C₂²×D₃	—	4	4	no	—
29	C₂²×D₃	C₂×C₄	1	8	yes	complemented
30	C₂²×D₃	C₂³	1	4	yes	—
31	C₂×C₃×C₄	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄	C₂³	1	4	yes	—
33	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×Dic₃	C₂³	1	4	yes	—
36	D₁₂	C₂×C₄	1	8	yes	complemented
37	D₁₂	C₂×C₄	1	8	yes	complemented
38	C₄×D₃	—	4	8	no	—
39	C₂⁴	—	3	2	no	—
40	C₂×D₄	—	3	8	no	—
41	C₂×D₄	—	3	2	no	—
42	C₂×D₄	—	3	2	no	—
43	C₂²×C₄	—	3	8	no	—
44	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
45	C₄⋊C₄	—	3	2	no	—
46	C₄⋊C₄	C₂×D₃	1	2	yes	complemented
47	C₂²⋊C₄	—	3	2	no	—
48	C₂²⋊C₄	—	3	8	no	—
49	C₂²⋊C₄	C₂×D₃	1	2	yes	—
50	C₄²	—	3	2	no	—
51	C₄²	C₂×D₃	1	2	yes	complemented
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₃	—	2	8	no	—
56	C₂×D₃	—	4	8	no	—
57	C₂×D₃	C₂²×C₄	1	8	yes	—
58	C₃×C₄	—	2	4	no	—
59	C₃×C₄	C₂²×C₄	1	2	yes	—
60	C₃×C₄	C₂²×C₄	1	2	yes	—
61	Dic₃	—	2	4	no	—
62	C₂³	—	3	4	no	—
63	C₂³	—	3	8	no	—
64	C₂³	—	12	4	no	—
65	C₂³	C₂²×D₃	1	1	yes	characteristic
66	D₄	—	3	8	no	—
67	D₄	—	3	8	no	—
68	C₂×C₄	—	3	4	no	—
69	C₂×C₄	—	12	8	no	—
70	C₂×C₄	C₄×D₃	1	4	yes	complemented
71	C₂×C₄	C₂²×D₃	1	4	yes	—
72	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
77	D₃	—	2	8	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	3	8	no	—
80	C₂²	—	6	8	no	—
81	C₂²	—	12	8	no	—
82	C₂²	C₂×C₄×D₃	1	2	yes	—
83	C₂²	C₂³×D₃	1	1	yes	characteristic
84	C₄	—	2	4	no	—
85	C₄	—	6	4	no	—
86	C₄	C₂×C₄×D₃	1	2	yes	—
87	C₄	C₂×C₄×D₃	1	2	yes	—
88	C₃	#518	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	2	2	no	—
90	C₂	—	6	8	no	—
91	C₂	C₂²×C₄×D₃	1	1	yes	characteristic
92	C₂	#1009	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄
Fitting	C₃×#117	C₂
socle, cosocle	C₂²×C₃	C₂⁴
prime radical	C₂²×D₁₂	C₂
hypercenter	#117	D₃
hypocenter	C₃	#518
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₃×#117	C₂	semidirect
3. #926	C₂	semidirect
4. #893	C₂	semidirect
5. D₃×C₂²⋊C₄	C₂	semidirect
6. C₄×D₁₂	C₂	semidirect
10. C₂×D₁₂	C₄	semidirect
13. C₂×C₄×D₃	C₂²	semidirect
14. C₃×C₄⋊C₄	C₂²	semidirect
16. C₃×C₄²	C₂²	semidirect
18. (C₂×D₃)⋊C₄	C₂²	semidirect
19. C₄⋊Dic₃	C₂²	semidirect
20. C₄×Dic₃	C₂²	semidirect
25. #117	D₃	semidirect
29. C₂²×D₃	C₂×C₄	semidirect
31. C₂×C₃×C₄	C₂×C₄	semidirect
36. D₁₂	C₂×C₄	semidirect
37. D₁₂	C₂×C₄	semidirect
46. C₄⋊C₄	C₂×D₃	semidirect
51. C₄²	C₂×D₃	semidirect
70. C₂×C₄	C₄×D₃	semidirect
88. 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×D₁₂	2
C₃×#117	2
#926	2
#893	2
D₃×C₂²⋊C₄	2
C₄×D₁₂	2
#518	3