Subgroups — #5500

Order 192.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	2	yes	maximal
3	#924	C₂	1	4	yes	maximal
4	#890	C₂	1	4	yes	maximal, complemented
5	#888	C₂	1	4	yes	maximal, complemented
6	#544	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	2	yes	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₆	C₂²	1	4	yes	—
10	C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	#217	C₂²	1	4	yes	—
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₄⋊Dic₃	C₂²	1	4	yes	—
14	C₄⋊Dic₃	C₂²	1	2	yes	—
15	Dic₃⋊C₄	C₂²	1	8	yes	—
16	C₄×Dic₃	C₂²	1	4	yes	—
17	C₄⋊Q₈	—	3	4	no	—
18	#125	—	3	4	no	—
19	C₂²⋊Q₈	—	3	4	no	—
20	#117	—	3	2	no	—
21	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂³×C₃	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄	—	4	2	no	—
24	C₂×C₃×C₄	Q₈	1	4	yes	complemented
25	C₂×C₃×C₄	C₂³	1	4	yes	—
26	C₂×C₃×C₄	C₂³	1	2	yes	—
27	C₂×Dic₃	C₂³	1	4	yes	—
28	C₂×Dic₃	C₂³	1	4	yes	—
29	Dic₆	—	4	4	no	—
30	C₂×Q₈	—	3	4	no	—
31	C₂²×C₄	C₂×D₃	1	2	yes	—
32	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₄⋊C₄	—	3	2	no	—
34	C₄⋊C₄	—	3	2	no	—
35	C₄⋊C₄	—	3	4	no	—
36	C₄⋊C₄	—	3	8	no	—
37	C₄⋊C₄	C₂×D₃	1	4	yes	—
38	C₂²⋊C₄	—	3	4	no	—
39	C₄²	—	3	4	no	—
40	C₂²×C₃	—	2	1	no	—
41	C₂²×C₃	—	2	1	no	—
42	C₂²×C₃	C₂×Q₈	1	2	yes	—
43	C₂²×C₃	C₂⁴	1	1	yes	characteristic
44	C₃×C₄	—	2	4	no	—
45	C₃×C₄	C₂×Q₈	1	4	yes	—
46	Dic₃	—	2	4	no	—
47	Dic₃	—	2	4	no	—
48	C₂³	C₂²×D₃	1	1	yes	characteristic
49	Q₈	—	12	4	no	—
50	C₂×C₄	—	3	4	no	—
51	C₂×C₄	—	3	4	no	—
52	C₂×C₄	—	4	2	no	—
53	C₂×C₄	Dic₆	1	4	yes	complemented
54	C₂×C₄	C₂²×D₃	1	4	yes	—
55	C₂×C₄	C₂²×D₃	1	2	yes	—
56	C₂×C₃	—	2	2	no	—
57	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
58	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
59	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	C₂×Dic₆	1	2	yes	—
63	C₂²	C₂³×D₃	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	6	4	no	—
66	C₄	—	6	4	no	—
67	C₄	C₂×Dic₆	1	4	yes	—
68	C₃	#544	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₆	1	1	yes	characteristic
71	C₂	#1004	1	1	yes	characteristic
72	C₂	#1007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₂×C₃×C₄⋊C₄	C₂
prime radical	C₂³×C₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #890	C₂	semidirect
5. #888	C₂	semidirect
24. C₂×C₃×C₄	Q₈	semidirect
53. C₂×C₄	Dic₆	semidirect
68. C₃	#544	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
#926	2
#924	2
#890	2
#888	2
#544	3