Subgroups — #5483

Order 192.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#217	C₂	1	2	yes	maximal
3	#924	C₂	1	4	yes	maximal, complemented
4	#878	C₂	1	8	yes	maximal, complemented
5	#543	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₂²	1	2	yes	—
8	C₂²×Dic₃	C₂²	1	4	yes	—
9	C₂×Dic₆	C₂²	1	4	yes	complemented
10	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
11	#217	C₂²	1	4	yes	—
12	#217	C₂²	1	4	yes	—
13	C₄⋊Dic₃	C₂²	1	4	yes	complemented
14	Dic₃⋊C₄	C₂²	1	8	yes	complemented
15	C₂²⋊Q₈	—	3	4	no	—
16	C₂²⋊Q₈	—	3	8	no	—
17	C₂×C₂²⋊C₄	—	3	2	no	—
18	C₂×C₂²⋊C₄	D₃	1	1	yes	characteristic
19	C₂³×C₃	—	4	2	no	—
20	C₂³×C₃	Q₈	1	4	yes	complemented
21	C₂³×C₃	C₂³	1	1	yes	characteristic
22	C₂³×C₃	C₂³	1	2	yes	—
23	C₂×C₃×C₄	—	4	2	no	—
24	C₂×C₃×C₄	C₂³	1	4	yes	—
25	C₂×Dic₃	—	4	4	no	—
26	C₂×Dic₃	C₂³	1	8	yes	—
27	Dic₆	—	4	4	no	—
28	C₂⁴	C₂×D₃	1	1	yes	characteristic
29	C₂×Q₈	—	3	4	no	—
30	C₂²×C₄	—	3	4	no	—
31	C₂²×C₄	C₂×D₃	1	2	yes	—
32	C₄⋊C₄	—	3	8	no	—
33	C₄⋊C₄	—	3	4	no	—
34	C₂²⋊C₄	—	3	4	no	—
35	C₂²⋊C₄	—	3	4	no	—
36	C₂²⋊C₄	C₂×D₃	1	4	yes	—
37	C₂²×C₃	—	2	4	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	4	4	no	—
40	C₂²×C₃	C₂×Q₈	1	2	yes	—
41	C₂²×C₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₃	C₂⁴	1	1	yes	characteristic
43	C₃×C₄	—	2	4	no	—
44	Dic₃	—	2	8	no	—
45	C₂³	—	4	2	no	—
46	C₂³	Dic₆	1	4	yes	complemented
47	C₂³	C₂²×D₃	1	1	yes	characteristic
48	C₂³	C₂²×D₃	1	2	yes	—
49	Q₈	—	12	4	no	—
50	C₂×C₄	—	3	8	no	—
51	C₂×C₄	—	4	2	no	—
52	C₂×C₄	—	12	4	no	—
53	C₂×C₄	C₂²×D₃	1	4	yes	—
54	C₂×C₃	—	2	4	no	—
55	C₂×C₃	—	2	2	no	—
56	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
57	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
58	C₂²	—	2	2	no	—
59	C₂²	—	2	4	no	—
60	C₂²	—	4	4	no	—
61	C₂²	C₂×Dic₆	1	2	yes	—
62	C₂²	C₂×Dic₆	1	4	yes	—
63	C₂²	C₂³×D₃	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	6	8	no	—
66	C₃	#543	1	1	yes	characteristic, complemented, Sylow
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₆	1	1	yes	characteristic
70	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
prime radical	C₂⁴×C₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₃
hypocenter	C₃	#543
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
3. #924	C₂	semidirect
4. #878	C₂	semidirect
9. C₂×Dic₆	C₂²	semidirect
13. C₄⋊Dic₃	C₂²	semidirect
14. Dic₃⋊C₄	C₂²	semidirect
20. C₂³×C₃	Q₈	semidirect
46. C₂³	Dic₆	semidirect
66. C₃	#543	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₂²⋊C₄	2
C₂×#217	2
#924	2
#878	2
#543	3