Subgroups — #69083

Order 384.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic
2	#4683	C₂	1	1	yes	maximal, characteristic
3	C₃×#346	C₂	1	1	yes	maximal, characteristic
4	#4503	C₂	1	4	yes	maximal
5	#2245	—	3	1	no	maximal, Sylow
6	C₂×Dic₁₂	—	2	1	no	—
7	C₂×Dic₁₂	C₂²	1	2	yes	—
8	#870	—	2	1	no	—
9	#869	—	2	1	no	—
10	C₂×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#818	C₂²	1	2	yes	—
13	#817	—	2	1	no	—
14	#499	—	3	1	no	—
15	#494	—	3	1	no	—
16	#358	—	3	4	no	—
17	#346	D₃	1	1	yes	characteristic
18	C₂×Dic₆	—	2	2	no	—
19	C₂×C₃×C₈	D₄	1	1	yes	characteristic
20	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
21	C₃×C₄²	D₄	1	1	yes	characteristic
22	C₄⋊Dic₃	—	2	2	no	—
23	C₄⋊Dic₃	—	4	2	no	—
24	Dic₃⋊C₄	—	4	2	no	—
25	Dic₁₂	—	2	4	no	—
26	Dic₁₂	—	4	2	no	—
27	C₃×C₁₆	—	2	2	no	—
28	C₂×Q₁₆	—	3	2	no	—
29	C₂×Q₁₆	—	6	1	no	—
30	C₄⋊Q₈	—	6	1	no	—
31	#125	—	6	1	no	—
32	C₂×C₁₆	C₂×D₃	1	2	yes	—
33	#107	—	3	2	no	—
34	#106	—	6	1	no	—
35	C₄×C₈	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄	D₈	1	2	yes	—
37	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₃	—	4	2	no	—
39	C₂×Dic₃	—	4	2	no	—
40	Dic₆	—	4	2	no	—
41	Dic₆	—	4	2	no	—
42	C₃×C₈	SD₁₆	1	2	yes	—
43	C₃×C₈	#41	1	2	yes	—
44	C₂×Q₈	—	6	2	no	—
45	Q₁₆	—	6	4	no	—
46	Q₁₆	—	12	2	no	—
47	C₂×C₈	D₁₂	1	1	yes	characteristic
48	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
49	C₄⋊C₄	—	6	2	no	—
50	C₄⋊C₄	—	12	2	no	—
51	C₄⋊C₄	—	12	2	no	—
52	C₄²	D₁₂	1	1	yes	characteristic
53	C₁₆	—	2	2	no	—
54	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	#124	1	1	yes	characteristic
57	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₃	—	8	2	no	—
59	Dic₃	—	8	2	no	—
60	Q₈	—	12	2	no	—
61	Q₈	—	12	2	no	—
62	C₂×C₄	—	12	2	no	—
63	C₂×C₄	—	12	2	no	—
64	C₂×C₄	D₂₄	1	2	yes	—
65	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
66	C₈	#204	1	2	yes	—
67	C₈	#235	1	2	yes	—
68	C₂×C₃	#486	1	1	yes	characteristic
69	C₂×C₃	#510	1	2	yes	—
70	C₂²	C₂×D₂₄	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	24	2	no	—
73	C₄	—	24	2	no	—
74	C₄	#875	1	1	yes	characteristic
75	C₄	C₂×#204	1	1	yes	characteristic
76	C₃	#2245	1	1	yes	characteristic, complemented, Sylow
77	C₂	#4694	1	1	yes	characteristic
78	C₂	#4909	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#346	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#346	D₃
hypocenter	C₃	#2245
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
76. C₃	#2245	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
#4683	2
C₃×#346	2
#4503	2
#2245	3