Subgroups — #71353

Order 384.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#499	C₂	1	1	yes	maximal, characteristic
2	#4699	C₂	1	1	yes	maximal, characteristic
3	#4492	C₂	1	4	yes	maximal
4	#4461	C₂	1	1	yes	maximal, characteristic
5	#2245	—	3	1	no	maximal, Sylow
6	C₃×#125	—	2	1	no	—
7	C₂×Dic₁₂	—	2	1	no	—
8	C₂×Dic₁₂	C₂²	1	2	yes	—
9	#869	—	2	1	no	—
10	C₃×#107	C₂²	1	2	yes	—
11	C₃×#106	—	2	1	no	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
14	#499	D₃	1	1	yes	characteristic
15	#494	—	3	1	no	—
16	#358	—	3	4	no	—
17	#346	—	3	1	no	—
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×C₄⋊C₄	—	2	2	no	—
22	C₃×C₄⋊C₄	—	4	2	no	—
23	C₃×C₄²	D₄	1	1	yes	characteristic
24	C₄⋊Dic₃	—	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	—	2	1	no	—
32	C₂×C₁₆	—	3	2	no	—
33	#107	C₂×D₃	1	2	yes	—
34	#106	—	2	1	no	—
35	C₄×C₈	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄	—	4	2	no	—
37	C₂×C₃×C₄	D₈	1	2	yes	—
38	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
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	#67	1	1	yes	characteristic
48	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	12	2	no	—
52	C₄²	#67	1	1	yes	characteristic
53	C₁₆	—	6	2	no	—
54	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	—	8	2	no	—
57	C₃×C₄	#124	1	1	yes	characteristic
58	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
59	Dic₃	—	8	2	no	—
60	Q₈	—	12	2	no	—
61	Q₈	—	12	2	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	12	2	no	—
64	C₂×C₄	#213	1	2	yes	—
65	C₂×C₄	C₂×#67	1	1	yes	characteristic
66	C₈	#215	1	2	yes	—
67	C₈	#239	1	2	yes	—
68	C₂×C₃	#486	1	1	yes	characteristic
69	C₂×C₃	#510	1	2	yes	—
70	C₂²	C₂×#213	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	8	2	no	—
73	C₄	—	24	2	no	—
74	C₄	C₂×#215	1	1	yes	characteristic
75	C₄	#947	1	1	yes	characteristic
76	C₃	#2245	1	1	yes	characteristic, complemented, Sylow
77	C₂	#5088	1	1	yes	characteristic
78	C₂	#5195	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#499	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#499	D₃
hypocenter	C₃	#2245
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

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
C₃×#499	2
#4699	2
#4492	2
#4461	2
#2245	3