Subgroups — #71338

Order 384.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#498	C₂	1	1	yes	maximal, characteristic
2	#5080	C₂	1	2	yes	maximal
3	#5079	C₂	1	2	yes	maximal
4	#4699	C₂	1	1	yes	maximal, characteristic
5	#4455	C₂	1	1	yes	maximal, characteristic
6	#1716	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×Dic₁₂	—	2	2	no	—
9	#869	C₂²	1	1	yes	characteristic
10	#869	C₂²	1	1	yes	characteristic
11	C₃×#106	—	2	2	no	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#835	—	2	4	no	—
14	#808	—	2	4	no	—
15	C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	#498	—	3	2	no	—
17	#498	D₃	1	1	yes	characteristic
18	#494	—	3	1	no	—
19	#487	—	3	2	no	—
20	#322	—	3	1	no	—
21	C₂×C₃×Q₈	—	2	2	no	—
22	C₂×Dic₆	—	2	1	no	—
23	C₂×Dic₆	—	2	1	no	—
24	C₂×C₃×C₈	D₄	1	2	yes	—
25	C₃×C₄⋊C₄	—	2	2	no	—
26	C₃×C₄⋊C₄	—	2	2	no	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₃	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₃⋊C₈	D₄	1	4	yes	—
31	Dic₁₂	—	4	4	no	—
32	C₂×Q₁₆	—	6	2	no	—
33	C₄⋊Q₈	—	3	1	no	—
34	C₄⋊Q₈	—	3	1	no	—
35	C₄⋊Q₈	C₂×D₃	1	2	yes	—
36	#106	—	2	2	no	—
37	#106	—	6	4	no	—
38	#103	—	6	4	no	—
39	C₄×C₈	—	3	2	no	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₃×Q₈	—	4	4	no	—
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
44	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₃	—	4	1	no	—
46	C₂×Dic₃	—	4	1	no	—
47	Dic₆	—	4	2	no	—
48	Dic₆	—	4	2	no	—
49	C₃×C₈	SD₁₆	1	4	yes	—
50	C₃⋊C₈	—	2	4	no	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	6	1	no	—
53	C₂×Q₈	—	6	1	no	—
54	Q₁₆	—	12	4	no	—
55	C₂×C₈	—	3	4	no	—
56	C₂×C₈	#67	1	2	yes	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	6	2	no	—
60	C₄⋊C₄	—	6	2	no	—
61	C₄²	C₂²×D₃	1	1	yes	characteristic
62	C₂²×C₃	#127	1	1	yes	characteristic
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	C₂×SD₁₆	1	2	yes	—
65	C₃×C₄	#137	1	4	yes	—
66	Dic₃	—	8	1	no	—
67	Dic₃	—	8	1	no	—
68	Q₈	—	4	4	no	—
69	Q₈	—	12	2	no	—
70	Q₈	—	12	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	12	1	no	—
73	C₂×C₄	—	12	1	no	—
74	C₂×C₄	D₃×D₄	1	2	yes	—
75	C₂×C₄	C₂×#67	1	1	yes	characteristic
76	C₈	—	6	4	no	—
77	C₈	#215	1	4	yes	—
78	C₂×C₃	#492	1	1	yes	characteristic
79	C₂×C₃	#497	1	2	yes	—
80	C₂²	#940	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	24	1	no	—
83	C₄	—	24	1	no	—
84	C₄	#915	1	4	yes	—
85	C₄	C₂×#215	1	2	yes	—
86	C₃	#1716	1	1	yes	characteristic, complemented, Sylow
87	C₂	#5085	1	1	yes	characteristic
88	C₂	#5167	1	2	yes	—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#498	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#498	D₃
hypocenter	C₃	#1716
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
86. C₃	#1716	semidirect

Maximal subgroups

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

subgroup	index
C₃×#498	2
#5080	2
#5079	2
#4699	2
#4455	2
#1716	3