Subgroups — #68721

Order 384.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5126	C₂	1	3	yes	maximal
2	#5125	C₂	1	3	yes	maximal, complemented
3	C₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	#2091	—	3	1	no	maximal, Sylow
5	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×M₁₆	C₂²	1	3	yes	—
7	C₂×#208	C₂²	1	3	yes	—
8	(C₂×D₃)⋊C₈	—	2	6	no	—
9	#819	—	2	6	no	—
10	#430	—	3	3	no	—
11	#407	—	3	3	no	—
12	#338	D₃	1	1	yes	characteristic, complemented
13	C₂³×D₃	Q₈	1	1	yes	characteristic
14	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
15	C₂²×Dic₃	Q₈	1	1	yes	characteristic
16	C₂×C₄×D₃	—	2	3	no	—
17	C₂×C₄×D₃	—	2	3	no	—
18	C₃×M₁₆	—	2	6	no	—
19	C₂×C₃×C₈	D₄	1	6	yes	—
20	#208	—	2	6	no	—
21	C₂×C₃⋊C₈	—	2	3	no	—
22	C₂³×C₄	—	3	1	no	—
23	C₂×M₁₆	—	3	3	no	—
24	C₂×M₁₆	C₂×D₃	1	3	yes	—
25	#108	—	6	6	no	—
26	C₂²⋊C₈	—	6	6	no	—
27	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
28	C₂²×D₃	—	2	3	no	—
29	C₂²×D₃	—	4	2	no	—
30	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
31	C₂×C₃×C₄	#41	1	3	yes	—
32	C₂×Dic₃	—	2	3	no	—
33	C₄×D₃	—	4	3	no	—
34	C₄×D₃	—	4	1	no	—
35	C₃×C₈	—	2	6	no	—
36	C₃⋊C₈	—	4	3	no	—
37	C₂⁴	—	3	1	no	—
38	C₂²×C₄	—	3	1	no	—
39	C₂²×C₄	—	6	3	no	—
40	C₂²×C₄	—	6	3	no	—
41	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
42	M₁₆	—	2	6	no	—
43	M₁₆	—	6	6	no	—
44	C₂×C₈	—	6	3	no	—
45	C₂×C₈	D₁₂	1	6	yes	—
46	C₂²×C₃	—	4	1	no	—
47	C₂²×C₃	C₂²⋊Q₈	1	3	yes	—
48	C₂×D₃	—	2	6	no	—
49	C₂×D₃	—	4	3	no	—
50	C₂×D₃	—	4	1	no	—
51	C₃×C₄	#123	1	3	yes	—
52	C₃×C₄	#127	1	1	yes	characteristic
53	Dic₃	—	4	1	no	—
54	C₂³	—	6	3	no	—
55	C₂³	—	12	2	no	—
56	C₂³	D₃×Q₈	1	1	yes	characteristic
57	C₂×C₄	—	6	3	no	—
58	C₂×C₄	—	12	1	no	—
59	C₂×C₄	—	12	3	no	—
60	C₂×C₄	C₂×D₁₂	1	3	yes	—
61	C₂×C₄	#237	1	3	yes	—
62	C₈	—	2	6	no	—
63	C₈	—	12	3	no	—
64	C₂×C₃	—	2	3	no	—
65	C₂×C₃	#399	1	1	yes	characteristic
66	D₃	—	4	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	6	6	no	—
69	C₂²	—	12	3	no	—
70	C₂²	—	12	1	no	—
71	C₂²	#897	1	3	yes	—
72	C₄	—	12	1	no	—
73	C₄	#874	1	1	yes	characteristic
74	C₄	#886	1	3	yes	—
75	C₃	#2091	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	3	no	—
77	C₂	—	12	2	no	—
78	C₂	#4677	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#874
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#338	C₂
socle	C₂×C₃	#399
prime radical	C₂³×D₃	Q₈
hypercenter	#338	D₃
hypocenter	C₃	#2091
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
2. #5125	C₂	semidirect
3. C₃×#338	C₂	semidirect
12. #338	D₃	semidirect
75. C₃	#2091	semidirect

Maximal subgroups

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

subgroup	index
#5126	2
#5125	2
C₃×#338	2
#2091	3