Subgroups — #69360

Order 384.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#798	C₂	1	1	yes	maximal, characteristic
2	D₃×#108	C₂	1	2	yes	maximal
3	C₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	#4506	C₂	1	1	yes	maximal, characteristic, complemented
5	#4493	C₂	1	2	yes	maximal, complemented
6	#2161	—	3	1	no	maximal, Sylow
7	D₃×M₁₆	—	2	2	no	—
8	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#108	C₂²	1	2	yes	—
11	#822	—	2	2	no	—
12	#819	C₂²	1	2	yes	—
13	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	#798	C₄	1	4	yes	—
15	C₂×M₃₂	—	3	1	no	—
16	C₂×#108	—	3	2	no	—
17	#368	—	3	1	no	—
18	#368	—	3	2	no	—
19	#368	D₃	1	1	yes	characteristic, complemented
20	C₂×C₄×D₃	D₄	1	1	yes	characteristic
21	C₃×M₁₆	—	2	2	no	—
22	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
23	#208	—	2	2	no	—
24	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
25	ΓL(1,𝔽₂₅)	—	4	2	no	—
26	D₃×C₈	—	4	2	no	—
27	D₃×C₈	C₂×C₄	1	2	yes	—
28	D₃×C₈	Q₈	1	2	yes	—
29	C₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₃⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	—	6	2	no	—
32	C₂²×C₈	—	3	1	no	—
33	M₃₂	—	3	4	no	—
34	C₂×C₁₆	—	3	1	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#108	—	3	2	no	—
37	#108	—	6	2	no	—
38	#108	C₂×D₃	1	2	yes	—
39	C₂²×D₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃	Q₁₆	1	1	yes	characteristic
42	C₄×D₃	C₄⋊C₄	1	2	yes	—
43	C₄×D₃	D₈	1	2	yes	—
44	C₃×C₈	—	4	2	no	—
45	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₃⋊C₈	—	4	2	no	—
48	C₃⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄	—	3	1	no	—
50	M₁₆	—	2	2	no	—
51	M₁₆	—	6	2	no	—
52	M₁₆	—	12	2	no	—
53	C₂×C₈	—	3	1	no	—
54	C₂×C₈	—	3	2	no	—
55	C₂×C₈	—	3	2	no	—
56	C₂×C₈	—	12	2	no	—
57	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
58	C₁₆	—	3	2	no	—
59	C₁₆	C₄×D₃	1	2	yes	—
60	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×D₃	—	4	1	no	—
62	C₂×D₃	#107	1	2	yes	—
63	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₄	C₂×D₈	1	1	yes	characteristic
65	Dic₃	#107	1	2	yes	—
66	C₂³	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	D₃×D₄	1	1	yes	characteristic
71	C₈	—	3	2	no	—
72	C₈	—	4	2	no	—
73	C₈	—	12	2	no	—
74	C₈	C₂×C₄×D₃	1	1	yes	characteristic
75	C₈	D₃×Q₈	1	1	yes	characteristic
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	C₂×#107	1	1	yes	characteristic
78	D₃	—	2	2	no	—
79	C₂²	—	3	2	no	—
80	C₂²	—	12	1	no	—
81	C₂²	D₃×Q₁₆	1	1	yes	characteristic
82	C₄	—	3	2	no	—
83	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
84	C₄	D₃×D₈	1	1	yes	characteristic
85	C₃	#2161	1	1	yes	characteristic, complemented, Sylow
86	C₂	—	2	1	no	—
87	C₂	—	6	2	no	—
88	C₂	D₃×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#368	C₂
socle	C₂×C₃	C₂×#107
prime radical	C₂²×D₃	Q₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#368	D₃
hypocenter	C₃	#2161
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
3. C₃×#368	C₂	semidirect
4. #4506	C₂	semidirect
5. #4493	C₂	semidirect
19. #368	D₃	semidirect
85. C₃	#2161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#798	2
D₃×#108	2
C₃×#368	2
#4506	2
#4493	2
#2161	3