Subgroups — #71429

Order 384.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#819	C₂	1	2	yes	maximal
3	#4506	C₂	1	4	yes	maximal, complemented
4	#2161	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₂×#208	—	2	2	no	—
7	C₃×M₃₂	C₄	1	4	yes	—
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	#819	—	2	2	no	—
10	#819	C₂²	1	4	yes	—
11	C₂×M₃₂	D₃	1	1	yes	characteristic
12	C₂×#108	—	3	2	no	—
13	#368	—	3	4	no	—
14	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
15	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	Q₈	1	2	yes	—
18	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
19	#208	—	2	4	no	—
20	#208	—	4	2	no	—
21	C₂×C₃⋊C₈	—	4	2	no	—
22	C₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₂×M₁₆	—	6	2	no	—
24	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
25	M₃₂	Dic₃	1	4	yes	—
26	C₂×C₁₆	C₂×D₃	1	2	yes	—
27	#108	—	3	4	no	—
28	#108	—	6	2	no	—
29	C₂³×C₃	Q₁₆	1	1	yes	characteristic
30	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
31	C₂×C₃×C₄	D₈	1	2	yes	—
32	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
34	C₃×C₈	C₄⋊C₄	1	2	yes	—
35	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₃⋊C₈	—	4	4	no	—
38	C₂²×C₄	D₁₂	1	1	yes	characteristic
39	M₁₆	—	6	4	no	—
40	M₁₆	—	12	2	no	—
41	C₂×C₈	—	12	2	no	—
42	C₂×C₈	Dic₆	1	2	yes	—
43	C₂×C₈	D₁₂	1	1	yes	characteristic
44	C₂×C₈	C₂×Dic₃	1	2	yes	—
45	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
46	C₁₆	C₂×Dic₃	1	4	yes	—
47	C₂²×C₃	—	4	1	no	—
48	C₂²×C₃	#107	1	2	yes	—
49	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
50	C₃×C₄	#107	1	2	yes	—
51	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₃×C₄	C₂×D₈	1	1	yes	characteristic
53	C₂³	Dic₁₂	1	1	yes	characteristic
54	C₂×C₄	D₂₄	1	2	yes	—
55	C₂×C₄	Dic₁₂	1	1	yes	characteristic
56	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
57	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
58	C₈	—	12	4	no	—
59	C₈	C₄⋊Dic₃	1	2	yes	—
60	C₈	C₂×Dic₆	1	1	yes	characteristic
61	C₈	C₂²×Dic₃	1	1	yes	characteristic
62	C₂×C₃	—	2	2	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	C₂×#107	1	1	yes	characteristic
65	C₂²	—	4	1	no	—
66	C₂²	#818	1	2	yes	—
67	C₂²	C₂×Dic₁₂	1	1	yes	characteristic
68	C₄	#818	1	2	yes	—
69	C₄	C₂×D₂₄	1	1	yes	characteristic
70	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
71	C₃	#2161	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#818	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂×C₃	C₂×#107
prime radical	C₂³×C₃	Q₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#2161
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #4506	C₂	semidirect
71. 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×C₃×M₃₂	2
C₂×#819	2
#4506	2
#2161	3