Subgroups — #69792

Order 384.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#819	C₂	1	4	yes	maximal
3	#4924	C₂	1	2	yes	maximal
4	#1848	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	C₂²	1	4	yes	—
8	C₃×C₄×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₈	C₄	1	2	yes	—
10	#819	—	2	8	no	—
11	C₄⋊(C₃⋊C₈)	—	2	4	no	—
12	C₂×#108	—	3	4	no	—
13	#423	—	3	2	no	—
14	C₂×C₄×C₈	D₃	1	1	yes	characteristic
15	C₂²×C₃×C₄	Q₈	1	2	yes	—
16	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
17	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
18	C₂×C₃×C₈	D₄	1	4	yes	—
19	C₂×C₃×C₈	Q₈	1	4	yes	—
20	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄²	D₄	1	2	yes	—
23	#208	—	2	8	no	—
24	C₂×C₃⋊C₈	—	2	4	no	—
25	C₂×M₁₆	—	3	4	no	—
26	C₂²×C₈	C₂×D₃	1	2	yes	—
27	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
28	#108	—	6	8	no	—
29	C₄⋊C₈	—	6	4	no	—
30	C₄×C₈	Dic₃	1	2	yes	—
31	C₄×C₈	Dic₃	1	2	yes	—
32	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
35	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
38	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
40	C₃×C₈	C₄⋊C₄	1	8	yes	—
41	C₃⋊C₈	—	4	4	no	—
42	C₂²×C₄	Dic₆	1	2	yes	—
43	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
44	M₁₆	—	6	8	no	—
45	C₂×C₈	—	6	4	no	—
46	C₂×C₈	Dic₆	1	4	yes	—
47	C₂×C₈	D₁₂	1	4	yes	—
48	C₂×C₈	C₂×Dic₃	1	4	yes	—
49	C₄²	D₁₂	1	2	yes	—
50	C₄²	C₂×Dic₃	1	1	yes	characteristic
51	C₄²	C₂×Dic₃	1	1	yes	characteristic
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	#108	1	4	yes	complemented
57	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
58	C₃×C₄	#127	1	1	yes	characteristic
59	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
60	C₂³	C₂×Dic₆	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
63	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
64	C₂×C₄	C₂×Dic₆	1	2	yes	—
65	C₂×C₄	C₂×D₁₂	1	2	yes	—
66	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
67	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
68	C₈	—	12	4	no	—
69	C₈	C₄⋊Dic₃	1	8	yes	—
70	C₂×C₃	—	2	2	no	—
71	C₂×C₃	#384	1	1	yes	characteristic
72	C₂×C₃	C₂×#108	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	#869	1	2	yes	—
75	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	#819	1	4	yes	complemented
78	C₄	#869	1	1	yes	characteristic
79	C₄	#874	1	1	yes	characteristic
80	C₄	C₂×C₄⋊Dic₃	1	2	yes	—
81	C₃	#1848	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	2	2	no	—
83	C₂	#4935	1	1	yes	characteristic
84	C₂	C₂×#819	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1848
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
56. C₃×C₄	#108	semidirect
77. C₄	#819	semidirect
81. C₃	#1848	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
C₂×#819	2
#4924	2
#1848	3