Subgroups — #67518

Order 384.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4679	C₂	1	1	yes	maximal, characteristic
2	C₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#1338	—	3	1	no	maximal, Sylow
5	C₄×Dic₆	C₄	1	1	yes	characteristic
6	C₂×C₃×C₁₆	—	2	1	no	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#815	—	2	1	no	—
9	Dic₃⋊C₈	—	2	1	no	—
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
12	#446	—	3	1	no	—
13	C₄⋊C₁₆	—	3	1	no	—
14	#346	D₃	1	1	yes	characteristic
15	C₂×Dic₆	C₈	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Dic₃	C₈	1	1	yes	characteristic
20	Dic₃⋊C₄	—	2	1	no	—
21	C₄×Dic₃	—	2	1	no	—
22	C₂×C₃⋊C₈	—	2	1	no	—
23	C₃×C₁₆	—	2	2	no	—
24	C₃⋊C₁₆	—	4	1	no	—
25	C₄×Q₈	—	3	1	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	6	1	no	—
28	C₄⋊C₈	—	3	1	no	—
29	C₄⋊C₈	—	6	1	no	—
30	#97	—	6	1	no	—
31	C₄×C₈	C₂×D₃	1	1	yes	characteristic
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
34	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
35	C₂×Dic₃	—	2	1	no	—
36	C₂×Dic₃	—	2	1	no	—
37	Dic₆	—	4	1	no	—
38	C₃×C₈	—	2	1	no	—
39	C₃×C₈	SD₁₆	1	1	yes	characteristic
40	C₃×C₈	Q₁₆	1	1	yes	characteristic
41	C₃⋊C₈	—	4	1	no	—
42	C₂×Q₈	—	3	1	no	—
43	C₂×C₈	—	6	1	no	—
44	C₂×C₈	D₁₂	1	1	yes	characteristic
45	C₂×C₈	#67	1	1	yes	characteristic
46	C₄⋊C₄	—	3	1	no	—
47	C₄⋊C₄	—	6	1	no	—
48	C₄²	—	6	1	no	—
49	C₄²	C₄×D₃	1	1	yes	characteristic
50	C₁₆	—	2	2	no	—
51	C₁₆	—	12	1	no	—
52	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₃×C₄	—	2	1	no	—
54	C₃×C₄	—	2	1	no	—
55	C₃×C₄	#103	1	1	yes	characteristic
56	C₃×C₄	#104	1	1	yes	characteristic
57	Dic₃	—	4	1	no	—
58	Dic₃	—	4	1	no	—
59	Q₈	—	12	1	no	—
60	C₂×C₄	—	6	1	no	—
61	C₂×C₄	—	6	1	no	—
62	C₂×C₄	D₃×C₈	1	1	yes	characteristic
63	C₂×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
64	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	12	1	no	—
67	C₈	#204	1	1	yes	characteristic
68	C₈	Dic₁₂	1	1	yes	characteristic
69	C₂×C₃	Q₈⋊C₈	1	1	yes	characteristic
70	C₂×C₃	#349	1	1	yes	characteristic
71	C₂×C₃	#350	1	1	yes	characteristic
72	C₂²	(C₂×D₃)⋊C₈	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	12	1	no	—
76	C₄	—	12	1	no	—
77	C₄	#805	1	1	yes	characteristic
78	C₄	#816	1	1	yes	characteristic
79	C₃	#1338	1	1	yes	characteristic, complemented, Sylow
80	C₂	#4456	1	1	yes	characteristic
81	C₂	#4514	1	1	yes	characteristic
82	C₂	#4515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#346	C₂
socle, prime radical	C₂²×C₃	C₂²⋊C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#346	D₃
hypocenter	C₃	#1338
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
79. C₃	#1338	semidirect

Maximal subgroups

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

subgroup	index
#4679	2
C₃×#346	2
C₄⋊(C₃⋊C₁₆)	2
#1338	3