Subgroups — #67516

Order 384.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	#4457	C₂	1	1	yes	maximal, characteristic, complemented
4	#1333	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	C₄	1	1	yes	characteristic
6	C₄×D₁₂	—	2	1	no	—
7	C₂×C₃×C₁₆	—	2	1	no	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	#821	—	2	1	no	—
10	#818	C₄	1	1	yes	characteristic
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₄×D₈	—	3	1	no	—
13	#346	D₃	1	1	yes	characteristic, complemented
14	#334	—	3	1	no	—
15	C₂×D₁₂	—	2	1	no	—
16	C₂×C₄×D₃	—	4	1	no	—
17	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄²	D₄	1	1	yes	characteristic
20	(C₂×D₃)⋊C₄	—	4	1	no	—
21	C₄⋊Dic₃	—	2	1	no	—
22	C₂×C₃⋊C₈	—	4	1	no	—
23	D₂₄	—	2	1	no	—
24	D₂₄	C₈	1	2	yes	complemented
25	C₃×C₁₆	—	2	2	no	—
26	C₂×D₈	—	3	1	no	—
27	C₄×D₄	—	6	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	#107	—	3	1	no	—
30	C₄⋊C₈	—	6	1	no	—
31	#102	—	6	1	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂²×D₃	—	4	1	no	—
34	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₄	D₈	1	1	yes	characteristic
36	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
37	C₂×Dic₃	—	4	1	no	—
38	D₁₂	—	2	2	no	—
39	D₁₂	—	4	1	no	—
40	C₄×D₃	—	8	2	no	—
41	C₃×C₈	—	2	1	no	—
42	C₃×C₈	C₂×C₈	1	1	yes	characteristic
43	C₃×C₈	M₁₆	1	1	yes	characteristic
44	C₃⋊C₈	—	8	1	no	—
45	C₂×D₄	—	6	1	no	—
46	C₂²×C₄	—	12	1	no	—
47	D₈	—	3	2	no	—
48	D₈	—	6	1	no	—
49	C₂×C₈	—	12	1	no	—
50	C₂×C₈	C₄×D₃	1	1	yes	characteristic
51	C₂×C₈	#67	1	1	yes	characteristic
52	C₄⋊C₄	—	6	1	no	—
53	C₂²⋊C₄	—	12	1	no	—
54	C₄²	D₁₂	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₂²×C₃	#102	1	1	yes	characteristic
57	C₂×D₃	—	4	2	no	—
58	C₂×D₃	—	8	1	no	—
59	C₂×D₃	—	8	1	no	—
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
63	C₃×C₄	#104	1	1	yes	characteristic
64	Dic₃	—	8	1	no	—
65	C₂³	—	12	1	no	—
66	D₄	—	6	2	no	—
67	D₄	—	12	1	no	—
68	C₂×C₄	—	12	1	no	—
69	C₂×C₄	—	24	2	no	—
70	C₂×C₄	#204	1	1	yes	characteristic
71	C₂×C₄	D₂₄	1	1	yes	characteristic
72	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	24	1	no	—
75	C₈	D₃×C₈	1	1	yes	characteristic
76	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
77	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
78	C₂×C₃	#360	1	1	yes	characteristic
79	C₂×C₃	#361	1	1	yes	characteristic
80	D₃	—	8	2	no	—
81	C₂²	—	12	2	no	—
82	C₂²	—	24	1	no	—
83	C₂²	—	24	1	no	—
84	C₂²	#821	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	24	1	no	—
88	C₄	#805	1	1	yes	characteristic
89	C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
90	C₃	#1333	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	24	2	no	—
92	C₂	#4459	1	1	yes	characteristic
93	C₂	#4516	1	1	yes	characteristic
94	C₂	#4518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#821
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#346	C₂
socle	C₂²×C₃	#102
prime radical	C₂×D₂₄	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#346	D₃
hypocenter	C₃	#1333
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₃×#346	C₂	semidirect
3. #4457	C₂	semidirect
13. #346	D₃	semidirect
24. D₂₄	C₈	semidirect
90. C₃	#1333	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₄	2
C₃×#346	2
#4457	2
#1333	3