Subgroups — #69080

Order 384.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4695	C₂	1	1	yes	maximal, characteristic
2	#4683	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	#4509	C₂	1	4	yes	maximal, complemented
5	#2244	—	3	1	no	maximal, Sylow
6	C₂×D₂₄	—	2	1	no	—
7	C₂×D₂₄	C₂²	1	2	yes	—
8	#874	—	2	1	no	—
9	#870	—	2	1	no	—
10	C₂×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#818	C₂²	1	2	yes	—
13	#817	—	2	1	no	—
14	#499	—	3	1	no	—
15	#493	—	3	1	no	—
16	#357	—	3	4	no	—
17	#346	D₃	1	1	yes	characteristic, complemented
18	C₂×D₁₂	—	2	2	no	—
19	C₂×D₁₂	—	4	2	no	—
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄²	D₄	1	1	yes	characteristic
23	C₄⋊Dic₃	—	2	2	no	—
24	Dic₃⋊C₄	—	4	2	no	—
25	D₂₄	—	2	4	no	—
26	D₂₄	—	4	2	no	—
27	C₃×C₁₆	—	2	2	no	—
28	C₂×D₈	—	3	2	no	—
29	C₂×D₈	—	6	1	no	—
30	#127	—	6	1	no	—
31	#125	—	6	1	no	—
32	C₂×C₁₆	C₂×D₃	1	2	yes	—
33	#107	—	3	2	no	—
34	#106	—	6	1	no	—
35	C₄×C₈	C₂×D₃	1	1	yes	characteristic
36	C₂²×D₃	—	4	2	no	—
37	C₂×C₃×C₄	D₈	1	2	yes	—
38	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₃	—	4	2	no	—
40	D₁₂	—	4	2	no	—
41	D₁₂	—	4	2	no	—
42	D₁₂	—	8	4	no	—
43	C₃×C₈	SD₁₆	1	2	yes	—
44	C₃×C₈	#41	1	2	yes	—
45	C₂×D₄	—	6	2	no	—
46	C₂×D₄	—	12	2	no	—
47	D₈	—	6	4	no	—
48	D₈	—	12	2	no	—
49	C₂×C₈	D₁₂	1	1	yes	characteristic
50	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
51	C₄⋊C₄	—	6	2	no	—
52	C₄⋊C₄	—	12	2	no	—
53	C₄²	D₁₂	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
56	C₂×D₃	—	8	2	no	—
57	C₂×D₃	—	8	4	no	—
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	#124	1	1	yes	characteristic
60	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
61	Dic₃	—	8	2	no	—
62	C₂³	—	12	2	no	—
63	D₄	—	12	2	no	—
64	D₄	—	12	2	no	—
65	D₄	—	24	4	no	—
66	C₂×C₄	—	12	2	no	—
67	C₂×C₄	D₂₄	1	2	yes	—
68	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
69	C₈	#204	1	2	yes	—
70	C₈	#235	1	2	yes	—
71	C₂×C₃	#486	1	1	yes	characteristic
72	C₂×C₃	#509	1	2	yes	—
73	D₃	—	16	2	no	—
74	C₂²	—	24	4	no	—
75	C₂²	—	24	2	no	—
76	C₂²	C₂×D₂₄	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	24	2	no	—
79	C₄	#875	1	1	yes	characteristic
80	C₄	C₂×#204	1	1	yes	characteristic
81	C₃	#2244	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	48	2	no	—
83	C₂	#4694	1	1	yes	characteristic
84	C₂	#4908	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#346	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	#4695	C₂
hypercenter	#346	D₃
hypocenter	C₃	#2244
LCS	G ▷ C₂×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. #4683	C₂	semidirect
3. C₃×#346	C₂	semidirect
4. #4509	C₂	semidirect
17. #346	D₃	semidirect
81. C₃	#2244	semidirect

Maximal subgroups

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

subgroup	index
#4695	2
#4683	2
C₃×#346	2
#4509	2
#2244	3