Subgroups — #69078

Order 384.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₂₄	C₂	1	2	yes	maximal, complemented
3	C₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	#4509	C₂	1	2	yes	maximal, complemented
5	#4505	C₂	1	1	yes	maximal, characteristic, complemented
6	#2177	—	3	1	no	maximal, Sylow
7	C₂×D₂₄	C₂²	1	2	yes	—
8	C₄×D₁₂	—	2	2	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#821	—	2	2	no	—
13	#818	C₂²	1	2	yes	—
14	D₄₈	C₄	1	4	yes	complemented
15	C₂×D₁₆	—	3	1	no	—
16	C₄×D₈	—	3	2	no	—
17	#367	—	3	1	no	—
18	#357	—	3	2	no	—
19	#346	D₃	1	1	yes	characteristic, complemented
20	C₂×D₁₂	—	2	2	no	—
21	C₂×C₄×D₃	—	4	2	no	—
22	C₂×C₃×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	(C₂×D₃)⋊C₄	—	4	2	no	—
26	C₄⋊Dic₃	—	2	2	no	—
27	D₂₄	—	2	2	no	—
28	D₂₄	C₂×C₄	1	4	yes	complemented
29	C₃×C₁₆	—	2	1	no	—
30	C₃×C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×D₈	—	3	2	no	—
32	C₄×D₄	—	6	2	no	—
33	D₁₆	—	3	4	no	—
34	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	—	3	2	no	—
37	#102	—	6	2	no	—
38	C₄×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂²×D₃	—	4	2	no	—
40	C₂×C₃×C₄	D₈	1	2	yes	—
41	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₃	—	4	2	no	—
43	D₁₂	—	2	4	no	—
44	D₁₂	—	4	2	no	—
45	C₄×D₃	—	8	4	no	—
46	C₃×C₈	—	2	1	no	—
47	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₈	#41	1	1	yes	characteristic
49	C₂×D₄	—	6	2	no	—
50	C₂²×C₄	—	12	2	no	—
51	D₈	—	3	4	no	—
52	D₈	—	6	2	no	—
53	C₂×C₈	D₁₂	1	1	yes	characteristic
54	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
55	C₄⋊C₄	—	6	2	no	—
56	C₂²⋊C₄	—	12	2	no	—
57	C₄²	D₁₂	1	1	yes	characteristic
58	C₁₆	—	2	1	no	—
59	C₁₆	C₄×D₃	1	2	yes	complemented
60	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
61	C₂×D₃	—	4	4	no	—
62	C₂×D₃	—	8	2	no	—
63	C₂×D₃	—	8	2	no	—
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	C₄×D₄	1	1	yes	characteristic
66	C₃×C₄	#135	1	1	yes	characteristic
67	Dic₃	—	8	2	no	—
68	C₂³	—	12	2	no	—
69	D₄	—	6	4	no	—
70	D₄	—	12	2	no	—
71	C₂×C₄	—	12	2	no	—
72	C₂×C₄	—	24	4	no	—
73	C₂×C₄	D₂₄	1	2	yes	—
74	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	C₂×C₄×D₃	1	1	yes	characteristic
77	C₈	#235	1	1	yes	characteristic
78	C₂×C₃	C₄×D₈	1	1	yes	characteristic
79	C₂×C₃	#509	1	2	yes	—
80	D₃	—	8	4	no	—
81	C₂²	—	12	4	no	—
82	C₂²	—	24	2	no	—
83	C₂²	—	24	2	no	—
84	C₂²	C₂×D₂₄	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	24	2	no	—
87	C₄	C₄×D₁₂	1	1	yes	characteristic
88	C₄	#904	1	1	yes	characteristic
89	C₃	#2177	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	24	4	no	—
91	C₂	C₄×D₂₄	1	1	yes	characteristic
92	C₂	#4908	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#346	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×D₄₈	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#346	D₃
hypocenter	C₃	#2177
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. C₄×D₂₄	C₂	semidirect
3. C₃×#346	C₂	semidirect
4. #4509	C₂	semidirect
5. #4505	C₂	semidirect
14. D₄₈	C₄	semidirect
19. #346	D₃	semidirect
28. D₂₄	C₂×C₄	semidirect
30. C₃×C₁₆	C₂×C₄	semidirect
59. C₁₆	C₄×D₃	semidirect
89. C₃	#2177	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₈	2
C₄×D₂₄	2
C₃×#346	2
#4509	2
#4505	2
#2177	3