Subgroups — #71461

Order 384.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	Dic₃×D₈	C₂	1	2	yes	maximal, complemented
3	#4562	C₂	1	2	yes	maximal, complemented
4	#4505	C₂	1	1	yes	maximal, characteristic, complemented
5	#4502	C₂	1	1	yes	maximal, characteristic, complemented
6	#2177	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₈	C₂²	1	2	yes	—
8	D₄×Dic₃	—	2	2	no	—
9	C₃×D₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	#832	—	2	2	no	—
12	#818	C₂²	1	2	yes	—
13	C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₁₆	D₃	1	1	yes	characteristic
16	C₄×D₈	—	3	2	no	—
17	#367	—	3	1	no	—
18	#357	—	3	2	no	—
19	#346	—	3	1	no	—
20	C₂×C₃×D₄	—	2	2	no	—
21	C₂²×Dic₃	—	4	2	no	—
22	C₃×D₈	—	2	2	no	—
23	C₃×D₈	C₂×C₄	1	4	yes	complemented
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	#217	—	4	2	no	—
26	C₄⋊Dic₃	—	2	2	no	—
27	C₄×Dic₃	D₄	1	1	yes	characteristic
28	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
29	C₃×C₁₆	C₂×C₄	1	2	yes	complemented
30	C₃⋊C₁₆	—	2	1	no	—
31	C₂×D₈	C₂×D₃	1	2	yes	—
32	C₄×D₄	—	6	2	no	—
33	D₁₆	Dic₃	1	4	yes	complemented
34	C₂×C₁₆	—	3	1	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	—	3	2	no	—
37	#102	—	6	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₂³×C₃	—	4	2	no	—
40	C₃×D₄	—	2	4	no	—
41	C₃×D₄	—	4	2	no	—
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₃	—	4	2	no	—
44	C₂×Dic₃	—	8	4	no	—
45	C₂×Dic₃	D₈	1	2	yes	—
46	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₈	#41	1	1	yes	characteristic
48	C₃⋊C₈	—	2	1	no	—
49	C₂×D₄	—	2	2	no	—
50	C₂²×C₄	—	12	2	no	—
51	D₈	—	2	2	no	—
52	D₈	C₂×Dic₃	1	4	yes	complemented
53	C₂×C₈	—	3	1	no	—
54	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
55	C₄⋊C₄	—	6	2	no	—
56	C₂²⋊C₄	—	12	2	no	—
57	C₄²	—	3	1	no	—
58	C₁₆	—	6	1	no	—
59	C₁₆	C₂×Dic₃	1	2	yes	complemented
60	C₂²×C₃	—	4	4	no	—
61	C₂²×C₃	—	8	2	no	—
62	C₂²×C₃	—	8	2	no	—
63	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
64	C₃×C₄	C₄×D₄	1	1	yes	characteristic
65	C₃×C₄	#135	1	1	yes	characteristic
66	Dic₃	—	2	2	no	—
67	Dic₃	—	8	2	no	—
68	C₂³	—	4	2	no	—
69	D₄	—	2	4	no	—
70	D₄	—	4	2	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	12	2	no	—
73	C₂×C₄	—	24	4	no	—
74	C₂×C₄	D₃×D₄	1	1	yes	characteristic
75	C₈	—	6	1	no	—
76	C₈	#237	1	1	yes	characteristic
77	C₈	C₂²×Dic₃	1	1	yes	characteristic
78	C₂×C₃	—	8	4	no	—
79	C₂×C₃	C₄×D₈	1	1	yes	characteristic
80	C₂×C₃	#509	1	2	yes	—
81	C₂²	—	4	4	no	—
82	C₂²	—	8	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	D₃×D₈	1	1	yes	characteristic
85	C₄	—	6	2	no	—
86	C₄	—	24	2	no	—
87	C₄	#912	1	1	yes	characteristic
88	C₄	D₄×Dic₃	1	1	yes	characteristic
89	C₃	#2177	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	8	4	no	—
91	C₂	#4911	1	2	yes	—
92	C₂	Dic₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₁₆	C₂
socle	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×D₁₆	D₃
hypocenter	C₃	#2177
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
2. Dic₃×D₈	C₂	semidirect
3. #4562	C₂	semidirect
4. #4505	C₂	semidirect
5. #4502	C₂	semidirect
9. C₃×D₁₆	C₄	semidirect
23. C₃×D₈	C₂×C₄	semidirect
29. C₃×C₁₆	C₂×C₄	semidirect
33. D₁₆	Dic₃	semidirect
52. D₈	C₂×Dic₃	semidirect
59. C₁₆	C₂×Dic₃	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×C₃×D₁₆	2
Dic₃×D₈	2
#4562	2
#4505	2
#4502	2
#2177	3