Subgroups — #69339

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	#4872	C₂	1	2	yes	maximal, complemented
3	C₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#4502	C₂	1	1	yes	maximal, characteristic, complemented
5	#4491	C₂	1	2	yes	maximal, complemented
6	#2177	—	3	1	no	maximal, Sylow
7	C₂×D₂₄	C₂²	1	2	yes	—
8	#893	—	2	2	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#107	C₂²	1	2	yes	—
11	C₈×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#809	—	2	2	no	—
14	D₄₈	C₄	1	4	yes	complemented
15	C₂×D₁₆	—	3	1	no	—
16	C₄×D₈	—	3	2	no	—
17	#367	D₃	1	1	yes	characteristic, complemented
18	#357	—	3	2	no	—
19	#346	—	3	1	no	—
20	C₂×D₁₂	—	2	2	no	—
21	C₂×C₄×D₃	—	4	2	no	—
22	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄⋊C₄	—	2	2	no	—
24	(C₂×D₃)⋊C₄	—	4	2	no	—
25	C₄×Dic₃	D₄	1	1	yes	characteristic
26	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
27	D₂₄	—	2	2	no	—
28	D₂₄	C₂×C₄	1	4	yes	complemented
29	C₃×C₁₆	C₂×C₄	1	2	yes	complemented
30	C₃⋊C₁₆	—	2	1	no	—
31	C₂×D₈	—	3	2	no	—
32	C₄×D₄	—	6	2	no	—
33	D₁₆	—	3	4	no	—
34	C₂×C₁₆	—	3	1	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	C₂×D₃	1	2	yes	—
37	#102	—	6	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₂²×D₃	—	4	2	no	—
40	C₂×C₃×C₄	—	4	2	no	—
41	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₃	D₈	1	2	yes	—
43	D₁₂	—	2	4	no	—
44	D₁₂	—	4	2	no	—
45	C₄×D₃	—	8	4	no	—
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	—	6	2	no	—
50	C₂²×C₄	—	12	2	no	—
51	D₈	—	3	4	no	—
52	D₈	—	6	2	no	—
53	C₂×C₈	—	3	1	no	—
54	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₂²⋊C₄	—	12	2	no	—
57	C₄²	—	3	1	no	—
58	C₁₆	—	6	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	—	8	2	no	—
65	C₃×C₄	C₄×D₄	1	1	yes	characteristic
66	C₃×C₄	#135	1	1	yes	characteristic
67	Dic₃	—	2	2	no	—
68	C₂³	—	12	2	no	—
69	D₄	—	6	4	no	—
70	D₄	—	12	2	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	24	4	no	—
74	C₂×C₄	D₃×D₄	1	1	yes	characteristic
75	C₈	—	6	1	no	—
76	C₈	C₂×C₄×D₃	1	1	yes	characteristic
77	C₈	#239	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	D₃×D₈	1	1	yes	characteristic
85	C₄	—	6	2	no	—
86	C₄	—	8	2	no	—
87	C₄	#893	1	1	yes	characteristic
88	C₄	#919	1	1	yes	characteristic
89	C₃	#2177	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	24	4	no	—
91	C₂	#4872	1	1	yes	characteristic
92	C₂	#4914	1	2	yes	—

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#367	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×D₄₈	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#367	D₃
hypocenter	C₃	#2177
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
2. #4872	C₂	semidirect
3. C₃×#367	C₂	semidirect
4. #4502	C₂	semidirect
5. #4491	C₂	semidirect
14. D₄₈	C₄	semidirect
17. #367	D₃	semidirect
28. D₂₄	C₂×C₄	semidirect
29. 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
#4872	2
C₃×#367	2
#4502	2
#4491	2
#2177	3