Subgroups — Q₈×#14519852 #30624360

Order 15616.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462235	C₂	1	6	yes	maximal, complemented
2	Q₈×#11758187	C₂	1	2	yes	maximal, complemented
3	#21462152	C₂	1	3	yes	maximal
4	C₄×#14519852	C₂	1	3	yes	maximal
5	C₈×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519852	C₂²	1	3	yes	—
7	C₂×#14519852	C₂²	1	1	yes	characteristic
8	#16840683	C₂²	1	6	yes	—
9	C₄⋊#11758187	C₂²	1	6	yes	—
10	#16840651	C₂²	1	6	yes	—
11	C₄×#11758187	C₂²	1	6	yes	—
12	C₄×Q₈×D₆₁	C₄	1	1	yes	characteristic
13	Q₈×#92418	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₂₂	C₄	1	3	yes	—
15	#16839886	C₂²	1	3	yes	—
16	C₄⋊C₈×D₆₁	C₄	1	3	yes	—
17	C₄×C₈×D₆₁	C₂²	1	3	yes	—
18	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	#14519852	—	2	6	no	—
20	#14519852	Q₈	1	4	yes	direct product factor
21	C₂×#11758187	C₂³	1	6	yes	—
22	C₂×#11758187	C₂³	1	2	yes	—
23	C₂×Q₈×D₆₁	C₈	1	1	yes	characteristic
24	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
25	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
26	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
27	C₄×Dic₁₂₂	C₂×C₄	1	3	yes	—
28	#14519742	C₈	1	3	yes	—
29	C₄⋊C₄×D₆₁	C₈	1	3	yes	—
30	C₄²×D₆₁	C₂×C₄	1	3	yes	—
31	C₄⋊#92418	C₂×C₄	1	3	yes	—
32	#14519706	C₂×C₄	1	6	yes	—
33	C₄×#92418	C₂³	1	3	yes	—
34	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
35	C₈×Dic₆₁	C₂³	1	3	yes	—
36	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
39	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
40	Q₈×D₆₁	—	2	4	no	—
41	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₂₂	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
48	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
49	C₄⋊Dic₆₁	C₂×C₈	1	3	yes	—
50	#11758196	C₂×C₈	1	6	yes	—
51	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
52	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
53	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
54	C₂×#92418	C₂²×C₄	1	3	yes	—
55	C₂×#92418	C₂⁴	1	1	yes	characteristic
56	C₈×D₆₁	—	2	6	no	—
57	C₈×D₆₁	C₂×Q₈	1	2	yes	—
58	C₈×D₆₁	#41	1	2	yes	—
59	#11758187	—	2	6	no	—
60	#11758187	C₂×Q₈	1	4	yes	complemented
61	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₆₁	M₃₂	1	4	yes	complemented
63	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
64	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
67	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
68	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
69	C₄×D₆₁	—	2	6	no	—
70	C₄×D₆₁	—	4	3	no	—
71	C₄×D₆₁	C₄×Q₈	1	2	yes	—
72	C₄×D₆₁	#131	1	2	yes	—
73	Dic₁₂₂	—	2	6	no	—
74	C₈×C₆₁	—	2	3	no	—
75	C₈×C₆₁	C₄×Q₈	1	2	yes	—
76	#92418	—	2	3	no	—
77	#92418	C₂²×Q₈	1	1	yes	characteristic
78	#92418	C₂×#41	1	1	yes	characteristic
79	Q₈×M₃₂	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
81	C₂×D₆₁	—	2	1	no	—
82	C₂×D₆₁	—	2	1	no	—
83	C₂×D₆₁	C₈×Q₈	1	2	yes	—
84	C₄×C₆₁	—	2	3	no	—
85	C₄×C₆₁	C₂×M₃₂	1	6	yes	—
86	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
88	Dic₆₁	—	2	3	no	—
89	Dic₆₁	—	2	3	no	—
90	Dic₆₁	C₈×Q₈	1	2	yes	—
91	C₂×C₈×Q₈	—	61	1	no	—
92	#2183	—	61	6	no	—
93	Q₈×C₁₆	—	61	2	no	—
94	#2150	—	61	3	no	—
95	C₄×M₃₂	—	61	3	no	—
96	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
97	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
98	C₂×C₆₁	#3407	1	1	yes	characteristic
99	D₆₁	—	2	2	no	—
100	C₂×C₄×Q₈	—	61	1	no	—
101	C₂×M₃₂	—	61	1	no	—
102	C₂×M₃₂	—	61	3	no	—
103	C₈×Q₈	—	61	1	no	—
104	C₈×Q₈	—	61	3	no	—
105	C₈×Q₈	—	61	3	no	—
106	C₈×Q₈	#6923	1	1	yes	characteristic
107	C₂×C₄⋊C₈	—	61	3	no	—
108	C₂×C₄×C₈	—	61	3	no	—
109	C₄⋊C₁₆	—	61	6	no	—
110	C₄⋊C₁₆	—	61	6	no	—
111	#346	—	61	6	no	—
112	C₄×C₁₆	—	61	6	no	—
113	C₆₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	61	1	no	—
115	C₂²×C₈	—	61	1	no	—
116	C₂²×C₈	—	61	3	no	—
117	C₄×Q₈	—	61	1	no	—
118	C₄×Q₈	—	61	3	no	—
119	C₄×Q₈	—	61	3	no	—
120	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	61	3	no	—
122	C₂×C₄²	—	61	3	no	—
123	M₃₂	—	61	4	no	—
124	M₃₂	—	122	6	no	—
125	C₂×C₁₆	—	61	2	no	—
126	C₂×C₁₆	—	61	6	no	—
127	C₄⋊C₈	—	61	6	no	—
128	C₄⋊C₈	—	61	3	no	—
129	C₄⋊C₈	C₂×#6923	1	3	yes	—
130	C₄×C₈	—	61	3	no	—
131	C₄×C₈	—	61	3	no	—
132	C₄×C₈	—	61	3	no	—
133	C₄×C₈	C₂×#6923	1	3	yes	—
134	C₂×Q₈	—	61	3	no	—
135	C₂×Q₈	—	122	4	no	—
136	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	—	61	1	no	—
138	C₂²×C₄	—	61	3	no	—
139	C₂²×C₄	—	61	3	no	—
140	C₂×C₈	—	61	1	no	—
141	C₂×C₈	—	61	2	no	—
142	C₂×C₈	—	61	2	no	—
143	C₂×C₈	—	61	3	no	—
144	C₂×C₈	—	122	6	no	—
145	C₂×C₈	C₂²×#6923	1	3	yes	—
146	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
147	C₄⋊C₄	—	61	6	no	—
148	C₄⋊C₄	—	61	3	no	—
149	C₄⋊C₄	D₆₁⋊C₈	1	3	yes	—
150	C₄²	—	61	3	no	—
151	C₄²	—	61	3	no	—
152	C₄²	—	61	3	no	—
153	C₄²	C₂²×#6923	1	3	yes	—
154	C₁₆	—	61	4	no	—
155	C₁₆	—	122	6	no	—
156	C₂³	—	61	1	no	—
157	Q₈	—	122	6	no	—
158	Q₈	#14519852	1	4	yes	direct product factor
159	C₂×C₄	—	61	1	no	—
160	C₂×C₄	—	61	2	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	61	3	no	—
163	C₂×C₄	—	61	3	no	—
164	C₂×C₄	—	122	6	no	—
165	C₂×C₄	—	244	3	no	—
166	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
167	C₂×C₄	C₂×D₆₁⋊C₈	1	3	yes	—
168	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	61	1	no	—
171	C₈	—	61	1	no	—
172	C₈	—	122	3	no	—
173	C₈	Q₈×#6923	1	2	yes	—
174	C₂²	—	61	2	no	—
175	C₂²	—	122	1	no	—
176	C₂²	—	122	1	no	—
177	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	61	2	no	—
180	C₄	—	122	3	no	—
181	C₄	—	122	3	no	—
182	C₄	C₂×#14519852	1	6	yes	—
183	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
184	C₄	D₆₁⋊#131	1	1	yes	characteristic
185	C₂	—	122	2	no	—
186	C₂	Q₈×D₆₁⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×#14519852	1	1	yes	characteristic
188	C₂	#21462787	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₈×Q₈×C₆₁	C₄
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #21462235	C₂	semidirect
2. Q₈×#11758187	C₂	semidirect
20. #14519852	Q₈	direct (N × H)
60. #11758187	C₂×Q₈	semidirect
62. Q₈×C₆₁	M₃₂	semidirect
113. C₆₁	Q₈×M₃₂	semidirect
158. Q₈	#14519852	direct (N × H)

Maximal subgroups

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

subgroup	index
#21462235	2
Q₈×#11758187	2
#21462152	2
C₄×#14519852	2
C₈×Q₈×D₆₁	2
Q₈×M₃₂	61

Subgroups — Q8×#14519852 #30624360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14519852 #30624360