Subgroups — Q₈×#14519716 #29833523

Order 15616.

530 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	3	yes	maximal, complemented
3	#21447724	C₂	1	3	yes	maximal, complemented
4	Q₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447573	C₂	1	3	yes	maximal
6	C₄×#14519716	C₂	1	3	yes	maximal
7	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₆₁	C₄	1	1	yes	characteristic
9	C₂×#14519716	C₂²	1	3	yes	—
10	C₂×#14519716	C₂²	1	1	yes	characteristic
11	Q₈×#92418	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₂	C₂²	1	3	yes	—
13	#16839886	C₄	1	3	yes	—
14	C₄⋊C₈×D₆₁	C₄	1	3	yes	—
15	C₄×C₈×D₆₁	C₂²	1	3	yes	—
16	C₄⋊#11758185	C₂²	1	3	yes	—
17	#16839580	C₂²	1	6	yes	—
18	#16839537	C₂²	1	3	yes	—
19	#16839537	C₂²	1	3	yes	—
20	C₄×#11758185	C₂²	1	3	yes	—
21	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×C₄⋊C₁₆	C₂²	1	3	yes	—
23	C₄×C₁₆×C₆₁	C₂²	1	3	yes	—
24	C₂×Q₈×D₆₁	C₈	1	1	yes	characteristic
25	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
27	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
28	C₄×Dic₁₂₂	C₂×C₄	1	3	yes	—
29	#14519742	C₈	1	3	yes	—
30	C₄⋊C₄×D₆₁	C₈	1	3	yes	—
31	C₄²×D₆₁	C₂×C₄	1	3	yes	—
32	#14519716	—	2	6	no	—
33	#14519716	Q₈	1	4	yes	direct product factor
34	C₂×#11758185	C₂³	1	3	yes	—
35	C₂×#11758185	C₂³	1	1	yes	characteristic
36	C₄⋊#92418	C₂×C₄	1	3	yes	—
37	#14519706	C₂×C₄	1	6	yes	—
38	C₄×#92418	C₂×C₄	1	3	yes	—
39	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
40	C₈×Dic₆₁	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₁	C₂³	1	3	yes	—
43	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
45	C₄×C₈×C₆₁	C₂³	1	3	yes	—
46	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
47	Q₈×D₆₁	—	2	4	no	—
48	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₆₁	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₁₂₂	C₂×C₈	1	3	yes	—
52	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
55	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
56	C₄⋊Dic₆₁	C₂×C₈	1	3	yes	—
57	#11758196	C₂×C₈	1	6	yes	—
58	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
59	C₄×Dic₆₁	C₂×C₈	1	3	yes	—
60	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
61	C₂×#92418	C₂²×C₄	1	3	yes	—
62	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₆₁	—	2	6	no	—
64	C₈×D₆₁	C₂×Q₈	1	2	yes	—
65	C₈×D₆₁	#41	1	2	yes	—
66	C₁₆×C₆₁	—	2	3	no	—
67	C₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
68	#11758185	—	2	3	no	—
69	#11758185	C₂×Q₈	1	2	yes	complemented
70	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₆₁	M₃₂	1	4	yes	complemented
72	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
76	C₂×Dic₆₁	C₂²×C₈	1	3	yes	—
77	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
78	C₄×D₆₁	—	2	6	no	—
79	C₄×D₆₁	—	4	3	no	—
80	C₄×D₆₁	C₄×Q₈	1	2	yes	—
81	C₄×D₆₁	#131	1	2	yes	—
82	Dic₁₂₂	—	2	6	no	—
83	C₈×C₆₁	—	2	3	no	—
84	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
86	#92418	—	2	3	no	—
87	#92418	C₄×Q₈	1	2	yes	—
88	Q₈×M₃₂	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
90	C₂×D₆₁	—	2	1	no	—
91	C₂×D₆₁	—	2	1	no	—
92	C₂×D₆₁	C₈×Q₈	1	2	yes	—
93	C₄×C₆₁	—	2	3	no	—
94	C₄×C₆₁	C₂×M₃₂	1	6	yes	—
95	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
97	Dic₆₁	—	2	3	no	—
98	Dic₆₁	—	2	3	no	—
99	Dic₆₁	C₈×Q₈	1	2	yes	—
100	C₂×C₈×Q₈	—	61	1	no	—
101	#2183	—	61	3	no	—
102	#2183	—	61	3	no	—
103	Q₈×C₁₆	—	61	1	no	—
104	Q₈×C₁₆	D₆₁	1	1	yes	characteristic, complemented
105	#2150	—	61	3	no	—
106	C₄×M₃₂	—	61	3	no	—
107	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
108	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
109	C₂×C₆₁	#3407	1	1	yes	characteristic
110	D₆₁	—	2	2	no	—
111	C₂×C₄×Q₈	—	61	1	no	—
112	C₂×M₃₂	—	61	1	no	—
113	C₂×M₃₂	—	61	3	no	—
114	C₈×Q₈	—	61	1	no	—
115	C₈×Q₈	—	61	3	no	—
116	C₈×Q₈	—	61	3	no	—
117	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	61	3	no	—
119	C₂×C₄×C₈	—	61	3	no	—
120	C₄⋊C₁₆	—	61	6	no	—
121	C₄⋊C₁₆	—	61	3	no	—
122	C₄⋊C₁₆	C₂×D₆₁	1	3	yes	—
123	#346	—	61	3	no	—
124	#346	—	61	3	no	—
125	C₄×C₁₆	—	61	3	no	—
126	C₄×C₁₆	C₂×D₆₁	1	3	yes	—
127	C₆₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	61	1	no	—
129	C₂²×C₈	—	61	1	no	—
130	C₂²×C₈	—	61	3	no	—
131	C₄×Q₈	—	61	1	no	—
132	C₄×Q₈	—	61	3	no	—
133	C₄×Q₈	—	61	3	no	—
134	C₄×Q₈	C₄×D₆₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	61	3	no	—
136	C₂×C₄²	—	61	3	no	—
137	M₃₂	—	61	4	no	—
138	M₃₂	—	122	6	no	—
139	C₂×C₁₆	—	61	1	no	—
140	C₂×C₁₆	—	61	3	no	—
141	C₂×C₁₆	C₂²×D₆₁	1	3	yes	—
142	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
143	C₄⋊C₈	—	61	6	no	—
144	C₄⋊C₈	—	61	3	no	—
145	C₄⋊C₈	C₄×D₆₁	1	3	yes	—
146	C₄×C₈	—	61	3	no	—
147	C₄×C₈	—	61	3	no	—
148	C₄×C₈	—	61	3	no	—
149	C₄×C₈	C₂²×D₆₁	1	3	yes	—
150	C₂×Q₈	—	61	3	no	—
151	C₂×Q₈	—	122	4	no	—
152	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
153	C₂²×C₄	—	61	1	no	—
154	C₂²×C₄	—	61	3	no	—
155	C₂²×C₄	—	61	3	no	—
156	C₂×C₈	—	61	1	no	—
157	C₂×C₈	—	61	2	no	—
158	C₂×C₈	—	61	2	no	—
159	C₂×C₈	—	61	3	no	—
160	C₂×C₈	—	122	6	no	—
161	C₂×C₈	C₂×C₄×D₆₁	1	3	yes	—
162	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
163	C₄⋊C₄	—	61	6	no	—
164	C₄⋊C₄	—	61	3	no	—
165	C₄⋊C₄	C₈×D₆₁	1	3	yes	—
166	C₄²	—	61	3	no	—
167	C₄²	—	61	3	no	—
168	C₄²	—	61	3	no	—
169	C₄²	C₂×C₄×D₆₁	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	61	2	no	—
172	C₁₆	—	122	3	no	—
173	C₁₆	Q₈×D₆₁	1	2	yes	complemented
174	C₂³	—	61	1	no	—
175	Q₈	—	122	6	no	—
176	Q₈	#14519716	1	4	yes	direct product factor
177	C₂×C₄	—	61	1	no	—
178	C₂×C₄	—	61	2	no	—
179	C₂×C₄	—	61	2	no	—
180	C₂×C₄	—	61	3	no	—
181	C₂×C₄	—	61	3	no	—
182	C₂×C₄	—	122	6	no	—
183	C₂×C₄	—	244	3	no	—
184	C₂×C₄	C₂×C₈×D₆₁	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₆₁	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	61	2	no	—
189	C₈	—	122	3	no	—
190	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
191	C₈	#41×D₆₁	1	1	yes	characteristic
192	C₂²	—	61	2	no	—
193	C₂²	—	122	1	no	—
194	C₂²	—	122	1	no	—
195	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	61	2	no	—
198	C₄	—	122	3	no	—
199	C₄	—	122	3	no	—
200	C₄	C₂×#14519716	1	6	yes	—
201	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
202	C₄	#131×D₆₁	1	1	yes	characteristic
203	C₂	—	122	2	no	—
204	C₂	C₈×Q₈×D₆₁	1	1	yes	characteristic
205	C₂	C₂²×#14519716	1	1	yes	characteristic
206	C₂	#21458920	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	Q₈×C₁₆×C₆₁	C₂
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	Q₈×C₁₆	D₆₁
hypocenter	C₆₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447724	C₂	semidirect
4. Q₈×#11758185	C₂	semidirect
7. Q₈×C₁₆×C₆₁	C₂	semidirect
33. #14519716	Q₈	direct (N × H)
67. C₁₆×C₆₁	C₂×Q₈	semidirect
69. #11758185	C₂×Q₈	semidirect
71. Q₈×C₆₁	M₃₂	semidirect
104. Q₈×C₁₆	D₆₁	semidirect
127. C₆₁	Q₈×M₃₂	semidirect
173. C₁₆	Q₈×D₆₁	semidirect
176. Q₈	#14519716	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₆₁	2
#21447727	2
#21447724	2
Q₈×#11758185	2
#21447573	2
C₄×#14519716	2
Q₈×C₁₆×C₆₁	2
Q₈×M₃₂	61

Subgroups — Q8×#14519716 #29833523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14519716 #29833523