Subgroups — D₆₁⋊(Q₈⋊C₁₆) #30604445

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#21460786	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#11758187	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	—	2	1	no	—
7	C₂×D₆₁⋊C₁₆	—	2	1	no	—
8	#16840683	—	2	1	no	—
9	C₄⋊#11758187	C₂²	1	2	yes	—
10	#16840648	—	2	1	no	—
11	C₄×#11758187	C₂²	1	2	yes	—
12	C₄×Q₈×D₆₁	C₄	1	1	yes	characteristic
13	Q₈×#92418	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₂₂	—	2	1	no	—
15	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
16	#16839886	—	2	1	no	—
17	#16839886	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×D₆₁	—	2	1	no	—
19	C₄⋊C₈×D₆₁	C₄	1	1	yes	characteristic
20	C₄×C₈×D₆₁	—	2	1	no	—
21	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
23	D₆₁⋊C₁₆	—	2	4	no	—
24	D₆₁⋊C₁₆	—	4	2	no	—
25	C₂×#11758187	—	2	2	no	—
26	C₂×#11758187	—	2	2	no	—
27	C₂×Q₈×D₆₁	C₈	1	1	yes	characteristic
28	C₂×C₈×D₆₁	—	2	1	no	—
29	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
30	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
31	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
32	C₄×Dic₁₂₂	—	2	1	no	—
33	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
34	#14519742	—	2	1	no	—
35	#14519742	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₆₁	—	2	1	no	—
37	C₄⋊C₄×D₆₁	C₈	1	1	yes	characteristic
38	C₄²×D₆₁	—	2	1	no	—
39	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#92418	—	2	1	no	—
41	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
42	#14519706	—	2	1	no	—
43	#14519706	—	2	1	no	—
44	#14519706	—	2	1	no	—
45	C₄×#92418	—	2	1	no	—
46	C₄×#92418	C₂³	1	1	yes	characteristic
47	C₈×Dic₆₁	—	2	1	no	—
48	C₈×Dic₆₁	—	2	1	no	—
49	C₈×Dic₆₁	D₄	1	1	yes	characteristic
50	C₈×Dic₆₁	D₄	1	1	yes	characteristic
51	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₆₁	—	2	1	no	—
53	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	—	2	1	no	—
55	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
57	Q₈×D₆₁	—	2	2	no	—
58	Q₈×D₆₁	C₁₆	1	4	yes	complemented
59	C₂×C₄×D₆₁	—	2	1	no	—
60	C₂×C₄×D₆₁	—	2	1	no	—
61	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
64	C₂×Dic₁₂₂	—	2	1	no	—
65	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₆₁	—	2	1	no	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₆₁	—	2	1	no	—
72	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊Dic₆₁	—	2	1	no	—
74	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
75	#11758196	—	2	1	no	—
76	#11758196	—	2	1	no	—
77	#11758196	—	2	1	no	—
78	C₄×Dic₆₁	—	2	1	no	—
79	C₄×Dic₆₁	—	2	1	no	—
80	C₄×Dic₆₁	—	2	1	no	—
81	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
83	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
84	C₂×#92418	—	2	1	no	—
85	C₂×#92418	C₂×D₄	1	1	yes	characteristic
86	C₂×#92418	C₂×D₄	1	1	yes	characteristic
87	C₈×D₆₁	—	2	2	no	—
88	C₈×D₆₁	—	4	2	no	—
89	C₈×D₆₁	SD₁₆	1	2	yes	—
90	C₈×D₆₁	Q₁₆	1	2	yes	—
91	#11758187	—	2	4	no	—
92	#11758187	—	4	2	no	—
93	C₂²×D₆₁	C₂²⋊C₈	1	1	yes	characteristic
94	Q₈×C₆₁	—	2	1	no	—
95	Q₈×C₆₁	C₂×C₁₆	1	2	yes	complemented
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₆₁	—	2	1	no	—
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×Dic₆₁	C₂²⋊C₈	1	1	yes	characteristic
106	C₄×D₆₁	—	2	4	no	—
107	C₄×D₆₁	—	2	2	no	—
108	C₄×D₆₁	—	4	2	no	—
109	C₄×D₆₁	#103	1	2	yes	—
110	C₄×D₆₁	#104	1	2	yes	—
111	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
112	C₄×D₆₁	M₃₂	1	2	yes	—
113	Dic₁₂₂	—	2	1	no	—
114	Dic₁₂₂	—	2	4	no	—
115	Dic₁₂₂	C₂×C₁₆	1	2	yes	complemented
116	C₈×C₆₁	—	2	1	no	—
117	C₈×C₆₁	—	4	1	no	—
118	C₈×C₆₁	#103	1	2	yes	—
119	#92418	—	2	1	no	—
120	#92418	—	4	1	no	—
121	#92418	C₂×SD₁₆	1	1	yes	characteristic
122	#92418	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×Q₈⋊C₁₆	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂×D₆₁	Q₈⋊C₈	1	2	yes	—
126	C₂×D₆₁	C₂²⋊C₁₆	1	2	yes	—
127	C₂×D₆₁	#350	1	2	yes	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	—	4	1	no	—
131	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
132	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
133	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
134	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
135	Dic₆₁	—	2	2	no	—
136	Dic₆₁	—	2	1	no	—
137	Dic₆₁	—	4	1	no	—
138	Dic₆₁	Q₈⋊C₈	1	2	yes	—
139	Dic₆₁	C₂²⋊C₁₆	1	2	yes	—
140	C₂×C₈×Q₈	—	61	1	no	—
141	C₂×C₄⋊C₁₆	—	61	1	no	—
142	C₂×C₄×C₁₆	—	61	1	no	—
143	Q₈⋊C₁₆	—	61	2	no	—
144	Q₈⋊C₁₆	—	61	2	no	—
145	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
147	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
148	D₆₁	Q₈⋊C₁₆	1	4	yes	complemented
149	C₂×C₄×Q₈	—	61	1	no	—
150	C₂²×C₁₆	—	122	1	no	—
151	C₂²×C₁₆	—	122	1	no	—
152	C₈×Q₈	—	61	1	no	—
153	C₈×Q₈	—	61	1	no	—
154	C₈×Q₈	—	61	1	no	—
155	C₈×Q₈	—	122	1	no	—
156	C₈×Q₈	—	122	1	no	—
157	C₈×Q₈	#6923	1	1	yes	characteristic
158	C₂×C₄⋊C₈	—	61	1	no	—
159	C₂×C₄⋊C₈	—	122	1	no	—
160	C₂×C₄×C₈	—	61	1	no	—
161	C₂×C₄×C₈	—	122	1	no	—
162	C₄⋊C₁₆	—	61	2	no	—
163	C₄⋊C₁₆	—	122	1	no	—
164	C₄×C₁₆	—	61	2	no	—
165	C₄×C₁₆	—	122	1	no	—
166	C₆₁	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	61	1	no	—
168	C₂²×C₈	—	61	1	no	—
169	C₂²×C₈	—	61	1	no	—
170	C₂²×C₈	—	122	1	no	—
171	C₄×Q₈	—	61	1	no	—
172	C₄×Q₈	—	61	1	no	—
173	C₄×Q₈	—	61	1	no	—
174	C₄×Q₈	—	122	1	no	—
175	C₄×Q₈	—	122	1	no	—
176	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	61	1	no	—
178	C₂×C₄⋊C₄	—	122	1	no	—
179	C₂×C₄²	—	61	1	no	—
180	C₂×C₄²	—	122	1	no	—
181	C₂×C₁₆	—	122	2	no	—
182	C₂×C₁₆	—	122	4	no	—
183	C₂×C₁₆	—	122	2	no	—
184	C₂×C₁₆	—	244	2	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	61	1	no	—
187	C₄⋊C₈	—	122	1	no	—
188	C₄⋊C₈	—	122	1	no	—
189	C₄⋊C₈	—	122	1	no	—
190	C₄⋊C₈	—	122	1	no	—
191	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	61	1	no	—
194	C₄×C₈	—	61	1	no	—
195	C₄×C₈	—	61	1	no	—
196	C₄×C₈	—	122	1	no	—
197	C₄×C₈	—	122	1	no	—
198	C₄×C₈	—	122	1	no	—
199	C₄×C₈	C₂×#6923	1	1	yes	characteristic
200	C₂×Q₈	—	61	4	no	—
201	C₂×Q₈	—	61	1	no	—
202	C₂×Q₈	—	122	1	no	—
203	C₂×Q₈	—	122	2	no	—
204	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
205	C₂²×C₄	—	61	1	no	—
206	C₂²×C₄	—	61	1	no	—
207	C₂²×C₄	—	61	1	no	—
208	C₂²×C₄	—	122	1	no	—
209	C₂²×C₄	—	122	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	61	1	no	—
212	C₂×C₈	—	61	2	no	—
213	C₂×C₈	—	61	2	no	—
214	C₂×C₈	—	61	1	no	—
215	C₂×C₈	—	122	2	no	—
216	C₂×C₈	—	122	1	no	—
217	C₂×C₈	—	244	2	no	—
218	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	61	1	no	—
222	C₄⋊C₄	—	122	1	no	—
223	C₄⋊C₄	—	122	1	no	—
224	C₄⋊C₄	—	122	1	no	—
225	C₄⋊C₄	—	122	1	no	—
226	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
227	C₄²	—	2	1	no	—
228	C₄²	—	61	1	no	—
229	C₄²	—	61	1	no	—
230	C₄²	—	61	1	no	—
231	C₄²	—	122	1	no	—
232	C₄²	—	122	1	no	—
233	C₄²	—	122	1	no	—
234	C₄²	C₂²×#6923	1	1	yes	characteristic
235	C₁₆	—	122	4	no	—
236	C₁₆	—	244	2	no	—
237	C₂³	—	61	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	61	2	no	—
240	Q₈	—	122	4	no	—
241	Q₈	—	122	1	no	—
242	Q₈	D₆₁⋊C₁₆	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	61	1	no	—
246	C₂×C₄	—	61	1	no	—
247	C₂×C₄	—	61	2	no	—
248	C₂×C₄	—	61	2	no	—
249	C₂×C₄	—	61	2	no	—
250	C₂×C₄	—	61	2	no	—
251	C₂×C₄	—	61	1	no	—
252	C₂×C₄	—	122	2	no	—
253	C₂×C₄	—	122	4	no	—
254	C₂×C₄	—	122	1	no	—
255	C₂×C₄	—	122	1	no	—
256	C₂×C₄	—	244	2	no	—
257	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
258	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
259	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	61	1	no	—
263	C₈	—	61	1	no	—
264	C₈	—	122	1	no	—
265	C₈	—	244	1	no	—
266	C₈	#14519840	1	2	yes	—
267	C₂²	—	61	2	no	—
268	C₂²	—	61	2	no	—
269	C₂²	—	61	2	no	—
270	C₂²	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	61	2	no	—
275	C₄	—	61	2	no	—
276	C₄	—	122	2	no	—
277	C₄	—	122	1	no	—
278	C₄	—	244	1	no	—
279	C₄	C₂×#14519840	1	1	yes	characteristic
280	C₄	D₆₁⋊#104	1	1	yes	characteristic
281	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
282	C₄	D₆₁⋊M₃₂	1	1	yes	characteristic
283	C₂	—	61	4	no	—
284	C₂	D₆₁⋊(Q₈⋊C₈)	1	1	yes	characteristic
285	C₂	D₆₁⋊(C₂²⋊C₁₆)	1	1	yes	characteristic
286	C₂	D₆₁⋊#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₈×Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₆₁	C₂²⋊C₈
cosocle	C₄×#92418	C₂³
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #21460786	C₂	semidirect
4. Q₈⋊#11758187	C₂	semidirect
58. Q₈×D₆₁	C₁₆	semidirect
95. Q₈×C₆₁	C₂×C₁₆	semidirect
115. Dic₁₂₂	C₂×C₁₆	semidirect
148. D₆₁	Q₈⋊C₁₆	semidirect
166. C₆₁	C₂×Q₈⋊C₁₆	semidirect
242. Q₈	D₆₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊(C₄⋊C₁₆)	2
C₄×D₆₁⋊C₁₆	2
#21460786	2
Q₈⋊#11758187	2
C₈×Q₈×D₆₁	2
C₂×Q₈⋊C₁₆	61

Subgroups — D61⋊(Q8⋊C16) #30604445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊(Q₈⋊C₁₆) #30604445