Subgroups — C₂²×C₆₁×Q₈⋊C₈ #29520865

Order 15616.

1454 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×Q₈⋊C₈	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₆₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₆₁	—	2	1	no	—
7	C₂³×C₈×C₆₁	—	2	1	no	—
8	C₂×C₄×Q₈×C₆₁	—	2	3	no	—
9	C₂×C₄×Q₈×C₆₁	—	2	3	no	—
10	C₂×C₄×Q₈×C₆₁	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₆₁	C₂²	1	6	yes	—
12	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₆₁	—	2	1	no	—
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₁	—	2	3	no	—
17	C₂×C₄⋊C₈×C₆₁	C₂²	1	6	yes	—
18	C₂×C₄×C₈×C₆₁	—	2	3	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	6	yes	—
20	C₆₁×Q₈⋊C₈	C₂²	1	16	yes	direct product factor
21	C₂²×Q₈×C₆₁	—	2	3	no	—
22	C₂²×Q₈×C₆₁	—	2	4	no	—
23	C₂²×Q₈×C₆₁	C₈	1	8	yes	complemented
24	C₂²×Q₈×C₆₁	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₆₁	—	2	1	no	—
26	C₂³×C₄×C₆₁	—	2	1	no	—
27	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	8	no	—
31	C₂²×C₈×C₆₁	—	2	6	no	—
32	C₂²×C₈×C₆₁	—	2	6	no	—
33	C₂²×C₈×C₆₁	—	4	4	no	—
34	C₄×Q₈×C₆₁	—	2	12	no	—
35	C₄×Q₈×C₆₁	—	2	6	no	—
36	C₄×Q₈×C₆₁	—	2	6	no	—
37	C₄×Q₈×C₆₁	C₂×C₄	1	12	yes	—
38	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₆₁	—	2	6	no	—
44	C₂×C₄²×C₆₁	—	2	6	no	—
45	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₆₁	D₄	1	3	yes	—
47	C₂×C₄²×C₆₁	D₄	1	3	yes	—
48	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₁	—	2	6	no	—
50	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
51	C₄×C₈×C₆₁	—	2	6	no	—
52	C₄×C₈×C₆₁	C₂³	1	4	yes	—
53	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₆₁	—	2	24	no	—
55	C₂×Q₈×C₆₁	—	2	6	no	—
56	C₂×Q₈×C₆₁	—	2	12	no	—
57	C₂×Q₈×C₆₁	C₂×C₈	1	24	yes	complemented
58	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₆₁	—	2	8	no	—
60	C₂²×C₄×C₆₁	—	2	4	no	—
61	C₂²×C₄×C₆₁	—	2	6	no	—
62	C₂²×C₄×C₆₁	—	2	6	no	—
63	C₂²×C₄×C₆₁	—	4	4	no	—
64	C₂²×C₄×C₆₁	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₆₁	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₆₁	C₂²⋊C₄	1	3	yes	—
67	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
69	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₆₁	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
74	C₂×C₈×C₆₁	—	2	24	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	4	12	no	—
78	C₄⋊C₄×C₆₁	—	2	12	no	—
79	C₄⋊C₄×C₆₁	—	2	6	no	—
80	C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
82	C₄²×C₆₁	—	2	12	no	—
83	C₄²×C₆₁	—	2	4	no	—
84	C₄²×C₆₁	C₂²⋊C₄	1	6	yes	—
85	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
86	C₄²×C₆₁	C₂×D₄	1	3	yes	—
87	C₄²×C₆₁	C₂×D₄	1	3	yes	—
88	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
89	C₂³×C₆₁	C₂²⋊C₈	1	4	yes	—
90	C₂³×C₆₁	#103	1	4	yes	—
91	C₂³×C₆₁	#104	1	4	yes	—
92	C₂³×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
93	Q₈×C₆₁	—	2	4	no	—
94	Q₈×C₆₁	—	2	24	no	—
95	Q₈×C₆₁	C₂²×C₈	1	8	yes	complemented
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	2	24	no	—
98	C₂×C₄×C₆₁	—	2	6	no	—
99	C₂×C₄×C₆₁	—	2	6	no	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	—	4	12	no	—
102	C₂×C₄×C₆₁	C₂²⋊C₈	1	12	yes	—
103	C₂×C₄×C₆₁	#103	1	12	yes	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
105	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
106	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
108	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
109	C₂×C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
110	C₂×C₄×C₆₁	C₂×Q₁₆	1	6	yes	—
111	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₆₁	—	2	8	no	—
115	C₈×C₆₁	—	4	4	no	—
116	C₂²×Q₈⋊C₈	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₆₁	Q₈⋊C₈	1	16	yes	direct product factor
118	C₂²×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
119	C₂²×C₆₁	C₂×#103	1	6	yes	—
120	C₂²×C₆₁	C₂×#104	1	6	yes	—
121	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₆₁	—	2	8	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	3	no	—
125	C₄×C₆₁	—	4	4	no	—
126	C₄×C₆₁	C₂×C₂²⋊C₈	1	6	yes	—
127	C₄×C₆₁	C₂×#103	1	6	yes	—
128	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
130	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
131	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂²×C₄×Q₈	C₂×C₆₁	1	1	yes	characteristic
133	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
134	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
135	C₂×Q₈⋊C₈	C₂×C₆₁	1	12	yes	direct product factor
136	C₂×C₆₁	C₂×Q₈⋊C₈	1	12	yes	direct product factor
137	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₆₁	C₂²×#103	1	1	yes	characteristic
139	C₂×C₆₁	C₂²×#104	1	1	yes	characteristic
140	C₂³×Q₈	C₄×C₆₁	1	1	yes	characteristic
141	C₂³×C₈	—	2	1	no	—
142	C₂³×C₈	—	2	1	no	—
143	C₂×C₄×Q₈	—	2	3	no	—
144	C₂×C₄×Q₈	—	2	3	no	—
145	C₂×C₄×Q₈	C₄×C₆₁	1	6	yes	—
146	C₂×C₄×Q₈	C₂²×C₆₁	1	6	yes	—
147	C₂²×C₄⋊C₄	—	2	1	no	—
148	C₂²×C₄⋊C₄	C₄×C₆₁	1	1	yes	characteristic
149	C₂²×C₄²	—	2	1	no	—
150	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	2	3	no	—
152	C₂×C₄⋊C₈	C₂²×C₆₁	1	6	yes	—
153	C₂×C₄×C₈	—	2	3	no	—
154	C₂×C₄×C₈	C₂²×C₆₁	1	6	yes	—
155	Q₈⋊C₈	C₂²×C₆₁	1	16	yes	direct product factor
156	C₆₁	C₂²×Q₈⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	—	2	3	no	—
158	C₂²×Q₈	—	2	4	no	—
159	C₂²×Q₈	C₈×C₆₁	1	8	yes	complemented
160	C₂²×Q₈	C₂×C₄×C₆₁	1	6	yes	—
161	C₂³×C₄	—	2	1	no	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
166	C₂²×C₈	—	2	8	no	—
167	C₂²×C₈	—	2	6	no	—
168	C₂²×C₈	—	2	6	no	—
169	C₂²×C₈	—	4	4	no	—
170	C₄×Q₈	—	2	12	no	—
171	C₄×Q₈	—	2	6	no	—
172	C₄×Q₈	—	2	6	no	—
173	C₄×Q₈	C₂×C₄×C₆₁	1	12	yes	—
174	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
175	C₂×C₄⋊C₄	—	2	6	no	—
176	C₂×C₄⋊C₄	—	2	3	no	—
177	C₂×C₄⋊C₄	—	2	6	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	6	yes	—
179	C₂×C₄²	—	2	6	no	—
180	C₂×C₄²	—	2	6	no	—
181	C₂×C₄²	C₂×C₄×C₆₁	1	3	yes	—
182	C₂×C₄²	D₄×C₆₁	1	3	yes	—
183	C₂×C₄²	D₄×C₆₁	1	3	yes	—
184	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
185	C₄⋊C₈	—	2	6	no	—
186	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
187	C₄×C₈	—	2	6	no	—
188	C₄×C₈	C₂³×C₆₁	1	4	yes	—
189	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
190	C₂×Q₈	—	2	24	no	—
191	C₂×Q₈	—	2	6	no	—
192	C₂×Q₈	—	2	12	no	—
193	C₂×Q₈	C₂×C₈×C₆₁	1	24	yes	complemented
194	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊C₄×C₆₁	1	3	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₆₁	1	3	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₆₁	1	3	yes	—
203	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
204	C₂²×C₄	M₁₆×C₆₁	1	4	yes	—
205	C₂²×C₄	SD₁₆×C₆₁	1	4	yes	—
206	C₂²×C₄	Q₁₆×C₆₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
208	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
209	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
210	C₂×C₈	—	2	24	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	12	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	6	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
218	C₄²	—	2	12	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₂²⋊C₄×C₆₁	1	6	yes	—
221	C₄²	C₂²×C₄×C₆₁	1	3	yes	—
222	C₄²	C₂×D₄×C₆₁	1	3	yes	—
223	C₄²	C₂×D₄×C₆₁	1	3	yes	—
224	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
225	C₂³	C₂²⋊C₈×C₆₁	1	4	yes	—
226	C₂³	#103×C₆₁	1	4	yes	—
227	C₂³	#104×C₆₁	1	4	yes	—
228	C₂³	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
229	Q₈	—	2	4	no	—
230	Q₈	—	2	24	no	—
231	Q₈	C₂²×C₈×C₆₁	1	8	yes	complemented
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	12	no	—
238	C₂×C₄	C₂²⋊C₈×C₆₁	1	12	yes	—
239	C₂×C₄	#103×C₆₁	1	12	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
243	C₂×C₄	C₂²×C₈×C₆₁	1	6	yes	—
244	C₂×C₄	C₂×M₁₆×C₆₁	1	6	yes	—
245	C₂×C₄	C₂×SD₁₆×C₆₁	1	6	yes	—
246	C₂×C₄	C₂×Q₁₆×C₆₁	1	6	yes	—
247	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
250	C₈	—	2	8	no	—
251	C₈	—	4	4	no	—
252	C₂²	C₆₁×Q₈⋊C₈	1	16	yes	direct product factor
253	C₂²	C₂×C₂²⋊C₈×C₆₁	1	6	yes	—
254	C₂²	C₂×#103×C₆₁	1	6	yes	—
255	C₂²	C₂×#104×C₆₁	1	6	yes	—
256	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
257	C₄	—	2	8	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	3	no	—
260	C₄	—	4	4	no	—
261	C₄	C₂×C₂²⋊C₈×C₆₁	1	6	yes	—
262	C₄	C₂×#103×C₆₁	1	6	yes	—
263	C₄	C₂³×C₈×C₆₁	1	1	yes	characteristic
264	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
265	C₄	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
266	C₄	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
267	C₂	C₂×C₆₁×Q₈⋊C₈	1	12	yes	direct product factor
268	C₂	C₂²×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
269	C₂	C₂²×#103×C₆₁	1	1	yes	characteristic
270	C₂	C₂²×#104×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₆₁	D₄
commutator	C₄	C₂³×C₈×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle, prime radical	C₂⁴×C₆₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₆₁×Q₈⋊C₈	C₂	direct (N × H)
20. C₆₁×Q₈⋊C₈	C₂²	direct (N × H)
23. C₂²×Q₈×C₆₁	C₈	semidirect
57. C₂×Q₈×C₆₁	C₂×C₈	semidirect
95. Q₈×C₆₁	C₂²×C₈	semidirect
116. C₂²×Q₈⋊C₈	C₆₁	direct (N × H)
117. C₂²×C₆₁	Q₈⋊C₈	direct (N × H)
135. C₂×Q₈⋊C₈	C₂×C₆₁	direct (N × H)
136. C₂×C₆₁	C₂×Q₈⋊C₈	direct (N × H)
155. Q₈⋊C₈	C₂²×C₆₁	direct (N × H)
156. C₆₁	C₂²×Q₈⋊C₈	direct (N × H)
159. C₂²×Q₈	C₈×C₆₁	semidirect
193. C₂×Q₈	C₂×C₈×C₆₁	semidirect
231. Q₈	C₂²×C₈×C₆₁	semidirect
252. C₂²	C₆₁×Q₈⋊C₈	direct (N × H)
267. C₂	C₂×C₆₁×Q₈⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₁	2
C₂²×C₄⋊C₈×C₆₁	2
C₂²×C₄×C₈×C₆₁	2
C₂×C₆₁×Q₈⋊C₈	2
C₂²×Q₈⋊C₈	61

Subgroups — C22×C61×Q8⋊C8 #29520865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×Q₈⋊C₈ #29520865