Subgroups — C₈×#14519701 #29570782

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519701	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92418	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	#21443658	C₂	1	1	yes	maximal, characteristic
6	#21443653	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
9	C₂×#14519701	C₄	1	2	yes	—
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₆₁	—	2	1	no	—
13	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
14	#16839524	C₄	1	2	yes	—
15	#16839516	—	2	1	no	—
16	Q₈⋊#92418	C₄	1	4	yes	complemented
17	C₈×#92418	—	2	2	no	—
18	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₈×Q₈×C₆₁	—	2	2	no	—
20	C₈×Q₈×C₆₁	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	—	2	1	no	—
24	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×#92418	—	2	1	no	—
26	C₂²×#92418	—	2	1	no	—
27	C₂²×#92418	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	—	2	1	no	—
30	C₄⋊#92418	—	2	1	no	—
31	C₄⋊#92418	C₂×C₄	1	2	yes	—
32	#14519701	C₈	1	8	yes	direct product factor
33	C₄×#92418	—	2	1	no	—
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	—	2	1	no	—
36	C₄×#92418	C₂×C₄	1	2	yes	—
37	C₈×Dic₆₁	—	4	2	no	—
38	#14519684	—	2	1	no	—
39	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	—	2	1	no	—
41	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₆₁	—	2	1	no	—
44	C₄×Q₈×C₆₁	—	2	1	no	—
45	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	—	2	1	no	—
50	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₁	—	2	4	no	—
52	C₄⋊C₈×C₆₁	—	2	1	no	—
53	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₆₁	—	2	4	no	—
55	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₆₁	D₄	1	2	yes	—
57	C₂×Q₈×C₆₁	—	2	1	no	—
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	C₂×Q₈×C₆₁	C₄²	1	2	yes	—
60	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
66	C₂²×Dic₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	—	2	1	no	—
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	—	2	2	no	—
70	C₂×C₈×C₆₁	—	4	1	no	—
71	C₂×C₈×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
75	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
76	C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₄⋊C₄×C₆₁	C₄²	1	2	yes	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	2	2	no	—
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₁	#41	1	1	yes	characteristic
86	C₄⋊Dic₆₁	—	2	2	no	—
87	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
88	C₄×Dic₆₁	—	4	1	no	—
89	C₄×Dic₆₁	—	4	1	no	—
90	C₂×#92418	—	2	4	no	—
91	C₂×#92418	—	2	2	no	—
92	C₂×#92418	—	2	2	no	—
93	C₂×#92418	—	2	2	no	—
94	C₂×#92418	—	2	2	no	—
95	C₂×#92418	—	4	1	no	—
96	C₂×#92418	—	4	1	no	—
97	C₂×#92418	C₂×C₈	1	4	yes	—
98	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₆₁	—	2	2	no	—
100	Q₈×C₆₁	—	2	4	no	—
101	Q₈×C₆₁	C₄×C₈	1	4	yes	complemented
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₆₁	#131	1	2	yes	—
115	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
119	C₂×Dic₆₁	—	2	4	no	—
120	C₂×Dic₆₁	—	4	2	no	—
121	C₂×Dic₆₁	—	4	2	no	—
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	—	4	2	no	—
124	C₈×C₆₁	#103	1	4	yes	complemented
125	#92418	—	2	4	no	—
126	#92418	—	2	4	no	—
127	#92418	—	4	2	no	—
128	C₈×#103	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₁	C₈×D₄	1	2	yes	—
131	C₂²×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
132	C₂²×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
133	C₂²×C₆₁	#443	1	2	yes	—
134	C₄×C₆₁	—	2	4	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	—	4	2	no	—
138	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
139	C₄×C₆₁	#405	1	1	yes	characteristic
140	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
141	C₄×C₆₁	#416	1	1	yes	characteristic
142	C₄×C₆₁	C₈×D₄	1	2	yes	—
143	C₄×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
144	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
145	Dic₆₁	—	4	4	no	—
146	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
147	C₈×C₄⋊C₄	—	61	1	no	—
148	C₄×#103	—	61	1	no	—
149	C₂×Q₈⋊C₈	—	61	1	no	—
150	C₂×C₈²	—	61	1	no	—
151	#1279	—	61	1	no	—
152	#1277	—	61	1	no	—
153	C₂×C₆₁	C₈×SD₁₆	1	2	yes	—
154	C₂×C₆₁	C₈×Q₁₆	1	2	yes	—
155	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₆₁	C₄×#103	1	1	yes	characteristic
157	C₂×C₆₁	#1764	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
159	C₈×Q₈	—	2	2	no	—
160	C₈×Q₈	Dic₆₁	1	4	yes	complemented
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×C₄⋊C₈	—	61	1	no	—
163	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
164	C₂×#103	—	61	2	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	—	61	1	no	—
167	C₂×C₄×C₈	—	61	1	no	—
168	C₂×C₄×C₈	—	122	1	no	—
169	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
170	C₄×C₄⋊C₄	—	61	1	no	—
171	#340	—	61	2	no	—
172	#336	—	122	1	no	—
173	Q₈⋊C₈	—	61	4	no	—
174	C₈²	—	122	2	no	—
175	C₆₁	C₈×#103	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	C₄×D₆₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	61	2	no	—
179	C₂²×C₈	—	122	1	no	—
180	C₂²×C₈	—	122	1	no	—
181	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₄×D₆₁	1	2	yes	—
186	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	61	2	no	—
189	C₂×C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	122	1	no	—
192	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	61	2	no	—
196	C₄⋊C₈	—	122	1	no	—
197	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
198	#103	—	61	8	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	—	61	2	no	—
201	C₄×C₈	—	122	1	no	—
202	C₄×C₈	—	122	1	no	—
203	C₄×C₈	—	122	1	no	—
204	C₄×C₈	—	244	2	no	—
205	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
206	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
207	#95	—	122	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₈×D₆₁	1	4	yes	—
211	C₂×Q₈	C₄×Dic₆₁	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	122	2	no	—
215	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	#11758222	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	61	4	no	—
224	C₂×C₈	—	122	4	no	—
225	C₂×C₈	—	122	2	no	—
226	C₂×C₈	—	122	2	no	—
227	C₂×C₈	—	122	2	no	—
228	C₂×C₈	—	122	2	no	—
229	C₂×C₈	—	244	1	no	—
230	C₂×C₈	—	244	1	no	—
231	C₂×C₈	#11758201	1	2	yes	—
232	C₂×C₈	#11758202	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
234	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	61	4	no	—
239	C₄⋊C₄	—	122	2	no	—
240	C₄⋊C₄	C₄×Dic₆₁	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	244	1	no	—
244	C₄²	—	244	1	no	—
245	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
246	C₄²	#11758222	1	1	yes	characteristic
247	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
248	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	2	4	no	—
252	Q₈	C₈×Dic₆₁	1	4	yes	complemented
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	122	4	no	—
259	C₂×C₄	—	244	2	no	—
260	C₂×C₄	—	244	2	no	—
261	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
263	C₂×C₄	#14519736	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
268	C₂×C₄	#14519783	1	2	yes	—
269	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
270	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
271	C₂×C₄	#14519797	1	1	yes	characteristic
272	C₂×C₄	#14519797	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	122	4	no	—
276	C₈	—	122	4	no	—
277	C₈	—	244	2	no	—
278	C₈	#14519701	1	4	yes	direct product factor
279	C₂²	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
280	C₂²	C₈×C₂²⋊D₆₁	1	2	yes	—
281	C₂²	C₄×#11758201	1	1	yes	characteristic
282	C₂²	C₄×#11758202	1	1	yes	characteristic
283	C₂²	#16839877	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	244	4	no	—
289	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
290	C₄	#16839717	1	1	yes	characteristic
291	C₄	C₂×#14519701	1	1	yes	characteristic
292	C₄	#16839758	1	1	yes	characteristic
293	C₄	C₈×C₂²⋊D₆₁	1	2	yes	—
294	C₄	C₄×#11758201	1	1	yes	characteristic
295	C₄	C₄×#11758202	1	1	yes	characteristic
296	C₂	C₈×#11758201	1	2	yes	—
297	C₂	C₈×#11758202	1	2	yes	—
298	C₂	C₈×C₂²⋊Dic₆₁	1	1	yes	characteristic
299	C₂	C₄×#14519701	1	1	yes	characteristic
300	C₂	#21445654	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈×Q₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈×Q₈	D₆₁
hypocenter	C₆₁	C₈×#103
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
16. Q₈⋊#92418	C₄	semidirect
20. C₈×Q₈×C₆₁	C₄	semidirect
32. #14519701	C₈	direct (N × H)
101. Q₈×C₆₁	C₄×C₈	semidirect
124. C₈×C₆₁	#103	semidirect
160. C₈×Q₈	Dic₆₁	semidirect
175. C₆₁	C₈×#103	semidirect
252. Q₈	C₈×Dic₆₁	semidirect
278. C₈	#14519701	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×C₄⋊Dic₆₁	2
C₄×#14519701	2
C₂×Q₈⋊#92418	2
C₂×C₈×#92418	2
#21443658	2
#21443653	2
C₂×C₈×Q₈×C₆₁	2
C₈×#103	61

Subgroups — C8×#14519701 #29570782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14519701 #29570782