Subgroups — #29649495

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454930	C₂	1	2	yes	maximal, complemented
2	SD₁₆⋊#92418	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#92418	C₂	1	2	yes	maximal, complemented
4	D₈×#92418	C₂	1	2	yes	maximal, complemented
5	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444206	C₂	1	2	yes	maximal, complemented
7	#21444102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
9	Q₈×#92418	C₂²	1	4	yes	—
10	D₄×#92418	—	2	4	no	—
11	D₄×#92418	C₂²	1	4	yes	—
12	#16839815	—	2	2	no	—
13	#16839815	C₂²	1	1	yes	characteristic
14	#16839815	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#92418	—	2	2	no	—
16	C₂×C₄⋊#92418	C₂²	1	2	yes	—
17	#16839515	C₂²	1	2	yes	—
18	#16839515	C₂²	1	2	yes	—
19	Q₈⋊#92418	C₂²	1	4	yes	—
20	D₄⋊#92418	C₂²	1	4	yes	—
21	#16839483	C₂²	1	2	yes	—
22	C₈×#92418	C₂²	1	2	yes	—
23	C₂×#135×C₆₁	C₄	1	1	yes	characteristic
24	C₄×#41×C₆₁	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₆₁	C₄	1	2	yes	—
28	C₄×SD₁₆×C₆₁	C₂²	1	2	yes	—
29	C₄×D₈×C₆₁	C₄	1	1	yes	characteristic
30	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#427	C₄	1	1	yes	characteristic
32	C₆₁×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₂²×#92418	—	2	2	no	—
35	C₂²×#92418	—	2	4	no	—
36	C₄⋊#92418	—	2	2	no	—
37	C₄⋊#92418	—	2	1	no	—
38	C₄⋊#92418	—	2	4	no	—
39	C₄⋊#92418	D₄	1	1	yes	characteristic
40	C₄⋊#92418	D₄	1	1	yes	characteristic
41	C₄⋊#92418	C₂³	1	4	yes	—
42	C₂²⋊#92418	—	2	1	no	—
43	C₂²⋊#92418	—	2	4	no	—
44	C₂²⋊#92418	D₄	1	2	yes	—
45	C₄×#92418	—	2	4	no	—
46	C₄×#92418	C₂³	1	2	yes	—
47	C₄×#92418	C₂³	1	2	yes	—
48	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
49	#135×C₆₁	C₈	1	8	yes	complemented
50	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
57	C₄×D₄×C₆₁	—	2	2	no	—
58	C₄×D₄×C₆₁	—	2	2	no	—
59	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₆₁	C₂³	1	2	yes	—
61	#117×C₆₁	—	2	2	no	—
62	#117×C₆₁	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₆₁	—	2	2	no	—
64	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
65	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
67	#106×C₆₁	C₂×C₄	1	2	yes	—
68	#103×C₆₁	C₂×C₄	1	2	yes	—
69	#103×C₆₁	C₂×C₄	1	2	yes	—
70	#102×C₆₁	C₂×C₄	1	2	yes	—
71	#102×C₆₁	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	#41×C₆₁	—	2	4	no	—
77	#41×C₆₁	C₂×C₈	1	8	yes	complemented
78	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₆₁	—	2	2	no	—
80	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	—	2	2	no	—
83	C₂²×C₄×C₆₁	—	2	2	no	—
84	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
87	Q₁₆×C₆₁	C₂×C₈	1	4	yes	complemented
88	SD₁₆×C₆₁	C₂×C₈	1	8	yes	complemented
89	D₈×C₆₁	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
99	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
100	C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂²⋊C₄×C₆₁	—	2	2	no	—
102	C₂²⋊C₄×C₆₁	—	2	2	no	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	2	2	no	—
106	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₁	#41	1	1	yes	characteristic
109	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
110	C₂×#92418	—	2	4	no	—
111	C₂×#92418	—	2	2	no	—
112	C₂×#92418	—	8	4	no	—
113	C₂×#92418	—	8	2	no	—
114	C₂×#92418	C₂×D₄	1	2	yes	—
115	C₂×#92418	C₂×D₄	1	2	yes	—
116	C₂³×C₆₁	—	2	2	no	—
117	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	C₂²×C₈	1	4	yes	—
120	D₄×C₆₁	—	2	8	no	—
121	D₄×C₆₁	—	2	2	no	—
122	D₄×C₆₁	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	1	no	—
125	C₂×C₄×C₆₁	—	2	1	no	—
126	C₂×C₄×C₆₁	—	2	8	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	8	2	no	—
133	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₆₁	#131	1	2	yes	—
138	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	C₂²×C₈	1	2	yes	—
144	C₈×C₆₁	C₂²×C₈	1	2	yes	—
145	#92418	—	2	2	no	—
146	#92418	—	2	2	no	—
147	#92418	—	4	2	no	—
148	#92418	—	4	4	no	—
149	#17857	—	61	1	no	maximal, Sylow
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	—	2	1	no	—
152	C₂²×C₆₁	—	2	4	no	—
153	C₂²×C₆₁	—	4	2	no	—
154	C₂²×C₆₁	—	4	2	no	—
155	C₂²×C₆₁	C₈×D₄	1	2	yes	—
156	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₁	—	2	4	no	—
158	C₄×C₆₁	—	2	1	no	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	2	1	no	—
161	C₄×C₆₁	—	4	2	no	—
162	C₄×C₆₁	—	4	2	no	—
163	C₄×C₆₁	C₈×D₄	1	2	yes	—
164	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
166	C₄×C₆₁	#576	1	1	yes	characteristic
167	C₄×C₆₁	#578	1	1	yes	characteristic
168	#2988	—	61	2	no	—
169	C₄×#135	D₆₁	1	1	yes	characteristic
170	SD₁₆⋊C₈	—	61	4	no	—
171	C₈×Q₁₆	—	61	2	no	—
172	C₈×D₈	—	61	2	no	—
173	C₂×#335	—	61	1	no	—
174	#1477	—	61	2	no	—
175	#1450	—	61	1	no	—
176	C₂×C₆₁	—	2	2	no	—
177	C₂×C₆₁	—	4	4	no	—
178	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
179	C₂×C₆₁	#2951	1	1	yes	characteristic
180	C₂×C₆₁	#2957	1	1	yes	characteristic
181	C₂×#135	Dic₆₁	1	1	yes	characteristic
182	C₄×#41	C₂×D₆₁	1	2	yes	—
183	C₈×Q₈	—	61	4	no	—
184	C₄×Q₁₆	Dic₆₁	1	1	yes	characteristic
185	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
186	C₄×SD₁₆	Dic₆₁	1	2	yes	—
187	C₄×SD₁₆	C₂×D₆₁	1	2	yes	—
188	C₄×D₈	Dic₆₁	1	1	yes	characteristic
189	C₄×D₈	C₂×D₆₁	1	1	yes	characteristic
190	C₈×D₄	—	61	4	no	—
191	C₈×D₄	—	122	4	no	—
192	#431	—	61	1	no	—
193	#431	—	61	1	no	—
194	#431	—	122	2	no	—
195	#427	Dic₆₁	1	1	yes	characteristic
196	C₂×C₄⋊C₈	—	61	2	no	—
197	C₂×C₄⋊C₈	—	122	2	no	—
198	#416	Dic₆₁	1	2	yes	—
199	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
200	#335	—	61	2	no	—
201	#335	—	61	2	no	—
202	Q₈⋊C₈	—	61	4	no	—
203	D₄⋊C₈	—	61	4	no	—
204	#322	—	61	2	no	—
205	C₈²	—	61	2	no	—
206	C₆₁	#17857	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	C₂×Dic₆₁	1	2	yes	—
208	#135	#92418	1	8	yes	complemented
209	C₂×Q₁₆	C₂×Dic₆₁	1	1	yes	characteristic
210	C₂×SD₁₆	C₂×Dic₆₁	1	2	yes	—
211	C₂×D₈	C₂×Dic₆₁	1	1	yes	characteristic
212	C₂²×C₈	—	122	4	no	—
213	C₂²×C₈	—	122	2	no	—
214	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
215	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
216	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
217	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	C₂×Dic₆₁	1	2	yes	—
221	C₄×D₄	C₂²×D₆₁	1	2	yes	—
222	#117	—	2	2	no	—
223	#117	C₂×Dic₆₁	1	2	yes	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
226	#107	C₂×Dic₆₁	1	1	yes	characteristic
227	#107	C₂×Dic₆₁	1	1	yes	characteristic
228	#106	C₂×Dic₆₁	1	2	yes	—
229	C₄⋊C₈	—	61	1	no	—
230	C₄⋊C₈	—	61	1	no	—
231	C₄⋊C₈	—	61	4	no	—
232	C₄⋊C₈	—	122	4	no	—
233	C₄⋊C₈	—	122	2	no	—
234	C₄⋊C₈	—	122	1	no	—
235	#103	C₂×Dic₆₁	1	2	yes	—
236	#103	C₂×Dic₆₁	1	2	yes	—
237	#102	C₂×Dic₆₁	1	2	yes	—
238	#102	C₂×Dic₆₁	1	2	yes	—
239	C₂²⋊C₈	—	61	2	no	—
240	C₂²⋊C₈	—	122	4	no	—
241	C₂²⋊C₈	—	122	1	no	—
242	C₄×C₈	—	61	2	no	—
243	C₄×C₈	—	61	2	no	—
244	C₄×C₈	—	122	4	no	—
245	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
246	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
249	#41	—	2	4	no	—
250	#41	C₂×#92418	1	8	yes	complemented
251	C₂×Q₈	C₂²×Dic₆₁	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×Dic₆₁	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
258	C₂²×C₄	#11758224	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
260	Q₁₆	C₂×#92418	1	4	yes	complemented
261	SD₁₆	C₂×#92418	1	8	yes	complemented
262	D₈	C₂×#92418	1	4	yes	complemented
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	61	2	no	—
265	C₂×C₈	—	61	2	no	—
266	C₂×C₈	—	122	4	no	—
267	C₂×C₈	—	122	2	no	—
268	C₂×C₈	—	488	4	no	—
269	C₂×C₈	—	488	2	no	—
270	C₂×C₈	C₂×#92418	1	4	yes	complemented
271	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	C₂²×Dic₆₁	1	2	yes	—
278	C₄⋊C₄	C₂²×Dic₆₁	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	D₄×D₆₁	1	1	yes	characteristic
286	C₄²	#11758224	1	1	yes	characteristic
287	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
288	C₄²	C₂³×D₆₁	1	1	yes	characteristic
289	C₂³	—	2	2	no	—
290	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
291	Q₈	—	2	2	no	—
292	Q₈	C₂²×#92418	1	4	yes	—
293	D₄	—	2	8	no	—
294	D₄	—	2	2	no	—
295	D₄	C₂²×#92418	1	4	yes	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
307	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
308	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×#92418	1	2	yes	—
310	C₂×C₄	#14519784	1	2	yes	—
311	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
313	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	122	2	no	—
317	C₈	—	122	2	no	—
318	C₈	—	244	2	no	—
319	C₈	—	244	4	no	—
320	C₈	C₂²×#92418	1	2	yes	—
321	C₈	C₂²×#92418	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	D₄×#92418	1	2	yes	—
328	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	4	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	D₄×#92418	1	2	yes	—
336	C₄	C₂³×#92418	1	1	yes	characteristic
337	C₄	C₂×#14519784	1	1	yes	characteristic
338	C₄	#16840564	1	1	yes	characteristic
339	C₄	#16840576	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	4	no	—
342	C₂	C₂×D₄×#92418	1	1	yes	characteristic
343	C₂	#21454695	1	1	yes	characteristic
344	C₂	#21454736	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₄×#135×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#135×C₆₁	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₄×#135	D₆₁
hypocenter	C₆₁	#17857
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #21454930	C₂	semidirect
2. SD₁₆⋊#92418	C₂	semidirect
3. Q₁₆×#92418	C₂	semidirect
4. D₈×#92418	C₂	semidirect
5. C₂×#16839515	C₂	semidirect
6. #21444206	C₂	semidirect
7. #21444102	C₂	semidirect
49. #135×C₆₁	C₈	semidirect
77. #41×C₆₁	C₂×C₈	semidirect
87. Q₁₆×C₆₁	C₂×C₈	semidirect
88. SD₁₆×C₆₁	C₂×C₈	semidirect
89. D₈×C₆₁	C₂×C₈	semidirect
91. C₂×C₈×C₆₁	C₂×C₈	semidirect
206. C₆₁	#17857	semidirect
208. #135	#92418	semidirect
250. #41	C₂×#92418	semidirect
260. Q₁₆	C₂×#92418	semidirect
261. SD₁₆	C₂×#92418	semidirect
262. D₈	C₂×#92418	semidirect
270. C₂×C₈	C₂×#92418	semidirect

Maximal subgroups

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

subgroup	index
#21454930	2
SD₁₆⋊#92418	2
Q₁₆×#92418	2
D₈×#92418	2
C₂×#16839515	2
#21444206	2
#21444102	2
C₄×#135×C₆₁	2
#17857	61