Subgroups — #30598853

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461736	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519844	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92420	C₂	1	2	yes	maximal
4	#21460655	C₂	1	2	yes	maximal
5	C₄×Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519844	—	2	1	no	—
7	C₂×C₄⋊#92420	C₂²	1	2	yes	—
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	#16840628	—	2	1	no	—
10	#16840628	—	2	1	no	—
11	#16840628	—	2	1	no	—
12	Q₈⋊#92420	—	2	4	no	—
13	C₂×Q₈×Dic₆₁	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₆₁	—	2	1	no	—
16	C₄⋊C₄×Dic₆₁	—	2	1	no	—
17	C₄⋊C₄×Dic₆₁	C₄	1	1	yes	characteristic
18	C₄⋊C₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊Dic₆₁	—	2	1	no	—
20	C₄×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
21	C₄²×Dic₆₁	—	2	1	no	—
22	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
24	C₂²×#92420	—	2	2	no	—
25	C₂²×#92420	—	2	2	no	—
26	#14519844	—	2	4	no	—
27	C₄⋊#92420	—	2	2	no	—
28	C₄⋊#92420	—	2	2	no	—
29	C₄×#92420	—	2	2	no	—
30	C₄×#92420	—	2	2	no	—
31	Q₈×Dic₆₁	—	2	2	no	—
32	Q₈×Dic₆₁	—	2	1	no	—
33	Q₈×Dic₆₁	—	2	1	no	—
34	Q₈×Dic₆₁	C₈	1	4	yes	complemented
35	Q₈×Dic₆₁	D₄	1	2	yes	—
36	Q₈×Dic₆₁	Q₈	1	2	yes	—
37	C₂×C₄⋊Dic₆₁	—	2	1	no	—
38	C₂×C₄⋊Dic₆₁	—	2	1	no	—
39	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₆₁	—	2	1	no	—
42	C₂×C₄×Dic₆₁	—	2	1	no	—
43	C₂×C₄×Dic₆₁	—	2	1	no	—
44	C₂×C₄×Dic₆₁	—	2	1	no	—
45	C₂×C₄×Dic₆₁	—	2	1	no	—
46	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
52	#14519684	—	2	1	no	—
53	#14519684	—	2	1	no	—
54	#14519684	—	2	1	no	—
55	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₆₁	—	2	2	no	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₁	—	2	1	no	—
61	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₆₁	—	2	1	no	—
63	C₂×Q₈×C₆₁	—	2	1	no	—
64	C₂×Q₈×C₆₁	—	2	1	no	—
65	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
66	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
67	C₂×Q₈×C₆₁	M₁₆	1	2	yes	—
68	C₂²×C₄×C₆₁	—	2	1	no	—
69	C₂²×C₄×C₆₁	—	2	1	no	—
70	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₆₁	—	2	1	no	—
74	C₂²×Dic₆₁	—	2	1	no	—
75	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂×#92420	—	2	4	no	—
80	C₂×#92420	—	4	2	no	—
81	C₂×#92420	—	4	2	no	—
82	C₂×#92420	—	4	2	no	—
83	C₂×#92420	—	4	2	no	—
84	C₂×#92420	—	4	2	no	—
85	C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₄⋊C₄×C₆₁	M₁₆	1	2	yes	—
88	C₄²×C₆₁	—	2	4	no	—
89	C₄²×C₆₁	—	2	1	no	—
90	C₄²×C₆₁	—	2	1	no	—
91	C₄⋊Dic₆₁	—	2	1	no	—
92	C₄⋊Dic₆₁	—	2	1	no	—
93	C₄⋊Dic₆₁	—	2	4	no	—
94	C₄⋊Dic₆₁	—	2	2	no	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
97	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
98	C₄×Dic₆₁	—	2	2	no	—
99	C₄×Dic₆₁	—	2	4	no	—
100	C₄×Dic₆₁	—	2	2	no	—
101	C₄×Dic₆₁	—	2	2	no	—
102	C₄×Dic₆₁	—	2	2	no	—
103	C₄×Dic₆₁	—	2	2	no	—
104	C₄×Dic₆₁	—	4	1	no	—
105	C₄×Dic₆₁	—	4	2	no	—
106	C₄×Dic₆₁	—	4	2	no	—
107	C₄×Dic₆₁	—	4	2	no	—
108	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
109	C₄×Dic₆₁	M₁₆	1	2	yes	—
110	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
112	C₄×Dic₆₁	#41	1	1	yes	characteristic
113	C₄×Dic₆₁	#41	1	1	yes	characteristic
114	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Q₈×C₆₁	—	2	2	no	—
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	—	2	2	no	—
118	Q₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	1	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	4	1	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	#104	1	2	yes	—
131	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
137	C₂×Dic₆₁	—	2	1	no	—
138	C₂×Dic₆₁	—	2	1	no	—
139	C₂×Dic₆₁	—	2	1	no	—
140	C₂×Dic₆₁	—	2	1	no	—
141	C₂×Dic₆₁	—	2	1	no	—
142	C₂×Dic₆₁	—	2	1	no	—
143	C₂×Dic₆₁	—	2	2	no	—
144	C₂×Dic₆₁	—	2	2	no	—
145	C₂×Dic₆₁	—	4	1	no	—
146	C₂×Dic₆₁	—	4	1	no	—
147	C₂×Dic₆₁	—	4	1	no	—
148	C₂×Dic₆₁	—	4	1	no	—
149	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
150	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
151	C₂×Dic₆₁	#121	1	1	yes	characteristic
152	C₂×Dic₆₁	#121	1	1	yes	characteristic
153	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×Dic₆₁	#135	1	2	yes	—
156	C₂×Dic₆₁	#137	1	2	yes	—
157	#92420	—	4	4	no	—
158	#92420	—	8	2	no	—
159	#8566	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	#380	1	1	yes	characteristic
161	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₆₁	#407	1	1	yes	characteristic
163	C₂²×C₆₁	#416	1	1	yes	characteristic
164	C₂²×C₆₁	#419	1	1	yes	characteristic
165	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
166	C₂²×C₆₁	#421	1	1	yes	characteristic
167	C₄×C₆₁	—	2	4	no	—
168	C₄×C₆₁	—	2	2	no	—
169	C₄×C₆₁	—	4	1	no	—
170	C₄×C₆₁	—	4	2	no	—
171	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₆₁	#423	1	1	yes	characteristic
173	C₄×C₆₁	#431	1	1	yes	characteristic
174	C₄×C₆₁	#432	1	1	yes	characteristic
175	Dic₆₁	—	2	1	no	—
176	Dic₆₁	—	2	1	no	—
177	Dic₆₁	—	2	2	no	—
178	Dic₆₁	—	2	2	no	—
179	Dic₆₁	—	4	2	no	—
180	Dic₆₁	—	4	2	no	—
181	Dic₆₁	#464	1	2	yes	—
182	Dic₆₁	#479	1	2	yes	—
183	C₄²×Q₈	—	61	1	no	—
184	#1831	—	61	1	no	—
185	C₄×C₄⋊C₈	—	61	1	no	—
186	C₂×Q₈⋊C₈	—	61	2	no	—
187	#1279	—	61	2	no	—
188	C₂×C₆₁	#1476	1	1	yes	characteristic
189	C₂×C₆₁	#1479	1	1	yes	characteristic
190	C₂×C₆₁	#1488	1	1	yes	characteristic
191	C₂×C₆₁	#1491	1	1	yes	characteristic
192	C₂×C₆₁	#1797	1	1	yes	characteristic
193	C₂×C₆₁	#1805	1	1	yes	characteristic
194	C₂×C₆₁	#1806	1	1	yes	characteristic
195	C₂×C₄×Q₈	—	61	1	no	—
196	C₂×C₄×Q₈	—	61	1	no	—
197	C₂×C₄×Q₈	#6923	1	1	yes	characteristic
198	C₂×C₄⋊C₈	—	61	2	no	—
199	C₂×C₄⋊C₈	—	122	1	no	—
200	C₂×C₄×C₈	—	61	2	no	—
201	C₄×C₄⋊C₄	—	61	1	no	—
202	C₄×C₄⋊C₄	—	61	1	no	—
203	C₄×C₄⋊C₄	—	61	1	no	—
204	C₄×C₄⋊C₄	—	122	1	no	—
205	C₄×C₄⋊C₄	—	122	1	no	—
206	C₄×C₄⋊C₄	—	122	1	no	—
207	C₄³	—	61	1	no	—
208	C₄³	—	122	1	no	—
209	#336	—	122	1	no	—
210	#336	—	122	1	no	—
211	#336	—	122	1	no	—
212	Q₈⋊C₈	—	122	4	no	—
213	C₆₁	#8566	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	C₂×#6923	1	1	yes	characteristic
215	C₂²×C₈	—	122	2	no	—
216	C₂²×C₈	—	122	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	61	4	no	—
220	C₄×Q₈	—	61	2	no	—
221	C₄×Q₈	—	61	2	no	—
222	C₄×Q₈	—	122	1	no	—
223	C₄×Q₈	—	122	1	no	—
224	C₄×Q₈	—	122	2	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	61	1	no	—
227	C₂×C₄⋊C₄	—	61	1	no	—
228	C₂×C₄⋊C₄	—	122	1	no	—
229	C₂×C₄⋊C₄	—	122	1	no	—
230	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	61	1	no	—
233	C₂×C₄²	—	61	1	no	—
234	C₂×C₄²	—	61	1	no	—
235	C₂×C₄²	—	61	1	no	—
236	C₂×C₄²	—	61	1	no	—
237	C₂×C₄²	—	61	1	no	—
238	C₂×C₄²	—	122	1	no	—
239	C₂×C₄²	—	122	1	no	—
240	C₂×C₄²	—	122	1	no	—
241	C₂×C₄²	—	122	1	no	—
242	C₂×C₄²	—	122	1	no	—
243	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
244	C₄⋊C₈	—	122	2	no	—
245	C₄⋊C₈	—	122	2	no	—
246	C₄⋊C₈	—	122	4	no	—
247	C₄×C₈	—	122	2	no	—
248	C₄×C₈	—	122	2	no	—
249	#95	—	122	1	no	—
250	#95	—	122	1	no	—
251	#95	—	122	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	D₆₁⋊C₈	1	2	yes	—
256	C₂×Q₈	C₄⋊#6923	1	2	yes	—
257	C₂×Q₈	#11758217	1	2	yes	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	61	1	no	—
261	C₂²×C₄	—	61	1	no	—
262	C₂²×C₄	—	61	1	no	—
263	C₂²×C₄	—	61	1	no	—
264	C₂²×C₄	—	122	1	no	—
265	C₂²×C₄	—	122	1	no	—
266	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
268	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
269	C₂×C₈	—	122	4	no	—
270	C₂×C₈	—	244	2	no	—
271	C₂×C₈	—	244	2	no	—
272	C₂×C₈	—	244	2	no	—
273	C₂×C₈	—	244	2	no	—
274	C₂×C₈	—	244	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	61	2	no	—
278	C₄⋊C₄	—	61	2	no	—
279	C₄⋊C₄	—	122	2	no	—
280	C₄⋊C₄	—	122	2	no	—
281	C₄⋊C₄	—	122	4	no	—
282	C₄⋊C₄	—	122	1	no	—
283	C₄⋊C₄	—	122	1	no	—
284	C₄⋊C₄	#11758217	1	2	yes	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	61	1	no	—
289	C₄²	—	61	1	no	—
290	C₄²	—	61	1	no	—
291	C₄²	—	61	1	no	—
292	C₄²	—	61	2	no	—
293	C₄²	—	61	2	no	—
294	C₄²	—	122	2	no	—
295	C₄²	—	122	2	no	—
296	C₄²	—	122	2	no	—
297	C₄²	—	122	2	no	—
298	C₄²	—	122	4	no	—
299	C₄²	—	122	2	no	—
300	C₄²	—	244	1	no	—
301	C₄²	—	244	2	no	—
302	C₄²	—	244	2	no	—
303	C₄²	—	244	2	no	—
304	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	#14519844	1	4	yes	complemented
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	61	2	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	61	1	no	—
326	C₂×C₄	—	61	2	no	—
327	C₂×C₄	—	61	2	no	—
328	C₂×C₄	—	122	1	no	—
329	C₂×C₄	—	122	1	no	—
330	C₂×C₄	—	122	1	no	—
331	C₂×C₄	—	122	1	no	—
332	C₂×C₄	—	122	1	no	—
333	C₂×C₄	—	122	1	no	—
334	C₂×C₄	—	122	2	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	244	1	no	—
337	C₂×C₄	—	244	1	no	—
338	C₂×C₄	—	244	1	no	—
339	C₂×C₄	—	244	1	no	—
340	C₂×C₄	#14519842	1	2	yes	—
341	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
342	C₂×C₄	#14519856	1	1	yes	characteristic
343	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
344	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
345	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
346	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
347	C₈	—	488	2	no	—
348	C₈	—	244	4	no	—
349	C₂²	#16840704	1	1	yes	characteristic
350	C₂²	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
351	C₂²	#16840732	1	1	yes	characteristic
352	C₂²	#16840752	1	1	yes	characteristic
353	C₂²	#16840755	1	1	yes	characteristic
354	C₂²	C₂×#14519842	1	1	yes	characteristic
355	C₂²	#16840759	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	4	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	61	2	no	—
361	C₄	—	61	2	no	—
362	C₄	—	122	2	no	—
363	C₄	—	122	2	no	—
364	C₄	—	122	1	no	—
365	C₄	—	122	1	no	—
366	C₄	—	244	2	no	—
367	C₄	—	244	2	no	—
368	C₄	C₂×#14519844	1	1	yes	characteristic
369	C₄	#16840765	1	1	yes	characteristic
370	C₄	#16840779	1	1	yes	characteristic
371	C₄	#16840782	1	1	yes	characteristic
372	C₂	#21461113	1	1	yes	characteristic
373	C₂	#21461122	1	1	yes	characteristic
374	C₂	#21461150	1	1	yes	characteristic
375	C₂	#21461161	1	1	yes	characteristic
376	C₂	#21461675	1	1	yes	characteristic
377	C₂	#21461690	1	1	yes	characteristic
378	C₂	#21461692	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×Q₈	#6923
hypocenter	C₆₁	#8566
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
34. Q₈×Dic₆₁	C₈	semidirect
118. Q₈×C₆₁	C₄⋊C₈	semidirect
213. C₆₁	#8566	semidirect
308. Q₈	#14519844	semidirect

Maximal subgroups

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

subgroup	index
#21461736	2
C₄×#14519844	2
C₂×Q₈⋊#92420	2
#21460655	2
C₄×Q₈×Dic₆₁	2
#8566	61