Subgroups — #30597564

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461605	C₂	1	1	yes	maximal, characteristic
2	#21461534	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×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×#14519832	—	2	1	no	—
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	C₄	1	8	yes	complemented
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	C₄⋊#92420	—	2	2	no	—
27	C₄⋊#92420	C₂×C₄	1	4	yes	—
28	#14519832	—	4	2	no	—
29	C₄×#92420	—	2	2	no	—
30	C₄×#92420	C₂×C₄	1	4	yes	—
31	Q₈×Dic₆₁	—	2	2	no	—
32	Q₈×Dic₆₁	—	2	2	no	—
33	Q₈×Dic₆₁	—	2	2	no	—
34	Q₈×Dic₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊Dic₆₁	—	2	1	no	—
36	C₂×C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	—	2	1	no	—
40	C₂×C₄×Dic₆₁	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₆₁	D₄	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	C₂³	1	1	yes	characteristic
50	#14519684	—	2	1	no	—
51	#14519684	—	2	1	no	—
52	#14519684	—	2	1	no	—
53	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₆₁	—	2	2	no	—
55	C₄×Q₈×C₆₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₁	—	2	1	no	—
59	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₆₁	—	2	1	no	—
61	C₂×Q₈×C₆₁	—	2	1	no	—
62	C₂×Q₈×C₆₁	—	2	1	no	—
63	C₂×Q₈×C₆₁	C₄²	1	2	yes	—
64	C₂×Q₈×C₆₁	M₁₆	1	2	yes	—
65	C₂×Q₈×C₆₁	M₁₆	1	2	yes	—
66	C₂²×C₄×C₆₁	—	2	1	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₆₁	—	2	1	no	—
72	C₂²×Dic₆₁	—	2	1	no	—
73	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×#92420	—	2	4	no	—
78	C₂×#92420	—	2	4	no	—
79	C₂×#92420	—	4	2	no	—
80	C₂×#92420	—	4	2	no	—
81	C₂×#92420	—	4	2	no	—
82	C₂×#92420	—	4	2	no	—
83	C₄⋊C₄×C₆₁	—	2	4	no	—
84	C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₄⋊C₄×C₆₁	M₁₆	1	2	yes	—
86	C₄²×C₆₁	—	2	4	no	—
87	C₄²×C₆₁	—	2	1	no	—
88	C₄²×C₆₁	—	2	1	no	—
89	C₄⋊Dic₆₁	—	2	4	no	—
90	C₄⋊Dic₆₁	—	2	1	no	—
91	C₄⋊Dic₆₁	—	2	1	no	—
92	C₄⋊Dic₆₁	—	2	4	no	—
93	C₄⋊Dic₆₁	C₄²	1	2	yes	—
94	C₄⋊Dic₆₁	M₁₆	1	2	yes	—
95	C₄×Dic₆₁	—	2	4	no	—
96	C₄×Dic₆₁	—	2	1	no	—
97	C₄×Dic₆₁	—	2	1	no	—
98	C₄×Dic₆₁	—	2	4	no	—
99	C₄×Dic₆₁	—	2	2	no	—
100	C₄×Dic₆₁	—	4	1	no	—
101	C₄×Dic₆₁	—	4	1	no	—
102	C₄×Dic₆₁	—	4	2	no	—
103	C₄×Dic₆₁	—	4	2	no	—
104	C₄×Dic₆₁	—	4	2	no	—
105	C₄×Dic₆₁	SD₁₆	1	2	yes	—
106	C₄×Dic₆₁	Q₁₆	1	2	yes	—
107	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
108	C₄×Dic₆₁	#41	1	2	yes	—
109	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₆₁	—	2	2	no	—
111	Q₈×C₆₁	—	2	2	no	—
112	Q₈×C₆₁	—	2	2	no	—
113	Q₈×C₆₁	#97	1	4	yes	complemented
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	4	1	no	—
124	C₂×C₄×C₆₁	—	4	1	no	—
125	C₂×C₄×C₆₁	#103	1	2	yes	—
126	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
132	C₂×Dic₆₁	—	2	1	no	—
133	C₂×Dic₆₁	—	2	1	no	—
134	C₂×Dic₆₁	—	2	1	no	—
135	C₂×Dic₆₁	—	2	1	no	—
136	C₂×Dic₆₁	—	2	2	no	—
137	C₂×Dic₆₁	—	2	1	no	—
138	C₂×Dic₆₁	—	2	2	no	—
139	C₂×Dic₆₁	—	2	1	no	—
140	C₂×Dic₆₁	—	2	1	no	—
141	C₂×Dic₆₁	—	4	1	no	—
142	C₂×Dic₆₁	—	4	1	no	—
143	C₂×Dic₆₁	—	4	1	no	—
144	C₂×Dic₆₁	—	4	1	no	—
145	C₂×Dic₆₁	#103	1	2	yes	—
146	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
147	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
148	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
150	C₂×Dic₆₁	#135	1	2	yes	—
151	#92420	—	4	4	no	—
152	#92420	—	4	4	no	—
153	#8175	—	61	1	no	maximal, Sylow
154	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₆₁	#407	1	1	yes	characteristic
156	C₂²×C₆₁	#407	1	1	yes	characteristic
157	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
158	C₂²×C₆₁	#416	1	1	yes	characteristic
159	C₂²×C₆₁	#421	1	1	yes	characteristic
160	C₂²×C₆₁	#421	1	1	yes	characteristic
161	C₄×C₆₁	—	2	2	no	—
162	C₄×C₆₁	—	2	4	no	—
163	C₄×C₆₁	—	4	1	no	—
164	C₄×C₆₁	—	4	2	no	—
165	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
166	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
167	C₄×C₆₁	#432	1	1	yes	characteristic
168	C₄×C₆₁	#432	1	1	yes	characteristic
169	Dic₆₁	—	2	2	no	—
170	Dic₆₁	—	2	2	no	—
171	Dic₆₁	—	4	1	no	—
172	Dic₆₁	—	4	2	no	—
173	Dic₆₁	—	4	2	no	—
174	Dic₆₁	C₄×SD₁₆	1	2	yes	—
175	Dic₆₁	C₄×Q₁₆	1	2	yes	—
176	C₄²×Q₈	—	61	1	no	—
177	#1767	—	61	1	no	—
178	#1744	—	61	1	no	—
179	C₂×Q₈⋊C₈	—	61	2	no	—
180	#1279	—	61	2	no	—
181	C₂×C₆₁	#1487	1	1	yes	characteristic
182	C₂×C₆₁	#1487	1	1	yes	characteristic
183	C₂×C₆₁	#1488	1	1	yes	characteristic
184	C₂×C₆₁	#1488	1	1	yes	characteristic
185	C₂×C₆₁	#1749	1	1	yes	characteristic
186	C₂×C₆₁	#1759	1	1	yes	characteristic
187	C₂×C₆₁	C₄×#103	1	1	yes	characteristic
188	C₂×C₄×Q₈	—	61	1	no	—
189	C₂×C₄×Q₈	—	61	1	no	—
190	C₂×C₄×Q₈	#6923	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	61	2	no	—
192	C₂×#97	—	122	1	no	—
193	C₂×C₄×C₈	—	61	2	no	—
194	C₄×C₄⋊C₄	—	61	1	no	—
195	C₄×C₄⋊C₄	—	61	1	no	—
196	C₄×C₄⋊C₄	—	61	1	no	—
197	C₄×C₄⋊C₄	—	122	1	no	—
198	C₄×C₄⋊C₄	—	122	1	no	—
199	C₄×C₄⋊C₄	—	122	1	no	—
200	C₄³	—	61	1	no	—
201	C₄³	—	122	1	no	—
202	#336	—	122	1	no	—
203	#336	—	122	1	no	—
204	#336	—	122	1	no	—
205	Q₈⋊C₈	—	61	8	no	—
206	C₆₁	#8175	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	C₂×#6923	1	1	yes	characteristic
208	C₂²×C₈	—	122	2	no	—
209	C₂²×C₈	—	122	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	61	4	no	—
213	C₄×Q₈	—	122	2	no	—
214	C₄×Q₈	—	122	2	no	—
215	C₄×Q₈	—	122	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	61	1	no	—
218	C₂×C₄⋊C₄	—	61	1	no	—
219	C₂×C₄⋊C₄	—	122	1	no	—
220	C₂×C₄⋊C₄	—	122	1	no	—
221	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	—	61	1	no	—
224	C₂×C₄²	—	61	1	no	—
225	C₂×C₄²	—	61	1	no	—
226	C₂×C₄²	—	61	1	no	—
227	C₂×C₄²	—	61	1	no	—
228	C₂×C₄²	—	61	1	no	—
229	C₂×C₄²	—	122	1	no	—
230	C₂×C₄²	—	122	1	no	—
231	C₂×C₄²	—	122	1	no	—
232	C₂×C₄²	—	122	1	no	—
233	C₂×C₄²	—	122	1	no	—
234	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
235	C₄⋊C₈	—	61	4	no	—
236	C₄⋊C₈	—	122	2	no	—
237	#97	—	244	2	no	—
238	C₄×C₈	—	61	4	no	—
239	C₄×C₈	—	122	2	no	—
240	#95	—	122	1	no	—
241	#95	—	122	1	no	—
242	#95	—	122	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	#11758213	1	2	yes	—
247	C₂×Q₈	C₄×#6923	1	2	yes	—
248	C₂×Q₈	#11758217	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	61	1	no	—
252	C₂²×C₄	—	61	1	no	—
253	C₂²×C₄	—	61	1	no	—
254	C₂²×C₄	—	61	1	no	—
255	C₂²×C₄	—	122	1	no	—
256	C₂²×C₄	—	122	1	no	—
257	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
259	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
260	C₂×C₈	—	122	4	no	—
261	C₂×C₈	—	122	4	no	—
262	C₂×C₈	—	244	2	no	—
263	C₂×C₈	—	244	2	no	—
264	C₂×C₈	—	244	2	no	—
265	C₂×C₈	—	244	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	61	2	no	—
269	C₄⋊C₄	—	61	2	no	—
270	C₄⋊C₄	—	122	4	no	—
271	C₄⋊C₄	—	122	4	no	—
272	C₄⋊C₄	—	122	1	no	—
273	C₄⋊C₄	—	122	1	no	—
274	C₄⋊C₄	#11758217	1	2	yes	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	61	2	no	—
279	C₄²	—	61	2	no	—
280	C₄²	—	61	2	no	—
281	C₄²	—	61	2	no	—
282	C₄²	—	122	2	no	—
283	C₄²	—	122	4	no	—
284	C₄²	—	122	4	no	—
285	C₄²	—	122	1	no	—
286	C₄²	—	122	1	no	—
287	C₄²	—	244	1	no	—
288	C₄²	—	244	1	no	—
289	C₄²	—	244	2	no	—
290	C₄²	—	244	2	no	—
291	C₄²	—	244	2	no	—
292	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	2	2	no	—
296	Q₈	#14519832	1	4	yes	complemented
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	61	1	no	—
309	C₂×C₄	—	61	1	no	—
310	C₂×C₄	—	61	2	no	—
311	C₂×C₄	—	61	2	no	—
312	C₂×C₄	—	61	2	no	—
313	C₂×C₄	—	61	2	no	—
314	C₂×C₄	—	122	1	no	—
315	C₂×C₄	—	122	1	no	—
316	C₂×C₄	—	122	1	no	—
317	C₂×C₄	—	122	1	no	—
318	C₂×C₄	—	122	2	no	—
319	C₂×C₄	—	122	1	no	—
320	C₂×C₄	—	122	1	no	—
321	C₂×C₄	—	122	1	no	—
322	C₂×C₄	—	122	2	no	—
323	C₂×C₄	—	244	1	no	—
324	C₂×C₄	—	244	1	no	—
325	C₂×C₄	—	244	1	no	—
326	C₂×C₄	—	244	1	no	—
327	C₂×C₄	#14519840	1	2	yes	—
328	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
329	C₂×C₄	#14519856	1	1	yes	characteristic
330	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
331	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
332	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
333	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
334	C₈	—	244	4	no	—
335	C₈	—	244	4	no	—
336	C₂²	C₄×C₂²⋊#6923	1	1	yes	characteristic
337	C₂²	#16840732	1	1	yes	characteristic
338	C₂²	#16840733	1	1	yes	characteristic
339	C₂²	C₂×#14519840	1	1	yes	characteristic
340	C₂²	#16840752	1	1	yes	characteristic
341	C₂²	#16840759	1	1	yes	characteristic
342	C₂²	#16840761	1	1	yes	characteristic
343	C₄	—	2	4	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	61	2	no	—
348	C₄	—	61	2	no	—
349	C₄	—	122	2	no	—
350	C₄	—	122	2	no	—
351	C₄	—	244	2	no	—
352	C₄	—	244	1	no	—
353	C₄	—	244	2	no	—
354	C₄	C₂×#14519832	1	1	yes	characteristic
355	C₄	C₄×#11758217	1	1	yes	characteristic
356	C₄	#16840782	1	1	yes	characteristic
357	C₄	#16840783	1	1	yes	characteristic
358	C₂	#21461147	1	1	yes	characteristic
359	C₂	#21461148	1	1	yes	characteristic
360	C₂	#21461150	1	1	yes	characteristic
361	C₂	#21461153	1	1	yes	characteristic
362	C₂	#21461545	1	1	yes	characteristic
363	C₂	#21461583	1	1	yes	characteristic
364	C₂	C₄×#14519840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
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	#8175
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
12. Q₈⋊#92420	C₄	semidirect
113. Q₈×C₆₁	#97	semidirect
206. C₆₁	#8175	semidirect
296. Q₈	#14519832	semidirect

Maximal subgroups

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

subgroup	index
#21461605	2
#21461534	2
C₂×Q₈⋊#92420	2
#21460655	2
C₄×Q₈×Dic₆₁	2
#8175	61