Subgroups — C₄×#16840612 #30597547

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840612	C₂	1	2	yes	maximal
4	#21460655	C₂	1	2	yes	maximal
5	C₄×#14519742	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	—	2	1	no	—
8	C₂×C₄×#92420	—	2	1	no	—
9	C₂×C₄×#92420	—	2	1	no	—
10	C₂×C₄×#92420	C₂²	1	2	yes	—
11	#16840628	—	2	1	no	—
12	#16840612	C₄	1	8	yes	direct product factor
13	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
14	C₂×#14519742	C₄	1	1	yes	characteristic
15	C₂×#14519742	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₆₁	C₄	1	1	yes	characteristic
17	C₄⋊C₄×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₄×#11758196	—	2	1	no	—
19	#16839613	—	2	1	no	—
20	#16839613	—	2	1	no	—
21	C₄²×Dic₆₁	—	2	1	no	—
22	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×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	C₄×#92420	—	2	4	no	—
29	C₄×#92420	—	2	4	no	—
30	C₄×#92420	—	2	2	no	—
31	C₄×#92420	—	4	2	no	—
32	C₄×#92420	C₂×C₄	1	4	yes	—
33	C₂²×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₂₂	—	2	2	no	—
35	C₄×Dic₁₂₂	C₈	1	4	yes	complemented
36	#14519742	—	2	2	no	—
37	#14519742	—	2	2	no	—
38	#14519742	C₈	1	4	yes	complemented
39	#14519742	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758196	—	2	1	no	—
42	C₂×#11758196	—	2	1	no	—
43	C₂×#11758196	—	2	1	no	—
44	C₂×C₄×Dic₆₁	—	2	1	no	—
45	C₂×C₄×Dic₆₁	—	2	1	no	—
46	C₂×C₄×Dic₆₁	—	2	1	no	—
47	C₂×C₄×Dic₆₁	—	2	1	no	—
48	C₂×C₄×Dic₆₁	—	2	1	no	—
49	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
54	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
55	#14519684	—	2	1	no	—
56	#14519684	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	—	2	1	no	—
60	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
61	#95×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₆₁	—	2	1	no	—
68	C₂²×Dic₆₁	—	2	1	no	—
69	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×Dic₁₂₂	—	2	1	no	—
74	C₂×Dic₁₂₂	—	2	1	no	—
75	C₂×Dic₁₂₂	—	2	1	no	—
76	C₂×Dic₁₂₂	C₄²	1	2	yes	—
77	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
78	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
79	C₂×#92420	—	2	4	no	—
80	C₂×#92420	—	2	4	no	—
81	C₂×#92420	—	2	4	no	—
82	C₂×#92420	—	2	4	no	—
83	C₂×#92420	—	4	2	no	—
84	C₂×#92420	—	4	2	no	—
85	C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₄⋊C₄×C₆₁	C₄²	1	2	yes	—
88	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
89	C₄²×C₆₁	—	4	2	no	—
90	C₄²×C₆₁	C₂×C₈	1	2	yes	—
91	C₄²×C₆₁	M₁₆	1	2	yes	—
92	C₄⋊Dic₆₁	—	2	1	no	—
93	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
94	#11758196	—	2	4	no	—
95	#11758196	—	2	4	no	—
96	#11758196	—	2	4	no	—
97	C₄×Dic₆₁	—	2	4	no	—
98	C₄×Dic₆₁	—	2	2	no	—
99	C₄×Dic₆₁	—	2	4	no	—
100	C₄×Dic₆₁	—	2	2	no	—
101	C₄×Dic₆₁	—	2	4	no	—
102	C₄×Dic₆₁	—	2	2	no	—
103	C₄×Dic₆₁	—	4	2	no	—
104	C₄×Dic₆₁	—	4	2	no	—
105	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
106	C₄×Dic₆₁	M₁₆	1	2	yes	—
107	C₄×Dic₆₁	SD₁₆	1	2	yes	—
108	C₄×Dic₆₁	Q₁₆	1	2	yes	—
109	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
110	C₄×Dic₆₁	#41	1	2	yes	—
111	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	4	1	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
122	C₂×C₄×C₆₁	#103	1	2	yes	—
123	C₂×C₄×C₆₁	#104	1	2	yes	—
124	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×C₈	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×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	2	2	no	—
133	C₂×Dic₆₁	—	2	1	no	—
134	C₂×Dic₆₁	—	2	1	no	—
135	C₂×Dic₆₁	—	2	1	no	—
136	C₂×Dic₆₁	—	2	1	no	—
137	C₂×Dic₆₁	—	2	2	no	—
138	C₂×Dic₆₁	—	4	1	no	—
139	C₂×Dic₆₁	—	4	1	no	—
140	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
141	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
142	C₂×Dic₆₁	#103	1	2	yes	—
143	C₂×Dic₆₁	#104	1	2	yes	—
144	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
145	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
146	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
148	C₂×Dic₆₁	#135	1	2	yes	—
149	Dic₁₂₂	—	2	2	no	—
150	Dic₁₂₂	—	2	2	no	—
151	Dic₁₂₂	—	2	2	no	—
152	Dic₁₂₂	C₄×C₈	1	4	yes	complemented
153	#92420	—	2	8	no	—
154	#92420	—	4	4	no	—
155	C₄×Q₈⋊C₈	—	61	1	no	maximal, Sylow
156	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
160	C₂²×C₆₁	#416	1	1	yes	characteristic
161	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
162	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
163	C₄×C₆₁	—	2	2	no	—
164	C₄×C₆₁	—	4	2	no	—
165	C₄×C₆₁	—	4	2	no	—
166	C₄×C₆₁	Q₈⋊C₈	1	4	yes	complemented
167	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
168	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
169	C₄×C₆₁	#431	1	1	yes	characteristic
170	C₄×C₆₁	#431	1	1	yes	characteristic
171	Dic₆₁	—	2	2	no	—
172	Dic₆₁	—	2	4	no	—
173	Dic₆₁	—	2	2	no	—
174	Dic₆₁	—	4	2	no	—
175	Dic₆₁	Q₈⋊C₈	1	4	yes	complemented
176	Dic₆₁	C₄×SD₁₆	1	2	yes	—
177	Dic₆₁	C₄×Q₁₆	1	2	yes	—
178	C₄²×Q₈	—	61	1	no	—
179	C₄×C₄⋊C₈	—	61	1	no	—
180	C₄²×C₈	—	61	1	no	—
181	C₂×Q₈⋊C₈	—	61	2	no	—
182	#1279	—	61	2	no	—
183	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
184	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
185	C₂×C₆₁	#1476	1	1	yes	characteristic
186	C₂×C₆₁	#1476	1	1	yes	characteristic
187	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
188	C₂×C₆₁	C₄×#104	1	1	yes	characteristic
189	C₂×C₆₁	C₄×#103	1	1	yes	characteristic
190	C₂×C₄×Q₈	—	61	1	no	—
191	C₂×C₄×Q₈	—	61	1	no	—
192	C₂×C₄×Q₈	—	61	1	no	—
193	C₂×C₄⋊C₈	—	61	2	no	—
194	C₂×C₄×C₈	—	61	2	no	—
195	C₂×C₄×C₈	—	122	1	no	—
196	C₂×C₄×C₈	—	122	1	no	—
197	C₂×C₄×C₈	—	122	1	no	—
198	C₄×C₄⋊C₄	—	61	1	no	—
199	C₄×C₄⋊C₄	—	61	1	no	—
200	C₄×C₄⋊C₄	—	122	1	no	—
201	C₄×C₄⋊C₄	—	122	1	no	—
202	C₄×C₄⋊C₄	—	122	1	no	—
203	C₄×C₄⋊C₄	#6923	1	1	yes	characteristic
204	C₄³	—	61	1	no	—
205	C₄³	—	122	1	no	—
206	#336	—	122	1	no	—
207	Q₈⋊C₈	—	61	8	no	—
208	C₆₁	C₄×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	61	1	no	—
210	C₂²×C₈	—	122	2	no	—
211	C₂²×C₈	—	122	2	no	—
212	C₄×Q₈	—	61	4	no	—
213	C₄×Q₈	—	61	4	no	—
214	C₄×Q₈	—	61	4	no	—
215	C₄×Q₈	—	122	2	no	—
216	C₄×Q₈	—	122	2	no	—
217	C₄×Q₈	—	122	2	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	—	122	1	no	—
222	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
224	C₂×C₄²	—	2	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	—	61	1	no	—
230	C₂×C₄²	—	61	1	no	—
231	C₂×C₄²	—	122	1	no	—
232	C₂×C₄²	—	122	1	no	—
233	C₂×C₄²	—	122	1	no	—
234	C₂×C₄²	—	122	1	no	—
235	C₂×C₄²	—	122	1	no	—
236	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
237	C₄⋊C₈	—	61	4	no	—
238	C₄⋊C₈	—	122	2	no	—
239	C₄×C₈	—	61	4	no	—
240	C₄×C₈	—	122	2	no	—
241	C₄×C₈	—	122	4	no	—
242	C₄×C₈	—	122	4	no	—
243	C₄×C₈	—	244	2	no	—
244	#95	—	2	1	no	—
245	#95	—	122	1	no	—
246	#95	—	122	1	no	—
247	C₂×Q₈	—	61	2	no	—
248	C₂×Q₈	—	61	2	no	—
249	C₂×Q₈	—	61	2	no	—
250	C₂×Q₈	—	122	1	no	—
251	C₂×Q₈	—	122	1	no	—
252	C₂×Q₈	—	122	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	61	1	no	—
256	C₂²×C₄	—	61	1	no	—
257	C₂²×C₄	—	61	1	no	—
258	C₂²×C₄	—	61	1	no	—
259	C₂²×C₄	—	122	1	no	—
260	C₂²×C₄	—	122	1	no	—
261	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
263	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
264	C₂×C₈	—	122	4	no	—
265	C₂×C₈	—	122	4	no	—
266	C₂×C₈	—	122	4	no	—
267	C₂×C₈	—	122	4	no	—
268	C₂×C₈	—	244	2	no	—
269	C₂×C₈	—	244	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	61	2	no	—
273	C₄⋊C₄	—	122	4	no	—
274	C₄⋊C₄	—	122	4	no	—
275	C₄⋊C₄	—	122	4	no	—
276	C₄⋊C₄	—	122	1	no	—
277	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
278	C₄⋊C₄	C₄×#6923	1	2	yes	—
279	C₄²	—	4	2	no	—
280	C₄²	—	61	2	no	—
281	C₄²	—	61	2	no	—
282	C₄²	—	61	2	no	—
283	C₄²	—	61	2	no	—
284	C₄²	—	61	2	no	—
285	C₄²	—	61	2	no	—
286	C₄²	—	122	2	no	—
287	C₄²	—	122	2	no	—
288	C₄²	—	122	4	no	—
289	C₄²	—	122	4	no	—
290	C₄²	—	122	4	no	—
291	C₄²	—	122	2	no	—
292	C₄²	—	244	2	no	—
293	C₄²	—	244	2	no	—
294	C₄²	D₆₁⋊C₈	1	2	yes	—
295	C₄²	#11758213	1	2	yes	—
296	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
297	Q₈	—	61	4	no	—
298	Q₈	—	122	2	no	—
299	Q₈	—	122	2	no	—
300	Q₈	—	122	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	61	2	no	—
310	C₂×C₄	—	61	2	no	—
311	C₂×C₄	—	61	1	no	—
312	C₂×C₄	—	61	1	no	—
313	C₂×C₄	—	61	2	no	—
314	C₂×C₄	—	61	2	no	—
315	C₂×C₄	—	61	2	no	—
316	C₂×C₄	—	61	2	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	122	2	no	—
319	C₂×C₄	—	122	2	no	—
320	C₂×C₄	—	122	1	no	—
321	C₂×C₄	—	122	1	no	—
322	C₂×C₄	—	122	1	no	—
323	C₂×C₄	—	122	1	no	—
324	C₂×C₄	—	122	2	no	—
325	C₂×C₄	—	122	2	no	—
326	C₂×C₄	—	244	1	no	—
327	C₂×C₄	—	244	1	no	—
328	C₂×C₄	#14519834	1	2	yes	—
329	C₂×C₄	#14519834	1	2	yes	—
330	C₂×C₄	#14519840	1	2	yes	—
331	C₂×C₄	#14519841	1	2	yes	—
332	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
333	C₂×C₄	#14519856	1	1	yes	characteristic
334	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
335	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
336	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
337	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
338	C₈	—	122	8	no	—
339	C₈	—	244	4	no	—
340	C₂²	C₄×C₂²⋊#6923	1	1	yes	characteristic
341	C₂²	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
342	C₂²	C₂×#14519834	1	1	yes	characteristic
343	C₂²	C₂×#14519840	1	1	yes	characteristic
344	C₂²	#16840752	1	1	yes	characteristic
345	C₂²	C₂×#14519841	1	1	yes	characteristic
346	C₂²	D₆₁⋊#104	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	61	2	no	—
351	C₄	—	61	2	no	—
352	C₄	—	61	4	no	—
353	C₄	—	122	4	no	—
354	C₄	—	122	2	no	—
355	C₄	—	122	2	no	—
356	C₄	—	244	2	no	—
357	C₄	#16840612	1	4	yes	direct product factor
358	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
359	C₄	C₄×#11758213	1	1	yes	characteristic
360	C₄	#16840780	1	1	yes	characteristic
361	C₄	#16840781	1	1	yes	characteristic
362	C₂	D₆₁⋊(Q₈⋊C₈)	1	1	yes	characteristic
363	C₂	C₂×#16840612	1	1	yes	characteristic
364	C₂	#21461111	1	1	yes	characteristic
365	C₂	#21461112	1	1	yes	characteristic
366	C₂	C₄×#14519834	1	1	yes	characteristic
367	C₂	C₄×#14519841	1	1	yes	characteristic
368	C₂	C₄×#14519840	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
12. #16840612	C₄	direct (N × H)
35. C₄×Dic₁₂₂	C₈	semidirect
38. #14519742	C₈	semidirect
152. Dic₁₂₂	C₄×C₈	semidirect
166. C₄×C₆₁	Q₈⋊C₈	semidirect
175. Dic₆₁	Q₈⋊C₈	semidirect
208. C₆₁	C₄×Q₈⋊C₈	semidirect
357. C₄	#16840612	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#92420	2
C₄²×#92420	2
C₂×#16840612	2
#21460655	2
C₄×#14519742	2
C₄×Q₈⋊C₈	61

Subgroups — C4×#16840612 #30597547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840612 #30597547