Subgroups — C₄⋊C₄×Dic₃₃₂ #35422837

Order 21248.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₃₂	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₆₆	C₂	1	2	yes	maximal
4	#24908196	C₂	1	2	yes	maximal
5	#24907720	C₂	1	4	yes	maximal
6	#24907623	C₂	1	1	yes	maximal, characteristic
7	#24907469	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₆₆	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₆₆	C₂²	1	2	yes	—
12	C₄×Dic₃₃₂	—	2	8	no	—
13	C₄×Dic₃₃₂	C₄	1	8	yes	complemented
14	C₂×#14581726	C₂²	1	2	yes	—
15	C₂×#14581726	C₂²	1	2	yes	—
16	C₂×#14581726	C₂²	1	1	yes	characteristic
17	C₂×#14581717	C₂²	1	4	yes	—
18	C₂×#14581717	C₂²	1	2	yes	—
19	#19127599	C₂²	1	4	yes	—
20	#19127598	—	2	4	no	—
21	#19127571	C₂²	1	2	yes	—
22	#19127570	—	2	2	no	—
23	C₄⋊C₄×Dic₈₃	—	2	2	no	—
24	#19127464	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
29	#336×C₈₃	C₂²	1	2	yes	—
30	C₂²×Dic₁₆₆	C₂³	1	2	yes	—
31	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₃₃₂	D₄	1	4	yes	—
33	C₂×Dic₃₃₂	Q₈	1	4	yes	—
34	C₄×Dic₁₆₆	—	2	8	no	—
35	C₄×Dic₁₆₆	—	2	4	no	—
36	C₄×Dic₁₆₆	—	4	4	no	—
37	C₄×Dic₁₆₆	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊Dic₈₃	—	2	2	no	—
39	C₂×C₄⋊Dic₈₃	—	2	2	no	—
40	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₈₃	C₂³	1	2	yes	—
42	C₂×Dic₈₃⋊C₄	—	2	2	no	—
43	C₂×Dic₈₃⋊C₄	—	2	4	no	—
44	C₂×Dic₈₃⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₈₃	—	2	2	no	—
46	C₂×C₄×Dic₈₃	—	2	4	no	—
47	#14581726	—	2	4	no	—
48	#14581726	—	4	2	no	—
49	#14581726	C₂×C₄	1	4	yes	—
50	#14581717	—	2	8	no	—
51	#14581717	C₂×C₄	1	8	yes	—
52	C₈₃⋊#95	—	2	4	no	—
53	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₃	—	2	2	no	—
61	C₄×C₈×C₈₃	—	2	4	no	—
62	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
63	#95×C₈₃	D₄	1	2	yes	—
64	C₂×Dic₁₆₆	—	2	2	no	—
65	C₂×Dic₁₆₆	—	2	2	no	—
66	C₂×Dic₁₆₆	—	2	2	no	—
67	C₂×Dic₁₆₆	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₆₆	C₂×D₄	1	4	yes	—
69	C₂×Dic₁₆₆	C₂×Q₈	1	4	yes	—
70	C₂²×Dic₈₃	—	2	2	no	—
71	C₂²×Dic₈₃	—	2	2	no	—
72	C₂²×Dic₈₃	—	2	4	no	—
73	Dic₃₃₂	C₄⋊C₄	1	16	yes	direct product factor
74	C₄⋊Dic₈₃	—	2	2	no	—
75	C₄⋊Dic₈₃	—	2	4	no	—
76	C₄⋊Dic₈₃	—	2	4	no	—
77	C₄⋊Dic₈₃	—	4	4	no	—
78	C₄⋊Dic₈₃	—	8	2	no	—
79	C₄⋊Dic₈₃	C₂²×C₄	1	4	yes	—
80	Dic₈₃⋊C₄	—	2	8	no	—
81	Dic₈₃⋊C₄	—	4	8	no	—
82	Dic₈₃⋊C₄	—	8	4	no	—
83	C₄×Dic₈₃	—	2	8	no	—
84	C₄×Dic₈₃	—	4	8	no	—
85	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₈₃	—	2	1	no	—
91	C₂×C₈×C₈₃	—	2	1	no	—
92	C₂×C₈×C₈₃	—	2	2	no	—
93	C₂×C₈×C₈₃	—	2	2	no	—
94	C₂×C₈×C₈₃	—	2	2	no	—
95	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
99	C₄⋊C₄×C₈₃	—	2	2	no	—
100	C₄⋊C₄×C₈₃	Q₁₆	1	4	yes	complemented
101	C₄²×C₈₃	—	2	2	no	—
102	C₄²×C₈₃	—	2	2	no	—
103	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₈₃	#41	1	2	yes	—
105	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₃	—	2	2	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	2	no	—
110	C₂×C₄×C₈₃	—	2	2	no	—
111	C₂×C₄×C₈₃	—	2	2	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
116	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₈₃	#135	1	2	yes	—
118	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Dic₈₃	—	2	4	no	—
125	C₂×Dic₈₃	—	2	4	no	—
126	C₂×Dic₈₃	—	2	4	no	—
127	C₂×Dic₈₃	—	2	4	no	—
128	C₂×Dic₈₃	—	4	2	no	—
129	C₂×Dic₈₃	—	4	2	no	—
130	C₂×Dic₈₃	—	4	4	no	—
131	C₂×Dic₈₃	—	4	4	no	—
132	C₂×Dic₈₃	—	4	4	no	—
133	Dic₁₆₆	—	2	4	no	—
134	Dic₁₆₆	—	2	4	no	—
135	Dic₁₆₆	—	2	4	no	—
136	Dic₁₆₆	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₈₃	—	2	2	no	—
138	C₈×C₈₃	—	4	2	no	—
139	C₈×C₈₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₈₃	D₄²	1	1	yes	characteristic
142	C₂²×C₈₃	D₄×Q₈	1	1	yes	characteristic
143	C₂²×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
144	C₂²×C₈₃	C₂×#135	1	1	yes	characteristic
145	C₂²×C₈₃	#576	1	1	yes	characteristic
146	C₂²×C₈₃	#578	1	1	yes	characteristic
147	C₄×C₈₃	—	2	2	no	—
148	C₄×C₈₃	—	2	4	no	—
149	C₄×C₈₃	—	4	2	no	—
150	C₄×C₈₃	C₄×Q₁₆	1	4	yes	complemented
151	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈₃	#520	1	1	yes	characteristic
153	C₄×C₈₃	#547	1	1	yes	characteristic
154	C₄×C₈₃	#554	1	1	yes	characteristic
155	Dic₈₃	—	2	8	no	—
156	Dic₈₃	—	4	4	no	—
157	Dic₈₃	—	8	4	no	—
158	C₄⋊C₄×Q₁₆	—	83	1	no	maximal, Sylow
159	C₂×C₈₃	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₈₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
161	C₂×C₈₃	#2951	1	1	yes	characteristic
162	C₂×C₈₃	D₄×Q₁₆	1	1	yes	characteristic
163	C₂×C₈₃	#3287	1	1	yes	characteristic
164	C₂×C₈₃	Q₈×Q₁₆	1	1	yes	characteristic
165	C₂×C₈₃	#3383	1	1	yes	characteristic
166	C₂×C₄×Q₁₆	—	83	1	no	—
167	C₂×C₄×Q₁₆	—	83	2	no	—
168	Q₈×C₄⋊C₄	—	83	2	no	—
169	#1942	—	83	2	no	—
170	#1863	—	83	4	no	—
171	#1845	—	83	1	no	—
172	#1803	—	83	2	no	—
173	C₈×C₄⋊C₄	D₈₃	1	1	yes	characteristic
174	C₈₃	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	83	1	no	—
176	C₂×C₄×Q₈	—	83	4	no	—
177	C₂×C₄×Q₈	—	83	2	no	—
178	C₄×Q₁₆	—	83	8	no	—
179	C₄×Q₁₆	—	166	8	no	—
180	C₂×#107	—	83	2	no	—
181	C₂×#107	—	83	1	no	—
182	C₂×#107	—	83	2	no	—
183	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
184	C₂×#103	—	83	2	no	—
185	C₂×#103	—	83	4	no	—
186	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
187	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
188	#389	—	83	4	no	—
189	#389	—	166	4	no	—
190	#384	—	83	2	no	—
191	#384	—	166	2	no	—
192	C₄×C₄⋊C₄	—	166	2	no	—
193	C₄×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
194	#340	—	83	4	no	—
195	#336	C₂×D₈₃	1	2	yes	—
196	C₂²×Q₈	—	83	2	no	—
197	C₂×Q₁₆	—	83	4	no	—
198	C₂×Q₁₆	—	83	4	no	—
199	C₂×Q₁₆	—	83	4	no	—
200	C₂²×C₈	C₂²×D₈₃	1	2	yes	—
201	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
203	C₄×Q₈	—	332	4	no	—
204	C₄×Q₈	—	83	8	no	—
205	C₄×Q₈	—	166	8	no	—
206	C₄×Q₈	—	166	4	no	—
207	C₂×C₄⋊C₄	—	83	2	no	—
208	C₂×C₄⋊C₄	—	83	4	no	—
209	C₂×C₄⋊C₄	—	166	4	no	—
210	C₂×C₄⋊C₄	—	166	2	no	—
211	C₂×C₄⋊C₄	—	166	2	no	—
212	C₂×C₄⋊C₄	—	166	4	no	—
213	C₂×C₄⋊C₄	—	166	2	no	—
214	C₂×C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
216	C₂×C₄²	—	166	4	no	—
217	C₂×C₄²	—	166	2	no	—
218	C₂×C₄²	C₂²×D₈₃	1	2	yes	—
219	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
220	#107	—	332	2	no	—
221	#107	—	83	4	no	—
222	#107	—	166	4	no	—
223	C₄⋊C₈	—	2	2	no	—
224	#103	—	83	8	no	—
225	#103	—	166	8	no	—
226	C₄×C₈	—	2	4	no	—
227	C₄×C₈	C₄×D₈₃	1	4	yes	—
228	#95	—	166	4	no	—
229	#95	D₃₃₂	1	2	yes	—
230	C₂×Q₈	—	83	4	no	—
231	C₂×Q₈	—	83	4	no	—
232	C₂×Q₈	—	83	4	no	—
233	C₂×Q₈	—	166	2	no	—
234	C₂×Q₈	—	166	2	no	—
235	C₂×Q₈	—	166	2	no	—
236	C₂²×C₄	—	166	4	no	—
237	C₂²×C₄	—	166	2	no	—
238	C₂²×C₄	—	166	2	no	—
239	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
240	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
241	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
244	Q₁₆	—	83	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
252	C₂×C₈	D₄×D₈₃	1	2	yes	—
253	C₂×C₈	Q₈×D₈₃	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	332	8	no	—
256	C₄⋊C₄	—	332	4	no	—
257	C₄⋊C₄	—	83	4	no	—
258	C₄⋊C₄	—	664	4	no	—
259	C₄⋊C₄	—	664	2	no	—
260	C₄⋊C₄	—	166	4	no	—
261	C₄⋊C₄	—	166	4	no	—
262	C₄⋊C₄	—	166	2	no	—
263	C₄⋊C₄	—	166	8	no	—
264	C₄⋊C₄	Dic₃₃₂	1	4	yes	direct product factor
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	332	8	no	—
268	C₄²	—	166	8	no	—
269	C₄²	C₂×C₄×D₈₃	1	2	yes	—
270	C₄²	#13050590	1	2	yes	—
271	C₂³	C₂²×D₃₃₂	1	1	yes	characteristic
272	Q₈	—	83	8	no	—
273	Q₈	—	166	4	no	—
274	Q₈	—	166	4	no	—
275	Q₈	—	166	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	332	2	no	—
284	C₂×C₄	—	332	2	no	—
285	C₂×C₄	—	332	4	no	—
286	C₂×C₄	—	332	4	no	—
287	C₂×C₄	—	332	4	no	—
288	C₂×C₄	—	166	4	no	—
289	C₂×C₄	—	166	4	no	—
290	C₂×C₄	—	166	4	no	—
291	C₂×C₄	—	166	4	no	—
292	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
293	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
294	C₂×C₄	C₂×Dic₃₃₂	1	4	yes	—
295	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
296	C₂×C₄	#14581814	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
301	C₂×C₄	#14581838	1	1	yes	characteristic
302	C₂×C₄	#14581839	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	C₄⋊C₄×D₈₃	1	4	yes	—
306	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
307	C₂²	D₄×D₃₃₂	1	1	yes	characteristic
308	C₂²	Q₈×D₃₃₂	1	1	yes	characteristic
309	C₂²	C₂²×Dic₃₃₂	1	1	yes	characteristic
310	C₂²	C₂×#14581814	1	1	yes	characteristic
311	C₂²	#19128506	1	1	yes	characteristic
312	C₂²	#19128518	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	332	4	no	—
317	C₄	—	664	4	no	—
318	C₄	—	166	8	no	—
319	C₄	C₄×Dic₃₃₂	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
321	C₄	#19128181	1	1	yes	characteristic
322	C₄	#19128356	1	1	yes	characteristic
323	C₄	#19128411	1	1	yes	characteristic
324	C₂	C₄⋊C₄×D₃₃₂	1	1	yes	characteristic
325	C₂	C₂×C₄×Dic₃₃₂	1	1	yes	characteristic
326	C₂	#24916340	1	1	yes	characteristic
327	C₂	D₄×Dic₃₃₂	1	1	yes	characteristic
328	C₂	#24919223	1	1	yes	characteristic
329	C₂	Q₈×Dic₃₃₂	1	1	yes	characteristic
330	C₂	#24920355	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈×C₄⋊C₄×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×D₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₈₃
hypocenter	C₈₃	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₃₃₂	C₄	semidirect
73. Dic₃₃₂	C₄⋊C₄	direct (N × H)
100. C₄⋊C₄×C₈₃	Q₁₆	semidirect
150. C₄×C₈₃	C₄×Q₁₆	semidirect
174. C₈₃	C₄⋊C₄×Q₁₆	semidirect
264. C₄⋊C₄	Dic₃₃₂	direct (N × H)
319. C₄	C₄×Dic₃₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₃₂	2
C₄⋊C₄×Dic₁₆₆	2
#24908196	2
#24907720	2
#24907623	2
#24907469	2
C₈×C₄⋊C₄×C₈₃	2
C₄⋊C₄×Q₁₆	83

Subgroups — C4⋊C4×Dic332 #35422837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×Dic₃₃₂ #35422837