Subgroups — D₄×#15667958 #36711154

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667958	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26022504	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022341	C₂	1	2	yes	maximal, complemented
5	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022299	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022186	C₂	1	2	yes	maximal, complemented
8	#26022185	C₂	1	2	yes	maximal, complemented
9	C₄×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×#13083549	C₂	1	1	yes	maximal, characteristic
11	D₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
13	D₄×Dic₁₇₈	—	2	1	no	—
14	C₂×#15667958	—	2	8	no	—
15	C₂×#15667958	C₂²	1	2	yes	—
16	C₂×#15667958	C₂²	1	2	yes	—
17	C₂×#15667958	C₂²	1	2	yes	—
18	C₂×#15667958	C₂²	1	2	yes	—
19	C₂×#15667958	C₂²	1	1	yes	characteristic
20	C₂²×#13083549	C₂²	1	2	yes	—
21	#19151978	C₂²	1	1	yes	characteristic
22	#19151839	C₂²	1	2	yes	—
23	#19151809	—	2	1	no	—
24	#19151756	—	2	2	no	—
25	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
26	#19151423	C₂²	1	1	yes	characteristic
27	C₄⋊#13083549	C₂²	1	1	yes	characteristic
28	#19151402	C₂²	1	1	yes	characteristic
29	#19151402	C₂²	1	2	yes	—
30	#19151401	C₂²	1	1	yes	characteristic
31	#19151401	C₂²	1	2	yes	—
32	C₂²⋊#13083549	C₂²	1	2	yes	—
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
35	D₄×Q₈×C₈₉	—	2	1	no	—
36	#494×C₈₉	C₂²	1	1	yes	characteristic
37	#467×C₈₉	C₂²	1	2	yes	—
38	#462×C₈₉	—	2	1	no	—
39	#451×C₈₉	—	2	2	no	—
40	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
42	C₂²×Dic₁₇₈	—	2	2	no	—
43	C₂×Dic₃₅₆	—	2	4	no	—
44	C₂×Dic₃₅₆	—	2	2	no	—
45	C₂×Dic₃₅₆	—	2	1	no	—
46	C₂×Dic₃₅₆	—	4	2	no	—
47	C₂×Dic₃₅₆	C₂³	1	2	yes	—
48	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
49	#15668012	—	2	1	no	—
50	#15668011	—	4	1	no	—
51	#15667991	—	2	2	no	—
52	#15667989	—	4	2	no	—
53	C₄×Dic₁₇₈	—	2	1	no	—
54	D₄×Dic₈₉	—	4	1	no	—
55	#15667958	—	2	4	no	—
56	#15667958	—	2	4	no	—
57	#15667958	—	2	4	no	—
58	#15667958	—	4	2	no	—
59	#15667958	D₄	1	4	yes	direct product factor
60	C₂×#13083549	—	2	4	no	—
61	C₂×#13083549	C₂³	1	1	yes	characteristic
62	C₂×#13083549	C₂³	1	2	yes	—
63	C₂×#13083549	C₂³	1	1	yes	characteristic
64	#15667946	C₂³	1	1	yes	characteristic
65	#15667937	—	2	1	no	—
66	#15667937	—	2	2	no	—
67	C₂²×Q₈×C₈₉	—	2	2	no	—
68	C₂×Q₁₆×C₈₉	—	2	4	no	—
69	C₂×Q₁₆×C₈₉	—	2	2	no	—
70	C₂×Q₁₆×C₈₉	—	2	1	no	—
71	C₂×Q₁₆×C₈₉	—	4	2	no	—
72	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₈₉	—	2	1	no	—
76	C₄⋊Q₈×C₈₉	—	4	1	no	—
77	C₂²⋊Q₈×C₈₉	—	2	2	no	—
78	C₂²⋊Q₈×C₈₉	—	4	2	no	—
79	C₄×Q₈×C₈₉	—	2	1	no	—
80	C₄×D₄×C₈₉	—	4	1	no	—
81	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
82	#107×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
84	#103×C₈₉	—	2	1	no	—
85	#103×C₈₉	—	2	2	no	—
86	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×Dic₁₇₈	—	2	1	no	—
89	C₂×Dic₁₇₈	—	2	2	no	—
90	C₂×Dic₁₇₈	—	4	4	no	—
91	C₂×Dic₁₇₈	—	4	2	no	—
92	C₂×Dic₁₇₈	—	8	2	no	—
93	C₂²×Dic₈₉	—	4	2	no	—
94	Dic₃₅₆	—	2	2	no	—
95	Dic₃₅₆	—	2	1	no	—
96	Dic₃₅₆	—	2	2	no	—
97	Dic₃₅₆	—	4	4	no	—
98	Dic₃₅₆	—	8	1	no	—
99	Dic₃₅₆	C₂×D₄	1	2	yes	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	1	no	—
103	#13083553	—	4	1	no	—
104	#13083553	—	4	2	no	—
105	C₂²⋊Dic₈₉	—	4	2	no	—
106	C₄×Dic₈₉	—	4	1	no	—
107	#13083549	—	2	2	no	—
108	#13083549	—	2	1	no	—
109	#13083549	C₂×D₄	1	2	yes	—
110	C₂×Q₈×C₈₉	—	2	1	no	—
111	C₂×Q₈×C₈₉	—	2	2	no	—
112	C₂×Q₈×C₈₉	—	4	4	no	—
113	C₂×Q₈×C₈₉	—	4	2	no	—
114	C₂×Q₈×C₈₉	—	8	2	no	—
115	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	—	4	2	no	—
117	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
118	Q₁₆×C₈₉	—	2	2	no	—
119	Q₁₆×C₈₉	—	2	1	no	—
120	Q₁₆×C₈₉	—	2	2	no	—
121	Q₁₆×C₈₉	—	4	4	no	—
122	Q₁₆×C₈₉	—	8	1	no	—
123	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	—	4	1	no	—
131	C₄⋊C₄×C₈₉	—	4	1	no	—
132	C₄⋊C₄×C₈₉	—	4	2	no	—
133	C₄⋊C₄×C₈₉	—	4	1	no	—
134	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
135	C₂²⋊C₄×C₈₉	—	4	2	no	—
136	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
137	C₄²×C₈₉	—	4	1	no	—
138	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂³×C₈₉	C₂×D₈	1	2	yes	—
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	4	1	no	—
142	Q₈×C₈₉	—	4	2	no	—
143	Q₈×C₈₉	—	4	2	no	—
144	Q₈×C₈₉	—	8	4	no	—
145	D₄×C₈₉	Q₃₂	1	4	yes	complemented
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	2	2	no	—
148	C₂×C₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	—	8	4	no	—
152	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	4	1	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	C₂×Dic₈₉	—	8	4	no	—
160	Dic₁₇₈	—	2	2	no	—
161	Dic₁₇₈	—	4	1	no	—
162	Dic₁₇₈	—	4	2	no	—
163	Dic₁₇₈	—	4	2	no	—
164	Dic₁₇₈	—	8	4	no	—
165	C₈×C₈₉	—	2	2	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
168	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂²×C₈₉	—	2	2	no	—
170	C₂²×C₈₉	—	2	2	no	—
171	C₂²×C₈₉	C₂×Q₃₂	1	4	yes	complemented
172	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	2	no	—
175	C₄×C₈₉	—	4	2	no	—
176	C₄×C₈₉	—	8	1	no	—
177	C₄×C₈₉	—	8	2	no	—
178	C₄×C₈₉	C₂×Q₃₂	1	2	yes	complemented
179	C₄×C₈₉	D₄²	1	1	yes	characteristic
180	C₄×C₈₉	#576	1	1	yes	characteristic
181	Dic₈₉	—	4	2	no	—
182	Dic₈₉	—	8	1	no	—
183	Dic₈₉	—	8	2	no	—
184	D₄×Q₃₂	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	4	no	—
186	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
187	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
188	C₂×C₈₉	#3417	1	1	yes	characteristic
189	C₂²×Q₃₂	—	89	2	no	—
190	D₄×Q₁₆	—	89	1	no	—
191	D₄×Q₁₆	D₈₉	1	1	yes	characteristic
192	#2247	—	89	1	no	—
193	#2216	—	89	2	no	—
194	#2209	—	89	1	no	—
195	#2209	—	89	1	no	—
196	#2188	—	89	2	no	—
197	#2188	—	89	2	no	—
198	C₄×Q₃₂	—	89	1	no	—
199	D₄×C₁₆	—	89	1	no	—
200	C₈₉	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₁₆	—	89	2	no	—
202	C₂²×Q₁₆	C₂×D₈₉	1	2	yes	—
203	D₄×Q₈	—	2	1	no	—
204	D₄×Q₈	—	178	1	no	—
205	C₂×Q₃₂	—	89	1	no	—
206	C₂×Q₃₂	—	89	2	no	—
207	C₂×Q₃₂	—	89	2	no	—
208	C₂×Q₃₂	—	89	2	no	—
209	C₂×Q₃₂	—	89	2	no	—
210	C₂×Q₃₂	—	178	8	no	—
211	C₂²×C₁₆	—	89	2	no	—
212	#494	—	89	1	no	—
213	#494	C₂×D₈₉	1	1	yes	characteristic
214	#467	—	89	2	no	—
215	#467	C₂×D₈₉	1	2	yes	—
216	#462	—	2	1	no	—
217	#462	—	178	1	no	—
218	#451	—	2	2	no	—
219	#451	—	178	2	no	—
220	C₄×Q₁₆	—	89	1	no	—
221	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
222	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
223	#366	—	89	1	no	—
224	C₄⋊C₁₆	—	89	1	no	—
225	#358	—	89	1	no	—
226	#358	—	89	2	no	—
227	#358	—	89	1	no	—
228	#358	—	89	2	no	—
229	C₂²⋊C₁₆	—	89	2	no	—
230	C₄×C₁₆	—	89	1	no	—
231	C₂²×Q₈	—	2	2	no	—
232	C₂²×Q₈	—	178	2	no	—
233	C₂×Q₁₆	—	2	4	no	—
234	C₂×Q₁₆	—	2	2	no	—
235	C₂×Q₁₆	—	2	1	no	—
236	C₂×Q₁₆	—	4	2	no	—
237	C₂×Q₁₆	—	89	1	no	—
238	C₂×Q₁₆	—	89	2	no	—
239	C₂×Q₁₆	—	356	2	no	—
240	C₂×Q₁₆	—	178	4	no	—
241	C₂×Q₁₆	—	178	2	no	—
242	C₂×Q₁₆	—	178	1	no	—
243	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
244	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	4	1	no	—
248	C₄⋊Q₈	—	356	1	no	—
249	C₄⋊Q₈	—	178	1	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	4	2	no	—
252	C₂²⋊Q₈	—	356	2	no	—
253	C₂²⋊Q₈	—	178	2	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	178	1	no	—
256	C₄×D₄	—	4	1	no	—
257	C₄×D₄	—	356	1	no	—
258	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
259	Q₃₂	—	89	4	no	—
260	Q₃₂	—	356	2	no	—
261	Q₃₂	—	178	4	no	—
262	Q₃₂	—	178	4	no	—
263	Q₃₂	—	178	4	no	—
264	C₂×C₁₆	—	89	1	no	—
265	C₂×C₁₆	—	89	1	no	—
266	C₂×C₁₆	—	89	2	no	—
267	C₂×C₁₆	—	178	4	no	—
268	#107	—	89	1	no	—
269	#107	C₂²×D₈₉	1	1	yes	characteristic
270	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
271	#103	—	2	1	no	—
272	#103	—	2	2	no	—
273	#103	—	178	1	no	—
274	#103	—	178	2	no	—
275	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
276	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	4	4	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂×Q₈	—	356	4	no	—
283	C₂×Q₈	—	356	2	no	—
284	C₂×Q₈	—	178	2	no	—
285	C₂×Q₈	—	178	1	no	—
286	C₂×Q₈	—	712	2	no	—
287	C₂×D₄	#13083559	1	1	yes	characteristic
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	—	356	2	no	—
290	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	1	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	4	4	no	—
295	Q₁₆	—	8	1	no	—
296	Q₁₆	—	89	2	no	—
297	Q₁₆	—	356	4	no	—
298	Q₁₆	—	178	2	no	—
299	Q₁₆	—	178	2	no	—
300	Q₁₆	—	178	1	no	—
301	Q₁₆	—	712	1	no	—
302	Q₁₆	D₄×D₈₉	1	2	yes	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
307	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	356	1	no	—
314	C₄⋊C₄	—	356	1	no	—
315	C₄⋊C₄	—	356	1	no	—
316	C₄⋊C₄	—	356	2	no	—
317	C₄⋊C₄	—	178	1	no	—
318	C₄⋊C₄	#13083559	1	1	yes	characteristic
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	356	2	no	—
321	C₂²⋊C₄	#13083559	1	2	yes	—
322	C₄²	—	4	1	no	—
323	C₄²	—	356	1	no	—
324	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
325	C₁₆	—	89	2	no	—
326	C₁₆	—	178	2	no	—
327	C₁₆	—	178	1	no	—
328	C₂³	C₂×#13083559	1	2	yes	—
329	Q₈	—	2	2	no	—
330	Q₈	—	4	1	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	8	4	no	—
334	Q₈	—	356	1	no	—
335	Q₈	—	356	2	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	178	2	no	—
338	Q₈	—	712	4	no	—
339	D₄	#15667958	1	4	yes	direct product factor
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	8	4	no	—
346	C₂×C₄	—	356	1	no	—
347	C₂×C₄	—	356	1	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	712	4	no	—
350	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
351	C₂×C₄	C₂×#13083559	1	2	yes	—
352	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	2	no	—
356	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
357	C₈	#15668057	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	C₂×#15667958	1	4	yes	complemented
361	C₂²	C₂²×#13083559	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	1	no	—
367	C₄	—	356	2	no	—
368	C₄	—	712	2	no	—
369	C₄	—	712	1	no	—
370	C₄	C₂×#15667958	1	2	yes	complemented
371	C₄	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
372	C₄	#19152401	1	1	yes	characteristic
373	C₂	—	2	4	no	—
374	C₂	D₄×#13083559	1	1	yes	characteristic
375	C₂	C₂²×#15667958	1	1	yes	characteristic
376	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	D₄×Q₁₆	D₈₉
hypocenter	C₈₉	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667958	C₂	semidirect
3. #26022504	C₂	semidirect
4. #26022341	C₂	semidirect
5. #26022302	C₂	semidirect
6. #26022299	C₂	semidirect
7. #26022186	C₂	semidirect
8. #26022185	C₂	semidirect
9. C₄×#15667958	C₂	semidirect
59. #15667958	D₄	direct (N × H)
145. D₄×C₈₉	Q₃₂	semidirect
171. C₂²×C₈₉	C₂×Q₃₂	semidirect
178. C₄×C₈₉	C₂×Q₃₂	semidirect
200. C₈₉	D₄×Q₃₂	semidirect
339. D₄	#15667958	direct (N × H)
360. C₂²	C₂×#15667958	semidirect
370. C₄	C₂×#15667958	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667958	2
D₄×Dic₃₅₆	2
#26022504	2
#26022341	2
#26022302	2
#26022299	2
#26022186	2
#26022185	2
C₄×#15667958	2
D₄×#13083549	2
D₄×Q₁₆×C₈₉	2
D₄×Q₃₂	89

Subgroups — D4×#15667958 #36711154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#15667958 #36711154