Subgroups — D₄×#88069 #16019645

Order 3328.

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×#88069	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13117088	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116925	C₂	1	2	yes	maximal, complemented
5	#13116886	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116883	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116770	C₂	1	2	yes	maximal, complemented
8	#13116769	C₂	1	2	yes	maximal, complemented
9	C₄×#88069	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×#6090	C₂	1	1	yes	maximal, characteristic
11	D₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	D₄×Q₈×C₁₃	—	2	1	no	—
14	C₂²×Dic₅₂	C₂²	1	2	yes	—
15	D₄×Dic₂₆	—	2	1	no	—
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#467	C₂²	1	2	yes	—
18	C₁₃×#462	—	2	1	no	—
19	C₁₃×#451	—	2	2	no	—
20	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×#88069	—	2	8	no	—
23	C₂×#88069	C₂²	1	2	yes	—
24	C₂×#88069	C₂²	1	2	yes	—
25	C₂×#88069	C₂²	1	2	yes	—
26	C₂×#88069	C₂²	1	2	yes	—
27	C₂×#88069	C₂²	1	1	yes	characteristic
28	#11718777	C₂²	1	2	yes	—
29	C₂²×#6090	C₂²	1	2	yes	—
30	#11718510	—	2	1	no	—
31	#11718400	—	2	2	no	—
32	#11718360	C₂²	1	1	yes	characteristic
33	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
34	#11718156	C₂²	1	1	yes	characteristic
35	#11718156	C₂²	1	2	yes	—
36	C₂²⋊#6090	C₂²	1	2	yes	—
37	#11718084	C₂²	1	1	yes	characteristic
38	#11718084	C₂²	1	2	yes	—
39	#11718081	C₂²	1	1	yes	characteristic
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₄×#6090	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₁₃	—	2	2	no	—
43	C₂²×Dic₂₆	—	2	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	4	no	—
45	C₂×C₁₃×Q₁₆	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	—	2	1	no	—
47	C₂×C₁₃×Q₁₆	—	4	2	no	—
48	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
51	C₁₃×C₄⋊Q₈	—	2	1	no	—
52	C₁₃×C₄⋊Q₈	—	4	1	no	—
53	C₁₃×C₂²⋊Q₈	—	2	2	no	—
54	C₁₃×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₁₃	—	2	1	no	—
56	C₄×D₄×C₁₃	—	4	1	no	—
57	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
58	D₄×Dic₁₃	—	4	1	no	—
59	#88178	—	2	2	no	—
60	C₂×Dic₅₂	—	2	4	no	—
61	C₂×Dic₅₂	—	2	2	no	—
62	C₂×Dic₅₂	—	2	1	no	—
63	C₂×Dic₅₂	—	4	2	no	—
64	C₂×Dic₅₂	C₂³	1	2	yes	—
65	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
66	#88142	—	4	1	no	—
67	#88132	—	4	2	no	—
68	#88123	—	2	1	no	—
69	C₄×Dic₂₆	—	2	1	no	—
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₃×#103	—	2	1	no	—
73	C₁₃×#103	—	2	2	no	—
74	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88069	—	2	4	no	—
77	#88069	—	2	4	no	—
78	#88069	—	2	4	no	—
79	#88069	—	4	2	no	—
80	#88069	D₄	1	4	yes	direct product factor
81	#88058	C₂³	1	1	yes	characteristic
82	#88056	—	2	1	no	—
83	#88056	—	2	2	no	—
84	C₂×#6090	—	2	4	no	—
85	C₂×#6090	C₂³	1	2	yes	—
86	C₂×#6090	C₂³	1	1	yes	characteristic
87	C₂×#6090	C₂³	1	1	yes	characteristic
88	D₄×Q₃₂	—	13	1	no	maximal, Sylow
89	C₂×Q₈×C₁₃	—	2	1	no	—
90	C₂×Q₈×C₁₃	—	2	2	no	—
91	C₂×Q₈×C₁₃	—	4	4	no	—
92	C₂×Q₈×C₁₃	—	4	2	no	—
93	C₂×Q₈×C₁₃	—	8	2	no	—
94	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	—	4	2	no	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²×Dic₁₃	—	4	2	no	—
98	C₂×Dic₂₆	—	2	1	no	—
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	—	4	4	no	—
101	C₂×Dic₂₆	—	4	2	no	—
102	C₂×Dic₂₆	—	8	2	no	—
103	C₁₃×Q₁₆	—	2	2	no	—
104	C₁₃×Q₁₆	—	2	1	no	—
105	C₁₃×Q₁₆	—	2	2	no	—
106	C₁₃×Q₁₆	—	4	4	no	—
107	C₁₃×Q₁₆	—	8	1	no	—
108	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	2	no	—
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	2	1	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₄⋊C₄	—	4	1	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₄⋊C₄	—	4	1	no	—
119	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
120	C₁₃×C₂²⋊C₄	—	4	2	no	—
121	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
122	C₄²×C₁₃	—	4	1	no	—
123	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂²⋊Dic₁₃	—	4	2	no	—
125	C₄⋊Dic₁₃	—	2	1	no	—
126	C₄⋊Dic₁₃	—	4	1	no	—
127	C₄⋊Dic₁₃	—	4	1	no	—
128	#6101	—	4	1	no	—
129	#6101	—	4	2	no	—
130	C₄×Dic₁₃	—	4	1	no	—
131	Dic₅₂	—	2	2	no	—
132	Dic₅₂	—	2	1	no	—
133	Dic₅₂	—	2	2	no	—
134	Dic₅₂	—	4	4	no	—
135	Dic₅₂	—	8	1	no	—
136	Dic₅₂	C₂×D₄	1	2	yes	—
137	#6090	—	2	2	no	—
138	#6090	—	2	1	no	—
139	#6090	C₂×D₄	1	2	yes	—
140	C₂²×Q₃₂	—	13	2	no	—
141	D₄×Q₁₆	—	13	1	no	—
142	D₄×Q₁₆	D₁₃	1	1	yes	characteristic
143	#2247	—	13	1	no	—
144	#2216	—	13	2	no	—
145	#2209	—	13	1	no	—
146	#2209	—	13	1	no	—
147	#2188	—	13	2	no	—
148	#2188	—	13	2	no	—
149	C₄×Q₃₂	—	13	1	no	—
150	D₄×C₁₆	—	13	1	no	—
151	C₂³×C₁₃	C₂×D₈	1	2	yes	—
152	Q₈×C₁₃	—	2	2	no	—
153	Q₈×C₁₃	—	4	1	no	—
154	Q₈×C₁₃	—	4	2	no	—
155	Q₈×C₁₃	—	4	2	no	—
156	Q₈×C₁₃	—	8	4	no	—
157	D₄×C₁₃	Q₃₂	1	4	yes	complemented
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	4	1	no	—
161	C₂×C₄×C₁₃	—	4	1	no	—
162	C₂×C₄×C₁₃	—	4	2	no	—
163	C₂×C₄×C₁₃	—	8	4	no	—
164	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
166	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×Dic₁₃	—	4	1	no	—
169	C₂×Dic₁₃	—	4	1	no	—
170	C₂×Dic₁₃	—	4	2	no	—
171	C₂×Dic₁₃	—	8	4	no	—
172	Dic₂₆	—	2	2	no	—
173	Dic₂₆	—	4	1	no	—
174	Dic₂₆	—	4	2	no	—
175	Dic₂₆	—	4	2	no	—
176	Dic₂₆	—	8	4	no	—
177	C₈×C₁₃	—	2	2	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
180	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂²×Q₁₆	—	13	2	no	—
182	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
183	D₄×Q₈	—	2	1	no	—
184	D₄×Q₈	—	26	1	no	—
185	C₂×Q₃₂	—	13	1	no	—
186	C₂×Q₃₂	—	13	2	no	—
187	C₂×Q₃₂	—	13	2	no	—
188	C₂×Q₃₂	—	13	2	no	—
189	C₂×Q₃₂	—	13	2	no	—
190	C₂×Q₃₂	—	26	8	no	—
191	C₂²×C₁₆	—	13	2	no	—
192	#494	—	13	1	no	—
193	#494	C₂×D₁₃	1	1	yes	characteristic
194	#467	—	13	2	no	—
195	#467	C₂×D₁₃	1	2	yes	—
196	#462	—	2	1	no	—
197	#462	—	26	1	no	—
198	#451	—	2	2	no	—
199	#451	—	26	2	no	—
200	C₄×Q₁₆	—	13	1	no	—
201	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
202	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
203	#366	—	13	1	no	—
204	C₄⋊C₁₆	—	13	1	no	—
205	#358	—	13	1	no	—
206	#358	—	13	2	no	—
207	#358	—	13	1	no	—
208	#358	—	13	2	no	—
209	C₂²⋊C₁₆	—	13	2	no	—
210	C₄×C₁₆	—	13	1	no	—
211	C₂²×C₁₃	—	2	2	no	—
212	C₂²×C₁₃	—	2	2	no	—
213	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	complemented
214	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
215	C₄×C₁₃	—	2	1	no	—
216	C₄×C₁₃	—	2	2	no	—
217	C₄×C₁₃	—	4	2	no	—
218	C₄×C₁₃	—	8	1	no	—
219	C₄×C₁₃	—	8	2	no	—
220	C₄×C₁₃	C₂×Q₃₂	1	2	yes	complemented
221	C₄×C₁₃	D₄²	1	1	yes	characteristic
222	C₄×C₁₃	#576	1	1	yes	characteristic
223	Dic₁₃	—	4	2	no	—
224	Dic₁₃	—	8	1	no	—
225	Dic₁₃	—	8	2	no	—
226	C₂²×Q₈	—	2	2	no	—
227	C₂²×Q₈	—	26	2	no	—
228	C₂×Q₁₆	—	2	4	no	—
229	C₂×Q₁₆	—	2	2	no	—
230	C₂×Q₁₆	—	2	1	no	—
231	C₂×Q₁₆	—	4	2	no	—
232	C₂×Q₁₆	—	13	1	no	—
233	C₂×Q₁₆	—	13	2	no	—
234	C₂×Q₁₆	—	26	4	no	—
235	C₂×Q₁₆	—	26	2	no	—
236	C₂×Q₁₆	—	26	1	no	—
237	C₂×Q₁₆	—	52	2	no	—
238	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
239	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	4	1	no	—
243	C₄⋊Q₈	—	26	1	no	—
244	C₄⋊Q₈	—	52	1	no	—
245	C₂²⋊Q₈	—	2	2	no	—
246	C₂²⋊Q₈	—	4	2	no	—
247	C₂²⋊Q₈	—	26	2	no	—
248	C₂²⋊Q₈	—	52	2	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	26	1	no	—
251	C₄×D₄	—	4	1	no	—
252	C₄×D₄	—	52	1	no	—
253	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
254	Q₃₂	—	13	4	no	—
255	Q₃₂	—	26	4	no	—
256	Q₃₂	—	26	4	no	—
257	Q₃₂	—	26	4	no	—
258	Q₃₂	—	52	2	no	—
259	C₂×C₁₆	—	13	1	no	—
260	C₂×C₁₆	—	13	1	no	—
261	C₂×C₁₆	—	13	2	no	—
262	C₂×C₁₆	—	26	4	no	—
263	#107	—	13	1	no	—
264	#107	C₂²×D₁₃	1	1	yes	characteristic
265	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
266	#103	—	2	1	no	—
267	#103	—	2	2	no	—
268	#103	—	26	1	no	—
269	#103	—	26	2	no	—
270	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
271	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
272	C₂×C₁₃	—	2	4	no	—
273	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
274	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
275	C₂×C₁₃	#3417	1	1	yes	characteristic
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	4	4	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	8	2	no	—
281	C₂×Q₈	—	26	1	no	—
282	C₂×Q₈	—	26	2	no	—
283	C₂×Q₈	—	52	4	no	—
284	C₂×Q₈	—	52	2	no	—
285	C₂×Q₈	—	104	2	no	—
286	C₂×D₄	#6104	1	1	yes	characteristic
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	52	2	no	—
289	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	2	1	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	4	4	no	—
294	Q₁₆	—	8	1	no	—
295	Q₁₆	—	13	2	no	—
296	Q₁₆	—	26	2	no	—
297	Q₁₆	—	26	1	no	—
298	Q₁₆	—	26	2	no	—
299	Q₁₆	—	52	4	no	—
300	Q₁₆	—	104	1	no	—
301	Q₁₆	D₄×D₁₃	1	2	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
306	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	26	1	no	—
313	C₄⋊C₄	—	52	1	no	—
314	C₄⋊C₄	—	52	1	no	—
315	C₄⋊C₄	—	52	2	no	—
316	C₄⋊C₄	—	52	1	no	—
317	C₄⋊C₄	#6104	1	1	yes	characteristic
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	52	2	no	—
320	C₂²⋊C₄	#6104	1	2	yes	—
321	C₄²	—	4	1	no	—
322	C₄²	—	52	1	no	—
323	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
324	C₁₆	—	13	2	no	—
325	C₁₆	—	26	2	no	—
326	C₁₆	—	26	1	no	—
327	C₁₃	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
328	C₂³	C₂×#6104	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	—	26	2	no	—
335	Q₈	—	52	1	no	—
336	Q₈	—	52	2	no	—
337	Q₈	—	52	2	no	—
338	Q₈	—	104	4	no	—
339	D₄	#88069	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	—	52	1	no	—
347	C₂×C₄	—	52	1	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	4	no	—
350	C₂×C₄	C₂×#6104	1	1	yes	characteristic
351	C₂×C₄	C₂×#6104	1	2	yes	—
352	C₂×C₄	C₂×#6104	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
354	C₈	—	2	2	no	—
355	C₈	—	2	1	no	—
356	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
357	C₈	#88251	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	C₂×#88069	1	4	yes	complemented
361	C₂²	C₂²×#6104	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	52	2	no	—
368	C₄	—	104	1	no	—
369	C₄	—	104	2	no	—
370	C₄	C₂×#88069	1	2	yes	complemented
371	C₄	D₄×C₂²⋊D₁₃	1	1	yes	characteristic
372	C₄	#11719540	1	1	yes	characteristic
373	C₂	—	2	4	no	—
374	C₂	D₄×#6104	1	1	yes	characteristic
375	C₂	C₂²×#88069	1	1	yes	characteristic
376	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₄×C₁₃	D₈
hypercenter	D₄×Q₁₆	D₁₃
hypocenter	C₁₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
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×#88069	C₂	semidirect
3. #13117088	C₂	semidirect
4. #13116925	C₂	semidirect
5. #13116886	C₂	semidirect
6. #13116883	C₂	semidirect
7. #13116770	C₂	semidirect
8. #13116769	C₂	semidirect
9. C₄×#88069	C₂	semidirect
80. #88069	D₄	direct (N × H)
157. D₄×C₁₃	Q₃₂	semidirect
213. C₂²×C₁₃	C₂×Q₃₂	semidirect
220. C₄×C₁₃	C₂×Q₃₂	semidirect
327. C₁₃	D₄×Q₃₂	semidirect
339. D₄	#88069	direct (N × H)
360. C₂²	C₂×#88069	semidirect
370. C₄	C₂×#88069	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88069	2
D₄×Dic₅₂	2
#13117088	2
#13116925	2
#13116886	2
#13116883	2
#13116770	2
#13116769	2
C₄×#88069	2
D₄×#6090	2
D₄×C₁₃×Q₁₆	2
D₄×Q₃₂	13

Subgroups — D4×#88069 #16019645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#88069 #16019645