Subgroups — D₄×#88106 #16807155

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal, complemented
2	#13131220	C₂	1	2	yes	maximal, complemented
3	#13131216	C₂	1	4	yes	maximal, complemented
4	D₄×#6092	C₂	1	2	yes	maximal, complemented
5	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131063	C₂	1	2	yes	maximal, complemented
7	C₄×#88106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₂×#88106	—	2	8	no	—
12	C₂×#88106	C₂²	1	2	yes	—
13	C₂×#88106	C₂²	1	1	yes	characteristic
14	C₂×#88106	C₂²	1	1	yes	characteristic
15	C₂×#88106	C₂²	1	4	yes	complemented
16	C₂²×#6092	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊#1055	C₄	1	2	yes	—
18	D₄×#1055	C₄	1	2	yes	—
19	D₄×#1055	C₄	1	1	yes	characteristic
20	D₄×#1055	C₄	1	1	yes	characteristic
21	D₄×#1055	C₂²	1	1	yes	characteristic
22	D₄×#1055	C₂²	1	2	yes	—
23	D₄×#1055	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	C₂²⋊#6092	C₂²	1	4	yes	complemented
27	C₂²⋊#6092	C₂²	1	4	yes	complemented
28	#11718231	C₂²	1	2	yes	complemented
29	C₄⋊#6092	C₂²	1	2	yes	complemented
30	C₄⋊#6092	C₂²	1	2	yes	complemented
31	C₄×#6092	C₂²	1	2	yes	complemented
32	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₁₃	C₈	1	2	yes	—
35	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
36	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
37	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
41	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	—	2	4	no	—
44	C₂²×#1055	—	2	4	no	—
45	C₂²×#1055	C₂×C₄	1	2	yes	—
46	C₂²×#1055	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₂²×#1055	C₂³	1	1	yes	characteristic
52	#88106	—	2	4	no	—
53	#88106	—	2	2	no	—
54	#88106	—	2	8	no	—
55	#88106	D₄	1	4	yes	direct product factor
56	C₂×#6092	—	2	8	no	—
57	C₂×#6092	C₂³	1	2	yes	—
58	C₂×#6092	C₂³	1	4	yes	—
59	C₂×#6092	C₂³	1	2	yes	—
60	C₂²⋊#1055	C₂×C₄	1	2	yes	—
61	C₂²⋊#1055	C₂×C₄	1	2	yes	—
62	C₂²⋊#1055	C₂×C₄	1	2	yes	—
63	C₂²⋊#1055	C₂×C₄	1	2	yes	—
64	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
65	C₄⋊#1055	C₂×C₄	1	2	yes	—
66	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
67	C₄×#1055	C₂×C₄	1	1	yes	characteristic
68	C₄×#1055	C₂×C₄	1	1	yes	characteristic
69	C₄×#1055	C₂³	1	1	yes	characteristic
70	C₄×#1055	C₂³	1	1	yes	characteristic
71	D₄×M₃₂	—	13	1	no	maximal, Sylow
72	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×D₄×C₁₃	—	2	4	no	—
74	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
92	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
93	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
94	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
108	C₂×#1055	C₂²×C₄	1	2	yes	—
109	C₂×#1055	C₂²×C₄	1	2	yes	—
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	C₂×#1055	C₂×D₄	1	2	yes	—
112	C₂×#1055	#41	1	2	yes	—
113	C₂×#1055	C₂⁴	1	1	yes	characteristic
114	#6092	—	2	4	no	—
115	#6092	—	2	2	no	—
116	#6092	C₂×D₄	1	4	yes	complemented
117	C₂²×M₃₂	—	13	2	no	—
118	C₂×C₈×D₄	—	13	1	no	—
119	#2169	—	13	2	no	—
120	#2168	—	13	4	no	—
121	D₄×C₁₆	—	13	2	no	—
122	#2150	—	13	1	no	—
123	#2112	—	13	2	no	—
124	C₄×M₃₂	—	13	1	no	—
125	C₂³×C₁₃	—	2	4	no	—
126	C₂³×C₁₃	—	2	4	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
129	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
131	D₄×C₁₃	—	2	4	no	—
132	D₄×C₁₃	—	2	2	no	—
133	D₄×C₁₃	M₃₂	1	4	yes	complemented
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#131	1	2	yes	—
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	#1055	—	2	2	no	—
154	#1055	—	2	2	no	—
155	#1055	—	2	1	no	—
156	#1055	—	2	1	no	—
157	#1055	C₄×D₄	1	2	yes	—
158	#1055	C₂²×D₄	1	1	yes	characteristic
159	#1055	C₂×#41	1	1	yes	characteristic
160	C₂³×C₈	—	13	2	no	—
161	C₂×C₄×D₄	#261	1	1	yes	characteristic
162	C₂×M₃₂	—	13	1	no	—
163	C₂×M₃₂	—	13	1	no	—
164	C₂×M₃₂	—	13	2	no	—
165	C₂×M₃₂	—	13	4	no	—
166	C₂×M₃₂	—	26	8	no	—
167	C₂²×C₁₆	—	13	4	no	—
168	C₈×D₄	—	13	2	no	—
169	C₈×D₄	—	13	1	no	—
170	C₈×D₄	—	13	1	no	—
171	C₈×D₄	—	13	1	no	—
172	C₈×D₄	—	13	2	no	—
173	C₈×D₄	—	13	1	no	—
174	C₂×C₄⋊C₈	—	13	1	no	—
175	C₂×C₂²⋊C₈	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₂²⋊C₁₆	—	13	4	no	—
180	C₂²⋊C₁₆	—	13	4	no	—
181	#346	—	13	2	no	—
182	C₄×C₁₆	—	13	2	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	4	8	no	—
191	C₂²×C₁₃	C₈×D₄	1	2	yes	—
192	C₂²×C₁₃	C₂×M₃₂	1	4	yes	complemented
193	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	C₈×D₄	1	2	yes	—
200	C₄×C₁₃	C₂×M₃₂	1	2	yes	complemented
201	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
203	C₂²×D₄	#1057	1	1	yes	characteristic
204	C₂³×C₄	C₂×#261	1	2	yes	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	13	2	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	26	4	no	—
213	C₂²×C₈	—	26	4	no	—
214	C₄×D₄	#1057	1	2	yes	—
215	C₄×D₄	#1057	1	1	yes	characteristic
216	C₄×D₄	#1057	1	1	yes	characteristic
217	C₄×D₄	C₂×#261	1	2	yes	—
218	C₄×D₄	C₂×#261	1	1	yes	characteristic
219	C₄×D₄	C₂×#261	1	1	yes	characteristic
220	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	#1057	1	2	yes	—
222	C₂×C₄²	C₂×#261	1	1	yes	characteristic
223	M₃₂	—	13	4	no	—
224	M₃₂	—	26	8	no	—
225	M₃₂	—	26	2	no	—
226	M₃₂	—	26	4	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	13	2	no	—
229	C₂×C₁₆	—	13	4	no	—
230	C₂×C₁₆	—	26	8	no	—
231	C₄⋊C₈	—	13	1	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	—	13	1	no	—
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	2	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	1	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₂×C₁₃	—	2	2	no	—
243	C₂×C₁₃	—	2	4	no	—
244	C₂×C₁₃	—	4	2	no	—
245	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
246	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
247	C₂×C₁₃	#3407	1	1	yes	characteristic
248	C₂⁴	C₂×#1057	1	2	yes	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	C₂×#1057	1	2	yes	—
251	C₂×D₄	C₂×#1057	1	1	yes	characteristic
252	C₂×D₄	C₂×#1057	1	1	yes	characteristic
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
256	C₂²×C₄	C₂×#1057	1	2	yes	—
257	C₂²×C₄	C₂×#1057	1	2	yes	—
258	C₂²×C₄	C₂×#1057	1	2	yes	—
259	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
260	C₂²×C₄	C₂²×#261	1	2	yes	—
261	C₂²×C₄	C₂²×#261	1	2	yes	—
262	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	2	no	—
275	C₂×C₈	—	26	4	no	—
276	C₂×C₈	—	26	4	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	26	4	no	—
279	C₄⋊C₄	C₂×#1057	1	2	yes	—
280	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
281	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
282	C₂²⋊C₄	C₂×#1057	1	2	yes	—
283	C₂²⋊C₄	C₂×#1057	1	2	yes	—
284	C₂²⋊C₄	C₂×#1057	1	2	yes	—
285	C₂²⋊C₄	C₂×#1057	1	2	yes	—
286	C₄²	C₂×#1057	1	1	yes	characteristic
287	C₄²	C₂×#1057	1	1	yes	characteristic
288	C₄²	C₂²×#261	1	1	yes	characteristic
289	C₄²	C₂²×#261	1	1	yes	characteristic
290	C₁₆	—	13	4	no	—
291	C₁₆	—	26	4	no	—
292	C₁₆	—	26	2	no	—
293	C₁₃	D₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	2	4	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	C₂²×#1057	1	2	yes	—
298	C₂³	C₂²×#1057	1	2	yes	—
299	C₂³	C₂²×#1057	1	1	yes	characteristic
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	D₄	#88106	1	4	yes	direct product factor
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	#88238	1	2	yes	—
313	C₂×C₄	D₄×#261	1	2	yes	—
314	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
315	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
316	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
317	C₂×C₄	C₂²×#1057	1	2	yes	—
318	C₂×C₄	C₂²×#1057	1	2	yes	—
319	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
320	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
321	C₂×C₄	C₂³×#261	1	1	yes	characteristic
322	C₈	—	13	1	no	—
323	C₈	—	13	1	no	—
324	C₈	—	13	2	no	—
325	C₈	—	26	1	no	—
326	C₈	—	26	1	no	—
327	C₈	—	26	2	no	—
328	C₈	—	26	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	8	no	—
337	C₂²	C₂×#88106	1	4	yes	complemented
338	C₂²	D₄×#1057	1	2	yes	—
339	C₂²	C₂³×#1057	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	C₂×#88106	1	2	yes	complemented
346	C₄	D₄×#1057	1	2	yes	—
347	C₄	C₂×#88238	1	1	yes	characteristic
348	C₄	C₂×D₄×#261	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	4	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×D₄×#1057	1	1	yes	characteristic
353	C₂	C₂²×#88106	1	1	yes	characteristic
354	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×D₄×C₁₃	C₄
prime radical	C₂²×D₄×C₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×D₄	#261
hypocenter	C₁₃	D₄×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88106	C₂	semidirect
2. #13131220	C₂	semidirect
3. #13131216	C₂	semidirect
4. D₄×#6092	C₂	semidirect
5. #13131168	C₂	semidirect
6. #13131063	C₂	semidirect
7. C₄×#88106	C₂	semidirect
15. C₂×#88106	C₂²	semidirect
16. C₂²×#6092	C₂²	semidirect
26. C₂²⋊#6092	C₂²	semidirect
27. C₂²⋊#6092	C₂²	semidirect
28. #11718231	C₂²	semidirect
29. C₄⋊#6092	C₂²	semidirect
30. C₄⋊#6092	C₂²	semidirect
31. C₄×#6092	C₂²	semidirect
55. #88106	D₄	direct (N × H)
116. #6092	C₂×D₄	semidirect
133. D₄×C₁₃	M₃₂	semidirect
192. C₂²×C₁₃	C₂×M₃₂	semidirect
200. C₄×C₁₃	C₂×M₃₂	semidirect
293. C₁₃	D₄×M₃₂	semidirect
302. D₄	#88106	direct (N × H)
337. C₂²	C₂×#88106	semidirect
345. C₄	C₂×#88106	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13131220	2
#13131216	2
D₄×#6092	2
#13131168	2
#13131063	2
C₄×#88106	2
C₂×D₄×#1055	2
D₄×M₃₂	13

Subgroups — D4×#88106 #16807155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#88106 #16807155