Subgroups — C₂×#13271 #495344

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂²×#364	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	#13271	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1389	C₂	1	2	yes	maximal
6	C₂×#1384	C₂	1	2	yes	maximal
7	C₂×#533	—	2	1	no	—
8	C₂²×M₃₂	C₂²	1	2	yes	—
9	#3160	—	2	2	no	—
10	#3160	—	2	2	no	—
11	#3160	C₄	1	4	yes	complemented
12	#3160	C₂²	1	4	yes	—
13	C₂×#500	—	2	1	no	—
14	C₂×#499	—	2	1	no	—
15	C₂×#499	C₄	1	2	yes	complemented
16	C₂×#498	C₄	1	2	yes	complemented
17	C₂²×#108	C₂²	1	1	yes	characteristic
18	C₂×#427	—	2	1	no	—
19	C₂×#427	C₂²	1	2	yes	—
20	C₂×#427	C₂²	1	1	yes	characteristic
21	C₂×#405	—	2	1	no	—
22	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×#364	—	2	1	no	—
24	C₂×#364	—	2	1	no	—
25	C₂×#364	—	2	2	no	—
26	C₂×#364	C₂²	1	4	yes	—
27	#1772	—	2	2	no	—
28	#1772	C₂²	1	4	yes	—
29	C₂×#339	—	2	1	no	—
30	#1389	C₂²	1	8	yes	—
31	#1384	C₂²	1	8	yes	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	#533	—	2	4	no	—
36	#533	—	2	4	no	—
37	#533	—	4	2	no	—
38	#533	—	4	2	no	—
39	C₂×C₄⋊Q₈	—	2	2	no	—
40	C₂×#125	—	2	2	no	—
41	C₂×C₂²⋊Q₈	—	4	2	no	—
42	C₂×C₄×Q₈	—	4	2	no	—
43	C₂×#117	—	2	1	no	—
44	C₂×#117	—	2	2	no	—
45	C₂²×C₄²	D₄	1	1	yes	characteristic
46	C₂×M₃₂	—	2	2	no	—
47	C₂×M₃₂	—	2	2	no	—
48	C₂×M₃₂	—	2	2	no	—
49	C₂×M₃₂	—	2	2	no	—
50	C₂×M₃₂	—	2	2	no	—
51	C₂×M₃₂	C₂³	1	4	yes	—
52	C₂²×C₁₆	—	2	2	no	—
53	#500	—	2	4	no	—
54	#500	—	2	4	no	—
55	#499	—	2	4	no	—
56	#499	—	2	4	no	—
57	#499	—	2	4	no	—
58	#499	C₂×C₄	1	8	yes	complemented
59	#498	—	2	4	no	—
60	#498	C₂×C₄	1	8	yes	complemented
61	C₂×#108	—	2	1	no	—
62	C₂×#108	—	2	1	no	—
63	C₂×#108	—	2	1	no	—
64	C₂×#108	Q₈	1	2	yes	complemented
65	C₂×#108	Q₈	1	2	yes	complemented
66	C₂×#108	C₂³	1	2	yes	—
67	#427	—	2	4	no	—
68	#427	—	2	2	no	—
69	#427	—	2	2	no	—
70	#427	—	2	4	no	—
71	#427	—	2	1	no	—
72	#427	—	2	1	no	—
73	#427	C₂×C₄	1	4	yes	—
74	#427	Q₈	1	2	yes	complemented
75	#427	C₂³	1	4	yes	—
76	#427	C₂³	1	2	yes	—
77	C₂×#107	—	2	2	no	—
78	C₂×#107	—	2	2	no	—
79	C₂×#107	—	2	1	no	—
80	C₂×#106	—	2	2	no	—
81	C₂×#106	C₂×C₄	1	4	yes	—
82	C₂×#106	Q₈	1	2	yes	complemented
83	#405	—	2	4	no	—
84	#405	—	4	2	no	—
85	C₂×#97	—	4	1	no	—
86	C₂×C₄×C₈	—	4	1	no	—
87	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
90	C₂×C₄×C₈	D₄	1	2	yes	—
91	C₂×C₄×C₈	D₄	1	1	yes	characteristic
92	C₂×C₄×C₈	D₄	1	1	yes	characteristic
93	C₂×C₄×C₈	D₄	1	2	yes	—
94	C₂×C₄×C₈	C₂³	1	2	yes	—
95	#364	—	2	4	no	—
96	#364	—	2	4	no	—
97	#339	—	2	4	no	—
98	C₂²×Q₈	—	4	2	no	—
99	C₂³×C₄	—	2	1	no	—
100	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂²×C₈	—	4	1	no	—
108	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
111	C₂²×C₈	SD₁₆	1	1	yes	characteristic
112	C₂²×C₈	SD₁₆	1	1	yes	characteristic
113	C₂²×C₈	SD₁₆	1	2	yes	—
114	C₂²×C₈	Q₁₆	1	1	yes	characteristic
115	C₂²×C₈	Q₁₆	1	1	yes	characteristic
116	C₂²×C₈	Q₁₆	1	2	yes	—
117	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄	1	2	yes	—
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
125	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
126	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	C₂²×C₈	#41	1	1	yes	characteristic
130	C₂²×C₈	#41	1	1	yes	characteristic
131	C₂²×C₈	C₂⁴	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	—	4	4	no	—
134	#125	—	2	8	no	—
135	#125	—	4	4	no	—
136	C₂²⋊Q₈	—	4	8	no	—
137	C₂²⋊Q₈	—	4	8	no	—
138	C₄×Q₈	—	4	8	no	—
139	C₄×Q₈	—	4	8	no	—
140	#117	—	2	4	no	—
141	#117	—	2	2	no	—
142	#117	—	2	4	no	—
143	#117	—	2	2	no	—
144	#117	—	4	1	no	—
145	#117	—	4	1	no	—
146	#117	—	4	2	no	—
147	#117	—	4	2	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	4	4	no	—
151	C₂×C₄⋊C₄	—	4	2	no	—
152	C₂×C₂²⋊C₄	—	4	1	no	—
153	C₂×C₂²⋊C₄	—	4	2	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	—	4	2	no	—
159	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
162	C₂×C₄²	C₂×D₄	1	2	yes	—
163	M₃₂	—	2	4	no	—
164	M₃₂	—	2	4	no	—
165	M₃₂	—	4	2	no	—
166	M₃₂	—	4	2	no	—
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	—	4	2	no	—
169	C₂×C₁₆	—	4	2	no	—
170	#108	—	2	2	no	—
171	#108	—	2	2	no	—
172	#108	—	2	2	no	—
173	#108	C₂×Q₈	1	4	yes	complemented
174	#107	—	2	8	no	—
175	#107	—	2	4	no	—
176	#107	—	2	2	no	—
177	#107	—	2	4	no	—
178	#107	—	2	2	no	—
179	#106	—	2	8	no	—
180	#106	—	2	4	no	—
181	#106	—	2	2	no	—
182	#106	C₂²×C₄	1	8	yes	—
183	#106	C₂×Q₈	1	4	yes	complemented
184	#97	—	4	4	no	—
185	C₄×C₈	—	4	4	no	—
186	C₄×C₈	C₂²⋊C₄	1	2	yes	—
187	C₄×C₈	C₂²⋊C₄	1	2	yes	—
188	C₄×C₈	C₂²⋊C₄	1	2	yes	—
189	C₄×C₈	C₂²⋊C₄	1	2	yes	—
190	C₄×C₈	C₂²×C₄	1	2	yes	—
191	C₄×C₈	C₂²×C₄	1	2	yes	—
192	C₄×C₈	C₂×D₄	1	2	yes	—
193	C₄×C₈	C₂×D₄	1	2	yes	—
194	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂×Q₈	—	4	8	no	—
196	C₂×Q₈	—	8	8	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
214	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	#103	1	1	yes	characteristic
226	C₂×C₈	#103	1	1	yes	characteristic
227	C₂×C₈	#103	1	2	yes	—
228	C₂×C₈	#103	1	1	yes	characteristic
229	C₂×C₈	#103	1	1	yes	characteristic
230	C₂×C₈	#103	1	2	yes	—
231	C₂×C₈	#103	1	2	yes	—
232	C₂×C₈	#103	1	2	yes	—
233	C₂×C₈	#103	1	2	yes	—
234	C₂×C₈	#103	1	2	yes	—
235	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
239	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
240	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
241	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
242	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
243	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
244	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
245	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
246	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	#124	1	1	yes	characteristic
252	C₂×C₈	#124	1	1	yes	characteristic
253	C₂×C₈	#124	1	1	yes	characteristic
254	C₂×C₈	#124	1	1	yes	characteristic
255	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₈	C₂×SD₁₆	1	2	yes	—
260	C₂×C₈	C₂×SD₁₆	1	2	yes	—
261	C₂×C₈	C₂×Q₁₆	1	2	yes	—
262	C₂×C₈	C₂×Q₁₆	1	2	yes	—
263	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
266	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
267	C₂×C₈	C₂×#41	1	1	yes	characteristic
268	C₂×C₈	C₂×#41	1	1	yes	characteristic
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	16	no	—
272	C₄⋊C₄	—	4	8	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	4	no	—
285	C₄²	—	4	4	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₂²⋊C₄	1	2	yes	—
288	C₄²	C₂×C₂²⋊C₄	1	2	yes	—
289	C₁₆	—	4	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	#418	1	1	yes	characteristic
292	C₂³	#418	1	1	yes	characteristic
293	C₂³	#476	1	1	yes	characteristic
294	C₂³	#476	1	1	yes	characteristic
295	C₂³	#477	1	1	yes	characteristic
296	C₂³	#477	1	1	yes	characteristic
297	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
298	Q₈	—	8	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	—	8	8	no	—
309	C₂×C₄	#391	1	1	yes	characteristic
310	C₂×C₄	#391	1	1	yes	characteristic
311	C₂×C₄	#391	1	1	yes	characteristic
312	C₂×C₄	#391	1	1	yes	characteristic
313	C₂×C₄	C₂×#103	1	1	yes	characteristic
314	C₂×C₄	C₂×#103	1	1	yes	characteristic
315	C₂×C₄	C₂×#103	1	1	yes	characteristic
316	C₂×C₄	C₂×#103	1	1	yes	characteristic
317	C₂×C₄	#418	1	1	yes	characteristic
318	C₂×C₄	#418	1	1	yes	characteristic
319	C₂×C₄	#476	1	1	yes	characteristic
320	C₂×C₄	#476	1	1	yes	characteristic
321	C₂×C₄	#476	1	1	yes	characteristic
322	C₂×C₄	#476	1	1	yes	characteristic
323	C₂×C₄	#476	1	1	yes	characteristic
324	C₂×C₄	#476	1	1	yes	characteristic
325	C₂×C₄	#477	1	1	yes	characteristic
326	C₂×C₄	#477	1	1	yes	characteristic
327	C₂×C₄	#477	1	1	yes	characteristic
328	C₂×C₄	#477	1	1	yes	characteristic
329	C₂×C₄	#477	1	1	yes	characteristic
330	C₂×C₄	#477	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
334	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
336	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₈	—	4	4	no	—
338	C₈	#391	1	1	yes	characteristic
339	C₈	#391	1	1	yes	characteristic
340	C₈	#391	1	1	yes	characteristic
341	C₈	#391	1	1	yes	characteristic
342	C₈	C₂×#103	1	2	yes	—
343	C₈	C₂×#103	1	2	yes	—
344	C₈	C₂×#103	1	2	yes	—
345	C₈	C₂×#103	1	2	yes	—
346	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
347	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
348	C₈	C₂×#124	1	1	yes	characteristic
349	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	#1987	1	1	yes	characteristic
358	C₂²	#1987	1	1	yes	characteristic
359	C₂²	#1987	1	1	yes	characteristic
360	C₂²	#1987	1	1	yes	characteristic
361	C₂²	C₂×#418	1	1	yes	characteristic
362	C₂²	C₂×#476	1	1	yes	characteristic
363	C₂²	C₂×#477	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	4	2	no	—
366	C₄	—	8	4	no	—
367	C₄	—	8	8	no	—
368	C₄	#1987	1	1	yes	characteristic
369	C₄	#1987	1	1	yes	characteristic
370	C₄	#1987	1	1	yes	characteristic
371	C₄	#1987	1	1	yes	characteristic
372	C₄	C₂×#391	1	1	yes	characteristic
373	C₄	C₂²×#103	1	1	yes	characteristic
374	C₄	C₂×#476	1	1	yes	characteristic
375	C₄	C₂×#477	1	1	yes	characteristic
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	#13271	1	2	yes	direct product factor
383	C₂	C₂×#1987	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#1987
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13271	C₂	direct (N × H)
11. #3160	C₄	semidirect
15. C₂×#499	C₄	semidirect
16. C₂×#498	C₄	semidirect
58. #499	C₂×C₄	semidirect
60. #498	C₂×C₄	semidirect
64. C₂×#108	Q₈	semidirect
65. C₂×#108	Q₈	semidirect
74. #427	Q₈	semidirect
82. C₂×#106	Q₈	semidirect
173. #108	C₂×Q₈	semidirect
183. #106	C₂×Q₈	semidirect
382. C₂	#13271	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#3160	2
C₂²×#364	2
C₂×#1772	2
#13271	2
C₂×#1389	2
C₂×#1384	2

Subgroups — C2×#13271 #495344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13271 #495344