Subgroups — C₂×#13188 #495079

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2155	C₂	1	2	yes	maximal
2	C₂²×#364	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	1	yes	maximal, characteristic
4	C₂×#1772	C₂	1	1	yes	maximal, characteristic
5	#13188	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1388	C₂	1	2	yes	maximal
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂²×#108	—	2	1	no	—
9	C₂²×#108	C₂²	1	2	yes	—
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂×#423	—	2	1	no	—
13	C₂×#405	—	2	1	no	—
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#2155	—	2	4	no	—
16	#2155	C₂²	1	8	yes	—
17	C₂×#365	—	2	4	no	—
18	C₂×#364	C₄	1	2	yes	complemented
19	C₂×#364	C₄	1	2	yes	complemented
20	C₂×#364	C₄	1	4	yes	complemented
21	C₂×#364	C₂²	1	4	yes	—
22	#1848	—	2	2	no	—
23	#1848	C₂²	1	4	yes	—
24	#1772	—	2	2	no	—
25	#1772	C₂²	1	4	yes	—
26	C₂×#339	—	2	1	no	—
27	#1388	C₂²	1	8	yes	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	2	no	—
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂×#117	—	2	1	no	—
33	C₂²×C₄²	D₄	1	1	yes	characteristic
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	C₂×C₄	1	4	yes	—
38	C₂×M₃₂	C₂×C₄	1	4	yes	—
39	C₂×M₃₂	C₂³	1	4	yes	—
40	C₂²×C₁₆	—	2	2	no	—
41	C₂×#108	—	2	1	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	—	2	1	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	2	2	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	2	4	no	—
50	C₂×#108	—	4	1	no	—
51	C₂×#108	—	4	1	no	—
52	C₂×#108	—	4	2	no	—
53	C₂×#108	Q₈	1	2	yes	—
54	C₂×#108	Q₈	1	2	yes	—
55	C₂×#108	C₂³	1	4	yes	—
56	C₂×#108	C₂³	1	2	yes	—
57	#427	—	2	1	no	—
58	#427	—	2	1	no	—
59	#427	D₄	1	2	yes	—
60	#427	C₂³	1	2	yes	—
61	C₂×#107	—	2	1	no	—
62	C₂×#106	D₄	1	2	yes	—
63	#423	—	2	4	no	—
64	#423	—	4	2	no	—
65	C₂×C₄⋊C₈	—	4	2	no	—
66	#405	—	2	4	no	—
67	#405	—	4	2	no	—
68	C₂×#97	—	4	1	no	—
69	C₂×C₄×C₈	—	4	1	no	—
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	2	yes	—
74	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈	Q₈	1	2	yes	—
76	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	2	yes	—
78	#365	—	2	16	no	—
79	#364	C₂×C₄	1	8	yes	complemented
80	#364	C₂×C₄	1	8	yes	complemented
81	#339	—	2	4	no	—
82	C₂³×C₄	—	2	1	no	—
83	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	4	no	—
88	C₂×M₁₆	—	4	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂×M₁₆	—	4	2	no	—
92	C₂×M₁₆	—	4	2	no	—
93	C₂×M₁₆	—	4	2	no	—
94	C₂×M₁₆	—	4	2	no	—
95	C₂×M₁₆	—	4	2	no	—
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	—	4	2	no	—
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	2	yes	—
101	C₂²×C₈	SD₁₆	1	1	yes	characteristic
102	C₂²×C₈	SD₁₆	1	1	yes	characteristic
103	C₂²×C₈	SD₁₆	1	2	yes	—
104	C₂²×C₈	SD₁₆	1	1	yes	characteristic
105	C₂²×C₈	SD₁₆	1	1	yes	characteristic
106	C₂²×C₈	SD₁₆	1	2	yes	—
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	2	yes	—
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
116	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	C₂²×C₈	#41	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	C₂²×C₈	C₂⁴	1	1	yes	characteristic
122	#117	—	2	2	no	—
123	#117	—	2	2	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₂²⋊C₄	—	4	1	no	—
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	4	1	no	—
132	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄²	C₄⋊C₄	1	2	yes	—
134	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂×D₄	1	2	yes	—
136	M₃₂	—	2	4	no	—
137	M₃₂	—	2	4	no	—
138	M₃₂	—	2	4	no	—
139	M₃₂	C₂²×C₄	1	8	yes	—
140	C₂×C₁₆	—	2	4	no	—
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	2	4	no	—
143	#108	—	2	2	no	—
144	#108	—	2	2	no	—
145	#108	—	2	4	no	—
146	#108	—	2	2	no	—
147	#108	—	2	4	no	—
148	#108	—	4	2	no	—
149	#108	—	4	2	no	—
150	#108	—	4	4	no	—
151	#108	—	4	4	no	—
152	#108	C₂×Q₈	1	4	yes	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	C₂×D₄	1	4	yes	—
157	C₄⋊C₈	—	4	8	no	—
158	#97	—	4	4	no	—
159	C₄×C₈	—	4	4	no	—
160	C₄×C₈	C₄⋊C₄	1	2	yes	—
161	C₄×C₈	C₄⋊C₄	1	2	yes	—
162	C₄×C₈	C₄⋊C₄	1	2	yes	—
163	C₄×C₈	C₄⋊C₄	1	2	yes	—
164	C₄×C₈	C₂²×C₄	1	2	yes	—
165	C₄×C₈	C₂²×C₄	1	2	yes	—
166	C₄×C₈	C₂×Q₈	1	2	yes	—
167	C₄×C₈	C₂×Q₈	1	2	yes	—
168	C₂⁴	#137	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	#121	1	1	yes	characteristic
178	C₂²×C₄	#121	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
185	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂²×C₄	#137	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	4	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	8	2	no	—
196	M₁₆	—	8	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	—	8	2	no	—
203	C₂×C₈	#106	1	1	yes	characteristic
204	C₂×C₈	#106	1	1	yes	characteristic
205	C₂×C₈	#106	1	2	yes	—
206	C₂×C₈	#106	1	1	yes	characteristic
207	C₂×C₈	#106	1	1	yes	characteristic
208	C₂×C₈	#106	1	2	yes	—
209	C₂×C₈	#106	1	2	yes	—
210	C₂×C₈	#106	1	2	yes	—
211	C₂×C₈	#106	1	2	yes	—
212	C₂×C₈	#106	1	2	yes	—
213	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
216	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
217	C₂×C₈	C₄×D₄	1	1	yes	characteristic
218	C₂×C₈	C₄×D₄	1	1	yes	characteristic
219	C₂×C₈	C₄×D₄	1	1	yes	characteristic
220	C₂×C₈	C₄×D₄	1	1	yes	characteristic
221	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
222	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
223	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
224	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
225	C₂×C₈	#121	1	1	yes	characteristic
226	C₂×C₈	#121	1	1	yes	characteristic
227	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₈	#125	1	1	yes	characteristic
230	C₂×C₈	#125	1	1	yes	characteristic
231	C₂×C₈	#125	1	1	yes	characteristic
232	C₂×C₈	#125	1	1	yes	characteristic
233	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
234	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
235	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
237	C₂×C₈	C₂×SD₁₆	1	2	yes	—
238	C₂×C₈	C₂×SD₁₆	1	2	yes	—
239	C₂×C₈	C₂×SD₁₆	1	2	yes	—
240	C₂×C₈	C₂×SD₁₆	1	2	yes	—
241	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂×#41	1	1	yes	characteristic
246	C₂×C₈	C₂×#41	1	1	yes	characteristic
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×C₄⋊C₄	1	2	yes	—
258	C₄²	C₂×C₄⋊C₄	1	2	yes	—
259	C₁₆	—	2	8	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#428	1	1	yes	characteristic
262	C₂³	#428	1	1	yes	characteristic
263	C₂³	#461	1	1	yes	characteristic
264	C₂³	#461	1	1	yes	characteristic
265	C₂³	#475	1	1	yes	characteristic
266	C₂³	#475	1	1	yes	characteristic
267	C₂³	C₂×#137	1	1	yes	characteristic
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	#389	1	1	yes	characteristic
277	C₂×C₄	#389	1	1	yes	characteristic
278	C₂×C₄	#389	1	1	yes	characteristic
279	C₂×C₄	#389	1	1	yes	characteristic
280	C₂×C₄	C₂×#106	1	1	yes	characteristic
281	C₂×C₄	C₂×#106	1	1	yes	characteristic
282	C₂×C₄	C₂×#106	1	1	yes	characteristic
283	C₂×C₄	C₂×#106	1	1	yes	characteristic
284	C₂×C₄	#428	1	1	yes	characteristic
285	C₂×C₄	#428	1	1	yes	characteristic
286	C₂×C₄	#460	1	1	yes	characteristic
287	C₂×C₄	#460	1	1	yes	characteristic
288	C₂×C₄	#460	1	1	yes	characteristic
289	C₂×C₄	#460	1	1	yes	characteristic
290	C₂×C₄	#461	1	1	yes	characteristic
291	C₂×C₄	#461	1	1	yes	characteristic
292	C₂×C₄	#475	1	1	yes	characteristic
293	C₂×C₄	#475	1	1	yes	characteristic
294	C₂×C₄	#476	1	1	yes	characteristic
295	C₂×C₄	#476	1	1	yes	characteristic
296	C₂×C₄	#476	1	1	yes	characteristic
297	C₂×C₄	#476	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×#121	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
302	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
303	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₈	—	4	4	no	—
305	C₈	—	8	4	no	—
306	C₈	#389	1	1	yes	characteristic
307	C₈	#389	1	1	yes	characteristic
308	C₈	#389	1	1	yes	characteristic
309	C₈	#389	1	1	yes	characteristic
310	C₈	C₂×#106	1	2	yes	—
311	C₈	C₂×#106	1	2	yes	—
312	C₈	C₂×#106	1	2	yes	—
313	C₈	C₂×#106	1	2	yes	—
314	C₈	C₂×C₄×D₄	1	1	yes	characteristic
315	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
316	C₈	C₂×#125	1	1	yes	characteristic
317	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#1948	1	1	yes	characteristic
326	C₂²	#1948	1	1	yes	characteristic
327	C₂²	#1948	1	1	yes	characteristic
328	C₂²	#1948	1	1	yes	characteristic
329	C₂²	C₂×#428	1	1	yes	characteristic
330	C₂²	C₂×#461	1	1	yes	characteristic
331	C₂²	C₂×#475	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	4	no	—
335	C₄	#1948	1	1	yes	characteristic
336	C₄	#1948	1	1	yes	characteristic
337	C₄	#1948	1	1	yes	characteristic
338	C₄	#1948	1	1	yes	characteristic
339	C₄	C₂×#389	1	1	yes	characteristic
340	C₄	C₂²×#106	1	1	yes	characteristic
341	C₄	C₂×#460	1	1	yes	characteristic
342	C₄	C₂×#476	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	#13188	1	2	yes	direct product factor
350	C₂	C₂×#1948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#1948
prime radical	C₂⁴	#137
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
5. #13188	C₂	direct (N × H)
18. C₂×#364	C₄	semidirect
19. C₂×#364	C₄	semidirect
20. C₂×#364	C₄	semidirect
79. #364	C₂×C₄	semidirect
80. #364	C₂×C₄	semidirect
349. C₂	#13188	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×#2155	2
C₂²×#364	2
C₂×#1848	2
C₂×#1772	2
#13188	2
C₂×#1388	2

Subgroups — C2×#13188 #495079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13188 #495079