Subgroups — C₂×#12667 #493131

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#2110	C₂	1	2	yes	maximal
3	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#1772	C₂	1	1	yes	maximal, characteristic
5	#12667	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1378	C₂	1	2	yes	maximal
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₄	1	1	yes	characteristic
11	C₂×#427	C₄	1	1	yes	characteristic
12	C₂×#405	C₂²	1	2	yes	—
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×#364	C₄	1	4	yes	complemented
15	C₂×#364	C₄	1	2	yes	complemented
16	C₂×#364	C₄	1	2	yes	complemented
17	C₂×#364	C₂²	1	4	yes	—
18	#2110	—	2	4	no	—
19	#2110	C₂²	1	8	yes	—
20	C₂×#347	—	2	4	no	—
21	C₄×M₃₂	C₄	1	4	yes	complemented
22	C₄×M₃₂	C₂²	1	4	yes	—
23	C₂×#346	C₄	1	2	yes	complemented
24	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
25	#1772	C₄	1	4	yes	—
26	#1772	C₂²	1	4	yes	—
27	C₂×#339	C₄	1	2	yes	—
28	#1378	C₂²	1	8	yes	—
29	C₂²×M₁₆	C₂×C₄	1	2	yes	—
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂×#117	C₂×C₄	1	2	yes	—
33	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	2	no	—
41	C₂×M₃₂	C₂×C₄	1	4	yes	—
42	C₂×M₃₂	C₂×C₄	1	4	yes	—
43	C₂×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×M₃₂	C₂×C₄	1	2	yes	—
45	C₂×M₃₂	D₄	1	2	yes	—
46	C₂×M₃₂	D₄	1	2	yes	—
47	C₂×M₃₂	Q₈	1	2	yes	—
48	C₂×M₃₂	Q₈	1	2	yes	—
49	C₂×M₃₂	C₂³	1	4	yes	—
50	C₂×M₃₂	C₂³	1	2	yes	—
51	C₂×M₃₂	C₂³	1	2	yes	—
52	C₂²×C₁₆	—	2	2	no	—
53	C₂²×C₁₆	C₂×C₄	1	2	yes	—
54	C₂²×C₁₆	C₂×C₄	1	2	yes	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	C₂×C₄	1	4	yes	—
59	#427	—	2	2	no	—
60	#427	C₂×C₄	1	4	yes	—
61	C₂×#107	—	2	1	no	—
62	C₂×#106	—	2	1	no	—
63	#405	—	2	2	no	—
64	#405	—	2	2	no	—
65	#405	C₂×C₄	1	4	yes	—
66	#405	C₂³	1	4	yes	—
67	C₂×#97	—	2	2	no	—
68	C₂×C₄×C₈	—	2	2	no	—
69	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
73	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
74	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂³	1	2	yes	—
77	#364	C₂×C₄	1	8	yes	complemented
78	#364	C₂×C₄	1	8	yes	complemented
79	#347	—	2	16	no	—
80	#346	C₂×C₄	1	8	yes	complemented
81	C₄×C₁₆	C₂×C₄	1	8	yes	complemented
82	#339	C₂×C₄	1	8	yes	—
83	C₂³×C₄	—	2	1	no	—
84	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	C₂²×C₄	1	4	yes	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	C₄²	1	1	yes	characteristic
93	C₂²×C₈	C₄²	1	1	yes	characteristic
94	C₂²×C₈	C₄²	1	2	yes	—
95	C₂²×C₈	M₁₆	1	1	yes	characteristic
96	C₂²×C₈	M₁₆	1	1	yes	characteristic
97	C₂²×C₈	M₁₆	1	2	yes	—
98	C₂²×C₈	M₁₆	1	1	yes	characteristic
99	C₂²×C₈	M₁₆	1	1	yes	characteristic
100	C₂²×C₈	M₁₆	1	2	yes	—
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	2	yes	—
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	C₂⁴	1	1	yes	characteristic
116	#117	—	2	2	no	—
117	#117	—	2	2	no	—
118	#117	—	2	2	no	—
119	#117	C₂²×C₄	1	4	yes	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₂²⋊C₄	—	2	2	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₄²	1	1	yes	characteristic
127	C₂×C₄²	C₄²	1	1	yes	characteristic
128	C₂×C₄²	C₄²	1	2	yes	—
129	C₂×C₄²	C₂²×C₄	1	2	yes	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	4	no	—
133	M₃₂	—	2	4	no	—
134	M₃₂	—	2	4	no	—
135	M₃₂	—	2	2	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	C₄⋊C₄	1	4	yes	—
138	M₃₂	C₄⋊C₄	1	4	yes	—
139	M₃₂	C₂²×C₄	1	8	yes	—
140	M₃₂	C₂×D₄	1	4	yes	—
141	M₃₂	C₂×Q₈	1	4	yes	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	—	2	4	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
148	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
149	C₂×C₁₆	C₂²×C₄	1	4	yes	—
150	C₂×C₁₆	C₂²×C₄	1	4	yes	—
151	#108	—	2	4	no	—
152	#108	—	2	4	no	—
153	#107	—	2	4	no	—
154	#106	—	2	4	no	—
155	#97	—	2	4	no	—
156	#97	—	2	4	no	—
157	C₄×C₈	—	2	4	no	—
158	C₄×C₈	—	2	4	no	—
159	C₄×C₈	C₄²	1	2	yes	—
160	C₄×C₈	C₄²	1	2	yes	—
161	C₄×C₈	C₄²	1	2	yes	—
162	C₄×C₈	C₄²	1	2	yes	—
163	C₄×C₈	C₂²×C₄	1	2	yes	—
164	C₄×C₈	C₂²×C₄	1	2	yes	—
165	C₄×C₈	C₂²×C₄	1	2	yes	—
166	C₄×C₈	C₂²×C₄	1	2	yes	—
167	C₂⁴	#131	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
176	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
184	C₂²×C₄	#131	1	1	yes	characteristic
185	C₂²×C₄	#131	1	1	yes	characteristic
186	C₂²×C₄	#131	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
188	M₁₆	—	2	4	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	#97	1	2	yes	—
196	C₂×C₈	#97	1	1	yes	characteristic
197	C₂×C₈	#97	1	1	yes	characteristic
198	C₂×C₈	#97	1	2	yes	—
199	C₂×C₈	#97	1	1	yes	characteristic
200	C₂×C₈	#97	1	1	yes	characteristic
201	C₂×C₈	#97	1	2	yes	—
202	C₂×C₈	#97	1	2	yes	—
203	C₂×C₈	#97	1	2	yes	—
204	C₂×C₈	#97	1	2	yes	—
205	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄²	1	2	yes	—
208	C₂×C₈	C₂×C₄²	1	2	yes	—
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₈	#117	1	1	yes	characteristic
214	C₂×C₈	#117	1	1	yes	characteristic
215	C₂×C₈	#117	1	1	yes	characteristic
216	C₂×C₈	#117	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
222	C₂×C₈	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
226	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
227	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
228	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
229	C₂×C₈	C₂×M₁₆	1	2	yes	—
230	C₂×C₈	C₂×M₁₆	1	2	yes	—
231	C₂×C₈	C₂×M₁₆	1	2	yes	—
232	C₂×C₈	C₂×M₁₆	1	2	yes	—
233	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
237	C₂×C₈	C₂×#41	1	1	yes	characteristic
238	C₂×C₈	C₂×#41	1	1	yes	characteristic
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	4	2	no	—
249	C₄²	C₂×C₄²	1	2	yes	—
250	C₄²	C₂×C₄²	1	2	yes	—
251	C₁₆	—	2	4	no	—
252	C₁₆	—	2	8	no	—
253	C₁₆	C₂×C₄⋊C₄	1	8	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	#405	1	1	yes	characteristic
256	C₂³	#405	1	1	yes	characteristic
257	C₂³	#436	1	1	yes	characteristic
258	C₂³	#436	1	1	yes	characteristic
259	C₂³	#446	1	1	yes	characteristic
260	C₂³	#446	1	1	yes	characteristic
261	C₂³	C₂×#131	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×#97	1	1	yes	characteristic
275	C₂×C₄	C₂×#97	1	1	yes	characteristic
276	C₂×C₄	C₂×#97	1	1	yes	characteristic
277	C₂×C₄	C₂×#97	1	1	yes	characteristic
278	C₂×C₄	#405	1	1	yes	characteristic
279	C₂×C₄	#405	1	1	yes	characteristic
280	C₂×C₄	#436	1	1	yes	characteristic
281	C₂×C₄	#436	1	1	yes	characteristic
282	C₂×C₄	#436	1	1	yes	characteristic
283	C₂×C₄	#436	1	1	yes	characteristic
284	C₂×C₄	#436	1	1	yes	characteristic
285	C₂×C₄	#436	1	1	yes	characteristic
286	C₂×C₄	#446	1	1	yes	characteristic
287	C₂×C₄	#446	1	1	yes	characteristic
288	C₂×C₄	#446	1	1	yes	characteristic
289	C₂×C₄	#446	1	1	yes	characteristic
290	C₂×C₄	#446	1	1	yes	characteristic
291	C₂×C₄	#446	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
296	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
297	C₂×C₄	C₂×#131	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
300	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
301	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
302	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
303	C₈	C₂×#97	1	2	yes	—
304	C₈	C₂×#97	1	2	yes	—
305	C₈	C₂×#97	1	2	yes	—
306	C₈	C₂×#97	1	2	yes	—
307	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
308	C₈	C₂×#117	1	1	yes	characteristic
309	C₈	C₂×C₄×D₄	1	1	yes	characteristic
310	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	#1770	1	1	yes	characteristic
319	C₂²	#1770	1	1	yes	characteristic
320	C₂²	#1770	1	1	yes	characteristic
321	C₂²	#1770	1	1	yes	characteristic
322	C₂²	C₂×#405	1	1	yes	characteristic
323	C₂²	C₂×#436	1	1	yes	characteristic
324	C₂²	C₂×#446	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	4	no	—
328	C₄	#1770	1	1	yes	characteristic
329	C₄	#1770	1	1	yes	characteristic
330	C₄	#1770	1	1	yes	characteristic
331	C₄	#1770	1	1	yes	characteristic
332	C₄	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
333	C₄	C₂²×#97	1	1	yes	characteristic
334	C₄	C₂×#436	1	1	yes	characteristic
335	C₄	C₂×#446	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	#12667	1	2	yes	direct product factor
343	C₂	C₂×#1770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#1770
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ 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. #12667	C₂	direct (N × H)
14. C₂×#364	C₄	semidirect
15. C₂×#364	C₄	semidirect
16. C₂×#364	C₄	semidirect
21. C₄×M₃₂	C₄	semidirect
23. C₂×#346	C₄	semidirect
24. C₂×C₄×C₁₆	C₄	semidirect
77. #364	C₂×C₄	semidirect
78. #364	C₂×C₄	semidirect
80. #346	C₂×C₄	semidirect
81. C₄×C₁₆	C₂×C₄	semidirect
342. C₂	#12667	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#364	2
C₂×#2110	2
C₂×C₄×M₃₂	2
C₂×#1772	2
#12667	2
C₂×#1378	2

Subgroups — C2×#12667 #493131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12667 #493131