Subgroups — #483064

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 415 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	C₂×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#11942	C₂	1	4	yes	maximal, complemented
5	#11941	C₂	1	2	yes	maximal, complemented
6	#11940	C₂	1	2	yes	maximal, complemented
7	#11939	C₂	1	4	yes	maximal, complemented
8	C₂×#533	—	2	1	no	—
9	C₂²×M₃₂	—	2	1	no	—
10	#3160	—	2	1	no	—
11	#3160	—	2	1	no	—
12	#3160	—	2	1	no	—
13	#3160	—	2	1	no	—
14	#3160	C₄	1	2	yes	—
15	#3160	C₄	1	2	yes	—
16	#3160	C₂²	1	2	yes	—
17	#3160	C₂²	1	2	yes	—
18	C₂×#500	C₂²	1	1	yes	characteristic
19	C₂×#500	C₂²	1	1	yes	characteristic
20	C₂×#499	—	2	1	no	—
21	C₂×#499	C₄	1	2	yes	—
22	C₂×#498	—	2	1	no	—
23	C₂×#431	—	2	1	no	—
24	C₂×#427	—	2	1	no	—
25	C₂×#427	C₄	1	2	yes	—
26	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
27	#2150	—	2	1	no	—
28	#2150	—	2	1	no	—
29	#2150	C₂²	1	2	yes	—
30	#2150	C₂²	1	2	yes	—
31	C₂×C₄⋊C₁₆	—	2	1	no	—
32	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
33	#1564	—	2	1	no	—
34	#1564	—	2	1	no	—
35	#1564	C₂²	1	2	yes	—
36	#1564	C₂²	1	2	yes	—
37	C₂×#335	C₂²	1	2	yes	—
38	C₂×#334	—	2	1	no	—
39	#1365	C₂²	1	8	yes	complemented
40	#1364	C₂²	1	8	yes	complemented
41	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₈	D₄	1	1	yes	characteristic
43	#533	—	2	2	no	—
44	#533	—	2	2	no	—
45	#533	—	2	2	no	—
46	#533	—	2	2	no	—
47	#533	—	4	1	no	—
48	#533	—	4	1	no	—
49	#533	—	4	1	no	—
50	#533	—	4	1	no	—
51	C₂×C₄⋊Q₈	—	2	1	no	—
52	C₂×C₄⋊Q₈	—	2	1	no	—
53	C₂×#125	—	2	2	no	—
54	C₂×C₂²⋊Q₈	—	4	1	no	—
55	C₂×C₂²⋊Q₈	—	4	1	no	—
56	C₂×C₄×Q₈	—	4	1	no	—
57	C₂×C₄×Q₈	—	4	1	no	—
58	C₂×#117	—	2	2	no	—
59	C₂²×C₄²	D₄	1	1	yes	characteristic
60	C₂×M₃₂	—	2	2	no	—
61	C₂×M₃₂	—	2	2	no	—
62	C₂×M₃₂	—	4	4	no	—
63	C₂²×C₁₆	—	2	2	no	—
64	#500	—	2	1	no	—
65	#500	—	2	1	no	—
66	#500	—	2	1	no	—
67	#500	—	2	1	no	—
68	#500	C₂×C₄	1	2	yes	—
69	#500	C₂×C₄	1	2	yes	—
70	#500	C₂³	1	2	yes	—
71	#500	C₂³	1	2	yes	—
72	#499	—	2	2	no	—
73	#499	—	2	2	no	—
74	#499	—	2	2	no	—
75	#499	—	2	2	no	—
76	#499	—	2	2	no	—
77	#499	—	2	2	no	—
78	#499	C₂×C₄	1	4	yes	—
79	#499	C₂×C₄	1	4	yes	—
80	#498	—	2	2	no	—
81	#498	—	2	2	no	—
82	#498	—	2	2	no	—
83	#498	—	2	2	no	—
84	#431	—	2	2	no	—
85	#431	—	2	2	no	—
86	#431	—	4	1	no	—
87	#431	—	4	1	no	—
88	#427	—	2	2	no	—
89	#427	—	2	2	no	—
90	#427	—	2	2	no	—
91	#427	—	2	2	no	—
92	#427	—	2	2	no	—
93	#427	—	2	2	no	—
94	#427	C₂×C₄	1	4	yes	—
95	#427	C₂×C₄	1	4	yes	—
96	C₂×#107	—	2	1	no	—
97	C₂×#107	—	2	1	no	—
98	C₂×#107	C₂×C₄	1	2	yes	—
99	C₂×#107	C₂×C₄	1	2	yes	—
100	C₂×#106	—	2	2	no	—
101	C₂×#106	—	2	2	no	—
102	C₂×C₄⋊C₈	—	2	2	no	—
103	C₂×C₂²⋊C₈	—	4	1	no	—
104	C₂×C₄×C₈	—	4	1	no	—
105	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈	D₄	1	1	yes	characteristic
109	C₂×C₄×C₈	D₄	1	1	yes	characteristic
110	C₂×C₄×C₈	D₄	1	1	yes	characteristic
111	C₂×C₄×C₈	D₄	1	1	yes	characteristic
112	C₂×C₄×C₈	D₄	1	1	yes	characteristic
113	C₂×C₄×C₈	D₄	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	C₄⋊C₁₆	—	2	2	no	—
119	C₄⋊C₁₆	—	2	2	no	—
120	C₄⋊C₁₆	C₂³	1	4	yes	—
121	#335	—	2	2	no	—
122	#335	C₂³	1	4	yes	—
123	#334	—	2	2	no	—
124	#334	—	2	2	no	—
125	C₂²×Q₈	—	4	1	no	—
126	C₂²×Q₈	—	4	1	no	—
127	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₄	D₈	1	1	yes	characteristic
129	C₂³×C₄	SD₁₆	1	1	yes	characteristic
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	4	2	no	—
132	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₈	SD₁₆	1	4	yes	—
139	C₂²×C₈	Q₁₆	1	4	yes	—
140	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	4	1	no	—
151	C₄⋊Q₈	—	4	1	no	—
152	C₄⋊Q₈	—	4	1	no	—
153	C₄⋊Q₈	—	4	1	no	—
154	#125	—	2	4	no	—
155	#125	—	2	4	no	—
156	#125	—	4	2	no	—
157	#125	—	4	2	no	—
158	C₂²⋊Q₈	—	4	2	no	—
159	C₂²⋊Q₈	—	4	2	no	—
160	C₂²⋊Q₈	—	4	2	no	—
161	C₂²⋊Q₈	—	4	2	no	—
162	C₂²⋊Q₈	—	4	2	no	—
163	C₂²⋊Q₈	—	4	2	no	—
164	C₂²⋊Q₈	—	4	2	no	—
165	C₂²⋊Q₈	—	4	2	no	—
166	C₄×Q₈	—	4	2	no	—
167	C₄×Q₈	—	4	2	no	—
168	C₄×Q₈	—	4	2	no	—
169	C₄×Q₈	—	4	2	no	—
170	C₄×Q₈	—	4	2	no	—
171	C₄×Q₈	—	4	2	no	—
172	C₄×Q₈	—	4	2	no	—
173	C₄×Q₈	—	4	2	no	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	#117	—	4	2	no	—
177	#117	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	4	2	no	—
181	C₂×C₄⋊C₄	—	4	2	no	—
182	C₂×C₄⋊C₄	—	4	1	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₂²⋊C₄	—	4	2	no	—
185	C₂×C₄²	—	4	2	no	—
186	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄²	D₈	1	1	yes	characteristic
190	C₂×C₄²	D₈	1	1	yes	characteristic
191	C₂×C₄²	D₈	1	1	yes	characteristic
192	C₂×C₄²	SD₁₆	1	1	yes	characteristic
193	C₂×C₄²	SD₁₆	1	1	yes	characteristic
194	C₂×C₄²	SD₁₆	1	1	yes	characteristic
195	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
196	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
197	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
198	M₃₂	—	4	4	no	—
199	M₃₂	—	4	4	no	—
200	C₂×C₁₆	—	2	4	no	—
201	C₂×C₁₆	—	4	4	no	—
202	#107	—	2	2	no	—
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	C₂²×C₄	1	4	yes	—
209	#107	C₂²×C₄	1	4	yes	—
210	#106	—	2	4	no	—
211	#106	—	2	4	no	—
212	#106	—	2	4	no	—
213	#106	—	2	4	no	—
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	4	2	no	—
216	C₂²⋊C₈	—	4	2	no	—
217	C₂²⋊C₈	—	4	2	no	—
218	C₄×C₈	—	4	2	no	—
219	C₄×C₈	—	4	2	no	—
220	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
226	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
227	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
228	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
229	C₄×C₈	C₂×D₄	1	1	yes	characteristic
230	C₄×C₈	C₂×D₄	1	1	yes	characteristic
231	C₄×C₈	C₂×D₄	1	1	yes	characteristic
232	C₄×C₈	C₂×D₄	1	1	yes	characteristic
233	C₄×C₈	C₂×D₄	1	1	yes	characteristic
234	C₄×C₈	C₂×D₄	1	1	yes	characteristic
235	C₄×C₈	C₂⁴	1	1	yes	characteristic
236	C₂⁴	#102	1	1	yes	characteristic
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	8	4	no	—
242	C₂×Q₈	—	8	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	#101	1	4	yes	—
248	C₂²×C₄	#102	1	1	yes	characteristic
249	C₂²×C₄	#102	1	1	yes	characteristic
250	C₂²×C₄	#102	1	1	yes	characteristic
251	C₂²×C₄	#102	1	1	yes	characteristic
252	C₂²×C₄	#102	1	1	yes	characteristic
253	C₂²×C₄	#102	1	1	yes	characteristic
254	C₂²×C₄	#102	1	1	yes	characteristic
255	C₂²×C₄	#102	1	1	yes	characteristic
256	C₂²×C₄	#102	1	1	yes	characteristic
257	C₂²×C₄	#103	1	4	yes	—
258	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
264	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
265	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
266	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	8	4	no	—
275	C₂×C₈	#103	1	4	yes	—
276	C₂×C₈	#103	1	4	yes	—
277	C₂×C₈	#103	1	4	yes	—
278	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₈	C₂×SD₁₆	1	2	yes	—
285	C₂×C₈	C₂×SD₁₆	1	2	yes	—
286	C₂×C₈	C₂×SD₁₆	1	2	yes	—
287	C₂×C₈	C₂×Q₁₆	1	2	yes	—
288	C₂×C₈	C₂×Q₁₆	1	2	yes	—
289	C₂×C₈	C₂×Q₁₆	1	2	yes	—
290	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
291	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄⋊C₄	—	4	4	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	4	no	—
305	C₂²⋊C₄	—	4	4	no	—
306	C₄²	—	4	4	no	—
307	C₄²	—	4	4	no	—
308	C₄²	#102	1	1	yes	characteristic
309	C₄²	#102	1	1	yes	characteristic
310	C₄²	#102	1	1	yes	characteristic
311	C₄²	#102	1	1	yes	characteristic
312	C₄²	#102	1	1	yes	characteristic
313	C₄²	#102	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
317	C₄²	C₂×D₈	1	1	yes	characteristic
318	C₄²	C₂×D₈	1	1	yes	characteristic
319	C₄²	C₂×D₈	1	1	yes	characteristic
320	C₄²	C₂×SD₁₆	1	1	yes	characteristic
321	C₄²	C₂×SD₁₆	1	1	yes	characteristic
322	C₄²	C₂×SD₁₆	1	1	yes	characteristic
323	C₄²	C₂²×D₄	1	1	yes	characteristic
324	C₁₆	—	4	4	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	4	1	no	—
327	C₂³	#332	1	4	yes	—
328	C₂³	C₂×#102	1	1	yes	characteristic
329	C₂³	C₂×#102	1	1	yes	characteristic
330	C₂³	C₂×#102	1	1	yes	characteristic
331	Q₈	—	8	4	no	—
332	Q₈	—	8	4	no	—
333	Q₈	—	8	4	no	—
334	Q₈	—	8	4	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	#332	1	4	yes	—
352	C₂×C₄	#332	1	4	yes	—
353	C₂×C₄	#332	1	4	yes	—
354	C₂×C₄	C₂×#101	1	2	yes	—
355	C₂×C₄	C₂×#101	1	2	yes	—
356	C₂×C₄	C₂×#101	1	2	yes	—
357	C₂×C₄	C₂×#102	1	1	yes	characteristic
358	C₂×C₄	C₂×#102	1	1	yes	characteristic
359	C₂×C₄	C₂×#102	1	1	yes	characteristic
360	C₂×C₄	C₂×#102	1	1	yes	characteristic
361	C₂×C₄	C₂×#102	1	1	yes	characteristic
362	C₂×C₄	C₂×#102	1	1	yes	characteristic
363	C₂×C₄	C₂×#102	1	1	yes	characteristic
364	C₂×C₄	C₂×#102	1	1	yes	characteristic
365	C₂×C₄	C₂×#102	1	1	yes	characteristic
366	C₂×C₄	C₂×#103	1	2	yes	—
367	C₂×C₄	C₂×#103	1	2	yes	—
368	C₂×C₄	C₂×#103	1	2	yes	—
369	C₂×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
370	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
371	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	—	8	4	no	—
377	C₈	C₂×#103	1	2	yes	—
378	C₈	C₂×#103	1	2	yes	—
379	C₈	C₂×#103	1	2	yes	—
380	C₈	C₂²×SD₁₆	1	1	yes	characteristic
381	C₈	C₂²×Q₁₆	1	1	yes	characteristic
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	4	4	no	—
389	C₂²	C₂×#332	1	2	yes	—
390	C₂²	C₂×#332	1	2	yes	—
391	C₂²	C₂×#332	1	2	yes	—
392	C₂²	C₂²×#102	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	8	4	no	—
400	C₄	—	8	4	no	—
401	C₄	C₂×#332	1	2	yes	—
402	C₄	C₂×#332	1	2	yes	—
403	C₄	C₂×#332	1	2	yes	—
404	C₄	#2148	1	2	yes	—
405	C₄	C₂²×#101	1	1	yes	characteristic
406	C₄	C₂²×#103	1	1	yes	characteristic
407	C₄	#3414	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	2	2	no	—
410	C₂	—	2	2	no	—
411	C₂	C₂²×#332	1	1	yes	characteristic
412	C₂	#33380	1	1	yes	characteristic
413	C₂	C₂×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #11942	C₂	semidirect
5. #11941	C₂	semidirect
6. #11940	C₂	semidirect
7. #11939	C₂	semidirect
39. #1365	C₂²	semidirect
40. #1364	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3160	2
C₂×#2150	2
C₂×#1564	2
#11942	2
#11941	2
#11940	2
#11939	2