Subgroups — #147186

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	C₂×#1952	C₂	1	1	yes	maximal, characteristic
3	C₂×#1387	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	1	yes	maximal, characteristic
5	C₂×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂×#1380	C₂	1	1	yes	maximal, characteristic
7	#10870	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂²×#108	C₂²	1	1	yes	characteristic
12	C₂×#428	—	2	1	no	—
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂²×#107	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2108	—	2	2	no	—
17	#2108	—	2	2	no	—
18	C₂×#346	—	2	1	no	—
19	C₂×C₄×C₁₆	—	2	1	no	—
20	#1952	—	2	2	no	—
21	#1952	C₄	1	4	yes	complemented
22	C₂×#343	—	2	1	no	—
23	C₂×#343	—	2	1	no	—
24	#1736	—	2	1	no	—
25	#1387	—	2	2	no	—
26	#1387	C₄	1	4	yes	complemented
27	#1381	C₄	1	4	yes	complemented
28	#1380	—	2	1	no	—
29	#1380	—	2	1	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	D₄	1	1	yes	characteristic
33	C₂³×C₈	C₂³	1	1	yes	characteristic
34	C₂×#117	D₄	1	1	yes	characteristic
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄⋊C₄	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×M₃₂	D₄	1	2	yes	—
42	C₂×M₃₂	Q₈	1	2	yes	—
43	C₂²×C₁₆	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	—	2	1	no	—
46	C₂²×C₁₆	—	2	1	no	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	C₂×C₄	1	2	yes	—
49	C₂²×C₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₁₆	D₄	1	1	yes	characteristic
51	C₂²×C₁₆	Q₈	1	2	yes	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	1	no	—
58	C₂×#108	—	2	1	no	—
59	C₂×#108	—	2	1	no	—
60	C₂×#108	C₂×C₄	1	2	yes	—
61	C₂×#108	C₂×C₄	1	2	yes	—
62	C₂×#108	C₂×C₄	1	2	yes	—
63	C₂×#108	C₂×C₄	1	2	yes	—
64	#428	—	2	4	no	—
65	#428	—	4	2	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	2	1	no	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	1	no	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	2	1	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	2	1	no	—
76	C₂×#107	—	4	1	no	—
77	C₂×#107	C₂×C₄	1	2	yes	—
78	C₂×#107	C₂×C₄	1	2	yes	—
79	C₂×#106	—	4	1	no	—
80	#405	—	2	1	no	—
81	#405	—	2	1	no	—
82	#405	C₂×C₄	1	2	yes	—
83	#405	D₄	1	2	yes	—
84	C₂×#97	—	2	1	no	—
85	C₂×C₄×C₈	—	2	1	no	—
86	#346	—	2	4	no	—
87	C₄×C₁₆	—	2	4	no	—
88	#343	—	2	4	no	—
89	#343	—	4	2	no	—
90	#340	—	4	1	no	—
91	#340	—	4	1	no	—
92	C₂³×C₄	—	4	1	no	—
93	C₂³×C₄	—	4	1	no	—
94	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
111	C₂×M₁₆	SD₁₆	1	2	yes	—
112	C₂×M₁₆	Q₁₆	1	2	yes	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	—	4	1	no	—
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	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	#117	—	2	1	no	—
131	#117	—	2	1	no	—
132	#117	C₂²⋊C₄	1	2	yes	—
133	#117	SD₁₆	1	2	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
147	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	—	2	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	C₄⋊C₄	1	4	yes	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
166	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	2	2	no	—
170	#108	—	2	2	no	—
171	#108	—	2	2	no	—
172	#108	—	4	1	no	—
173	#108	—	4	1	no	—
174	#108	C₄²	1	4	yes	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	—	4	1	no	—
180	#107	—	4	1	no	—
181	#107	—	4	1	no	—
182	#107	—	4	1	no	—
183	#107	—	4	4	no	—
184	#106	—	4	4	no	—
185	#97	—	2	2	no	—
186	#97	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₂⁴	#135	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	8	1	no	—
195	C₂²×C₄	—	8	1	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	1	no	—
198	C₂²×C₄	—	16	1	no	—
199	C₂²×C₄	—	16	1	no	—
200	C₂²×C₄	—	16	1	no	—
201	C₂²×C₄	—	16	1	no	—
202	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂²×C₄	#121	1	1	yes	characteristic
206	C₂²×C₄	#121	1	1	yes	characteristic
207	C₂²×C₄	#124	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₈	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	#135	1	1	yes	characteristic
214	C₂²×C₄	#135	1	1	yes	characteristic
215	C₂²×C₄	#135	1	1	yes	characteristic
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	8	1	no	—
226	M₁₆	—	8	1	no	—
227	M₁₆	#103	1	4	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	8	1	no	—
236	C₂×C₈	—	8	1	no	—
237	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₈	#117	1	1	yes	characteristic
241	C₂×C₈	#117	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×D₄	1	1	yes	characteristic
249	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
250	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
251	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
252	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
253	C₂×C₈	#120	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	#123	1	1	yes	characteristic
259	C₂×C₈	#123	1	1	yes	characteristic
260	C₂×C₈	#123	1	1	yes	characteristic
261	C₂×C₈	#125	1	1	yes	characteristic
262	C₂×C₈	#125	1	1	yes	characteristic
263	C₂×C₈	#126	1	1	yes	characteristic
264	C₂×C₈	#126	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	8	1	no	—
272	C₄⋊C₄	—	8	1	no	—
273	C₄⋊C₄	—	8	1	no	—
274	C₄⋊C₄	—	8	1	no	—
275	C₄⋊C₄	#102	1	2	yes	—
276	C₄⋊C₄	#102	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	4	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	#416	1	1	yes	characteristic
289	C₂³	C₄×SD₁₆	1	1	yes	characteristic
290	C₂³	C₄×Q₁₆	1	1	yes	characteristic
291	C₂³	#465	1	1	yes	characteristic
292	C₂³	#466	1	1	yes	characteristic
293	C₂³	#488	1	1	yes	characteristic
294	C₂³	#499	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	—	16	1	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	16	1	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	16	2	no	—
311	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
312	C₂×C₄	#386	1	1	yes	characteristic
313	C₂×C₄	#388	1	1	yes	characteristic
314	C₂×C₄	#388	1	1	yes	characteristic
315	C₂×C₄	#389	1	1	yes	characteristic
316	C₂×C₄	#389	1	1	yes	characteristic
317	C₂×C₄	#391	1	1	yes	characteristic
318	C₂×C₄	C₂×#102	1	1	yes	characteristic
319	C₂×C₄	C₂×#103	1	1	yes	characteristic
320	C₂×C₄	#416	1	1	yes	characteristic
321	C₂×C₄	C₄×D₈	1	1	yes	characteristic
322	C₂×C₄	C₄×D₈	1	1	yes	characteristic
323	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
324	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
325	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
326	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
327	C₂×C₄	#465	1	1	yes	characteristic
328	C₂×C₄	#465	1	1	yes	characteristic
329	C₂×C₄	#465	1	1	yes	characteristic
330	C₂×C₄	#466	1	1	yes	characteristic
331	C₂×C₄	#467	1	1	yes	characteristic
332	C₂×C₄	#467	1	1	yes	characteristic
333	C₂×C₄	#486	1	1	yes	characteristic
334	C₂×C₄	#487	1	1	yes	characteristic
335	C₂×C₄	#488	1	1	yes	characteristic
336	C₂×C₄	#498	1	1	yes	characteristic
337	C₂×C₄	#499	1	1	yes	characteristic
338	C₂×C₄	#500	1	1	yes	characteristic
339	C₈	—	2	4	no	—
340	C₈	—	4	4	no	—
341	C₈	—	8	2	no	—
342	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
343	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
344	C₈	#385	1	1	yes	characteristic
345	C₈	#385	1	1	yes	characteristic
346	C₈	#387	1	1	yes	characteristic
347	C₈	#387	1	1	yes	characteristic
348	C₈	#387	1	1	yes	characteristic
349	C₈	#387	1	1	yes	characteristic
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	C₄×#103	1	1	yes	characteristic
363	C₂²	#1893	1	1	yes	characteristic
364	C₂²	#1933	1	1	yes	characteristic
365	C₂²	#1934	1	1	yes	characteristic
366	C₂²	#1941	1	1	yes	characteristic
367	C₂²	#1942	1	1	yes	characteristic
368	C₂²	#1991	1	1	yes	characteristic
369	C₂²	#2221	1	1	yes	characteristic
370	C₂²	#2222	1	1	yes	characteristic
371	C₂²	#2223	1	1	yes	characteristic
372	C₂²	#2223	1	1	yes	characteristic
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	16	2	no	—
376	C₄	—	16	2	no	—
377	C₄	#1436	1	1	yes	characteristic
378	C₄	C₄×#102	1	1	yes	characteristic
379	C₄	#1892	1	1	yes	characteristic
380	C₄	#1933	1	1	yes	characteristic
381	C₄	#1934	1	1	yes	characteristic
382	C₄	#1941	1	1	yes	characteristic
383	C₄	#1942	1	1	yes	characteristic
384	C₄	#1990	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	#10880	1	1	yes	characteristic
390	C₂	#13072	1	1	yes	characteristic
391	C₂	#13073	1	1	yes	characteristic
392	C₂	#13144	1	1	yes	characteristic
393	C₂	#13145	1	1	yes	characteristic
394	C₂	#13171	1	1	yes	characteristic
395	C₂	#13172	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
21. #1952	C₄	semidirect
26. #1387	C₄	semidirect
27. #1381	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
C₂×#1952	2
C₂×#1387	2
C₂×#1381	2
C₂×#1380	2
#10870	2