Subgroups — #147179

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1945	C₂	1	1	yes	maximal, characteristic
3	C₂×#1389	C₂	1	1	yes	maximal, characteristic
4	C₂×#1388	C₂	1	1	yes	maximal, characteristic
5	#12569	C₂	1	1	yes	maximal, characteristic
6	#12567	C₂	1	1	yes	maximal, characteristic
7	#10859	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂²×#108	C₂²	1	1	yes	characteristic
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂×#424	—	2	1	no	—
15	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄×M₃₂	—	2	2	no	—
17	C₄×M₃₂	—	2	2	no	—
18	C₂×#346	—	2	1	no	—
19	C₂×C₄×C₁₆	—	2	1	no	—
20	#1945	—	2	1	no	—
21	#1945	—	2	1	no	—
22	#1945	C₄	1	2	yes	complemented
23	#1945	C₄	1	2	yes	complemented
24	C₂×#339	—	2	1	no	—
25	C₂×#339	—	2	1	no	—
26	#1729	—	2	1	no	—
27	#1389	—	2	2	no	—
28	#1388	—	2	2	no	—
29	#1387	C₄	1	4	yes	complemented
30	#1386	C₄	1	4	yes	complemented
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	—	2	1	no	—
36	C₂×#117	—	2	1	no	—
37	C₂²×C₄²	D₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	C₂×C₄	1	2	yes	—
45	C₂×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	D₄	1	2	yes	—
47	C₂×M₃₂	D₄	1	2	yes	—
48	C₂×M₃₂	Q₈	1	2	yes	—
49	C₂×M₃₂	Q₈	1	2	yes	—
50	C₂²×C₁₆	—	2	1	no	—
51	C₂²×C₁₆	—	2	1	no	—
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	—	2	1	no	—
61	C₂×#108	—	2	1	no	—
62	C₂×#108	C₂×C₄	1	2	yes	—
63	C₂×#108	C₂×C₄	1	2	yes	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	—	2	1	no	—
67	#427	—	2	1	no	—
68	#427	—	2	1	no	—
69	#427	—	2	1	no	—
70	#427	C₂×C₄	1	2	yes	—
71	#427	C₂×C₄	1	2	yes	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	C₂×C₄	1	2	yes	—
74	C₂×#106	—	2	1	no	—
75	C₂×#106	C₂×C₄	1	2	yes	—
76	#424	—	2	2	no	—
77	#424	—	2	2	no	—
78	#424	—	4	2	no	—
79	C₂×C₄⋊C₈	—	4	2	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	C₂×C₄×C₈	—	2	1	no	—
83	C₂×C₄×C₈	—	2	1	no	—
84	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×C₈	D₄	1	1	yes	characteristic
87	C₂×C₄×C₈	D₄	1	1	yes	characteristic
88	#346	—	2	4	no	—
89	C₄×C₁₆	—	2	4	no	—
90	#339	—	2	2	no	—
91	#339	—	2	2	no	—
92	#339	—	4	2	no	—
93	#336	—	4	2	no	—
94	C₂³×C₄	—	2	1	no	—
95	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	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	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	4	1	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₈	D₈	1	2	yes	—
117	C₂²×C₈	SD₁₆	1	2	yes	—
118	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₄	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	#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	C₂²×C₈	#41	1	1	yes	characteristic
131	C₂²×C₈	#41	1	1	yes	characteristic
132	#117	—	2	2	no	—
133	#117	—	2	2	no	—
134	#117	—	4	1	no	—
135	#117	—	4	1	no	—
136	#117	—	4	1	no	—
137	#117	—	4	1	no	—
138	#117	—	4	1	no	—
139	#117	—	4	1	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₂²⋊C₄	—	4	1	no	—
143	C₂×C₂²⋊C₄	—	4	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄²	SD₁₆	1	2	yes	—
152	C₂×C₄²	Q₁₆	1	2	yes	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	C₄⋊C₄	1	4	yes	—
160	M₃₂	C₄⋊C₄	1	4	yes	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	2	2	no	—
170	#108	—	2	2	no	—
171	#108	—	4	1	no	—
172	#108	—	4	1	no	—
173	#108	—	4	1	no	—
174	#108	—	4	1	no	—
175	#107	—	2	2	no	—
176	#107	—	4	1	no	—
177	#107	—	4	1	no	—
178	#107	C₄²	1	4	yes	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	#106	—	2	2	no	—
183	C₄⋊C₈	—	4	4	no	—
184	C₄⋊C₈	—	4	4	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₂⁴	#135	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	#121	1	1	yes	characteristic
205	C₂²×C₄	#121	1	1	yes	characteristic
206	C₂²×C₄	#124	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
210	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
212	C₂²×C₄	#135	1	1	yes	characteristic
213	C₂²×C₄	#135	1	1	yes	characteristic
214	C₂²×C₄	#135	1	1	yes	characteristic
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	8	1	no	—
220	M₁₆	—	8	1	no	—
221	M₁₆	—	8	1	no	—
222	M₁₆	—	8	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
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	—	8	1	no	—
234	C₂×C₈	—	8	1	no	—
235	C₂×C₈	—	8	1	no	—
236	C₂×C₈	—	8	1	no	—
237	C₂×C₈	#102	1	2	yes	—
238	C₂×C₈	#102	1	2	yes	—
239	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	#117	1	1	yes	characteristic
244	C₂×C₈	#117	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×D₄	1	1	yes	characteristic
250	C₂×C₈	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
254	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
255	C₂×C₈	#120	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	C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
260	C₂×C₈	#123	1	1	yes	characteristic
261	C₂×C₈	#123	1	1	yes	characteristic
262	C₂×C₈	#123	1	1	yes	characteristic
263	C₂×C₈	#125	1	1	yes	characteristic
264	C₂×C₈	#125	1	1	yes	characteristic
265	C₂×C₈	#126	1	1	yes	characteristic
266	C₂×C₈	#126	1	1	yes	characteristic
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	8	1	no	—
274	C₂²⋊C₄	—	8	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	8	1	no	—
282	C₄²	—	8	1	no	—
283	C₄²	#103	1	2	yes	—
284	C₄²	#103	1	2	yes	—
285	C₁₆	—	2	4	no	—
286	C₁₆	—	2	4	no	—
287	C₂³	—	4	2	no	—
288	C₂³	#416	1	1	yes	characteristic
289	C₂³	C₄×D₈	1	1	yes	characteristic
290	C₂³	C₄×SD₁₆	1	1	yes	characteristic
291	C₂³	#465	1	1	yes	characteristic
292	C₂³	#467	1	1	yes	characteristic
293	C₂³	#488	1	1	yes	characteristic
294	C₂³	#499	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄	#386	1	1	yes	characteristic
307	C₂×C₄	#388	1	1	yes	characteristic
308	C₂×C₄	#388	1	1	yes	characteristic
309	C₂×C₄	#389	1	1	yes	characteristic
310	C₂×C₄	#389	1	1	yes	characteristic
311	C₂×C₄	#391	1	1	yes	characteristic
312	C₂×C₄	C₂×#102	1	1	yes	characteristic
313	C₂×C₄	C₂×#103	1	1	yes	characteristic
314	C₂×C₄	#416	1	1	yes	characteristic
315	C₂×C₄	C₄×D₈	1	1	yes	characteristic
316	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
317	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
318	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
319	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
320	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
321	C₂×C₄	#465	1	1	yes	characteristic
322	C₂×C₄	#465	1	1	yes	characteristic
323	C₂×C₄	#465	1	1	yes	characteristic
324	C₂×C₄	#466	1	1	yes	characteristic
325	C₂×C₄	#466	1	1	yes	characteristic
326	C₂×C₄	#467	1	1	yes	characteristic
327	C₂×C₄	#486	1	1	yes	characteristic
328	C₂×C₄	#487	1	1	yes	characteristic
329	C₂×C₄	#488	1	1	yes	characteristic
330	C₂×C₄	#498	1	1	yes	characteristic
331	C₂×C₄	#499	1	1	yes	characteristic
332	C₂×C₄	#500	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	8	2	no	—
336	C₈	—	8	2	no	—
337	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
338	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
339	C₈	#385	1	1	yes	characteristic
340	C₈	#385	1	1	yes	characteristic
341	C₈	#387	1	1	yes	characteristic
342	C₈	#387	1	1	yes	characteristic
343	C₈	#387	1	1	yes	characteristic
344	C₈	#387	1	1	yes	characteristic
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	C₄×#102	1	1	yes	characteristic
353	C₂²	#1892	1	1	yes	characteristic
354	C₂²	#1933	1	1	yes	characteristic
355	C₂²	#1934	1	1	yes	characteristic
356	C₂²	#1941	1	1	yes	characteristic
357	C₂²	#1942	1	1	yes	characteristic
358	C₂²	#1991	1	1	yes	characteristic
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	8	4	no	—
362	C₄	—	16	2	no	—
363	C₄	#1436	1	1	yes	characteristic
364	C₄	C₄×#103	1	1	yes	characteristic
365	C₄	#1893	1	1	yes	characteristic
366	C₄	#1933	1	1	yes	characteristic
367	C₄	#1934	1	1	yes	characteristic
368	C₄	#1941	1	1	yes	characteristic
369	C₄	#1942	1	1	yes	characteristic
370	C₄	#1990	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	#10880	1	1	yes	characteristic
378	C₂	#12646	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
22. #1945	C₄	semidirect
23. #1945	C₄	semidirect
29. #1387	C₄	semidirect
30. #1386	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1945	2
C₂×#1389	2
C₂×#1388	2
#12569	2
#12567	2
#10859	2