Subgroups — #147290

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#365	C₂	1	1	yes	maximal, characteristic
2	C₂×#1778	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×#1383	C₂	1	1	yes	maximal, characteristic
6	C₂×#1377	C₂	1	1	yes	maximal, characteristic
7	#10863	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	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#106	C₂²	1	1	yes	characteristic
14	C₂×C₄×M₁₆	—	2	1	no	—
15	C₂²×#97	C₂²	1	1	yes	characteristic
16	C₂×#365	—	2	2	no	—
17	C₂×#365	—	2	2	no	—
18	C₂×#365	C₄	1	4	yes	complemented
19	#1778	—	2	2	no	—
20	#1778	—	2	2	no	—
21	C₂×#341	—	2	1	no	—
22	C₂×#340	—	2	1	no	—
23	C₄⋊#95	—	2	1	no	—
24	#1387	—	2	2	no	—
25	#1387	C₄	1	4	yes	complemented
26	#1383	—	2	2	no	—
27	#1377	C₄	1	4	yes	complemented
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄⋊C₄	—	2	1	no	—
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄²	D₄	1	1	yes	characteristic
35	C₂×M₃₂	—	2	1	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	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×M₃₂	C₂×C₄	1	2	yes	—
45	C₂×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	—	2	1	no	—
49	C₂×#108	—	2	1	no	—
50	C₂×#108	—	2	1	no	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	C₂×C₄	1	2	yes	—
56	C₂×#108	C₂×C₄	1	2	yes	—
57	C₂×#108	D₄	1	2	yes	—
58	C₂×#108	D₄	1	2	yes	—
59	C₂×#108	D₄	1	2	yes	—
60	C₂×#108	Q₈	1	2	yes	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	C₂×C₄	1	2	yes	—
67	C₂×#106	—	2	1	no	—
68	C₂×#106	—	2	1	no	—
69	C₂×#106	—	2	1	no	—
70	C₂×#106	—	2	1	no	—
71	C₂×#106	—	2	1	no	—
72	C₂×#106	C₂×C₄	1	2	yes	—
73	C₄×M₁₆	—	2	4	no	—
74	C₄×M₁₆	—	4	2	no	—
75	C₂×#97	—	2	1	no	—
76	C₂×#97	—	2	1	no	—
77	C₂×#97	—	2	1	no	—
78	C₂×#97	—	2	1	no	—
79	C₂×#97	—	4	1	no	—
80	C₂×#97	C₂×C₄	1	2	yes	—
81	C₂×#97	Q₈	1	2	yes	—
82	C₂×C₄×C₈	—	4	1	no	—
83	C₂×#95	—	4	2	no	—
84	#365	—	2	8	no	—
85	#341	—	2	4	no	—
86	#340	—	4	2	no	—
87	C₂³×C₄	—	4	1	no	—
88	C₂³×C₄	—	4	1	no	—
89	C₂³×C₄	SD₁₆	1	2	yes	—
90	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	4	1	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	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	4	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₄⋊C₄	1	2	yes	—
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	C₂²×C₈	#41	1	1	yes	characteristic
122	C₂²×C₈	#41	1	1	yes	characteristic
123	C₂²×C₈	#41	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄²	—	2	1	no	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₄⋊C₄	1	2	yes	—
143	C₂×C₄²	SD₁₆	1	2	yes	—
144	M₃₂	—	2	2	no	—
145	M₃₂	—	2	2	no	—
146	M₃₂	—	2	2	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	4	1	no	—
150	M₃₂	—	4	1	no	—
151	M₃₂	C₄²	1	4	yes	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	4	1	no	—
157	C₂×C₁₆	—	4	1	no	—
158	#108	—	2	2	no	—
159	#108	—	2	2	no	—
160	#108	—	2	2	no	—
161	#108	—	2	2	no	—
162	#108	—	2	2	no	—
163	#108	—	2	2	no	—
164	#108	C₂²⋊C₄	1	4	yes	—
165	#108	C₄⋊C₄	1	4	yes	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	—	4	1	no	—
169	#107	—	4	1	no	—
170	#106	—	2	2	no	—
171	#106	—	2	2	no	—
172	#106	—	4	1	no	—
173	#106	—	4	1	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	#97	—	4	4	no	—
179	C₄×C₈	—	4	4	no	—
180	#95	—	8	4	no	—
181	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	8	2	no	—
189	C₂²×C₄	—	8	2	no	—
190	C₂²×C₄	—	8	2	no	—
191	C₂²×C₄	—	8	2	no	—
192	C₂²×C₄	—	8	1	no	—
193	C₂²×C₄	—	8	1	no	—
194	C₂²×C₄	—	8	1	no	—
195	C₂²×C₄	—	8	1	no	—
196	C₂²×C₄	#106	1	4	yes	—
197	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	#120	1	1	yes	characteristic
201	C₂²×C₄	#121	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
203	C₂²×C₄	#123	1	1	yes	characteristic
204	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
205	C₂²×C₄	#135	1	1	yes	characteristic
206	C₂²×C₄	#135	1	1	yes	characteristic
207	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₂²×C₄	#137	1	1	yes	characteristic
210	C₂²×C₄	#137	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₈	#117	1	1	yes	characteristic
236	C₂×C₈	#117	1	1	yes	characteristic
237	C₂×C₈	#117	1	1	yes	characteristic
238	C₂×C₈	C₄×D₄	1	1	yes	characteristic
239	C₂×C₈	C₄×D₄	1	1	yes	characteristic
240	C₂×C₈	C₄×D₄	1	1	yes	characteristic
241	C₂×C₈	C₄×D₄	1	1	yes	characteristic
242	C₂×C₈	C₄×D₄	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
245	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
246	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
247	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
248	C₂×C₈	#121	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	#123	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#124	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	#126	1	1	yes	characteristic
258	C₂×C₈	#126	1	1	yes	characteristic
259	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	8	1	no	—
265	C₄⋊C₄	—	8	1	no	—
266	C₄⋊C₄	—	8	1	no	—
267	C₄⋊C₄	—	8	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₁₆	—	2	4	no	—
273	C₁₆	—	4	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₂×#106	1	1	yes	characteristic
277	C₂³	C₄×SD₁₆	1	1	yes	characteristic
278	C₂³	C₄×SD₁₆	1	1	yes	characteristic
279	C₂³	#448	1	1	yes	characteristic
280	C₂³	#460	1	1	yes	characteristic
281	C₂³	#475	1	1	yes	characteristic
282	C₂³	#481	1	1	yes	characteristic
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	16	1	no	—
293	C₂×C₄	—	16	1	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	16	1	no	—
296	C₂×C₄	—	16	1	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#385	1	1	yes	characteristic
300	C₂×C₄	#385	1	1	yes	characteristic
301	C₂×C₄	#386	1	1	yes	characteristic
302	C₂×C₄	#387	1	1	yes	characteristic
303	C₂×C₄	#388	1	1	yes	characteristic
304	C₂×C₄	#389	1	1	yes	characteristic
305	C₂×C₄	#427	1	1	yes	characteristic
306	C₂×C₄	#428	1	1	yes	characteristic
307	C₂×C₄	#428	1	1	yes	characteristic
308	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
309	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
310	C₂×C₄	#440	1	1	yes	characteristic
311	C₂×C₄	#440	1	1	yes	characteristic
312	C₂×C₄	#441	1	1	yes	characteristic
313	C₂×C₄	#442	1	1	yes	characteristic
314	C₂×C₄	#449	1	1	yes	characteristic
315	C₂×C₄	#450	1	1	yes	characteristic
316	C₂×C₄	#452	1	1	yes	characteristic
317	C₂×C₄	#461	1	1	yes	characteristic
318	C₂×C₄	#463	1	1	yes	characteristic
319	C₂×C₄	#464	1	1	yes	characteristic
320	C₂×C₄	#476	1	1	yes	characteristic
321	C₂×C₄	#478	1	1	yes	characteristic
322	C₂×C₄	#479	1	1	yes	characteristic
323	C₂×C₄	#482	1	1	yes	characteristic
324	C₂×C₄	#483	1	1	yes	characteristic
325	C₂×C₄	#485	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	4	4	no	—
328	C₈	—	4	4	no	—
329	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
330	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
331	C₈	#387	1	1	yes	characteristic
332	C₈	#387	1	1	yes	characteristic
333	C₈	#387	1	1	yes	characteristic
334	C₈	#388	1	1	yes	characteristic
335	C₈	#389	1	1	yes	characteristic
336	C₈	#391	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₄×#106	1	1	yes	characteristic
346	C₂²	#1853	1	1	yes	characteristic
347	C₂²	#1865	1	1	yes	characteristic
348	C₂²	#1880	1	1	yes	characteristic
349	C₂²	#1919	1	1	yes	characteristic
350	C₂²	#1926	1	1	yes	characteristic
351	C₂²	#1948	1	1	yes	characteristic
352	C₂²	#2238	1	2	yes	—
353	C₂²	#2239	1	2	yes	—
354	C₄	—	4	4	no	—
355	C₄	—	16	2	no	—
356	C₄	—	16	2	no	—
357	C₄	#1436	1	1	yes	characteristic
358	C₄	#1776	1	1	yes	characteristic
359	C₄	#1854	1	1	yes	characteristic
360	C₄	#1864	1	1	yes	characteristic
361	C₄	#1877	1	1	yes	characteristic
362	C₄	#1920	1	1	yes	characteristic
363	C₄	#1927	1	1	yes	characteristic
364	C₄	#1949	1	1	yes	characteristic
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	#10908	1	1	yes	characteristic
370	C₂	#12931	1	2	yes	—
371	C₂	#13114	1	2	yes	—
372	C₂	#13138	1	1	yes	characteristic
373	C₂	#13139	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. C₂×#365	C₄	semidirect
25. #1387	C₄	semidirect
27. #1377	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#365	2
C₂×#1778	2
C₂×#1387	2
C₂×#1383	2
C₂×#1377	2
#10863	2