Subgroups — #126763

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#20437	C₂	1	1	yes	maximal, characteristic
4	#7169	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	—	2	1	no	—
6	C₂²×M₃₂	—	2	1	no	—
7	C₂×#423	C₄	1	1	yes	characteristic
8	C₂×#405	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2285	C₄	1	1	yes	characteristic
11	#2150	—	2	1	no	—
12	#2150	—	2	1	no	—
13	#2150	C₄	1	2	yes	complemented
14	#2150	C₂²	1	2	yes	—
15	C₂×C₄⋊C₁₆	—	2	1	no	—
16	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
17	C₄×M₃₂	—	2	1	no	—
18	C₄×M₃₂	—	2	1	no	—
19	C₄×M₃₂	C₄	1	2	yes	complemented
20	C₄×M₃₂	C₂²	1	2	yes	—
21	C₂×#346	C₄	1	2	yes	complemented
22	C₂×C₄×C₁₆	—	2	1	no	—
23	#1770	—	2	1	no	—
24	#1770	C₄	1	2	yes	—
25	C₈×C₄⋊C₄	—	2	1	no	—
26	C₈×C₄⋊C₄	C₂²	1	2	yes	—
27	#1729	—	2	1	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂×#117	—	2	1	no	—
32	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×M₃₂	—	2	4	no	—
34	C₂×M₃₂	—	2	4	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	2	no	—
38	C₂×M₃₂	—	2	2	no	—
39	C₂×M₃₂	—	4	2	no	—
40	C₂×M₃₂	—	4	2	no	—
41	C₂²×C₁₆	—	2	2	no	—
42	C₂²×C₁₆	—	2	2	no	—
43	#423	—	2	1	no	—
44	#423	—	2	1	no	—
45	#423	C₈	1	2	yes	—
46	#423	C₈	1	2	yes	—
47	C₂×C₄⋊C₈	—	2	1	no	—
48	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	#405	—	4	2	no	—
50	#405	—	4	2	no	—
51	C₂×#97	—	2	2	no	—
52	C₂×C₄×C₈	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#384	—	2	1	no	—
66	#382	C₈	1	2	yes	—
67	#380	—	2	1	no	—
68	#380	C₈	1	2	yes	—
69	C₄×C₄⋊C₄	—	2	1	no	—
70	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
75	#346	—	2	2	no	—
76	#346	C₂×C₄	1	4	yes	—
77	C₄×C₁₆	—	2	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	#336	—	2	2	no	—
80	#336	—	2	2	no	—
81	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂³×C₄	M₁₆	1	1	yes	characteristic
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	D₈	1	2	yes	—
102	C₂²×C₈	SD₁₆	1	2	yes	—
103	C₂²×C₈	SD₁₆	1	2	yes	—
104	C₂²×C₈	Q₁₆	1	2	yes	—
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	#117	—	4	2	no	—
108	#117	—	4	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	C₄²	1	1	yes	characteristic
115	C₂×C₄²	C₄²	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
120	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
122	C₂×C₄²	M₁₆	1	1	yes	characteristic
123	C₂×C₄²	M₁₆	1	1	yes	characteristic
124	C₂×C₄²	M₁₆	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
126	M₃₂	—	2	8	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	—	4	4	no	—
130	M₃₂	—	4	2	no	—
131	M₃₂	—	4	2	no	—
132	C₂×C₁₆	—	2	8	no	—
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	4	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	C₂×C₈	1	4	yes	—
140	#97	—	4	4	no	—
141	C₄×C₈	—	4	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	C₄²	1	1	yes	characteristic
144	C₄×C₈	C₄²	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄	1	1	yes	characteristic
158	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
159	#95	—	2	2	no	—
160	#95	C₂×C₈	1	4	yes	—
161	C₂⁴	C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	#95	1	1	yes	characteristic
166	C₂²×C₄	#95	1	1	yes	characteristic
167	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
168	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
169	C₂²×C₄	#97	1	1	yes	characteristic
170	C₂²×C₄	#97	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	#102	1	2	yes	—
177	C₂²×C₄	#103	1	2	yes	—
178	C₂²×C₄	#104	1	2	yes	—
179	C₂²×C₄	#104	1	2	yes	—
180	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
187	M₁₆	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	#95	1	1	yes	characteristic
201	C₂×C₈	#95	1	1	yes	characteristic
202	C₂×C₈	#95	1	1	yes	characteristic
203	C₂×C₈	#95	1	1	yes	characteristic
204	C₂×C₈	#102	1	2	yes	—
205	C₂×C₈	#102	1	2	yes	—
206	C₂×C₈	#102	1	2	yes	—
207	C₂×C₈	#103	1	2	yes	—
208	C₂×C₈	#103	1	2	yes	—
209	C₂×C₈	#103	1	2	yes	—
210	C₂×C₈	#106	1	2	yes	—
211	C₂×C₈	#106	1	2	yes	—
212	C₂×C₈	#107	1	2	yes	—
213	C₂×C₈	#107	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₂×D₈	1	1	yes	characteristic
219	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
220	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
221	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	4	no	—
226	C₄²	—	4	2	no	—
227	C₄²	—	4	2	no	—
228	C₄²	#95	1	1	yes	characteristic
229	C₄²	#95	1	1	yes	characteristic
230	C₄²	C₄×C₈	1	1	yes	characteristic
231	C₄²	C₄×C₈	1	1	yes	characteristic
232	C₄²	#97	1	1	yes	characteristic
233	C₄²	#97	1	1	yes	characteristic
234	C₄²	C₂²⋊C₈	1	1	yes	characteristic
235	C₄²	C₂²⋊C₈	1	1	yes	characteristic
236	C₄²	C₄⋊C₈	1	1	yes	characteristic
237	C₄²	C₄⋊C₈	1	1	yes	characteristic
238	C₄²	C₄⋊C₈	1	1	yes	characteristic
239	C₄²	C₄⋊C₈	1	1	yes	characteristic
240	C₄²	C₂×C₄²	1	1	yes	characteristic
241	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
242	C₄²	C₂²×C₈	1	1	yes	characteristic
243	C₄²	C₂×M₁₆	1	1	yes	characteristic
244	C₁₆	—	2	8	no	—
245	C₁₆	—	4	4	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	1	no	—
248	C₂³	D₄⋊C₈	1	2	yes	—
249	C₂³	Q₈⋊C₈	1	2	yes	—
250	C₂³	#336	1	1	yes	characteristic
251	C₂³	#336	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	D₄⋊C₈	1	2	yes	—
269	C₂×C₄	D₄⋊C₈	1	2	yes	—
270	C₂×C₄	D₄⋊C₈	1	2	yes	—
271	C₂×C₄	Q₈⋊C₈	1	2	yes	—
272	C₂×C₄	Q₈⋊C₈	1	2	yes	—
273	C₂×C₄	Q₈⋊C₈	1	2	yes	—
274	C₂×C₄	#336	1	1	yes	characteristic
275	C₂×C₄	#336	1	1	yes	characteristic
276	C₂×C₄	#336	1	1	yes	characteristic
277	C₂×C₄	#336	1	1	yes	characteristic
278	C₂×C₄	#336	1	1	yes	characteristic
279	C₂×C₄	#336	1	1	yes	characteristic
280	C₂×C₄	#339	1	2	yes	—
281	C₂×C₄	#339	1	2	yes	—
282	C₂×C₄	#340	1	2	yes	—
283	C₂×C₄	#340	1	2	yes	—
284	C₂×C₄	C₂×#95	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×#97	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#102	1	1	yes	characteristic
289	C₂×C₄	C₂×#103	1	1	yes	characteristic
290	C₂×C₄	C₂×#104	1	1	yes	characteristic
291	C₂×C₄	C₂×#104	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	4	no	—
299	C₈	#340	1	2	yes	—
300	C₈	#340	1	2	yes	—
301	C₈	C₂×#102	1	1	yes	characteristic
302	C₈	C₂×#103	1	1	yes	characteristic
303	C₈	C₂×#106	1	1	yes	characteristic
304	C₈	C₂×#107	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#1279	1	2	yes	—
314	C₂²	#1279	1	2	yes	—
315	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
316	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
317	C₂²	C₂×#336	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	4	no	—
325	C₄	#1279	1	2	yes	—
326	C₄	#1279	1	2	yes	—
327	C₄	#1378	1	2	yes	—
328	C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
329	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
330	C₄	C₂×#339	1	1	yes	characteristic
331	C₄	C₂×#340	1	1	yes	characteristic
332	C₄	#2153	1	2	yes	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	2	2	no	—
336	C₂	—	2	2	no	—
337	C₂	C₂×#1279	1	1	yes	characteristic
338	C₂	#12549	1	1	yes	characteristic
339	C₂	C₂×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7169	C₂	semidirect
13. #2150	C₄	semidirect
16. C₂×C₄⋊C₁₆	C₄	semidirect
19. C₄×M₃₂	C₄	semidirect
21. C₂×#346	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2150	2
C₂×C₄×M₃₂	2
#20437	2
#7169	2