Subgroups — #479981

Order 512.

465 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33484	C₂	1	1	yes	maximal, characteristic
2	#33484	C₂	1	1	yes	maximal, characteristic
3	#17330	C₂	1	1	yes	maximal, characteristic
4	#11416	C₂	1	1	yes	maximal, characteristic, complemented
5	#11416	C₂	1	1	yes	maximal, characteristic, complemented
6	#11415	C₂	1	1	yes	maximal, characteristic, complemented
7	#11414	C₂	1	1	yes	maximal, characteristic, complemented
8	#11414	C₂	1	1	yes	maximal, characteristic, complemented
9	#11413	C₂	1	1	yes	maximal, characteristic, complemented
10	#11413	C₂	1	1	yes	maximal, characteristic, complemented
11	#11413	C₂	1	1	yes	maximal, characteristic, complemented
12	#11413	C₂	1	1	yes	maximal, characteristic, complemented
13	#11406	C₂	1	1	yes	maximal, characteristic, complemented
14	#11405	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#1313	C₂	1	1	yes	maximal, characteristic, complemented
16	#2965	C₄	1	1	yes	characteristic
17	C₈×#41	C₂²	1	1	yes	characteristic
18	#2922	C₄	1	1	yes	characteristic
19	#2183	C₄	1	2	yes	—
20	#2183	C₄	1	2	yes	—
21	#2168	C₄	1	2	yes	—
22	#2168	C₄	1	2	yes	—
23	#2168	C₄	1	2	yes	—
24	#2168	C₄	1	2	yes	—
25	D₄×C₁₆	C₄	1	2	yes	—
26	D₄×C₁₆	C₄	1	2	yes	—
27	#2163	C₂²	1	1	yes	characteristic
28	#2163	C₂²	1	1	yes	characteristic
29	#2163	C₂²	1	1	yes	characteristic
30	#2163	C₂²	1	1	yes	characteristic
31	#2163	C₂²	1	1	yes	characteristic
32	#2163	C₂²	1	1	yes	characteristic
33	C₄×M₃₂	C₂²	1	1	yes	characteristic
34	C₄×M₃₂	C₂²	1	1	yes	characteristic
35	C₂×#346	C₂²	1	1	yes	characteristic
36	C₂×#346	C₂²	1	1	yes	characteristic
37	C₂×#346	C₂²	1	1	yes	characteristic
38	C₂×#346	C₂²	1	1	yes	characteristic
39	#1454	C₄	1	1	yes	characteristic
40	#1454	C₄	1	1	yes	characteristic
41	#1453	C₄	1	1	yes	characteristic
42	#1453	C₄	1	1	yes	characteristic
43	#1451	C₂²	1	1	yes	characteristic
44	#1451	C₂²	1	1	yes	characteristic
45	#1451	C₂²	1	1	yes	characteristic
46	#1450	C₄	1	1	yes	characteristic
47	#1450	C₂²	1	1	yes	characteristic
48	#1450	C₂²	1	1	yes	characteristic
49	C₈×M₁₆	C₄	1	1	yes	characteristic
50	C₂×#322	C₂²	1	1	yes	characteristic
51	#1313	C₂²	1	1	yes	characteristic
52	#1313	C₂²	1	1	yes	characteristic
53	#1313	C₂²	1	1	yes	characteristic
54	#1313	C₂²	1	1	yes	characteristic
55	#1313	C₂²	1	1	yes	characteristic
56	#1313	C₂²	1	1	yes	characteristic
57	#1313	C₂²	1	1	yes	characteristic
58	#1313	C₂²	1	1	yes	characteristic
59	#1313	C₂²	1	1	yes	characteristic
60	#1313	C₂²	1	1	yes	characteristic
61	#1313	C₂²	1	1	yes	characteristic
62	#1313	C₂²	1	1	yes	characteristic
63	#1313	C₂²	1	1	yes	characteristic
64	#1313	C₂²	1	1	yes	characteristic
65	#1313	C₂²	1	1	yes	characteristic
66	#1313	C₂²	1	1	yes	characteristic
67	C₄×#41	C₂×C₄	1	1	yes	characteristic
68	C₂×M₃₂	C₂×C₄	1	2	yes	—
69	C₂×M₃₂	C₂×C₄	1	2	yes	—
70	C₂²×C₁₆	C₂×C₄	1	2	yes	—
71	C₂²×C₁₆	C₂×C₄	1	2	yes	—
72	C₂²×C₁₆	C₂×C₄	1	2	yes	—
73	C₂²×C₁₆	C₂×C₄	1	2	yes	—
74	#446	—	2	1	no	—
75	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
76	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
77	#436	—	2	1	no	—
78	#435	—	2	1	no	—
79	#435	—	2	1	no	—
80	C₈×D₄	C₂×C₄	1	1	yes	characteristic
81	C₈×D₄	C₂×C₄	1	1	yes	characteristic
82	C₈×D₄	C₂×C₄	1	1	yes	characteristic
83	C₈×D₄	C₂×C₄	1	1	yes	characteristic
84	C₈×D₄	C₂×C₄	1	1	yes	characteristic
85	C₈×D₄	C₂×C₄	1	1	yes	characteristic
86	#433	—	2	1	no	—
87	#433	—	2	1	no	—
88	#431	C₂×C₄	1	1	yes	characteristic
89	#431	C₂×C₄	1	1	yes	characteristic
90	#431	C₂×C₄	1	1	yes	characteristic
91	#431	C₂×C₄	1	1	yes	characteristic
92	#431	C₂×C₄	1	1	yes	characteristic
93	#431	C₂×C₄	1	1	yes	characteristic
94	#431	C₂×C₄	1	1	yes	characteristic
95	#431	C₂×C₄	1	1	yes	characteristic
96	#431	C₂×C₄	1	1	yes	characteristic
97	#431	C₂×C₄	1	1	yes	characteristic
98	#424	—	2	1	no	—
99	#408	—	2	1	no	—
100	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
101	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
102	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
109	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
110	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
111	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
112	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
113	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
114	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
115	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
116	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
117	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
118	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
119	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
120	#346	C₂×C₄	1	2	yes	—
121	#346	C₂×C₄	1	2	yes	—
122	#346	C₂×C₄	1	2	yes	—
123	#346	C₂×C₄	1	2	yes	—
124	#346	C₂³	1	1	yes	characteristic
125	#346	C₂³	1	1	yes	characteristic
126	#346	C₂³	1	1	yes	characteristic
127	#346	C₂³	1	1	yes	characteristic
128	#346	C₂³	1	1	yes	characteristic
129	#346	C₂³	1	1	yes	characteristic
130	#346	C₂³	1	1	yes	characteristic
131	#346	C₂³	1	1	yes	characteristic
132	C₄×C₁₆	C₂×C₄	1	2	yes	—
133	C₄×C₁₆	C₂×C₄	1	2	yes	—
134	#322	C₂×C₄	1	1	yes	characteristic
135	#322	C₂×C₄	1	1	yes	characteristic
136	#322	C₂×C₄	1	1	yes	characteristic
137	#322	C₂×C₄	1	1	yes	characteristic
138	#322	C₂×C₄	1	1	yes	characteristic
139	#322	C₂×C₄	1	1	yes	characteristic
140	#322	C₂×C₄	1	1	yes	characteristic
141	#322	C₂×C₄	1	1	yes	characteristic
142	#322	C₂³	1	1	yes	characteristic
143	#322	C₂³	1	1	yes	characteristic
144	#322	C₂³	1	1	yes	characteristic
145	#322	C₂³	1	1	yes	characteristic
146	C₈²	C₂×C₄	1	1	yes	characteristic
147	C₈²	C₂×C₄	1	1	yes	characteristic
148	C₈²	C₂×C₄	1	1	yes	characteristic
149	C₈²	C₂×C₄	1	1	yes	characteristic
150	C₂×#41	M₁₆	1	1	yes	characteristic
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	1	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
162	C₄×Q₈	C₄²	1	1	yes	characteristic
163	C₄×Q₈	M₁₆	1	1	yes	characteristic
164	C₄×D₄	C₄²	1	1	yes	characteristic
165	C₄×D₄	C₄²	1	1	yes	characteristic
166	C₄×D₄	C₄²	1	1	yes	characteristic
167	C₄×D₄	M₁₆	1	1	yes	characteristic
168	C₄×D₄	M₁₆	1	1	yes	characteristic
169	C₄×D₄	M₁₆	1	1	yes	characteristic
170	#117	M₁₆	1	1	yes	characteristic
171	#117	M₁₆	1	1	yes	characteristic
172	#117	M₁₆	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
176	M₃₂	—	2	4	no	—
177	M₃₂	—	4	2	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	4	2	no	—
181	C₂×C₁₆	—	4	2	no	—
182	C₂×C₁₆	C₂²×C₄	1	2	yes	—
183	C₂×C₁₆	C₂²×C₄	1	2	yes	—
184	C₂×C₁₆	C₂²×C₄	1	2	yes	—
185	C₂×C₁₆	C₂²×C₄	1	2	yes	—
186	C₂×C₁₆	C₂²×C₄	1	2	yes	—
187	C₂×C₁₆	C₂²×C₄	1	2	yes	—
188	C₂×C₁₆	C₂²×C₄	1	2	yes	—
189	C₂×C₁₆	C₂²×C₄	1	2	yes	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	C₄²	1	1	yes	characteristic
198	C₄⋊C₈	C₄²	1	1	yes	characteristic
199	C₄⋊C₈	C₄²	1	1	yes	characteristic
200	C₄⋊C₈	C₄²	1	1	yes	characteristic
201	C₄⋊C₈	C₄²	1	1	yes	characteristic
202	C₄⋊C₈	C₄²	1	1	yes	characteristic
203	C₂²⋊C₈	—	2	1	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	C₄²	1	1	yes	characteristic
211	C₂²⋊C₈	C₄²	1	1	yes	characteristic
212	C₂²⋊C₈	C₄²	1	1	yes	characteristic
213	C₂²⋊C₈	C₄²	1	1	yes	characteristic
214	C₂²⋊C₈	C₄²	1	1	yes	characteristic
215	C₂²⋊C₈	C₄²	1	1	yes	characteristic
216	#97	—	2	1	no	—
217	#97	—	2	1	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
230	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
231	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
232	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
233	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
234	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
235	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
236	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
237	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
238	C₄×C₈	C₂⁴	1	1	yes	characteristic
239	#41	—	4	2	no	—
240	C₂×Q₈	#97	1	1	yes	characteristic
241	C₂×D₄	#97	1	1	yes	characteristic
242	C₂×D₄	#97	1	1	yes	characteristic
243	C₂×D₄	#97	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
247	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
248	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
249	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
250	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
252	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
253	M₁₆	—	4	2	no	—
254	M₁₆	—	8	1	no	—
255	M₁₆	—	8	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	8	1	no	—
272	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
277	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
278	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
280	C₄⋊C₄	#97	1	1	yes	characteristic
281	C₄⋊C₄	#97	1	1	yes	characteristic
282	C₄⋊C₄	#97	1	1	yes	characteristic
283	C₄⋊C₄	#97	1	1	yes	characteristic
284	C₄⋊C₄	#97	1	1	yes	characteristic
285	C₄⋊C₄	#97	1	1	yes	characteristic
286	C₂²⋊C₄	#97	1	1	yes	characteristic
287	C₂²⋊C₄	#97	1	1	yes	characteristic
288	C₂²⋊C₄	#97	1	1	yes	characteristic
289	C₂²⋊C₄	#97	1	1	yes	characteristic
290	C₂²⋊C₄	#97	1	1	yes	characteristic
291	C₂²⋊C₄	#97	1	1	yes	characteristic
292	C₄²	C₂×C₄²	1	1	yes	characteristic
293	C₄²	C₂×C₄²	1	1	yes	characteristic
294	C₄²	C₂×C₄²	1	1	yes	characteristic
295	C₄²	C₂×M₁₆	1	1	yes	characteristic
296	C₄²	C₂×M₁₆	1	1	yes	characteristic
297	C₄²	C₂×M₁₆	1	1	yes	characteristic
298	C₄²	C₂×M₁₆	1	1	yes	characteristic
299	C₄²	C₂×M₁₆	1	1	yes	characteristic
300	C₄²	C₂×M₁₆	1	1	yes	characteristic
301	C₄²	C₂³×C₄	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	2	2	no	—
306	C₁₆	—	2	2	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	#131	1	4	yes	—
310	C₂³	C₂×#97	1	1	yes	characteristic
311	C₂³	C₂×#97	1	1	yes	characteristic
312	C₂³	C₂×#97	1	1	yes	characteristic
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	C₂×#97	1	1	yes	characteristic
330	C₂×C₄	C₂×#97	1	1	yes	characteristic
331	C₂×C₄	C₂×#97	1	1	yes	characteristic
332	C₂×C₄	C₂×#97	1	1	yes	characteristic
333	C₂×C₄	C₂×#97	1	1	yes	characteristic
334	C₂×C₄	C₂×#97	1	1	yes	characteristic
335	C₂×C₄	C₂×#97	1	1	yes	characteristic
336	C₂×C₄	C₂×#97	1	1	yes	characteristic
337	C₂×C₄	C₂×#97	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
339	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
340	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	1	no	—
353	C₈	—	4	1	no	—
354	C₈	—	4	1	no	—
355	C₈	C₂×#131	1	2	yes	—
356	C₈	#568	1	2	yes	—
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	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	#2110	1	2	yes	—
366	C₂²	C₂²×#97	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	C₄×#131	1	1	yes	characteristic
379	C₄	#2875	1	1	yes	characteristic
380	C₂	—	2	2	no	—
381	C₂	—	4	1	no	—
382	C₂	—	4	1	no	—
383	C₂	#17320	1	1	yes	characteristic
384	C₂	C₂×#2110	1	1	yes	characteristic
385	C₂	#33329	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #11416	C₂	semidirect
5. #11416	C₂	semidirect
6. #11415	C₂	semidirect
7. #11414	C₂	semidirect
8. #11414	C₂	semidirect
9. #11413	C₂	semidirect
10. #11413	C₂	semidirect
11. #11413	C₂	semidirect
12. #11413	C₂	semidirect
13. #11406	C₂	semidirect
14. #11405	C₂	semidirect
15. C₂×#1313	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33484	2
#17330	2
#11416	2
#11415	2
#11414	2
#11413	2
#11406	2
#11405	2
C₂×#1313	2