Subgroups — #146343

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 401 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×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#1389	C₂	1	1	yes	maximal, characteristic
5	C₂×#1388	C₂	1	1	yes	maximal, characteristic
6	#12569	C₂	1	1	yes	maximal, characteristic
7	#12567	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×#405	—	2	1	no	—
15	C₂×#405	—	2	1	no	—
16	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄×M₃₂	—	2	2	no	—
18	C₄×M₃₂	C₄	1	4	yes	complemented
19	C₂×#346	—	2	1	no	—
20	C₂×C₄×C₁₆	—	2	1	no	—
21	#1772	—	2	1	no	—
22	#1772	—	2	1	no	—
23	#1772	—	2	2	no	—
24	#1772	C₄	1	2	yes	complemented
25	#1772	C₄	1	2	yes	complemented
26	#1772	C₄	1	4	yes	complemented
27	C₂×#339	—	2	1	no	—
28	C₂×#339	—	2	1	no	—
29	#1389	C₄	1	4	yes	complemented
30	#1388	C₄	1	4	yes	complemented
31	#1387	C₄	1	4	yes	complemented
32	#1386	C₄	1	4	yes	complemented
33	C₂²×M₁₆	—	2	1	no	—
34	C₂²×M₁₆	—	2	1	no	—
35	C₂³×C₈	D₄	1	1	yes	characteristic
36	C₂³×C₈	C₂³	1	1	yes	characteristic
37	C₂×#117	—	2	1	no	—
38	C₂×#117	—	2	1	no	—
39	C₂²×C₄²	D₄	1	1	yes	characteristic
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	—	2	1	no	—
45	C₂×M₃₂	—	2	1	no	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂×M₃₂	C₂×C₄	1	2	yes	—
49	C₂×M₃₂	C₂×C₄	1	2	yes	—
50	C₂×M₃₂	C₂×C₄	1	2	yes	—
51	C₂×M₃₂	C₂×C₄	1	2	yes	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	—	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	C₂×#108	C₂×C₄	1	2	yes	—
65	C₂×#108	C₂×C₄	1	2	yes	—
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	#427	C₂×C₄	1	2	yes	—
73	#427	C₂×C₄	1	2	yes	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	C₂×C₄	1	2	yes	—
76	C₂×#106	—	2	1	no	—
77	C₂×#106	C₂×C₄	1	2	yes	—
78	#405	—	2	4	no	—
79	#405	—	2	2	no	—
80	#405	—	2	2	no	—
81	#405	—	4	1	no	—
82	#405	—	4	1	no	—
83	#405	—	4	2	no	—
84	C₂×#97	—	4	1	no	—
85	C₂×#97	—	4	1	no	—
86	C₂×C₄×C₈	—	4	1	no	—
87	C₂×C₄×C₈	—	4	1	no	—
88	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
89	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
90	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄×C₈	D₄	1	1	yes	characteristic
93	C₂×C₄×C₈	D₄	1	1	yes	characteristic
94	C₂×C₄×C₈	D₄	1	2	yes	—
95	C₂×C₄×C₈	Q₈	1	2	yes	—
96	#346	—	2	4	no	—
97	C₄×C₁₆	—	2	4	no	—
98	#339	—	2	2	no	—
99	#339	—	2	2	no	—
100	#339	—	2	4	no	—
101	C₂³×C₄	—	2	1	no	—
102	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂²×C₈	—	4	1	no	—
116	C₂²×C₈	—	4	1	no	—
117	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
119	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₈	C₄⋊C₄	1	2	yes	—
121	C₂²×C₈	D₈	1	1	yes	characteristic
122	C₂²×C₈	D₈	1	1	yes	characteristic
123	C₂²×C₈	SD₁₆	1	1	yes	characteristic
124	C₂²×C₈	SD₁₆	1	2	yes	—
125	C₂²×C₈	SD₁₆	1	1	yes	characteristic
126	C₂²×C₈	Q₁₆	1	2	yes	—
127	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
137	C₂²×C₈	#41	1	1	yes	characteristic
138	C₂²×C₈	#41	1	1	yes	characteristic
139	C₂²×C₈	#41	1	1	yes	characteristic
140	C₂²×C₈	#41	1	1	yes	characteristic
141	#117	—	2	2	no	—
142	#117	—	2	2	no	—
143	#117	—	2	2	no	—
144	#117	—	2	2	no	—
145	#117	—	4	1	no	—
146	#117	—	4	1	no	—
147	#117	—	4	1	no	—
148	#117	—	4	1	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₂²⋊C₄	—	4	1	no	—
152	C₂×C₂²⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
161	C₂×C₄²	C₄⋊C₄	1	2	yes	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	2	2	no	—
165	M₃₂	—	2	2	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	C₄²	1	4	yes	—
169	M₃₂	C₄²	1	4	yes	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	#108	—	2	2	no	—
177	#108	—	2	2	no	—
178	#108	—	2	2	no	—
179	#108	—	2	2	no	—
180	#108	—	2	2	no	—
181	#108	—	2	2	no	—
182	#108	C₄²	1	4	yes	—
183	#108	C₄²	1	4	yes	—
184	#107	—	2	2	no	—
185	#107	—	2	2	no	—
186	#107	—	2	2	no	—
187	#107	C₄²	1	4	yes	—
188	#106	—	2	2	no	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	#106	C₄²	1	4	yes	—
192	#97	—	4	4	no	—
193	#97	—	4	2	no	—
194	#97	—	4	2	no	—
195	C₄×C₈	—	4	4	no	—
196	C₄×C₈	—	4	2	no	—
197	C₄×C₈	—	4	2	no	—
198	C₄×C₈	C₄²	1	2	yes	—
199	C₄×C₈	C₄²	1	2	yes	—
200	C₄×C₈	C₂²⋊C₄	1	2	yes	—
201	C₄×C₈	C₂²⋊C₄	1	2	yes	—
202	C₄×C₈	C₄⋊C₄	1	2	yes	—
203	C₄×C₈	C₄⋊C₄	1	2	yes	—
204	C₄×C₈	C₄⋊C₄	1	2	yes	—
205	C₄×C₈	C₄⋊C₄	1	2	yes	—
206	C₂⁴	#135	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
223	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
225	C₂²×C₄	#135	1	1	yes	characteristic
226	C₂²×C₄	#135	1	1	yes	characteristic
227	C₂²×C₄	#135	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	#95	1	2	yes	—
243	C₂×C₈	#95	1	2	yes	—
244	C₂×C₈	#102	1	1	yes	characteristic
245	C₂×C₈	#102	1	2	yes	—
246	C₂×C₈	#102	1	2	yes	—
247	C₂×C₈	#102	1	1	yes	characteristic
248	C₂×C₈	#102	1	1	yes	characteristic
249	C₂×C₈	#102	1	1	yes	characteristic
250	C₂×C₈	#103	1	2	yes	—
251	C₂×C₈	#103	1	2	yes	—
252	C₂×C₈	#103	1	2	yes	—
253	C₂×C₈	#103	1	2	yes	—
254	C₂×C₈	#106	1	2	yes	—
255	C₂×C₈	#106	1	2	yes	—
256	C₂×C₈	#107	1	2	yes	—
257	C₂×C₈	#107	1	2	yes	—
258	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₈	#117	1	1	yes	characteristic
269	C₂×C₈	#117	1	1	yes	characteristic
270	C₂×C₈	#117	1	1	yes	characteristic
271	C₂×C₈	#117	1	1	yes	characteristic
272	C₂×C₈	#117	1	1	yes	characteristic
273	C₂×C₈	#117	1	1	yes	characteristic
274	C₂×C₈	C₄×D₄	1	1	yes	characteristic
275	C₂×C₈	C₄×D₄	1	1	yes	characteristic
276	C₂×C₈	C₄×D₄	1	1	yes	characteristic
277	C₂×C₈	C₄×D₄	1	1	yes	characteristic
278	C₂×C₈	C₄×D₄	1	1	yes	characteristic
279	C₂×C₈	C₄×D₄	1	1	yes	characteristic
280	C₂×C₈	C₄×D₄	1	1	yes	characteristic
281	C₂×C₈	C₄×D₄	1	1	yes	characteristic
282	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
283	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
284	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
285	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	#95	1	2	yes	—
303	C₄²	#95	1	2	yes	—
304	C₁₆	—	2	4	no	—
305	C₁₆	—	2	4	no	—
306	C₂³	—	4	2	no	—
307	C₂³	#416	1	1	yes	characteristic
308	C₂³	#416	1	1	yes	characteristic
309	C₂³	#427	1	1	yes	characteristic
310	C₂³	C₄×D₈	1	1	yes	characteristic
311	C₂³	C₄×SD₁₆	1	1	yes	characteristic
312	C₂³	C₄×SD₁₆	1	1	yes	characteristic
313	C₂³	C₄×Q₁₆	1	1	yes	characteristic
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	C₂×#95	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
327	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
331	C₂×C₄	C₂×#102	1	1	yes	characteristic
332	C₂×C₄	C₂×#102	1	1	yes	characteristic
333	C₂×C₄	C₂×#103	1	1	yes	characteristic
334	C₂×C₄	C₂×#103	1	1	yes	characteristic
335	C₂×C₄	#416	1	1	yes	characteristic
336	C₂×C₄	#416	1	1	yes	characteristic
337	C₂×C₄	C₂×#106	1	1	yes	characteristic
338	C₂×C₄	C₂×#107	1	1	yes	characteristic
339	C₂×C₄	#427	1	1	yes	characteristic
340	C₂×C₄	C₄×D₈	1	1	yes	characteristic
341	C₂×C₄	C₄×D₈	1	1	yes	characteristic
342	C₂×C₄	C₄×D₈	1	1	yes	characteristic
343	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
344	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
345	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
346	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
347	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
348	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
349	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
350	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
351	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
352	C₈	—	4	4	no	—
353	C₈	—	4	4	no	—
354	C₈	#340	1	2	yes	—
355	C₈	#340	1	2	yes	—
356	C₈	#340	1	2	yes	—
357	C₈	#340	1	2	yes	—
358	C₈	C₄³	1	1	yes	characteristic
359	C₈	#376	1	1	yes	characteristic
360	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
361	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
362	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
363	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
364	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
365	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	4	1	no	—
372	C₂²	—	4	1	no	—
373	C₂²	#1735	1	1	yes	characteristic
374	C₂²	C₄×#102	1	1	yes	characteristic
375	C₂²	C₄×#102	1	1	yes	characteristic
376	C₂²	C₄×#103	1	1	yes	characteristic
377	C₂²	C₄×#103	1	1	yes	characteristic
378	C₂²	C₄×#106	1	1	yes	characteristic
379	C₂²	C₄×#107	1	1	yes	characteristic
380	C₄	—	2	4	no	—
381	C₄	—	4	2	no	—
382	C₄	—	8	4	no	—
383	C₄	—	8	4	no	—
384	C₄	C₄×#95	1	1	yes	characteristic
385	C₄	C₂×#340	1	1	yes	characteristic
386	C₄	C₄×#102	1	1	yes	characteristic
387	C₄	C₄×#102	1	1	yes	characteristic
388	C₄	C₄×#103	1	1	yes	characteristic
389	C₄	C₄×#103	1	1	yes	characteristic
390	C₄	C₄×#106	1	1	yes	characteristic
391	C₄	C₄×#107	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	C₄×#340	1	1	yes	characteristic
399	C₂	#12646	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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
18. C₄×M₃₂	C₄	semidirect
24. #1772	C₄	semidirect
25. #1772	C₄	semidirect
26. #1772	C₄	semidirect
29. #1389	C₄	semidirect
30. #1388	C₄	semidirect
31. #1387	C₄	semidirect
32. #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×#1772	2
C₂×#1389	2
C₂×#1388	2
#12569	2
#12567	2