Subgroups — #109764

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21177	C₂	1	1	yes	maximal, characteristic
2	#17314	C₂	1	1	yes	maximal, characteristic
3	#10851	C₂	1	1	yes	maximal, characteristic
4	#8095	C₂	1	1	yes	maximal, characteristic
5	C₂×#1303	C₂	1	1	yes	maximal, characteristic
6	#7957	C₂	1	1	yes	maximal, characteristic
7	#7901	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#430	—	2	1	no	—
11	C₂×#423	C₂²	1	1	yes	characteristic
12	C₂×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	#2285	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#1950	—	2	2	no	—
17	#1950	C₄	1	4	yes	complemented
18	#1738	—	2	1	no	—
19	C₂×#339	—	2	1	no	—
20	C₂×#339	—	2	1	no	—
21	#1450	—	2	2	no	—
22	C₈×M₁₆	—	2	2	no	—
23	C₈×M₁₆	—	2	2	no	—
24	#1303	—	2	2	no	—
25	#1303	C₄	1	4	yes	complemented
26	#1286	C₄	1	4	yes	complemented
27	#1279	—	2	2	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂×#117	—	2	1	no	—
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄²	—	2	1	no	—
35	C₂²×C₄²	C₂³	1	1	yes	characteristic
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	C₂×C₄	1	2	yes	—
41	#431	C₂×C₄	1	2	yes	—
42	#431	D₄	1	2	yes	—
43	#431	Q₈	1	2	yes	—
44	#430	—	4	2	no	—
45	#430	—	4	2	no	—
46	C₂×#108	—	2	4	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	—	2	1	no	—
50	#423	—	2	1	no	—
51	#423	—	2	1	no	—
52	#423	C₂×C₄	1	2	yes	—
53	#423	C₂×C₄	1	2	yes	—
54	#423	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₈	—	2	1	no	—
56	C₂×C₄⋊C₈	—	2	1	no	—
57	C₂×C₄⋊C₈	—	2	1	no	—
58	C₂×C₄⋊C₈	—	2	1	no	—
59	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₈	Q₈	1	2	yes	—
61	C₂×C₂²⋊C₈	—	2	1	no	—
62	C₂×C₂²⋊C₈	—	2	1	no	—
63	C₄×M₁₆	—	2	1	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₄×M₁₆	D₄	1	2	yes	—
67	C₂×#97	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	D₄	1	2	yes	—
71	#384	—	2	1	no	—
72	#382	C₂×C₄	1	2	yes	—
73	#380	—	2	1	no	—
74	#380	—	2	1	no	—
75	C₄×C₄⋊C₄	—	2	1	no	—
76	C₄×C₄⋊C₄	—	2	1	no	—
77	C₄×C₄⋊C₄	—	2	1	no	—
78	C₄×C₄⋊C₄	—	2	1	no	—
79	C₄×C₄⋊C₄	—	2	1	no	—
80	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
81	C₂×#95	—	2	1	no	—
82	#341	—	2	4	no	—
83	#339	—	4	2	no	—
84	#339	—	4	2	no	—
85	#322	—	2	4	no	—
86	C₈²	—	2	4	no	—
87	C₂³×C₄	—	4	1	no	—
88	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	2	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	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
110	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
111	C₂×M₁₆	D₈	1	2	yes	—
112	C₂×M₁₆	SD₁₆	1	2	yes	—
113	C₂×M₁₆	SD₁₆	1	2	yes	—
114	C₂×M₁₆	Q₁₆	1	2	yes	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	4	1	no	—
121	#117	—	4	1	no	—
122	#117	—	4	1	no	—
123	#117	—	4	1	no	—
124	#117	—	4	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	4	4	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₂²⋊C₄	—	4	1	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	—	4	1	no	—
135	C₂×C₄²	—	4	1	no	—
136	C₂×C₄²	—	4	1	no	—
137	C₂×C₄²	—	4	1	no	—
138	C₂×C₄²	—	4	1	no	—
139	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
146	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
147	C₂×C₄²	#41	1	1	yes	characteristic
148	C₂×C₄²	#41	1	1	yes	characteristic
149	C₂×C₄²	#41	1	1	yes	characteristic
150	C₂×C₄²	#41	1	1	yes	characteristic
151	#108	—	4	8	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	4	1	no	—
161	C₄⋊C₈	—	4	1	no	—
162	C₄⋊C₈	C₄²	1	4	yes	—
163	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	#97	—	2	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	C₄⋊C₄	1	4	yes	—
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	#95	—	4	1	no	—
180	C₂⁴	#124	1	1	yes	characteristic
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	8	1	no	—
186	C₂²×C₄	—	8	2	no	—
187	C₂²×C₄	—	8	1	no	—
188	C₂²×C₄	—	8	1	no	—
189	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	#117	1	1	yes	characteristic
191	C₂²×C₄	#117	1	1	yes	characteristic
192	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
198	C₂²×C₄	#121	1	1	yes	characteristic
199	C₂²×C₄	#121	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₄	#123	1	1	yes	characteristic
203	C₂²×C₄	#123	1	1	yes	characteristic
204	C₂²×C₄	#126	1	1	yes	characteristic
205	C₂²×C₄	#126	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
209	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
211	C₂²×C₄	#135	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	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	8	1	no	—
228	M₁₆	—	8	1	no	—
229	M₁₆	#102	1	4	yes	—
230	M₁₆	#103	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	8	2	no	—
244	C₂×C₈	—	8	2	no	—
245	C₂×C₈	—	8	1	no	—
246	C₂×C₈	—	8	1	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	8	2	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₂²⋊C₄	—	8	1	no	—
255	C₂²⋊C₄	—	8	1	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	8	1	no	—
259	C₄²	—	8	1	no	—
260	C₄²	—	8	1	no	—
261	C₄²	—	8	1	no	—
262	C₄²	C₂×C₄²	1	1	yes	characteristic
263	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
264	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
265	C₄²	#117	1	1	yes	characteristic
266	C₄²	C₄×D₄	1	1	yes	characteristic
267	C₄²	C₄×D₄	1	1	yes	characteristic
268	C₄²	C₄×D₄	1	1	yes	characteristic
269	C₄²	C₄×D₄	1	1	yes	characteristic
270	C₄²	C₄×Q₈	1	1	yes	characteristic
271	C₄²	C₄×Q₈	1	1	yes	characteristic
272	C₄²	#120	1	1	yes	characteristic
273	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
274	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
275	C₄²	#123	1	1	yes	characteristic
276	C₄²	#125	1	1	yes	characteristic
277	C₄²	#125	1	1	yes	characteristic
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#388	1	1	yes	characteristic
281	C₂³	#388	1	1	yes	characteristic
282	C₂³	#391	1	1	yes	characteristic
283	C₂³	#486	1	1	yes	characteristic
284	C₂³	#487	1	1	yes	characteristic
285	C₂³	#488	1	1	yes	characteristic
286	C₂³	#488	1	1	yes	characteristic
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
310	C₂×C₄	#385	1	1	yes	characteristic
311	C₂×C₄	#385	1	1	yes	characteristic
312	C₂×C₄	#386	1	1	yes	characteristic
313	C₂×C₄	#387	1	1	yes	characteristic
314	C₂×C₄	#387	1	1	yes	characteristic
315	C₂×C₄	#387	1	1	yes	characteristic
316	C₂×C₄	#387	1	1	yes	characteristic
317	C₂×C₄	#389	1	1	yes	characteristic
318	C₂×C₄	#389	1	1	yes	characteristic
319	C₂×C₄	C₂×#102	1	1	yes	characteristic
320	C₂×C₄	C₂×#103	1	1	yes	characteristic
321	C₂×C₄	#416	1	1	yes	characteristic
322	C₂×C₄	#416	1	1	yes	characteristic
323	C₂×C₄	C₄×D₈	1	1	yes	characteristic
324	C₂×C₄	C₄×D₈	1	1	yes	characteristic
325	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
326	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
327	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
328	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
329	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
330	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
331	C₂×C₄	#465	1	1	yes	characteristic
332	C₂×C₄	#465	1	1	yes	characteristic
333	C₂×C₄	#465	1	1	yes	characteristic
334	C₂×C₄	#465	1	1	yes	characteristic
335	C₂×C₄	#466	1	1	yes	characteristic
336	C₂×C₄	#466	1	1	yes	characteristic
337	C₂×C₄	#467	1	1	yes	characteristic
338	C₂×C₄	#467	1	1	yes	characteristic
339	C₂×C₄	#498	1	1	yes	characteristic
340	C₂×C₄	#499	1	1	yes	characteristic
341	C₂×C₄	#499	1	1	yes	characteristic
342	C₂×C₄	#500	1	1	yes	characteristic
343	C₈	—	2	4	no	—
344	C₈	—	2	4	no	—
345	C₈	—	4	4	no	—
346	C₈	—	4	4	no	—
347	C₈	—	4	4	no	—
348	C₈	—	8	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	#1436	1	1	yes	characteristic
360	C₂²	#1933	1	1	yes	characteristic
361	C₂²	#1933	1	1	yes	characteristic
362	C₂²	#1934	1	1	yes	characteristic
363	C₂²	#1934	1	1	yes	characteristic
364	C₂²	#1990	1	1	yes	characteristic
365	C₂²	#1991	1	1	yes	characteristic
366	C₄	—	2	1	no	—
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	—	8	4	no	—
374	C₄	—	16	2	no	—
375	C₄	C₄×#102	1	1	yes	characteristic
376	C₄	C₄×#103	1	1	yes	characteristic
377	C₄	#1861	1	2	yes	—
378	C₄	#1892	1	1	yes	characteristic
379	C₄	#1893	1	1	yes	characteristic
380	C₄	#1941	1	1	yes	characteristic
381	C₄	#1941	1	1	yes	characteristic
382	C₄	#1942	1	1	yes	characteristic
383	C₄	#1942	1	1	yes	characteristic
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	#10880	1	1	yes	characteristic
391	C₂	#10901	1	1	yes	characteristic
392	C₂	#10905	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
17. #1950	C₄	semidirect
25. #1303	C₄	semidirect
26. #1286	C₄	semidirect

Maximal subgroups

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

subgroup	index
#21177	2
#17314	2
#10851	2
#8095	2
C₂×#1303	2
#7957	2
#7901	2