Subgroups — #121079

Order 512.

468 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#341	C₂	1	1	yes	maximal, characteristic
2	#7932	C₂	1	1	yes	maximal, characteristic
3	#7918	C₂	1	1	yes	maximal, characteristic
4	#7882	C₂	1	1	yes	maximal, characteristic
5	#7872	C₂	1	1	yes	maximal, characteristic
6	#7860	C₂	1	1	yes	maximal, characteristic
7	#7849	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#431	C₂²	1	1	yes	characteristic
11	C₂×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#341	C₄	1	1	yes	characteristic
16	C₂×#341	C₄	1	1	yes	characteristic
17	C₂×#341	C₄	1	1	yes	characteristic
18	C₂×#341	C₄	1	1	yes	characteristic
19	#1283	—	2	2	no	—
20	#1281	—	2	2	no	—
21	#1278	—	2	2	no	—
22	#1277	—	2	2	no	—
23	#1275	—	2	2	no	—
24	#1273	—	2	2	no	—
25	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄²	C₂³	1	1	yes	characteristic
32	#431	—	2	1	no	—
33	#431	—	2	1	no	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	—	2	1	no	—
44	#431	—	2	1	no	—
45	#431	—	2	1	no	—
46	#431	—	2	1	no	—
47	#431	D₄	1	2	yes	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	D₄	1	2	yes	—
52	C₂×C₂²⋊C₈	—	2	1	no	—
53	C₂×C₂²⋊C₈	—	2	1	no	—
54	C₂×C₂²⋊C₈	—	2	1	no	—
55	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
56	C₄×M₁₆	—	2	1	no	—
57	C₄×M₁₆	—	2	1	no	—
58	C₄×M₁₆	—	2	1	no	—
59	C₄×M₁₆	—	2	1	no	—
60	C₄×M₁₆	—	2	1	no	—
61	C₄×M₁₆	—	2	1	no	—
62	C₄×M₁₆	—	2	1	no	—
63	C₄×M₁₆	—	2	1	no	—
64	C₂×#97	—	2	1	no	—
65	C₂×#97	—	2	1	no	—
66	C₂×C₄×C₈	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	—	2	1	no	—
76	C₂×C₄×C₈	—	2	1	no	—
77	C₂×C₄×C₈	D₄	1	2	yes	—
78	#341	—	2	2	no	—
79	#341	—	2	2	no	—
80	#341	—	2	2	no	—
81	#341	C₈	1	4	yes	complemented
82	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂³×C₄	#41	1	1	yes	characteristic
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	1	no	—
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	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	C₂×C₈	1	2	yes	—
106	C₂×M₁₆	C₂×C₈	1	2	yes	—
107	C₂×M₁₆	M₁₆	1	2	yes	—
108	C₂×M₁₆	M₁₆	1	2	yes	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	4	1	no	—
129	C₂²×C₈	C₂×C₈	1	2	yes	—
130	C₂²×C₈	M₁₆	1	2	yes	—
131	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄²	#41	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	C₂×C₄²	#41	1	1	yes	characteristic
137	C₂×C₄²	#41	1	1	yes	characteristic
138	C₂×C₄²	#41	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₂×C₄²	#41	1	1	yes	characteristic
141	C₂×C₄²	#41	1	1	yes	characteristic
142	C₂×C₄²	#41	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	C₄⋊C₈	—	4	1	no	—
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	4	1	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	4	1	no	—
155	C₂²⋊C₈	—	4	1	no	—
156	C₂²⋊C₈	—	4	1	no	—
157	C₂²⋊C₈	—	4	1	no	—
158	C₂²⋊C₈	—	4	1	no	—
159	#97	—	4	1	no	—
160	#97	—	4	1	no	—
161	#97	—	4	1	no	—
162	#97	—	4	1	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	4	1	no	—
167	C₄×C₈	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	C₄×C₈	—	4	1	no	—
170	C₄×C₈	—	4	1	no	—
171	C₄×C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	—	4	1	no	—
174	C₄×C₈	—	4	1	no	—
175	C₄×C₈	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₂⁴	C₄×Q₈	1	1	yes	characteristic
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	#117	1	1	yes	characteristic
183	C₂²×C₄	#117	1	1	yes	characteristic
184	C₂²×C₄	#117	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	#123	1	1	yes	characteristic
192	C₂²×C₄	#123	1	1	yes	characteristic
193	C₂²×C₄	#123	1	1	yes	characteristic
194	C₂²×C₄	#125	1	1	yes	characteristic
195	C₂²×C₄	#125	1	1	yes	characteristic
196	C₂²×C₄	#125	1	1	yes	characteristic
197	C₂²×C₄	#126	1	1	yes	characteristic
198	C₂²×C₄	#126	1	1	yes	characteristic
199	C₂²×C₄	#126	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
204	C₂²×C₄	#131	1	1	yes	characteristic
205	C₂²×C₄	#131	1	1	yes	characteristic
206	C₂²×C₄	#131	1	1	yes	characteristic
207	C₂²×C₄	#131	1	1	yes	characteristic
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	8	2	no	—
257	C₂×C₈	—	8	2	no	—
258	C₄²	#121	1	1	yes	characteristic
259	C₄²	#121	1	1	yes	characteristic
260	C₄²	#121	1	1	yes	characteristic
261	C₄²	#123	1	1	yes	characteristic
262	C₄²	#123	1	1	yes	characteristic
263	C₄²	#123	1	1	yes	characteristic
264	C₄²	#124	1	1	yes	characteristic
265	C₄²	#124	1	1	yes	characteristic
266	C₄²	#124	1	1	yes	characteristic
267	C₄²	#126	1	1	yes	characteristic
268	C₄²	#126	1	1	yes	characteristic
269	C₄²	#126	1	1	yes	characteristic
270	C₄²	#126	1	1	yes	characteristic
271	C₄²	#126	1	1	yes	characteristic
272	C₄²	#126	1	1	yes	characteristic
273	C₄²	#126	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	#387	1	1	yes	characteristic
277	C₂³	#387	1	1	yes	characteristic
278	C₂³	#387	1	1	yes	characteristic
279	C₂³	C₈×Q₈	1	1	yes	characteristic
280	C₂³	#446	1	1	yes	characteristic
281	C₂³	#446	1	1	yes	characteristic
282	C₂³	#446	1	1	yes	characteristic
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	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	#383	1	1	yes	characteristic
296	C₂×C₄	#388	1	1	yes	characteristic
297	C₂×C₄	#388	1	1	yes	characteristic
298	C₂×C₄	#388	1	1	yes	characteristic
299	C₂×C₄	#396	1	1	yes	characteristic
300	C₂×C₄	#397	1	1	yes	characteristic
301	C₂×C₄	#397	1	1	yes	characteristic
302	C₂×C₄	#397	1	1	yes	characteristic
303	C₂×C₄	#400	1	1	yes	characteristic
304	C₂×C₄	#400	1	1	yes	characteristic
305	C₂×C₄	#400	1	1	yes	characteristic
306	C₂×C₄	#401	1	1	yes	characteristic
307	C₂×C₄	#431	1	1	yes	characteristic
308	C₂×C₄	#431	1	1	yes	characteristic
309	C₂×C₄	#431	1	1	yes	characteristic
310	C₂×C₄	#432	1	1	yes	characteristic
311	C₂×C₄	#432	1	1	yes	characteristic
312	C₂×C₄	#432	1	1	yes	characteristic
313	C₂×C₄	#433	1	1	yes	characteristic
314	C₂×C₄	#433	1	1	yes	characteristic
315	C₂×C₄	#433	1	1	yes	characteristic
316	C₂×C₄	#433	1	1	yes	characteristic
317	C₂×C₄	#433	1	1	yes	characteristic
318	C₂×C₄	#433	1	1	yes	characteristic
319	C₂×C₄	C₈×D₄	1	1	yes	characteristic
320	C₂×C₄	C₈×D₄	1	1	yes	characteristic
321	C₂×C₄	C₈×D₄	1	1	yes	characteristic
322	C₂×C₄	#435	1	1	yes	characteristic
323	C₂×C₄	#435	1	1	yes	characteristic
324	C₂×C₄	#435	1	1	yes	characteristic
325	C₂×C₄	#435	1	1	yes	characteristic
326	C₂×C₄	#435	1	1	yes	characteristic
327	C₂×C₄	#435	1	1	yes	characteristic
328	C₂×C₄	#436	1	1	yes	characteristic
329	C₂×C₄	#436	1	1	yes	characteristic
330	C₂×C₄	#436	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	8	2	no	—
338	C₈	—	8	2	no	—
339	C₈	—	8	2	no	—
340	C₈	—	8	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	#1443	1	1	yes	characteristic
352	C₂²	#1916	1	1	yes	characteristic
353	C₂²	#1916	1	1	yes	characteristic
354	C₂²	#1916	1	1	yes	characteristic
355	C₂²	#1917	1	1	yes	characteristic
356	C₂²	#1917	1	1	yes	characteristic
357	C₂²	#1917	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	#1839	1	1	yes	characteristic
365	C₄	#1840	1	1	yes	characteristic
366	C₄	#1923	1	1	yes	characteristic
367	C₄	#1923	1	1	yes	characteristic
368	C₄	#1923	1	1	yes	characteristic
369	C₄	#1924	1	1	yes	characteristic
370	C₄	#1924	1	1	yes	characteristic
371	C₄	#1924	1	1	yes	characteristic
372	C₄	#2080	1	2	yes	—
373	C₄	#2103	1	2	yes	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	2	1	no	—
380	C₂	#11298	1	1	yes	characteristic
381	C₂	#11305	1	1	yes	characteristic
382	C₂	#11306	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
81. #341	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#341	2
#7932	2
#7918	2
#7882	2
#7872	2
#7860	2
#7849	2