Subgroups — #495804

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33441	C₂	1	2	yes	maximal, complemented
2	C₂×#2170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#504	C₂	1	1	yes	maximal, characteristic, complemented
4	#21558	C₂	1	1	yes	maximal, characteristic
5	#13197	C₂	1	1	yes	maximal, characteristic, complemented
6	#13164	C₂	1	1	yes	maximal, characteristic, complemented
7	#13123	C₂	1	2	yes	maximal, complemented
8	#12987	C₂	1	2	yes	maximal, complemented
9	#12792	C₂	1	1	yes	maximal, characteristic, complemented
10	#12664	C₂	1	1	yes	maximal, characteristic, complemented
11	#12613	C₂	1	2	yes	maximal, complemented
12	C₂×#504	C₂²	1	1	yes	characteristic
13	C₂×#504	C₂²	1	2	yes	—
14	C₂×#504	C₂²	1	1	yes	characteristic
15	#3067	C₄	1	1	yes	characteristic
16	#2955	C₂²	1	2	yes	—
17	C₂×#443	C₂²	1	1	yes	characteristic
18	C₄×#131	C₂²	1	1	yes	characteristic
19	#2171	—	2	4	no	—
20	#2170	—	2	4	no	—
21	#2170	—	2	1	no	—
22	#2170	—	2	1	no	—
23	#2170	—	2	1	no	—
24	#2170	—	2	1	no	—
25	#2153	C₂²	1	2	yes	—
26	C₂×#364	C₂²	1	1	yes	characteristic
27	C₂×#364	C₂²	1	1	yes	characteristic
28	#2117	C₂²	1	2	yes	—
29	#2117	C₂²	1	2	yes	—
30	C₂×#350	C₂²	1	2	yes	—
31	#2115	C₂²	1	2	yes	—
32	#2108	—	2	1	no	—
33	#2108	—	2	1	no	—
34	#2108	C₂²	1	1	yes	characteristic
35	#2108	C₂²	1	1	yes	characteristic
36	#2108	C₂²	1	2	yes	—
37	C₄×M₃₂	—	2	1	no	—
38	C₄×M₃₂	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₁₆	—	2	1	no	—
40	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
41	#1976	C₄	1	1	yes	characteristic
42	#1953	C₄	1	1	yes	characteristic
43	#1932	C₄	1	2	yes	—
44	#1906	C₄	1	2	yes	—
45	#1814	C₄	1	1	yes	characteristic
46	#1772	C₂²	1	1	yes	characteristic
47	#1757	C₂²	1	2	yes	—
48	#1379	C₂²	1	2	yes	—
49	#1378	C₂²	1	2	yes	—
50	#1378	C₂²	1	2	yes	—
51	#575	C₂×C₄	1	2	yes	—
52	C₂×#135	C₂×C₄	1	1	yes	characteristic
53	C₂×#131	C₂×C₄	1	1	yes	characteristic
54	C₂×#131	C₂³	1	2	yes	—
55	C₂×#131	C₂³	1	1	yes	characteristic
56	C₄×#41	C₂×C₄	1	1	yes	characteristic
57	#504	—	2	2	no	—
58	#504	—	2	2	no	—
59	#504	—	2	1	no	—
60	#504	—	2	1	no	—
61	#504	—	2	2	no	—
62	#504	—	2	2	no	—
63	#504	—	2	2	no	—
64	#504	—	2	2	no	—
65	#504	D₄	1	2	yes	—
66	#504	D₄	1	2	yes	—
67	C₂×M₃₂	—	2	1	no	—
68	C₂×M₃₂	—	2	1	no	—
69	C₂×M₃₂	—	2	2	no	—
70	C₂×M₃₂	C₂³	1	2	yes	—
71	C₂×M₃₂	C₂³	1	2	yes	—
72	C₂×M₃₂	C₂³	1	1	yes	characteristic
73	C₂×M₃₂	C₂³	1	1	yes	characteristic
74	C₂²×C₁₆	—	2	1	no	—
75	C₂²×C₁₆	—	2	1	no	—
76	C₂²×C₁₆	—	2	2	no	—
77	C₂²×C₁₆	C₂³	1	1	yes	characteristic
78	C₂²×C₁₆	C₂³	1	1	yes	characteristic
79	#473	C₈	1	2	yes	—
80	#472	C₈	1	2	yes	—
81	#471	C₈	1	4	yes	—
82	#444	—	2	4	no	—
83	#443	—	2	4	no	—
84	#443	—	2	1	no	—
85	#443	—	2	1	no	—
86	#443	—	2	1	no	—
87	#443	—	2	1	no	—
88	#430	C₂×C₄	1	2	yes	—
89	C₂×#108	C₂×C₄	1	1	yes	characteristic
90	C₂×#108	C₂×C₄	1	1	yes	characteristic
91	#427	C₂×C₄	1	1	yes	characteristic
92	#421	C₂×C₄	1	2	yes	—
93	#421	C₂×C₄	1	2	yes	—
94	C₂×#104	C₂×C₄	1	2	yes	—
95	#413	C₂×C₄	1	2	yes	—
96	#410	C₂×C₄	1	2	yes	—
97	#405	—	2	1	no	—
98	#405	—	2	1	no	—
99	#405	C₂×C₄	1	1	yes	characteristic
100	#405	C₂³	1	1	yes	characteristic
101	#405	C₂³	1	2	yes	—
102	C₄×M₁₆	—	2	1	no	—
103	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄×C₈	—	2	1	no	—
105	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
106	#364	—	2	4	no	—
107	#364	—	2	1	no	—
108	#364	—	2	1	no	—
109	#364	—	2	1	no	—
110	#364	—	2	1	no	—
111	#350	—	2	4	no	—
112	#350	—	2	4	no	—
113	#350	—	2	2	no	—
114	#350	—	2	2	no	—
115	#349	—	2	4	no	—
116	#346	—	2	1	no	—
117	#346	—	2	1	no	—
118	#346	—	2	2	no	—
119	#346	—	2	1	no	—
120	C₄×C₁₆	—	2	1	no	—
121	C₄×C₁₆	—	2	1	no	—
122	C₄×C₁₆	—	2	2	no	—
123	C₄×C₁₆	—	2	1	no	—
124	C₄×C₁₆	D₄	1	1	yes	characteristic
125	C₄×C₁₆	D₄	1	1	yes	characteristic
126	C₄×C₁₆	D₄	1	1	yes	characteristic
127	C₄×C₁₆	D₄	1	1	yes	characteristic
128	#344	C₂×C₄	1	2	yes	—
129	#344	C₂×C₄	1	2	yes	—
130	#339	C₂×C₄	1	2	yes	—
131	C₂×#41	C₂²×C₄	1	2	yes	—
132	C₂×#41	C₂²×C₄	1	1	yes	characteristic
133	#137	—	2	2	no	—
134	#137	C₂×C₈	1	4	yes	—
135	AGL(1,ℤ/8)	—	2	2	no	—
136	AGL(1,ℤ/8)	C₂×C₈	1	4	yes	—
137	#135	—	2	4	no	—
138	#135	—	2	1	no	—
139	#135	—	2	1	no	—
140	#135	—	2	1	no	—
141	#135	—	2	1	no	—
142	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
143	C₂×SD₁₆	C₂×C₈	1	2	yes	—
144	C₂×D₈	C₂×C₈	1	1	yes	characteristic
145	#131	—	2	1	no	—
146	#131	—	2	1	no	—
147	#131	—	2	1	no	—
148	#131	—	2	1	no	—
149	#131	—	2	2	no	—
150	#131	—	2	2	no	—
151	#131	—	2	2	no	—
152	#131	—	2	2	no	—
153	#131	C₂×C₈	1	2	yes	—
154	#131	C₂×C₈	1	2	yes	—
155	#131	C₂×D₄	1	2	yes	—
156	#131	#41	1	2	yes	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	C₂²×C₄	1	2	yes	—
161	C₂×M₁₆	C₂²×C₄	1	1	yes	characteristic
162	C₂×M₁₆	C₂²×C₄	1	1	yes	characteristic
163	C₂×M₁₆	C₂²×C₄	1	2	yes	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₈	C₂⁴	1	1	yes	characteristic
169	C₄×Q₈	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	2	no	—
172	#117	—	2	1	no	—
173	#117	C₂²×C₄	1	2	yes	—
174	#117	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
177	M₃₂	—	2	4	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	4	2	no	—
187	M₃₂	—	4	2	no	—
188	M₃₂	C₂×D₄	1	2	yes	—
189	M₃₂	#41	1	2	yes	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	2	2	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	4	2	no	—
200	C₂×C₁₆	—	4	2	no	—
201	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
202	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
203	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
204	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
205	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
206	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
207	C₂×C₁₆	#41	1	1	yes	characteristic
208	C₂×C₁₆	#41	1	1	yes	characteristic
209	#108	—	2	4	no	—
210	#108	—	2	1	no	—
211	#108	—	2	1	no	—
212	#108	C₂×C₈	1	2	yes	—
213	#108	C₂×C₈	1	2	yes	—
214	#107	—	2	1	no	—
215	#106	—	2	1	no	—
216	#104	—	2	4	no	—
217	#104	—	2	4	no	—
218	#104	—	2	2	no	—
219	#104	—	2	2	no	—
220	#101	C₂×C₈	1	4	yes	—
221	#100	C₂×C₈	1	4	yes	—
222	#99	—	2	4	no	—
223	#97	—	2	1	no	—
224	#97	—	2	1	no	—
225	#97	—	2	2	no	—
226	#97	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	C₂×D₄	1	1	yes	characteristic
232	C₄×C₈	C₂×D₄	1	1	yes	characteristic
233	C₄×C₈	#41	1	1	yes	characteristic
234	C₄×C₈	#41	1	1	yes	characteristic
235	#41	—	2	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	#41	C₄×D₄	1	2	yes	—
242	#41	C₂²×C₈	1	2	yes	—
243	C₂×Q₈	C₂²×C₈	1	2	yes	—
244	C₂×Q₈	#131	1	1	yes	characteristic
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×C₈	1	2	yes	—
248	C₂×D₄	#131	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	SD₁₆	—	2	4	no	—
258	SD₁₆	—	2	4	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	D₈	—	2	4	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	1	no	—
264	M₁₆	—	2	1	no	—
265	M₁₆	—	2	1	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	4	2	no	—
275	M₁₆	C₄×D₄	1	2	yes	—
276	M₁₆	C₂²×C₈	1	2	yes	—
277	M₁₆	C₂²×C₈	1	4	yes	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	C₄×D₄	1	1	yes	characteristic
289	C₂×C₈	C₄×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
292	C₂×C₈	#131	1	1	yes	characteristic
293	C₂×C₈	#131	1	1	yes	characteristic
294	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₈	C₂×#41	1	1	yes	characteristic
297	C₂×C₈	C₂×#41	1	1	yes	characteristic
298	C₂×C₈	C₂×#41	1	1	yes	characteristic
299	C₂×C₈	C₂×#41	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	#131	1	2	yes	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	2	2	no	—
310	C₄²	C₄×D₄	1	1	yes	characteristic
311	C₄²	C₄×D₄	1	1	yes	characteristic
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	4	2	no	—
317	C₁₆	—	4	2	no	—
318	C₁₆	#121	1	1	yes	characteristic
319	C₁₆	#121	1	1	yes	characteristic
320	C₁₆	#121	1	1	yes	characteristic
321	C₁₆	#121	1	1	yes	characteristic
322	C₂³	—	2	1	no	—
323	C₂³	—	2	2	no	—
324	C₂³	C₂×#131	1	1	yes	characteristic
325	Q₈	—	2	1	no	—
326	Q₈	—	2	2	no	—
327	Q₈	—	2	2	no	—
328	Q₈	C₈×D₄	1	2	yes	—
329	D₄	—	2	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	2	1	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	C₈×D₄	1	2	yes	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
351	C₂×C₄	C₄×#41	1	1	yes	characteristic
352	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
353	C₂×C₄	C₂×#131	1	1	yes	characteristic
354	C₂×C₄	C₂×#131	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	4	no	—
357	C₈	—	2	2	no	—
358	C₈	—	4	2	no	—
359	C₈	C₈×D₄	1	1	yes	characteristic
360	C₈	C₈×D₄	1	1	yes	characteristic
361	C₈	C₈×D₄	1	1	yes	characteristic
362	C₈	C₈×D₄	1	1	yes	characteristic
363	C₈	C₂×#121	1	1	yes	characteristic
364	C₈	#525	1	1	yes	characteristic
365	C₈	#529	1	1	yes	characteristic
366	C₈	#532	1	1	yes	characteristic
367	C₂²	—	2	2	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	#2929	1	1	yes	characteristic
373	C₂²	#2931	1	1	yes	characteristic
374	C₂²	C₈×#41	1	1	yes	characteristic
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	2	no	—
381	C₄	C₄×#121	1	1	yes	characteristic
382	C₄	C₂×C₈×D₄	1	1	yes	characteristic
383	C₄	C₂×C₈×D₄	1	1	yes	characteristic
384	C₄	#2966	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	4	2	no	—
389	C₂	—	8	2	no	—
390	C₂	C₈×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₈×#121
prime radical	#3067	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33441	C₂	semidirect
2. C₂×#2170	C₂	semidirect
3. C₄×#504	C₂	semidirect
5. #13197	C₂	semidirect
6. #13164	C₂	semidirect
7. #13123	C₂	semidirect
8. #12987	C₂	semidirect
9. #12792	C₂	semidirect
10. #12664	C₂	semidirect
11. #12613	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33441	2
C₂×#2170	2
C₄×#504	2
#21558	2
#13197	2
#13164	2
#13123	2
#12987	2
#12792	2
#12664	2
#12613	2