Subgroups — #483754

Order 512.

448 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33503	C₂	1	1	yes	maximal, characteristic, complemented
2	#33502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#135	C₂	1	1	yes	maximal, characteristic
4	#12057	C₂	1	2	yes	maximal, complemented
5	#12056	C₂	1	2	yes	maximal, complemented
6	#12053	C₂	1	2	yes	maximal, complemented
7	#12038	C₂	1	2	yes	maximal, complemented
8	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
9	#11621	C₂	1	1	yes	maximal, characteristic, complemented
10	#11621	C₂	1	1	yes	maximal, characteristic, complemented
11	#11411	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×#41	C₂²	1	1	yes	characteristic
13	C₈×#41	C₂²	1	1	yes	characteristic
14	C₄×#135	C₄	1	1	yes	characteristic
15	#2183	C₂²	1	2	yes	—
16	Q₈×C₁₆	C₂²	1	2	yes	—
17	#2169	C₂²	1	2	yes	—
18	#2168	—	2	2	no	—
19	D₄×C₁₆	—	2	2	no	—
20	D₄×C₁₆	C₂²	1	2	yes	—
21	#2163	—	2	1	no	—
22	#2162	—	2	1	no	—
23	#2162	C₂²	1	1	yes	characteristic
24	#2162	C₂²	1	1	yes	characteristic
25	#2150	—	2	1	no	—
26	C₂×C₄⋊C₁₆	—	2	1	no	—
27	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₈×Q₁₆	C₄	1	1	yes	characteristic
30	C₈×Q₁₆	C₂²	1	1	yes	characteristic
31	C₈×SD₁₆	C₄	1	1	yes	characteristic
32	C₈×SD₁₆	C₄	1	1	yes	characteristic
33	C₈×SD₁₆	C₂²	1	1	yes	characteristic
34	C₈×SD₁₆	C₂²	1	1	yes	characteristic
35	C₈×D₈	C₄	1	1	yes	characteristic
36	C₈×D₈	C₂²	1	1	yes	characteristic
37	#1564	C₄	1	1	yes	characteristic
38	#1476	C₄	1	1	yes	characteristic
39	#1476	C₄	1	1	yes	characteristic
40	C₂×C₈²	C₂²	1	1	yes	characteristic
41	#1370	C₂²	1	2	yes	—
42	#1370	C₂²	1	2	yes	—
43	Q₈⋊C₁₆	C₂²	1	2	yes	—
44	Q₈⋊C₁₆	C₂²	1	2	yes	—
45	D₄⋊C₁₆	C₂²	1	2	yes	—
46	D₄⋊C₁₆	C₂²	1	2	yes	—
47	#1312	C₂²	1	2	yes	—
48	#1311	C₂²	1	2	yes	—
49	C₂×#135	C₈	1	1	yes	characteristic
50	C₄×#41	C₂×C₄	1	1	yes	characteristic
51	C₄×#41	C₂×C₄	1	1	yes	characteristic
52	C₂×M₃₂	—	2	2	no	—
53	C₂²×C₁₆	—	2	1	no	—
54	C₂²×C₁₆	—	2	1	no	—
55	C₂²×C₁₆	—	2	2	no	—
56	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
57	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
58	C₈×Q₈	C₂³	1	1	yes	characteristic
59	C₈×Q₈	C₂³	1	1	yes	characteristic
60	C₄×Q₁₆	C₈	1	1	yes	characteristic
61	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×SD₁₆	C₈	1	1	yes	characteristic
63	C₄×SD₁₆	C₈	1	1	yes	characteristic
64	C₄×SD₁₆	C₂×C₄	1	1	yes	characteristic
65	C₄×SD₁₆	C₂×C₄	1	1	yes	characteristic
66	C₄×D₈	C₈	1	1	yes	characteristic
67	C₄×D₈	C₂×C₄	1	1	yes	characteristic
68	C₈×D₄	—	2	1	no	—
69	C₈×D₄	—	2	1	no	—
70	C₈×D₄	—	2	1	no	—
71	C₈×D₄	—	2	1	no	—
72	C₈×D₄	C₂×C₄	1	1	yes	characteristic
73	C₈×D₄	C₂×C₄	1	1	yes	characteristic
74	C₈×D₄	C₂³	1	1	yes	characteristic
75	C₈×D₄	C₂³	1	1	yes	characteristic
76	#431	—	2	1	no	—
77	#431	—	2	1	no	—
78	#431	C₂×C₄	1	1	yes	characteristic
79	#431	C₂×C₄	1	1	yes	characteristic
80	#427	C₈	1	1	yes	characteristic
81	#416	C₈	1	1	yes	characteristic
82	#416	C₈	1	1	yes	characteristic
83	C₂×C₄×C₈	—	2	1	no	—
84	C₂×C₄×C₈	—	2	1	no	—
85	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	2	no	—
89	C₄⋊C₁₆	—	2	2	no	—
90	C₄⋊C₁₆	—	2	1	no	—
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	D₄	1	1	yes	characteristic
94	C₄⋊C₁₆	D₄	1	1	yes	characteristic
95	C₄⋊C₁₆	C₂³	1	2	yes	—
96	C₄⋊C₁₆	C₂³	1	2	yes	—
97	C₂²⋊C₁₆	—	2	1	no	—
98	C₂²⋊C₁₆	—	2	2	no	—
99	C₂²⋊C₁₆	—	2	2	no	—
100	C₂²⋊C₁₆	D₄	1	2	yes	—
101	#346	—	2	2	no	—
102	C₄×C₁₆	—	2	2	no	—
103	C₄×C₁₆	C₂³	1	2	yes	—
104	C₄×C₁₆	C₂³	1	2	yes	—
105	#335	C₂×C₄	1	1	yes	characteristic
106	#335	C₂×C₄	1	1	yes	characteristic
107	#334	C₂×C₄	1	1	yes	characteristic
108	#334	C₂×C₄	1	1	yes	characteristic
109	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
110	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
111	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
112	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
113	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
114	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
115	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
116	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
117	C₈²	C₂×C₄	1	1	yes	characteristic
118	C₈²	C₂×C₄	1	1	yes	characteristic
119	C₈²	C₂³	1	1	yes	characteristic
120	C₈²	C₂³	1	1	yes	characteristic
121	C₂×#41	C₂×C₈	1	1	yes	characteristic
122	C₂×#41	C₂×C₈	1	1	yes	characteristic
123	#135	—	2	4	no	—
124	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
125	C₂×SD₁₆	C₂×C₈	1	1	yes	characteristic
126	C₂×SD₁₆	C₂×C₈	1	1	yes	characteristic
127	C₂×D₈	C₂×C₈	1	1	yes	characteristic
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₈	#41	1	1	yes	characteristic
137	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
138	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
139	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	C₂×C₈	1	1	yes	characteristic
146	C₄×D₄	C₂×C₈	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
149	#117	—	2	1	no	—
150	#117	—	2	1	no	—
151	#117	C₂×C₈	1	1	yes	characteristic
152	#117	C₂×C₈	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
156	M₃₂	—	8	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	8	2	no	—
161	C₂×C₁₆	—	8	1	no	—
162	C₂×C₁₆	—	8	1	no	—
163	C₂×C₁₆	C₂×D₄	1	2	yes	—
164	C₂×C₁₆	C₂×D₄	1	2	yes	—
165	#107	C₂×C₈	1	1	yes	characteristic
166	#107	C₂×C₈	1	1	yes	characteristic
167	#106	C₂×C₈	1	1	yes	characteristic
168	#106	C₂×C₈	1	1	yes	characteristic
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
175	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
176	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
177	#103	C₂×C₈	1	1	yes	characteristic
178	#103	C₂×C₈	1	1	yes	characteristic
179	#103	C₂×C₈	1	1	yes	characteristic
180	#103	C₂×C₈	1	1	yes	characteristic
181	#102	C₂×C₈	1	1	yes	characteristic
182	#102	C₂×C₈	1	1	yes	characteristic
183	#102	C₂×C₈	1	1	yes	characteristic
184	#102	C₂×C₈	1	1	yes	characteristic
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊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	—	2	1	no	—
198	C₄×C₈	C₂×C₈	1	1	yes	characteristic
199	C₄×C₈	C₂×C₈	1	1	yes	characteristic
200	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
201	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
202	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
203	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
204	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
205	C₄×C₈	C₂×D₄	1	1	yes	characteristic
206	C₄×C₈	#41	1	1	yes	characteristic
207	C₄×C₈	C₂⁴	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	#41	—	2	2	no	—
211	#41	M₃₂	1	4	yes	complemented
212	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
213	C₂×Q₈	C₂²×C₈	1	1	yes	characteristic
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₂²×C₈	1	1	yes	characteristic
217	C₂×D₄	C₂²×C₈	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
226	C₂²×C₄	#131	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	SD₁₆	—	2	2	no	—
229	SD₁₆	—	2	2	no	—
230	D₈	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	2	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	—	8	1	no	—
248	C₂×C₈	—	8	1	no	—
249	C₂×C₈	C₄×D₄	1	1	yes	characteristic
250	C₂×C₈	C₄×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
252	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
253	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×#41	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×C₈	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	C₄×D₄	1	1	yes	characteristic
278	C₄²	C₂²×C₈	1	1	yes	characteristic
279	C₄²	#131	1	1	yes	characteristic
280	C₄²	C₂³×C₄	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	4	2	no	—
284	C₁₆	—	4	2	no	—
285	C₁₆	—	4	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	C₈×D₄	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	C₂×M₃₂	1	2	yes	—
293	D₄	—	2	4	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	2	no	—
298	D₄	C₂×M₃₂	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	C₈×D₄	1	1	yes	characteristic
317	C₂×C₄	C₈×D₄	1	1	yes	characteristic
318	C₂×C₄	C₈×D₄	1	1	yes	characteristic
319	C₂×C₄	C₂×M₃₂	1	2	yes	—
320	C₂×C₄	#504	1	2	yes	—
321	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
323	C₂×C₄	C₂×#131	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	#576	1	1	yes	characteristic
338	C₈	#578	1	1	yes	characteristic
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	#2168	1	2	yes	—
348	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	#2168	1	2	yes	—
359	C₄	#2951	1	1	yes	characteristic
360	C₄	#2957	1	1	yes	characteristic
361	C₄	C₂²×M₃₂	1	1	yes	characteristic
362	C₄	C₂×#504	1	1	yes	characteristic
363	C₂	—	2	2	no	—
364	C₂	—	4	2	no	—
365	C₂	—	8	1	no	—
366	C₂	#17857	1	1	yes	characteristic
367	C₂	C₂×#2168	1	1	yes	characteristic
368	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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
1. #33503	C₂	semidirect
2. #33502	C₂	semidirect
4. #12057	C₂	semidirect
5. #12056	C₂	semidirect
6. #12053	C₂	semidirect
7. #12038	C₂	semidirect
8. C₂×#1370	C₂	semidirect
9. #11621	C₂	semidirect
10. #11621	C₂	semidirect
11. #11411	C₂	semidirect
211. #41	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33503	2
#33502	2
C₈×#135	2
#12057	2
#12056	2
#12053	2
#12038	2
C₂×#1370	2
#11621	2
#11411	2