Subgroups — #483464

Order 512.

403 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33484	C₂	1	1	yes	maximal, characteristic
2	#17834	C₂	1	1	yes	maximal, characteristic
3	#17827	C₂	1	1	yes	maximal, characteristic
4	#12031	C₂	1	1	yes	maximal, characteristic, complemented
5	#12031	C₂	1	1	yes	maximal, characteristic, complemented
6	#12030	C₂	1	1	yes	maximal, characteristic, complemented
7	#12029	C₂	1	1	yes	maximal, characteristic, complemented
8	#12028	C₂	1	1	yes	maximal, characteristic, complemented
9	#12028	C₂	1	1	yes	maximal, characteristic, complemented
10	#12021	C₂	1	1	yes	maximal, characteristic, complemented
11	#12019	C₂	1	1	yes	maximal, characteristic, complemented
12	#12019	C₂	1	1	yes	maximal, characteristic, complemented
13	#12000	C₂	1	1	yes	maximal, characteristic, complemented
14	#11999	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#1368	C₂	1	1	yes	maximal, characteristic, complemented
16	#2990	—	2	1	no	—
17	C₈×#41	C₂²	1	1	yes	characteristic
18	#2925	—	2	1	no	—
19	#2183	C₄	1	1	yes	characteristic
20	#2183	C₄	1	1	yes	characteristic
21	#2168	C₄	1	1	yes	characteristic
22	#2168	C₄	1	1	yes	characteristic
23	#2168	C₄	1	1	yes	characteristic
24	#2168	C₄	1	1	yes	characteristic
25	D₄×C₁₆	C₄	1	1	yes	characteristic
26	D₄×C₁₆	C₄	1	1	yes	characteristic
27	#2163	C₂²	1	1	yes	characteristic
28	#2163	C₂²	1	1	yes	characteristic
29	#2163	C₂²	1	1	yes	characteristic
30	C₄×M₃₂	C₂²	1	1	yes	characteristic
31	C₂×#346	C₂²	1	1	yes	characteristic
32	C₂×#346	C₂²	1	1	yes	characteristic
33	#1574	—	2	1	no	—
34	#1573	—	2	1	no	—
35	#1573	—	2	1	no	—
36	#1572	—	2	1	no	—
37	#1572	—	2	1	no	—
38	#1571	—	2	1	no	—
39	#1568	C₂²	1	1	yes	characteristic
40	#1568	C₂²	1	1	yes	characteristic
41	#1568	C₂²	1	1	yes	characteristic
42	#1568	C₂²	1	1	yes	characteristic
43	#1566	C₂²	1	1	yes	characteristic
44	#1566	C₂²	1	1	yes	characteristic
45	#1564	C₂²	1	1	yes	characteristic
46	#1564	C₂²	1	1	yes	characteristic
47	#1564	C₂²	1	1	yes	characteristic
48	C₂×#334	C₂²	1	1	yes	characteristic
49	C₂×#334	C₂²	1	1	yes	characteristic
50	C₂×#334	C₂²	1	1	yes	characteristic
51	#1368	C₂²	1	1	yes	characteristic
52	#1368	C₂²	1	1	yes	characteristic
53	#1368	C₂²	1	1	yes	characteristic
54	#1368	C₂²	1	1	yes	characteristic
55	#1368	C₂²	1	1	yes	characteristic
56	#1368	C₂²	1	1	yes	characteristic
57	#1368	C₂²	1	1	yes	characteristic
58	#1368	C₂²	1	1	yes	characteristic
59	#1368	C₂²	1	1	yes	characteristic
60	#1368	C₂²	1	1	yes	characteristic
61	#1368	C₂²	1	1	yes	characteristic
62	#1368	C₂²	1	1	yes	characteristic
63	#1368	C₂²	1	1	yes	characteristic
64	#1368	C₂²	1	1	yes	characteristic
65	#1368	C₂²	1	1	yes	characteristic
66	#1368	C₂²	1	1	yes	characteristic
67	C₄×#41	D₄	1	1	yes	characteristic
68	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
69	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
74	#446	—	4	1	no	—
75	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
76	C₈×Q₈	Q₈	1	1	yes	characteristic
77	#435	—	4	1	no	—
78	#435	—	4	1	no	—
79	C₈×D₄	—	4	1	no	—
80	C₈×D₄	C₂×C₄	1	1	yes	characteristic
81	C₈×D₄	C₂×C₄	1	1	yes	characteristic
82	C₈×D₄	C₂×C₄	1	1	yes	characteristic
83	C₈×D₄	Q₈	1	1	yes	characteristic
84	C₈×D₄	Q₈	1	1	yes	characteristic
85	C₈×D₄	Q₈	1	1	yes	characteristic
86	#433	—	4	1	no	—
87	#433	—	4	1	no	—
88	#432	—	2	1	no	—
89	#432	—	2	1	no	—
90	#432	—	2	1	no	—
91	#432	—	2	1	no	—
92	#431	—	2	1	no	—
93	#431	—	2	1	no	—
94	#431	—	2	1	no	—
95	#431	D₄	1	1	yes	characteristic
96	#431	D₄	1	1	yes	characteristic
97	#431	D₄	1	1	yes	characteristic
98	#423	—	2	1	no	—
99	#423	—	2	1	no	—
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	#408	—	4	1	no	—
104	#405	—	4	1	no	—
105	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
111	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
112	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
113	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
114	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
115	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
116	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
117	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
118	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
119	C₂²⋊C₁₆	C₂×C₄	1	1	yes	characteristic
120	#346	C₂×C₄	1	1	yes	characteristic
121	#346	C₂×C₄	1	1	yes	characteristic
122	#346	C₂×C₄	1	1	yes	characteristic
123	#346	C₂×C₄	1	1	yes	characteristic
124	#346	C₂³	1	1	yes	characteristic
125	#346	C₂³	1	1	yes	characteristic
126	#346	C₂³	1	1	yes	characteristic
127	#346	C₂³	1	1	yes	characteristic
128	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
129	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
130	#335	—	2	1	no	—
131	#335	—	2	1	no	—
132	#335	—	2	1	no	—
133	#335	—	2	1	no	—
134	#335	—	2	1	no	—
135	#335	—	2	1	no	—
136	#335	—	2	1	no	—
137	#334	—	2	1	no	—
138	#334	—	2	1	no	—
139	#334	—	2	1	no	—
140	#334	—	2	1	no	—
141	#334	—	2	1	no	—
142	#334	C₂³	1	1	yes	characteristic
143	#334	C₂³	1	1	yes	characteristic
144	#334	C₂³	1	1	yes	characteristic
145	#334	C₂³	1	1	yes	characteristic
146	#334	C₂³	1	1	yes	characteristic
147	#334	C₂³	1	1	yes	characteristic
148	#334	C₂³	1	1	yes	characteristic
149	#334	C₂³	1	1	yes	characteristic
150	C₂×#41	SD₁₆	1	1	yes	characteristic
151	C₂×M₁₆	—	4	1	no	—
152	C₂×M₁₆	—	4	1	no	—
153	C₂²×C₈	—	4	1	no	—
154	C₂²×C₈	—	4	1	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
160	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
161	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
162	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
163	C₄×Q₈	SD₁₆	1	1	yes	characteristic
164	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
165	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
166	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
167	C₄×D₄	SD₁₆	1	1	yes	characteristic
168	C₄×D₄	SD₁₆	1	1	yes	characteristic
169	C₄×D₄	SD₁₆	1	1	yes	characteristic
170	#117	SD₁₆	1	1	yes	characteristic
171	#117	SD₁₆	1	1	yes	characteristic
172	#117	SD₁₆	1	1	yes	characteristic
173	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
174	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
175	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
176	M₃₂	—	2	2	no	—
177	M₃₂	—	4	1	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	4	1	no	—
181	C₂×C₁₆	—	4	1	no	—
182	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
188	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊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	—	4	1	no	—
199	C₄⋊C₈	—	4	1	no	—
200	C₄⋊C₈	—	4	1	no	—
201	C₄⋊C₈	—	4	1	no	—
202	C₄⋊C₈	—	4	1	no	—
203	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
204	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
205	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
206	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
207	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
208	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
209	C₂²⋊C₈	—	4	1	no	—
210	C₂²⋊C₈	—	4	1	no	—
211	C₂²⋊C₈	—	4	1	no	—
212	C₂²⋊C₈	—	4	1	no	—
213	C₂²⋊C₈	—	4	1	no	—
214	C₂²⋊C₈	—	4	1	no	—
215	C₂²⋊C₈	—	4	1	no	—
216	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
217	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
218	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
219	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
220	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
221	C₂²⋊C₈	C₄⋊C₄	1	1	yes	characteristic
222	#97	—	4	1	no	—
223	#97	—	4	1	no	—
224	#97	—	4	1	no	—
225	#97	—	4	1	no	—
226	C₄×C₈	—	4	1	no	—
227	C₄×C₈	—	4	1	no	—
228	C₄×C₈	—	4	1	no	—
229	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
230	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
231	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
232	C₄×C₈	C₂×D₄	1	1	yes	characteristic
233	C₄×C₈	C₂×D₄	1	1	yes	characteristic
234	C₄×C₈	C₂×D₄	1	1	yes	characteristic
235	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
236	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
237	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
238	C₄×C₈	C₂⁴	1	1	yes	characteristic
239	#41	—	4	2	no	—
240	C₂×Q₈	#106	1	1	yes	characteristic
241	C₂×D₄	#106	1	1	yes	characteristic
242	C₂×D₄	#106	1	1	yes	characteristic
243	C₂×D₄	#106	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
248	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
249	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
250	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
252	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
253	M₁₆	—	8	2	no	—
254	M₁₆	—	16	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	8	2	no	—
273	C₂×C₈	—	16	1	no	—
274	C₂×C₈	—	16	1	no	—
275	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
282	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
283	C₄⋊C₄	#106	1	1	yes	characteristic
284	C₄⋊C₄	#106	1	1	yes	characteristic
285	C₄⋊C₄	#106	1	1	yes	characteristic
286	C₄⋊C₄	#106	1	1	yes	characteristic
287	C₄⋊C₄	#106	1	1	yes	characteristic
288	C₄⋊C₄	#106	1	1	yes	characteristic
289	C₂²⋊C₄	#106	1	1	yes	characteristic
290	C₂²⋊C₄	#106	1	1	yes	characteristic
291	C₂²⋊C₄	#106	1	1	yes	characteristic
292	C₂²⋊C₄	#106	1	1	yes	characteristic
293	C₂²⋊C₄	#106	1	1	yes	characteristic
294	C₂²⋊C₄	#106	1	1	yes	characteristic
295	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
296	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
297	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
298	C₄²	C₂×SD₁₆	1	1	yes	characteristic
299	C₄²	C₂×SD₁₆	1	1	yes	characteristic
300	C₄²	C₂×SD₁₆	1	1	yes	characteristic
301	C₄²	C₂×SD₁₆	1	1	yes	characteristic
302	C₄²	C₂×SD₁₆	1	1	yes	characteristic
303	C₄²	C₂×SD₁₆	1	1	yes	characteristic
304	C₄²	C₂²×D₄	1	1	yes	characteristic
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	1	no	—
309	C₁₆	—	2	1	no	—
310	C₁₆	—	2	1	no	—
311	C₁₆	—	2	1	no	—
312	C₁₆	#131	1	2	yes	—
313	C₂³	C₂×#106	1	1	yes	characteristic
314	C₂³	C₂×#106	1	1	yes	characteristic
315	C₂³	C₂×#106	1	1	yes	characteristic
316	Q₈	—	4	1	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	C₂×#106	1	1	yes	characteristic
333	C₂×C₄	C₂×#106	1	1	yes	characteristic
334	C₂×C₄	C₂×#106	1	1	yes	characteristic
335	C₂×C₄	C₂×#106	1	1	yes	characteristic
336	C₂×C₄	C₂×#106	1	1	yes	characteristic
337	C₂×C₄	C₂×#106	1	1	yes	characteristic
338	C₂×C₄	C₂×#106	1	1	yes	characteristic
339	C₂×C₄	C₂×#106	1	1	yes	characteristic
340	C₂×C₄	C₂×#106	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
342	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
343	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	8	1	no	—
354	C₈	—	8	1	no	—
355	C₈	—	8	1	no	—
356	C₈	—	8	1	no	—
357	C₈	—	8	1	no	—
358	C₈	—	8	1	no	—
359	C₈	C₂×#131	1	1	yes	characteristic
360	C₈	#568	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	#2155	1	2	yes	—
370	C₂²	C₂²×#106	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	#2906	1	1	yes	characteristic
383	C₄	#2916	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	4	1	no	—
386	C₂	—	4	1	no	—
387	C₂	#17826	1	1	yes	characteristic
388	C₂	C₂×#2155	1	1	yes	characteristic
389	C₂	#33404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ 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
4. #12031	C₂	semidirect
5. #12031	C₂	semidirect
6. #12030	C₂	semidirect
7. #12029	C₂	semidirect
8. #12028	C₂	semidirect
9. #12028	C₂	semidirect
10. #12021	C₂	semidirect
11. #12019	C₂	semidirect
12. #12019	C₂	semidirect
13. #12000	C₂	semidirect
14. #11999	C₂	semidirect
15. C₂×#1368	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33484	2
#17834	2
#17827	2
#12031	2
#12030	2
#12029	2
#12028	2
#12021	2
#12019	2
#12000	2
#11999	2
C₂×#1368	2