Subgroups — #226469

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22823	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#9606	C₂	1	2	yes	maximal
4	#9605	C₂	1	2	yes	maximal
5	#9603	C₂	1	2	yes	maximal
6	#8785	C₂	1	1	yes	maximal, characteristic
7	#8785	C₂	1	1	yes	maximal, characteristic
8	#8784	C₂	1	1	yes	maximal, characteristic
9	#8274	C₂	1	1	yes	maximal, characteristic
10	#8273	C₂	1	1	yes	maximal, characteristic
11	C₄×#334	C₂	1	1	yes	maximal, characteristic
12	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
14	C₄²×Q₈	C₂²	1	1	yes	characteristic
15	#1985	—	2	2	no	—
16	#1917	—	2	2	no	—
17	#1916	—	2	2	no	—
18	#1842	C₂²	1	2	yes	—
19	#1831	C₂²	1	2	yes	—
20	#1831	C₂²	1	2	yes	—
21	#1771	—	2	2	no	—
22	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
25	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
26	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
27	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
28	#1767	C₂²	1	2	yes	—
29	#1767	C₂²	1	2	yes	—
30	C₄×C₄⋊C₈	C₂²	1	2	yes	—
31	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₄²×C₈	C₂²	1	1	yes	characteristic
35	C₄²×C₈	C₂²	1	1	yes	characteristic
36	C₄²×C₈	C₂²	1	1	yes	characteristic
37	C₂×#335	C₂²	1	2	yes	—
38	C₂×#334	C₂²	1	2	yes	—
39	C₂×#334	C₂²	1	2	yes	—
40	#1277	C₂²	1	1	yes	characteristic
41	#1277	C₂²	1	1	yes	characteristic
42	#1277	C₂²	1	1	yes	characteristic
43	#1277	C₂²	1	2	yes	—
44	#1277	C₂²	1	1	yes	characteristic
45	#1277	C₂²	1	2	yes	—
46	#1277	C₂²	1	2	yes	—
47	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
49	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
50	C₈×Q₈	—	2	4	no	—
51	C₈×Q₈	—	2	4	no	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	—	2	2	no	—
54	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₈	C₂³	1	2	yes	—
61	C₂×C₄⋊C₈	C₂³	1	2	yes	—
62	C₂×C₄⋊C₈	C₂³	1	2	yes	—
63	C₂×C₄⋊C₈	C₂³	1	2	yes	—
64	C₂×#97	—	2	2	no	—
65	C₂×#97	—	2	2	no	—
66	C₂×C₄×C₈	—	2	2	no	—
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
85	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
86	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
90	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
91	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
92	C₄³	C₂³	1	1	yes	characteristic
93	C₄³	C₂³	1	1	yes	characteristic
94	C₄³	C₂³	1	1	yes	characteristic
95	#336	—	2	2	no	—
96	#336	—	2	2	no	—
97	#336	—	2	2	no	—
98	#336	—	2	2	no	—
99	#336	—	2	2	no	—
100	#336	—	2	2	no	—
101	#336	—	2	2	no	—
102	#336	C₂×C₄	1	1	yes	characteristic
103	#336	C₂×C₄	1	1	yes	characteristic
104	#336	C₂×C₄	1	1	yes	characteristic
105	#336	C₂×C₄	1	1	yes	characteristic
106	#336	C₂×C₄	1	1	yes	characteristic
107	#336	C₂×C₄	1	1	yes	characteristic
108	#335	—	2	4	no	—
109	#334	—	2	4	no	—
110	#334	—	2	4	no	—
111	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₄×Q₈	—	2	4	no	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
151	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
152	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
153	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
154	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
155	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
156	C₂×C₄²	C₂⁴	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	4	4	no	—
170	C₄⋊C₈	—	8	2	no	—
171	C₄⋊C₈	M₁₆	1	4	yes	—
172	C₄⋊C₈	#41	1	4	yes	—
173	#97	—	4	4	no	—
174	#97	—	8	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	4	1	no	—
188	C₄×C₈	—	4	1	no	—
189	C₄×C₈	—	4	1	no	—
190	C₄×C₈	—	8	2	no	—
191	C₄×C₈	M₁₆	1	4	yes	—
192	#95	C₄⋊C₄	1	1	yes	characteristic
193	#95	C₄⋊C₄	1	1	yes	characteristic
194	#95	C₄⋊C₄	1	1	yes	characteristic
195	#95	C₄⋊C₄	1	1	yes	characteristic
196	#95	C₄⋊C₄	1	1	yes	characteristic
197	#95	C₄⋊C₄	1	1	yes	characteristic
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	C₂×M₁₆	1	2	yes	—
248	C₂×C₈	C₂×M₁₆	1	2	yes	—
249	C₂×C₈	C₂×M₁₆	1	2	yes	—
250	C₂×C₈	C₂×M₁₆	1	2	yes	—
251	C₂×C₈	#131	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	#108	1	4	yes	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	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	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	C₂×M₁₆	1	2	yes	—
285	C₄²	C₂×M₁₆	1	2	yes	—
286	C₄²	#135	1	2	yes	—
287	C₄²	#135	1	2	yes	—
288	C₄²	C₂×#41	1	2	yes	—
289	C₄²	2₋¹⁺⁴	1	2	yes	—
290	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	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	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	#423	1	2	yes	—
324	C₂×C₄	#423	1	2	yes	—
325	C₂×C₄	#427	1	2	yes	—
326	C₂×C₄	#427	1	2	yes	—
327	C₂×C₄	C₂×#108	1	2	yes	—
328	C₂×C₄	C₂×#108	1	2	yes	—
329	C₂×C₄	C₂×#108	1	2	yes	—
330	C₂×C₄	C₄×#41	1	1	yes	characteristic
331	C₂×C₄	#519	1	1	yes	characteristic
332	C₂×C₄	#548	1	2	yes	—
333	C₂×C₄	#557	1	2	yes	—
334	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
335	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
336	C₂×C₄	C₂×#131	1	1	yes	characteristic
337	C₂×C₄	#568	1	1	yes	characteristic
338	C₂×C₄	C₂×#135	1	1	yes	characteristic
339	C₂×C₄	C₂×#135	1	1	yes	characteristic
340	C₂×C₄	#577	1	2	yes	—
341	C₈	—	2	2	no	—
342	C₈	—	2	2	no	—
343	C₈	—	2	2	no	—
344	C₈	—	2	2	no	—
345	C₈	—	2	2	no	—
346	C₈	—	4	4	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	8	2	no	—
351	C₈	—	8	2	no	—
352	C₈	—	8	2	no	—
353	C₂²	#2351	1	1	yes	characteristic
354	C₂²	C₂×#423	1	1	yes	characteristic
355	C₂²	#2906	1	1	yes	characteristic
356	C₂²	C₂×#427	1	1	yes	characteristic
357	C₂²	#2912	1	1	yes	characteristic
358	C₂²	C₂²×#108	1	1	yes	characteristic
359	C₂²	#2916	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	#2965	1	1	yes	characteristic
374	C₄	#2970	1	1	yes	characteristic
375	C₄	#2983	1	1	yes	characteristic
376	C₄	#2984	1	1	yes	characteristic
377	C₄	#3360	1	2	yes	—
378	C₄	#3365	1	2	yes	—
379	C₂	#17793	1	1	yes	characteristic
380	C₂	#17798	1	1	yes	characteristic
381	C₂	#17829	1	1	yes	characteristic
382	C₂	#17834	1	1	yes	characteristic
383	C₂	#22736	1	1	yes	characteristic
384	C₂	#22741	1	1	yes	characteristic
385	C₂	#22746	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂²×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#22823	2
C₄×C₈×Q₈	2
#9606	2
#9605	2
#9603	2
#8785	2
#8784	2
#8274	2
#8273	2
C₄×#334	2