Subgroups — #184400

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21340	C₂	1	1	yes	maximal, characteristic
2	#21339	C₂	1	1	yes	maximal, characteristic
3	#21339	C₂	1	1	yes	maximal, characteristic
4	#8265	C₂	1	1	yes	maximal, characteristic
5	#8265	C₂	1	1	yes	maximal, characteristic
6	#8264	C₂	1	1	yes	maximal, characteristic
7	#8264	C₂	1	1	yes	maximal, characteristic
8	#8264	C₂	1	1	yes	maximal, characteristic
9	#8264	C₂	1	1	yes	maximal, characteristic
10	#8263	C₂	1	1	yes	maximal, characteristic
11	#8260	C₂	1	1	yes	maximal, characteristic
12	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
13	#7845	C₂	1	1	yes	maximal, characteristic
14	#7843	C₂	1	1	yes	maximal, characteristic
15	C₈×#97	C₂	1	1	yes	maximal, characteristic
16	C₄²×Q₈	C₂²	1	1	yes	characteristic
17	#1985	C₄	1	2	yes	—
18	#1984	C₄	1	2	yes	—
19	#1984	C₄	1	2	yes	—
20	#1940	C₄	1	2	yes	—
21	#1939	C₄	1	2	yes	—
22	#1939	C₄	1	2	yes	—
23	#1917	C₄	1	2	yes	—
24	#1917	C₄	1	2	yes	—
25	#1916	C₄	1	2	yes	—
26	#1916	C₄	1	2	yes	—
27	#1916	C₄	1	2	yes	—
28	#1916	C₄	1	2	yes	—
29	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	#1744	C₂²	1	1	yes	characteristic
39	#1744	C₂²	1	1	yes	characteristic
40	#1744	C₂²	1	1	yes	characteristic
41	#1744	C₂²	1	1	yes	characteristic
42	#1744	C₂²	1	1	yes	characteristic
43	#1744	C₂²	1	1	yes	characteristic
44	C₄×#97	C₂²	1	1	yes	characteristic
45	C₄²×C₈	C₂²	1	1	yes	characteristic
46	C₄²×C₈	C₂²	1	1	yes	characteristic
47	C₂×#322	C₂²	1	1	yes	characteristic
48	C₂×#322	C₂²	1	1	yes	characteristic
49	C₂×#322	C₂²	1	1	yes	characteristic
50	C₂×#322	C₂²	1	1	yes	characteristic
51	C₂×#322	C₂²	1	1	yes	characteristic
52	C₂×#322	C₂²	1	1	yes	characteristic
53	C₂×#322	C₂²	1	1	yes	characteristic
54	C₂×C₈²	C₂²	1	1	yes	characteristic
55	C₂×C₈²	C₂²	1	1	yes	characteristic
56	C₂×C₈²	C₂²	1	1	yes	characteristic
57	#1273	C₂²	1	1	yes	characteristic
58	#1273	C₂²	1	1	yes	characteristic
59	#1273	C₂²	1	1	yes	characteristic
60	#1273	C₂²	1	1	yes	characteristic
61	#1273	C₂²	1	1	yes	characteristic
62	#1273	C₂²	1	1	yes	characteristic
63	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
71	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
73	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
74	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
75	C₂×#97	C₂×C₄	1	2	yes	—
76	C₂×#97	C₂×C₄	1	2	yes	—
77	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
78	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
79	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
80	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
81	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
100	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
101	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
102	C₄³	C₂³	1	1	yes	characteristic
103	C₄³	C₂³	1	1	yes	characteristic
104	C₄³	C₂³	1	1	yes	characteristic
105	#336	C₂×C₄	1	2	yes	—
106	#336	C₂×C₄	1	2	yes	—
107	#336	C₂×C₄	1	2	yes	—
108	#336	C₂×C₄	1	2	yes	—
109	#336	C₂×C₄	1	2	yes	—
110	#336	C₂×C₄	1	2	yes	—
111	#336	C₂×C₄	1	2	yes	—
112	#336	C₂×C₄	1	2	yes	—
113	#336	C₂×C₄	1	2	yes	—
114	#336	C₂×C₄	1	2	yes	—
115	#336	C₂×C₄	1	2	yes	—
116	#336	C₂×C₄	1	2	yes	—
117	#336	C₂×C₄	1	2	yes	—
118	#336	C₂×C₄	1	2	yes	—
119	#336	C₂×C₄	1	2	yes	—
120	#336	C₂×C₄	1	2	yes	—
121	#336	C₂×C₄	1	2	yes	—
122	#336	C₂×C₄	1	2	yes	—
123	#336	C₂×C₄	1	2	yes	—
124	#336	C₂×C₄	1	2	yes	—
125	#336	C₂×C₄	1	2	yes	—
126	#322	—	2	2	no	—
127	#322	—	2	2	no	—
128	#322	—	2	2	no	—
129	#322	—	2	2	no	—
130	#322	—	2	2	no	—
131	#322	—	2	2	no	—
132	#322	Q₈	1	4	yes	complemented
133	C₈²	—	2	2	no	—
134	C₈²	—	2	2	no	—
135	C₈²	—	2	2	no	—
136	C₂²×Q₈	C₄²	1	1	yes	characteristic
137	C₂²×C₈	C₂²×C₄	1	2	yes	—
138	C₂²×C₈	C₂²×C₄	1	2	yes	—
139	C₂²×C₈	C₂²×C₄	1	2	yes	—
140	C₂²×C₈	C₂²×C₄	1	2	yes	—
141	C₂²×C₈	C₂²×C₄	1	2	yes	—
142	C₂²×C₈	C₂²×C₄	1	2	yes	—
143	C₂²×C₈	C₂²×C₄	1	2	yes	—
144	C₂²×C₈	C₂²×C₄	1	2	yes	—
145	C₂²×C₈	C₂²×C₄	1	2	yes	—
146	C₂²×C₈	C₂²×C₄	1	2	yes	—
147	C₂²×C₈	C₂²×C₄	1	2	yes	—
148	C₂²×C₈	C₂²×C₄	1	2	yes	—
149	C₄×Q₈	—	2	4	no	—
150	C₄×Q₈	—	2	4	no	—
151	C₄×Q₈	—	4	2	no	—
152	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₄²	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₄²	C₂⁴	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	2	4	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	4	2	no	—
187	C₄⋊C₈	—	4	2	no	—
188	C₄⋊C₈	—	4	2	no	—
189	#97	—	2	4	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	4	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	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	4	2	no	—
212	C₄×C₈	—	4	2	no	—
213	C₄×C₈	C₂×Q₈	1	2	yes	—
214	C₄×C₈	C₂×Q₈	1	2	yes	—
215	C₄×C₈	C₂×Q₈	1	2	yes	—
216	C₄×C₈	#41	1	2	yes	—
217	C₄×C₈	#41	1	2	yes	—
218	C₄×C₈	#41	1	2	yes	—
219	#95	C₄²	1	1	yes	characteristic
220	#95	C₄²	1	1	yes	characteristic
221	#95	C₄²	1	1	yes	characteristic
222	#95	C₄²	1	1	yes	characteristic
223	#95	C₄²	1	1	yes	characteristic
224	#95	C₄²	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
240	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
241	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
242	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₄×Q₈	1	4	yes	—
275	C₂×C₈	C₄×Q₈	1	4	yes	—
276	C₂×C₈	C₄×Q₈	1	4	yes	—
277	C₂×C₈	#131	1	4	yes	—
278	C₂×C₈	#131	1	4	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	#131	1	4	yes	—
305	C₄²	#131	1	4	yes	—
306	C₄²	C₂²×Q₈	1	1	yes	characteristic
307	C₄²	C₂×#41	1	1	yes	characteristic
308	C₄²	C₂×#41	1	1	yes	characteristic
309	C₄²	C₂×#41	1	1	yes	characteristic
310	C₂³	C₂²×C₄²	1	1	yes	characteristic
311	Q₈	—	4	4	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	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	#405	1	2	yes	—
345	C₂×C₄	#405	1	2	yes	—
346	C₂×C₄	#405	1	2	yes	—
347	C₂×C₄	#405	1	2	yes	—
348	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
351	C₂×C₄	C₄×#41	1	1	yes	characteristic
352	C₂×C₄	C₄×#41	1	1	yes	characteristic
353	C₂×C₄	C₄×#41	1	1	yes	characteristic
354	C₂×C₄	C₂×#131	1	2	yes	—
355	C₂×C₄	C₂×#131	1	2	yes	—
356	C₂×C₄	C₂×#131	1	2	yes	—
357	C₂×C₄	C₂×#131	1	2	yes	—
358	C₂×C₄	#568	1	2	yes	—
359	C₂×C₄	#568	1	2	yes	—
360	C₈	—	2	4	no	—
361	C₈	—	2	4	no	—
362	C₈	—	2	4	no	—
363	C₈	—	2	4	no	—
364	C₈	—	2	4	no	—
365	C₈	—	4	2	no	—
366	C₈	—	4	2	no	—
367	C₈	—	4	2	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	2	no	—
371	C₈	—	4	2	no	—
372	C₂²	C₄²×Q₈	1	1	yes	characteristic
373	C₂²	C₂×#405	1	1	yes	characteristic
374	C₂²	C₂×#405	1	1	yes	characteristic
375	C₂²	#2873	1	1	yes	characteristic
376	C₂²	C₄×#131	1	1	yes	characteristic
377	C₂²	C₄×#131	1	1	yes	characteristic
378	C₂²	#2875	1	1	yes	characteristic
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	4	1	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	4	1	no	—
390	C₄	—	4	1	no	—
391	C₄	#2960	1	2	yes	—
392	C₄	#2962	1	2	yes	—
393	C₄	#2962	1	2	yes	—
394	C₄	#2966	1	2	yes	—
395	C₄	#2968	1	2	yes	—
396	C₄	#2969	1	2	yes	—
397	C₂	#17326	1	1	yes	characteristic
398	C₂	#17328	1	1	yes	characteristic
399	C₂	#17331	1	1	yes	characteristic
400	C₂	#17340	1	1	yes	characteristic
401	C₂	#20516	1	1	yes	characteristic
402	C₂	#20519	1	1	yes	characteristic
403	C₂	#20521	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
132. #322	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#21340	2
#21339	2
#8265	2
#8264	2
#8263	2
#8260	2
C₈×C₄⋊C₈	2
#7845	2
#7843	2
C₈×#97	2