Subgroups — #110021

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1567	C₂	1	1	yes	maximal, characteristic
3	#10853	C₂	1	1	yes	maximal, characteristic
4	C₂×#1301	C₂	1	1	yes	maximal, characteristic
5	#7901	C₂	1	1	yes	maximal, characteristic
6	C₂×#1279	C₂	1	1	yes	maximal, characteristic
7	C₂×#1278	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#423	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×M₁₆	—	2	1	no	—
13	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#1771	—	2	2	no	—
17	#1770	—	2	1	no	—
18	C₈×C₄⋊C₄	—	2	1	no	—
19	C₂×#341	—	2	1	no	—
20	C₂×#341	—	2	1	no	—
21	#1728	—	2	1	no	—
22	#1567	—	2	2	no	—
23	#1567	C₄	1	4	yes	complemented
24	C₂×#335	—	2	2	no	—
25	#1301	—	2	2	no	—
26	#1279	—	2	1	no	—
27	#1279	—	2	1	no	—
28	#1279	C₄	1	2	yes	—
29	#1279	C₄	1	2	yes	—
30	#1278	C₄	1	4	yes	complemented
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×M₁₆	D₄	1	1	yes	characteristic
35	C₂³×C₈	—	2	1	no	—
36	C₂³×C₈	—	2	1	no	—
37	C₂²×C₄⋊C₄	Q₈	1	2	yes	—
38	C₂²×C₄²	—	2	1	no	—
39	C₂²×C₄²	C₂³	1	1	yes	characteristic
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	C₂×C₄	1	2	yes	—
44	#423	—	2	1	no	—
45	#423	—	2	1	no	—
46	#423	—	2	1	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	C₂×C₄	1	2	yes	—
50	#423	C₂×C₄	1	2	yes	—
51	#423	D₄	1	2	yes	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	1	no	—
56	C₂×C₄⋊C₈	—	2	1	no	—
57	C₂×C₄⋊C₈	—	2	1	no	—
58	C₂×C₄⋊C₈	—	2	1	no	—
59	C₂×C₄⋊C₈	—	2	1	no	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₈	D₄	1	2	yes	—
63	C₂×C₂²⋊C₈	—	2	1	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	—	4	2	no	—
67	C₄×M₁₆	—	4	2	no	—
68	C₄×M₁₆	C₂×C₄	1	2	yes	—
69	C₄×M₁₆	Q₈	1	2	yes	—
70	C₂×#97	—	2	1	no	—
71	C₂×#97	—	2	2	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	—	2	1	no	—
76	C₂×C₄×C₈	—	2	1	no	—
77	C₂×C₄×C₈	—	2	2	no	—
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	1	yes	characteristic
81	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
82	C₄×C₄⋊C₄	—	2	1	no	—
83	C₄×C₄⋊C₄	—	2	1	no	—
84	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
85	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₂×#95	—	2	1	no	—
88	#341	—	4	2	no	—
89	#341	—	4	2	no	—
90	#336	—	2	2	no	—
91	#336	—	2	2	no	—
92	#335	—	2	8	no	—
93	C₂³×C₄	—	2	2	no	—
94	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
96	C₂³×C₄	#41	1	1	yes	characteristic
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂×M₁₆	—	4	1	no	—
116	C₂×M₁₆	—	4	1	no	—
117	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
118	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
119	C₂×M₁₆	D₈	1	2	yes	—
120	C₂×M₁₆	Q₁₆	1	2	yes	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	1	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	4	4	no	—
130	C₂²×C₈	—	4	1	no	—
131	C₂²×C₈	—	4	1	no	—
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	—	4	1	no	—
134	C₂²×C₈	—	4	1	no	—
135	C₂²×C₈	—	4	1	no	—
136	C₂²×C₈	—	4	1	no	—
137	C₂²×C₈	—	4	1	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	4	2	no	—
141	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
142	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	C₂×C₄²	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
156	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
157	C₂×C₄²	#41	1	1	yes	characteristic
158	C₂×C₄²	#41	1	1	yes	characteristic
159	C₂×C₄²	#41	1	1	yes	characteristic
160	C₂×C₄²	#41	1	1	yes	characteristic
161	C₂×C₄²	#41	1	1	yes	characteristic
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	—	2	2	no	—
170	C₄⋊C₈	—	4	1	no	—
171	C₄⋊C₈	—	4	1	no	—
172	C₄⋊C₈	—	4	1	no	—
173	C₄⋊C₈	—	4	1	no	—
174	C₄⋊C₈	—	4	1	no	—
175	C₄⋊C₈	C₂²⋊C₄	1	4	yes	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	4	1	no	—
178	#97	—	2	2	no	—
179	#97	—	2	2	no	—
180	#97	—	4	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	4	4	no	—
189	C₄×C₈	—	4	1	no	—
190	C₄×C₈	C₄²	1	2	yes	—
191	C₄×C₈	C₄²	1	2	yes	—
192	#95	—	2	2	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	8	2	no	—
205	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	#117	1	1	yes	characteristic
208	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
214	C₂²×C₄	#120	1	1	yes	characteristic
215	C₂²×C₄	#121	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
217	C₂²×C₄	#123	1	1	yes	characteristic
218	C₂²×C₄	#123	1	1	yes	characteristic
219	C₂²×C₄	#125	1	1	yes	characteristic
220	C₂²×C₄	#125	1	1	yes	characteristic
221	C₂²×C₄	#126	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
225	C₂²×C₄	#135	1	1	yes	characteristic
226	C₂²×C₄	#135	1	1	yes	characteristic
227	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂²×C₄	#137	1	1	yes	characteristic
230	C₂²×C₄	#137	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	8	1	no	—
245	M₁₆	—	8	1	no	—
246	M₁₆	#107	1	4	yes	—
247	C₂×C₈	—	2	4	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	4	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	—	4	4	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	8	2	no	—
268	C₂×C₈	—	8	1	no	—
269	C₂×C₈	—	8	1	no	—
270	C₂×C₈	—	8	1	no	—
271	C₂×C₈	—	8	1	no	—
272	C₂×C₈	—	8	1	no	—
273	C₂×C₈	—	8	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	#108	1	4	yes	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×C₄²	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₄²	#117	1	1	yes	characteristic
288	C₄²	#117	1	1	yes	characteristic
289	C₄²	C₄×D₄	1	1	yes	characteristic
290	C₄²	C₄×D₄	1	1	yes	characteristic
291	C₄²	C₄×D₄	1	1	yes	characteristic
292	C₄²	C₄×D₄	1	1	yes	characteristic
293	C₄²	C₄×Q₈	1	1	yes	characteristic
294	C₄²	C₄×Q₈	1	1	yes	characteristic
295	C₄²	#121	1	1	yes	characteristic
296	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
297	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
298	C₄²	#123	1	1	yes	characteristic
299	C₄²	#124	1	1	yes	characteristic
300	C₄²	#126	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	#385	1	1	yes	characteristic
304	C₂³	#387	1	1	yes	characteristic
305	C₂³	#389	1	1	yes	characteristic
306	C₂³	#474	1	1	yes	characteristic
307	C₂³	#477	1	1	yes	characteristic
308	C₂³	#478	1	1	yes	characteristic
309	C₂³	#479	1	1	yes	characteristic
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	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	4	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	4	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
334	C₂×C₄	#385	1	1	yes	characteristic
335	C₂×C₄	#386	1	1	yes	characteristic
336	C₂×C₄	#387	1	1	yes	characteristic
337	C₂×C₄	#387	1	1	yes	characteristic
338	C₂×C₄	#387	1	1	yes	characteristic
339	C₂×C₄	#388	1	1	yes	characteristic
340	C₂×C₄	#388	1	1	yes	characteristic
341	C₂×C₄	#389	1	1	yes	characteristic
342	C₂×C₄	#391	1	1	yes	characteristic
343	C₂×C₄	C₂×#107	1	1	yes	characteristic
344	C₂×C₄	#427	1	1	yes	characteristic
345	C₂×C₄	#428	1	1	yes	characteristic
346	C₂×C₄	#428	1	1	yes	characteristic
347	C₂×C₄	C₂×#108	1	2	yes	—
348	C₂×C₄	#430	1	2	yes	—
349	C₂×C₄	C₄×D₈	1	1	yes	characteristic
350	C₂×C₄	C₄×D₈	1	1	yes	characteristic
351	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
352	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
353	C₂×C₄	#440	1	1	yes	characteristic
354	C₂×C₄	#440	1	1	yes	characteristic
355	C₂×C₄	#441	1	1	yes	characteristic
356	C₂×C₄	#442	1	1	yes	characteristic
357	C₂×C₄	#447	1	1	yes	characteristic
358	C₂×C₄	#449	1	1	yes	characteristic
359	C₂×C₄	#451	1	1	yes	characteristic
360	C₂×C₄	#452	1	1	yes	characteristic
361	C₂×C₄	#459	1	1	yes	characteristic
362	C₂×C₄	#462	1	1	yes	characteristic
363	C₂×C₄	#463	1	1	yes	characteristic
364	C₂×C₄	#464	1	1	yes	characteristic
365	C₂×C₄	#480	1	1	yes	characteristic
366	C₂×C₄	#482	1	1	yes	characteristic
367	C₂×C₄	#484	1	1	yes	characteristic
368	C₂×C₄	#485	1	1	yes	characteristic
369	C₈	—	2	4	no	—
370	C₈	—	2	4	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	4	no	—
374	C₈	—	8	2	no	—
375	C₈	—	8	2	no	—
376	C₈	—	8	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	#1436	1	1	yes	characteristic
386	C₂²	#1855	1	2	yes	—
387	C₂²	#1861	1	2	yes	—
388	C₂²	#1862	1	1	yes	characteristic
389	C₂²	#1863	1	1	yes	characteristic
390	C₂²	#1918	1	1	yes	characteristic
391	C₂²	#1921	1	1	yes	characteristic
392	C₂²	#1947	1	1	yes	characteristic
393	C₂²	#1949	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	4	no	—
401	C₄	—	8	2	no	—
402	C₄	C₄×#107	1	1	yes	characteristic
403	C₄	#1776	1	1	yes	characteristic
404	C₄	C₄×#108	1	1	yes	characteristic
405	C₄	#1778	1	1	yes	characteristic
406	C₄	#1852	1	1	yes	characteristic
407	C₄	#1854	1	1	yes	characteristic
408	C₄	#1878	1	1	yes	characteristic
409	C₄	#1879	1	1	yes	characteristic
410	C₄	#1925	1	1	yes	characteristic
411	C₄	#1928	1	1	yes	characteristic
412	C₄	#1950	1	2	yes	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	#8906	1	2	yes	—
418	C₂	#8907	1	2	yes	—
419	C₂	#10894	1	1	yes	characteristic
420	C₂	#10907	1	1	yes	characteristic
421	C₂	#10911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
23. #1567	C₄	semidirect
30. #1278	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×M₁₆	2
C₂×#1567	2
#10853	2
C₂×#1301	2
#7901	2
C₂×#1279	2
C₂×#1278	2