Subgroups — #109418

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20797	C₂	1	1	yes	maximal, characteristic
2	#20437	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#8095	C₂	1	2	yes	maximal
5	#7981	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂×#427	—	2	1	no	—
8	C₂×#427	—	2	1	no	—
9	C₂×#423	C₂²	1	1	yes	characteristic
10	C₂×#405	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2285	C₂²	1	2	yes	—
13	#2285	C₂²	1	1	yes	characteristic
14	#1846	C₄	1	2	yes	complemented
15	#1846	C₄	1	2	yes	complemented
16	#1845	—	2	1	no	—
17	#1844	—	2	1	no	—
18	#1770	—	2	1	no	—
19	#1770	C₄	1	2	yes	complemented
20	C₈×C₄⋊C₄	—	2	1	no	—
21	C₈×C₄⋊C₄	C₄	1	2	yes	complemented
22	#1729	—	2	1	no	—
23	#1564	—	2	1	no	—
24	#1564	—	2	1	no	—
25	#1564	—	2	1	no	—
26	#1564	—	2	1	no	—
27	C₂×#335	C₄	1	2	yes	complemented
28	C₂×#334	C₄	1	2	yes	complemented
29	#1303	—	2	4	no	—
30	#1289	—	2	4	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂×#117	—	2	1	no	—
35	C₂×#117	—	2	2	no	—
36	C₂²×C₄²	C₂³	1	1	yes	characteristic
37	#431	—	2	2	no	—
38	#431	—	2	2	no	—
39	#431	—	2	2	no	—
40	#431	—	2	2	no	—
41	#427	—	4	2	no	—
42	#427	—	4	2	no	—
43	#427	—	4	2	no	—
44	#427	—	4	2	no	—
45	C₂×#107	—	2	2	no	—
46	C₂×#107	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	C₂×#106	—	2	2	no	—
49	#423	—	2	1	no	—
50	#423	—	2	1	no	—
51	#423	D₄	1	2	yes	—
52	#423	Q₈	1	2	yes	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₈	—	2	2	no	—
57	#405	—	4	2	no	—
58	#405	—	4	2	no	—
59	C₂×#97	—	2	2	no	—
60	C₂×C₄×C₈	—	2	2	no	—
61	C₂×C₄×C₈	—	2	2	no	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
74	#384	—	2	2	no	—
75	#384	—	2	1	no	—
76	#382	C₂×C₄	1	4	yes	—
77	#382	Q₈	1	2	yes	—
78	#380	—	2	1	no	—
79	#380	—	2	2	no	—
80	#380	—	2	2	no	—
81	#380	D₄	1	2	yes	—
82	C₄×C₄⋊C₄	—	2	1	no	—
83	C₄×C₄⋊C₄	—	2	2	no	—
84	C₄×C₄⋊C₄	—	2	2	no	—
85	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
86	#336	—	2	2	no	—
87	#336	—	2	2	no	—
88	#335	—	2	2	no	—
89	#335	—	2	2	no	—
90	#334	—	2	2	no	—
91	#334	—	2	2	no	—
92	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₄	#41	1	1	yes	characteristic
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₈	D₈	1	2	yes	—
120	C₂²×C₈	SD₁₆	1	2	yes	—
121	C₂²×C₈	SD₁₆	1	2	yes	—
122	C₂²×C₈	Q₁₆	1	2	yes	—
123	#117	—	4	1	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	#117	—	4	1	no	—
127	#117	—	4	2	no	—
128	#117	—	4	2	no	—
129	#117	—	4	2	no	—
130	#117	—	4	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	—	4	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₂²⋊C₄	—	2	2	no	—
138	C₂×C₂²⋊C₄	—	4	2	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	4	2	no	—
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×Q₈	1	1	yes	characteristic
148	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
149	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
150	C₂×C₄²	#41	1	1	yes	characteristic
151	C₂×C₄²	#41	1	1	yes	characteristic
152	C₂×C₄²	#41	1	1	yes	characteristic
153	#107	—	4	4	no	—
154	#107	—	4	4	no	—
155	#106	—	4	4	no	—
156	#106	—	4	4	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
163	C₂²⋊C₈	—	4	2	no	—
164	C₂²⋊C₈	—	4	2	no	—
165	#97	—	4	4	no	—
166	C₄×C₈	—	4	4	no	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	—	4	2	no	—
169	C₄×C₈	C₄²	1	1	yes	characteristic
170	C₄×C₈	C₄²	1	1	yes	characteristic
171	C₄×C₈	C₄²	1	1	yes	characteristic
172	C₄×C₈	C₄²	1	1	yes	characteristic
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	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
179	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
183	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
185	#95	—	2	2	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	C₄⋊C₄	1	4	yes	—
189	C₂⁴	#121	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	#117	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
202	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
203	C₂²×C₄	#121	1	1	yes	characteristic
204	C₂²×C₄	#121	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂²×C₄	#124	1	1	yes	characteristic
209	C₂²×C₄	#125	1	1	yes	characteristic
210	C₂²×C₄	#127	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
216	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²×C₄	#137	1	1	yes	characteristic
219	C₂²×C₄	#137	1	1	yes	characteristic
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	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	1	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	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	8	4	no	—
241	C₂×C₈	#95	1	1	yes	characteristic
242	C₂×C₈	#95	1	1	yes	characteristic
243	C₂×C₈	#95	1	1	yes	characteristic
244	C₂×C₈	#95	1	1	yes	characteristic
245	C₂×C₈	#95	1	1	yes	characteristic
246	C₂×C₈	#95	1	1	yes	characteristic
247	C₂×C₈	#95	1	1	yes	characteristic
248	C₂×C₈	#95	1	1	yes	characteristic
249	C₂×C₈	#102	1	2	yes	—
250	C₂×C₈	#102	1	2	yes	—
251	C₂×C₈	#102	1	2	yes	—
252	C₂×C₈	#102	1	2	yes	—
253	C₂×C₈	#103	1	2	yes	—
254	C₂×C₈	#103	1	2	yes	—
255	C₂×C₈	#103	1	2	yes	—
256	C₂×C₈	#103	1	2	yes	—
257	C₂×C₈	#106	1	2	yes	—
258	C₂×C₈	#106	1	2	yes	—
259	C₂×C₈	#107	1	2	yes	—
260	C₂×C₈	#107	1	2	yes	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	8	4	no	—
268	C₄⋊C₄	—	8	4	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	8	2	no	—
272	C₂²⋊C₄	—	8	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	8	2	no	—
276	C₄²	—	8	2	no	—
277	C₄²	C₂×C₄²	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
280	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
281	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
282	C₄²	#117	1	1	yes	characteristic
283	C₄²	C₄×D₄	1	1	yes	characteristic
284	C₄²	C₄×Q₈	1	1	yes	characteristic
285	C₄²	C₄×Q₈	1	1	yes	characteristic
286	C₄²	C₄×Q₈	1	1	yes	characteristic
287	C₄²	#121	1	1	yes	characteristic
288	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
289	C₄²	#124	1	1	yes	characteristic
290	C₄²	#125	1	1	yes	characteristic
291	C₄²	C₄⋊Q₈	1	1	yes	characteristic
292	C₄²	C₄⋊Q₈	1	1	yes	characteristic
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	#380	1	1	yes	characteristic
296	C₂³	#389	1	1	yes	characteristic
297	C₂³	#390	1	1	yes	characteristic
298	C₂³	#459	1	1	yes	characteristic
299	C₂³	#460	1	1	yes	characteristic
300	C₂³	#461	1	1	yes	characteristic
301	C₂³	#462	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	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	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	4	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	C₂×#95	1	1	yes	characteristic
321	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
323	C₂×C₄	#380	1	1	yes	characteristic
324	C₂×C₄	#382	1	1	yes	characteristic
325	C₂×C₄	#384	1	1	yes	characteristic
326	C₂×C₄	#389	1	1	yes	characteristic
327	C₂×C₄	#389	1	1	yes	characteristic
328	C₂×C₄	#389	1	1	yes	characteristic
329	C₂×C₄	#390	1	1	yes	characteristic
330	C₂×C₄	#391	1	1	yes	characteristic
331	C₂×C₄	#391	1	1	yes	characteristic
332	C₂×C₄	C₂×#102	1	1	yes	characteristic
333	C₂×C₄	C₂×#102	1	1	yes	characteristic
334	C₂×C₄	C₂×#103	1	1	yes	characteristic
335	C₂×C₄	C₂×#103	1	1	yes	characteristic
336	C₂×C₄	#417	1	1	yes	characteristic
337	C₂×C₄	#417	1	1	yes	characteristic
338	C₂×C₄	#418	1	1	yes	characteristic
339	C₂×C₄	#419	1	1	yes	characteristic
340	C₂×C₄	C₂×#106	1	1	yes	characteristic
341	C₂×C₄	C₂×#107	1	1	yes	characteristic
342	C₂×C₄	#428	1	1	yes	characteristic
343	C₂×C₄	#428	1	1	yes	characteristic
344	C₂×C₄	#459	1	1	yes	characteristic
345	C₂×C₄	#460	1	1	yes	characteristic
346	C₂×C₄	#461	1	1	yes	characteristic
347	C₂×C₄	#462	1	1	yes	characteristic
348	C₂×C₄	#474	1	1	yes	characteristic
349	C₂×C₄	#474	1	1	yes	characteristic
350	C₂×C₄	#475	1	1	yes	characteristic
351	C₂×C₄	#475	1	1	yes	characteristic
352	C₂×C₄	#476	1	1	yes	characteristic
353	C₂×C₄	#476	1	1	yes	characteristic
354	C₂×C₄	#477	1	1	yes	characteristic
355	C₂×C₄	#477	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	4	no	—
361	C₈	—	8	4	no	—
362	C₈	#340	1	2	yes	—
363	C₈	#340	1	2	yes	—
364	C₈	#340	1	2	yes	—
365	C₈	#340	1	2	yes	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	4	1	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	C₄⋊#95	1	1	yes	characteristic
377	C₂²	#1802	1	1	yes	characteristic
378	C₂²	#1803	1	1	yes	characteristic
379	C₂²	#1947	1	1	yes	characteristic
380	C₂²	#1948	1	1	yes	characteristic
381	C₂²	#1981	1	1	yes	characteristic
382	C₂²	#1982	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	8	4	no	—
390	C₄	—	16	4	no	—
391	C₄	C₂×#340	1	1	yes	characteristic
392	C₄	#1736	1	1	yes	characteristic
393	C₄	#1802	1	1	yes	characteristic
394	C₄	#1803	1	1	yes	characteristic
395	C₄	#1833	1	2	yes	—
396	C₄	#1947	1	1	yes	characteristic
397	C₄	#1948	1	1	yes	characteristic
398	C₄	#1976	1	2	yes	—
399	C₄	#1987	1	1	yes	characteristic
400	C₄	#1988	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	#10849	1	1	yes	characteristic
408	C₂	#10868	1	1	yes	characteristic
409	C₂	#10871	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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
14. #1846	C₄	semidirect
15. #1846	C₄	semidirect
19. #1770	C₄	semidirect
21. C₈×C₄⋊C₄	C₄	semidirect
27. C₂×#335	C₄	semidirect
28. C₂×#334	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20797	2
#20437	2
C₂×#1564	2
#8095	2
#7981	2