Subgroups — #105477

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20476	C₂	1	1	yes	maximal, characteristic
2	#17323	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#8124	C₂	1	1	yes	maximal, characteristic
5	#8118	C₂	1	1	yes	maximal, characteristic
6	#8113	C₂	1	1	yes	maximal, characteristic
7	#8106	C₂	1	1	yes	maximal, characteristic
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	#2920	C₂²	1	1	yes	characteristic
12	C₂×#432	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#2283	C₂²	1	1	yes	characteristic
15	#1770	—	2	1	no	—
16	C₈×C₄⋊C₄	—	2	1	no	—
17	#1752	C₄	1	2	yes	—
18	C₈×C₂²⋊C₄	C₄	1	2	yes	—
19	#1746	—	2	1	no	—
20	#1726	—	2	1	no	—
21	#1452	—	2	1	no	—
22	#1452	C₄	1	2	yes	complemented
23	#1451	—	2	1	no	—
24	#1451	—	2	1	no	—
25	#1451	—	2	1	no	—
26	#1451	C₄	1	2	yes	complemented
27	#1450	—	2	1	no	—
28	#1450	C₄	1	2	yes	complemented
29	C₈×M₁₆	—	2	1	no	—
30	C₈×M₁₆	—	2	1	no	—
31	C₈×M₁₆	—	2	1	no	—
32	C₈×M₁₆	C₄	1	2	yes	complemented
33	#1308	C₄	1	4	yes	complemented
34	#1307	C₄	1	4	yes	complemented
35	#1306	C₄	1	4	yes	—
36	#1305	C₄	1	4	yes	—
37	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄²	C₂³	1	1	yes	characteristic
40	#432	—	2	1	no	—
41	#432	—	2	1	no	—
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	#432	—	2	1	no	—
45	#432	—	2	1	no	—
46	#432	—	2	1	no	—
47	#432	C₂×C₄	1	2	yes	—
48	#432	C₂×C₄	1	2	yes	—
49	#432	C₂×C₄	1	2	yes	—
50	#432	C₂×C₄	1	2	yes	—
51	#432	C₂×C₄	1	2	yes	—
52	#431	—	2	1	no	—
53	#431	—	2	1	no	—
54	#431	—	2	1	no	—
55	#431	—	2	1	no	—
56	#431	—	2	1	no	—
57	#431	C₂×C₄	1	2	yes	—
58	#431	C₂×C₄	1	2	yes	—
59	#431	C₂×C₄	1	2	yes	—
60	#423	—	2	1	no	—
61	#423	—	2	1	no	—
62	#423	—	2	1	no	—
63	#423	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	#407	C₂×C₄	1	2	yes	—
66	#407	C₂×C₄	1	2	yes	—
67	#407	C₂×C₄	1	2	yes	—
68	#407	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₄×M₁₆	—	2	1	no	—
71	C₄×M₁₆	—	2	1	no	—
72	C₄×M₁₆	—	2	1	no	—
73	C₄×M₁₆	C₂×C₄	1	2	yes	—
74	C₂×#97	—	2	1	no	—
75	C₂×#97	—	2	2	no	—
76	C₂×C₄×C₈	—	2	2	no	—
77	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈	D₄	1	1	yes	characteristic
82	C₂×C₄×C₈	D₄	1	1	yes	characteristic
83	C₂×C₄×C₈	D₄	1	1	yes	characteristic
84	C₂×C₄×C₈	D₄	1	1	yes	characteristic
85	C₂×C₄×C₈	D₄	1	1	yes	characteristic
86	C₂×C₄×C₈	D₄	1	1	yes	characteristic
87	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
88	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
89	#383	—	2	1	no	—
90	#382	—	2	1	no	—
91	#381	C₂×C₄	1	2	yes	—
92	#380	C₂×C₄	1	2	yes	—
93	C₄×C₄⋊C₄	—	2	1	no	—
94	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
95	#376	—	2	1	no	—
96	#336	—	2	2	no	—
97	#336	—	2	2	no	—
98	#322	—	2	2	no	—
99	#322	—	2	2	no	—
100	#322	—	2	2	no	—
101	#322	—	2	2	no	—
102	#322	—	2	2	no	—
103	#322	—	2	2	no	—
104	C₈²	—	2	2	no	—
105	C₈²	—	2	2	no	—
106	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	C₄²	1	4	yes	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	4	2	no	—
115	C₂²×C₈	—	4	2	no	—
116	C₂²×C₈	C₄²	1	1	yes	characteristic
117	C₂²×C₈	C₄²	1	1	yes	characteristic
118	C₂²×C₈	C₄²	1	1	yes	characteristic
119	C₂²×C₈	C₄²	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	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₂²⋊C₄	C₄²	1	4	yes	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₂×C₄²	#41	1	1	yes	characteristic
141	C₂×C₄²	#41	1	1	yes	characteristic
142	C₂×C₄²	#41	1	1	yes	characteristic
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	2	2	no	—
158	C₂²⋊C₈	—	2	2	no	—
159	C₂²⋊C₈	C₄²	1	4	yes	—
160	C₂²⋊C₈	C₄²	1	4	yes	—
161	C₂²⋊C₈	C₄²	1	4	yes	—
162	C₂²⋊C₈	C₄²	1	4	yes	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	#97	—	2	2	no	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	#97	—	2	2	no	—
169	#97	—	4	4	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	4	4	no	—
175	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
183	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
190	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	C₂⁴	C₂×C₄²	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	#117	1	1	yes	characteristic
210	C₂²×C₄	#117	1	1	yes	characteristic
211	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
218	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
219	M₁₆	—	4	4	no	—
220	M₁₆	—	4	4	no	—
221	M₁₆	—	4	4	no	—
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
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	#95	1	1	yes	characteristic
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₄²	#117	1	1	yes	characteristic
259	C₄²	#117	1	1	yes	characteristic
260	C₄²	#117	1	1	yes	characteristic
261	C₄²	#117	1	1	yes	characteristic
262	C₄²	C₄×D₄	1	1	yes	characteristic
263	C₄²	C₄×D₄	1	1	yes	characteristic
264	C₄²	C₄×D₄	1	1	yes	characteristic
265	C₄²	C₄×D₄	1	1	yes	characteristic
266	C₄²	C₄×D₄	1	1	yes	characteristic
267	C₄²	C₄×D₄	1	1	yes	characteristic
268	C₄²	C₄×Q₈	1	1	yes	characteristic
269	C₄²	C₄×Q₈	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	C₄³	1	1	yes	characteristic
274	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
275	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	C₂×#95	1	1	yes	characteristic
301	C₂×C₄	#376	1	1	yes	characteristic
302	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	4	no	—
319	C₈	—	4	4	no	—
320	C₈	—	4	4	no	—
321	C₈	—	4	4	no	—
322	C₈	—	4	4	no	—
323	C₈	#344	1	2	yes	—
324	C₈	#344	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	C₄×#95	1	1	yes	characteristic
335	C₂²	#1757	1	2	yes	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	4	no	—
347	C₄	C₂×#344	1	1	yes	characteristic
348	C₄	C₂×#344	1	1	yes	characteristic
349	C₄	#1757	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	1	no	—
352	C₂	—	4	1	no	—
353	C₂	C₄×#344	1	1	yes	characteristic
354	C₂	#10711	1	1	yes	characteristic
355	C₂	#10711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
22. #1452	C₄	semidirect
26. #1451	C₄	semidirect
28. #1450	C₄	semidirect
32. C₈×M₁₆	C₄	semidirect
33. #1308	C₄	semidirect
34. #1307	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20476	2
#17323	2
#8124	2
#8118	2
#8113	2
#8106	2