Subgroups — #109174

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	#7924	C₂	1	2	yes	maximal
5	#7866	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#427	C₄	1	2	yes	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×#117	C₂²	1	2	yes	—
12	C₄×#107	—	2	1	no	—
13	C₄×#107	—	2	1	no	—
14	C₄×#106	—	2	1	no	—
15	C₄×#106	—	2	1	no	—
16	#1735	C₄	1	2	yes	—
17	#1564	—	2	1	no	—
18	#1564	—	2	1	no	—
19	#1564	—	2	1	no	—
20	#1564	—	2	1	no	—
21	C₂×#335	C₄	1	2	yes	complemented
22	C₂×#334	C₄	1	2	yes	complemented
23	#1450	—	2	1	no	—
24	#1450	—	2	1	no	—
25	#1450	C₄	1	2	yes	complemented
26	#1450	C₄	1	2	yes	complemented
27	C₂×#322	—	2	1	no	—
28	C₂×C₈²	—	2	1	no	—
29	#1282	—	2	4	no	—
30	#1276	—	2	4	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×#117	C₂×C₄	1	4	yes	—
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	2	no	—
40	#431	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	—	2	2	no	—
43	#431	C₂×C₄	1	2	yes	—
44	#431	C₂×C₄	1	2	yes	—
45	#431	D₄	1	2	yes	—
46	#431	Q₈	1	2	yes	—
47	#427	—	2	4	no	—
48	#427	C₈	1	8	yes	complemented
49	C₂×#107	—	2	2	no	—
50	C₂×#106	—	2	2	no	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₈	Q₈	1	2	yes	—
54	C₂×C₂²⋊C₈	—	2	1	no	—
55	C₂×C₂²⋊C₈	—	2	2	no	—
56	C₂×C₂²⋊C₈	D₄	1	2	yes	—
57	C₂×C₄×C₈	—	2	1	no	—
58	C₂×C₄×C₈	—	2	1	no	—
59	C₂×C₄×C₈	—	2	2	no	—
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	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	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	2	2	no	—
73	C₄×C₄⋊C₄	—	2	2	no	—
74	C₄×C₂²⋊C₄	—	2	2	no	—
75	C₄×C₂²⋊C₄	—	2	2	no	—
76	#376	—	2	2	no	—
77	C₄³	—	2	2	no	—
78	#340	—	2	2	no	—
79	#340	—	2	2	no	—
80	#335	—	2	2	no	—
81	#335	—	2	2	no	—
82	#334	—	2	2	no	—
83	#334	—	2	2	no	—
84	#322	—	2	2	no	—
85	#322	—	2	2	no	—
86	C₈²	—	2	2	no	—
87	C₈²	—	2	2	no	—
88	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	2	4	no	—
96	C₂²×C₈	C₄²	1	1	yes	characteristic
97	C₂²×C₈	C₄²	1	1	yes	characteristic
98	C₂²×C₈	C₄²	1	1	yes	characteristic
99	C₂²×C₈	C₄²	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₈	1	4	yes	—
109	C₂²×C₈	M₁₆	1	4	yes	—
110	#117	—	2	4	no	—
111	#117	—	2	4	no	—
112	#117	—	2	4	no	—
113	#117	C₂×C₈	1	8	yes	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₂²⋊C₄	—	2	4	no	—
116	C₂×C₄²	—	2	4	no	—
117	C₂×C₄²	—	4	2	no	—
118	C₂×C₄²	—	4	2	no	—
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
126	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
127	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₄²	#41	1	1	yes	characteristic
129	C₂×C₄²	#41	1	1	yes	characteristic
130	C₂×C₄²	#41	1	1	yes	characteristic
131	#107	—	2	8	no	—
132	#106	—	2	8	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	4	2	no	—
144	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
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	—	4	2	no	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
167	#95	—	4	2	no	—
168	#95	—	4	2	no	—
169	C₂⁴	C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	#117	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
183	C₂²×C₄	#121	1	1	yes	characteristic
184	C₂²×C₄	#121	1	1	yes	characteristic
185	C₂²×C₄	#121	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₄	#124	1	1	yes	characteristic
188	C₂²×C₄	#125	1	1	yes	characteristic
189	C₂²×C₄	#127	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₈	1	2	yes	—
192	C₂²×C₄	C₂×M₁₆	1	2	yes	—
193	C₂²×C₄	#131	1	2	yes	—
194	C₂²×C₄	#131	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	8	4	no	—
209	C₂×C₈	#95	1	1	yes	characteristic
210	C₂×C₈	#95	1	1	yes	characteristic
211	C₂×C₈	#95	1	1	yes	characteristic
212	C₂×C₈	#95	1	1	yes	characteristic
213	C₂×C₈	#95	1	1	yes	characteristic
214	C₂×C₈	#95	1	1	yes	characteristic
215	C₂×C₈	#95	1	1	yes	characteristic
216	C₂×C₈	#95	1	1	yes	characteristic
217	C₂×C₈	C₄×C₈	1	2	yes	—
218	C₂×C₈	C₄×C₈	1	2	yes	—
219	C₂×C₈	#97	1	2	yes	—
220	C₂×C₈	#97	1	2	yes	—
221	C₂×C₈	C₂²⋊C₈	1	4	yes	—
222	C₂×C₈	C₂²⋊C₈	1	4	yes	—
223	C₂×C₈	C₄⋊C₈	1	4	yes	—
224	C₂×C₈	C₄⋊C₈	1	4	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	4	4	no	—
231	C₄²	—	8	4	no	—
232	C₄²	—	8	4	no	—
233	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
238	C₄²	#117	1	1	yes	characteristic
239	C₄²	C₄×D₄	1	1	yes	characteristic
240	C₄²	C₄×Q₈	1	1	yes	characteristic
241	C₄²	#121	1	1	yes	characteristic
242	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
243	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
244	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
245	C₄²	#124	1	1	yes	characteristic
246	C₄²	#125	1	1	yes	characteristic
247	C₄²	C₄⋊Q₈	1	1	yes	characteristic
248	C₄²	C₄⋊Q₈	1	1	yes	characteristic
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
252	C₂³	#389	1	1	yes	characteristic
253	C₂³	#390	1	1	yes	characteristic
254	C₂³	C₈×D₄	1	2	yes	—
255	C₂³	#436	1	2	yes	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₂×#95	1	1	yes	characteristic
272	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	#380	1	1	yes	characteristic
274	C₂×C₄	#380	1	1	yes	characteristic
275	C₂×C₄	#382	1	1	yes	characteristic
276	C₂×C₄	#384	1	1	yes	characteristic
277	C₂×C₄	#389	1	1	yes	characteristic
278	C₂×C₄	#389	1	1	yes	characteristic
279	C₂×C₄	#389	1	1	yes	characteristic
280	C₂×C₄	#390	1	1	yes	characteristic
281	C₂×C₄	#391	1	1	yes	characteristic
282	C₂×C₄	#391	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
284	C₂×C₄	C₂×#97	1	1	yes	characteristic
285	C₂×C₄	#405	1	1	yes	characteristic
286	C₂×C₄	#405	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
288	C₂×C₄	#408	1	2	yes	—
289	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
290	C₂×C₄	#424	1	2	yes	—
291	C₂×C₄	C₈×D₄	1	2	yes	—
292	C₂×C₄	#436	1	2	yes	—
293	C₂×C₄	C₈×Q₈	1	2	yes	—
294	C₂×C₄	C₈×Q₈	1	2	yes	—
295	C₂×C₄	#446	1	2	yes	—
296	C₂×C₄	#446	1	2	yes	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	4	no	—
302	C₈	—	4	4	no	—
303	C₈	—	8	4	no	—
304	C₈	#336	1	2	yes	—
305	C₈	#336	1	2	yes	—
306	C₈	#336	1	2	yes	—
307	C₈	#336	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₄⋊#95	1	1	yes	characteristic
317	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
318	C₂²	#1770	1	1	yes	characteristic
319	C₂²	#1939	1	2	yes	—
320	C₂²	#1954	1	2	yes	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	8	8	no	—
328	C₄	C₂×#336	1	1	yes	characteristic
329	C₄	#1729	1	1	yes	characteristic
330	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
331	C₄	#1770	1	1	yes	characteristic
332	C₄	#1814	1	2	yes	—
333	C₄	#1939	1	2	yes	—
334	C₄	#1976	1	2	yes	—
335	C₄	#1984	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	#10834	1	1	yes	characteristic
341	C₂	#10871	1	1	yes	characteristic
342	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
21. C₂×#335	C₄	semidirect
22. C₂×#334	C₄	semidirect
25. #1450	C₄	semidirect
26. #1450	C₄	semidirect
48. #427	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#427	2
C₂×#1564	2
C₂×#1450	2
#7924	2
#7866	2