Subgroups — #495476

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	#20826	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#106	C₂	1	1	yes	maximal, characteristic
4	#13185	C₂	1	4	yes	maximal, complemented
5	#12929	C₂	1	4	yes	maximal
6	C₄×#365	C₂	1	4	yes	maximal, complemented
7	#2920	—	2	1	no	—
8	C₂²×#106	C₂²	1	2	yes	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄⋊C₄	—	2	1	no	—
11	#2163	C₄	1	2	yes	complemented
12	#2163	C₄	1	2	yes	complemented
13	#2162	C₄	1	2	yes	complemented
14	#2162	C₄	1	2	yes	complemented
15	C₂×#365	C₂²	1	8	yes	—
16	#2150	C₂²	1	2	yes	—
17	#2112	C₂²	1	2	yes	—
18	C₄×M₃₂	C₂²	1	2	yes	—
19	#1848	C₂²	1	2	yes	—
20	#1837	—	2	1	no	—
21	#1810	—	2	1	no	—
22	#1810	C₂²	1	2	yes	—
23	C₄×#108	—	2	1	no	—
24	C₄×#108	C₂²	1	2	yes	—
25	C₄×#106	—	2	1	no	—
26	C₄×#106	—	2	1	no	—
27	C₄×#106	C₂²	1	2	yes	—
28	C₄×#106	C₂²	1	2	yes	—
29	C₂×#340	—	2	1	no	—
30	#1383	C₂²	1	8	yes	—
31	C₂³×C₈	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄⋊C₄	—	2	2	no	—
34	C₂²×C₄²	—	4	1	no	—
35	C₂²×C₄²	D₄	1	1	yes	characteristic
36	C₂×M₃₂	C₂³	1	4	yes	—
37	#432	—	4	1	no	—
38	#432	—	4	1	no	—
39	#431	—	4	1	no	—
40	#431	—	4	1	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	C₂³	1	4	yes	—
43	C₂×#106	—	2	2	no	—
44	C₂×#106	—	2	8	no	—
45	C₂×#106	C₂³	1	4	yes	—
46	#423	—	2	2	no	—
47	#407	—	2	2	no	—
48	C₄×M₁₆	—	2	2	no	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	—	2	2	no	—
63	C₄×C₄⋊C₄	—	4	1	no	—
64	C₄×C₄⋊C₄	—	4	1	no	—
65	C₂×#95	—	4	1	no	—
66	#365	—	2	16	no	—
67	C₄⋊C₁₆	C₂×C₄	1	4	yes	complemented
68	C₄⋊C₁₆	C₂×C₄	1	4	yes	complemented
69	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
70	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
71	#346	C₂×C₄	1	4	yes	complemented
72	C₄×C₁₆	C₂×C₄	1	4	yes	complemented
73	#341	—	2	2	no	—
74	#341	—	2	2	no	—
75	#340	—	2	2	no	—
76	#340	—	2	2	no	—
77	C₂³×C₄	—	4	2	no	—
78	C₂³×C₄	SD₁₆	1	1	yes	characteristic
79	C₂³×C₄	SD₁₆	1	1	yes	characteristic
80	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×M₁₆	—	2	4	no	—
82	C₂²×C₈	—	2	4	no	—
83	C₂²×C₈	—	4	2	no	—
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂⁴	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	8	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	4	4	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₄²	—	4	2	no	—
101	C₂×C₄²	—	4	2	no	—
102	C₂×C₄²	—	8	4	no	—
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	SD₁₆	1	1	yes	characteristic
108	C₂×C₄²	SD₁₆	1	1	yes	characteristic
109	C₂×C₄²	SD₁₆	1	1	yes	characteristic
110	C₂×C₄²	SD₁₆	1	1	yes	characteristic
111	C₂×C₄²	SD₁₆	1	1	yes	characteristic
112	C₂×C₄²	SD₁₆	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	M₃₂	—	2	8	no	—
116	C₂×C₁₆	C₂²×C₄	1	8	yes	—
117	#108	—	2	8	no	—
118	#108	—	4	4	no	—
119	#106	—	2	8	no	—
120	#106	—	2	4	no	—
121	#106	#41	1	8	yes	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₂²⋊C₈	—	4	2	no	—
125	C₂²⋊C₈	—	4	2	no	—
126	#97	—	4	2	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
141	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
142	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	#95	—	4	2	no	—
145	#95	—	4	2	no	—
146	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	—	4	4	no	—
153	C₂²×C₄	—	8	4	no	—
154	C₂²×C₄	—	8	4	no	—
155	C₂²×C₄	#106	1	1	yes	characteristic
156	C₂²×C₄	#106	1	1	yes	characteristic
157	C₂²×C₄	#106	1	1	yes	characteristic
158	C₂²×C₄	#106	1	1	yes	characteristic
159	C₂²×C₄	#106	1	1	yes	characteristic
160	C₂²×C₄	#106	1	1	yes	characteristic
161	C₂²×C₄	#106	1	1	yes	characteristic
162	C₂²×C₄	#106	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
168	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
171	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
173	M₁₆	—	4	4	no	—
174	M₁₆	—	4	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₈	C₂×#41	1	4	yes	—
196	C₄⋊C₄	—	2	8	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄²	—	8	4	no	—
200	C₄²	—	8	4	no	—
201	C₄²	#106	1	1	yes	characteristic
202	C₄²	#106	1	1	yes	characteristic
203	C₄²	#106	1	1	yes	characteristic
204	C₄²	#106	1	1	yes	characteristic
205	C₄²	#106	1	1	yes	characteristic
206	C₄²	#106	1	1	yes	characteristic
207	C₄²	#106	1	1	yes	characteristic
208	C₄²	#106	1	1	yes	characteristic
209	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₄²	C₂×SD₁₆	1	1	yes	characteristic
214	C₄²	C₂×SD₁₆	1	1	yes	characteristic
215	C₄²	C₂×SD₁₆	1	1	yes	characteristic
216	C₄²	C₂×SD₁₆	1	1	yes	characteristic
217	C₁₆	—	2	8	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×#106	1	1	yes	characteristic
222	C₂³	C₂×#106	1	1	yes	characteristic
223	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	16	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	#365	1	4	yes	complemented
247	C₂×C₄	C₂×#106	1	1	yes	characteristic
248	C₂×C₄	C₂×#106	1	1	yes	characteristic
249	C₂×C₄	C₂×#106	1	1	yes	characteristic
250	C₂×C₄	C₂×#106	1	1	yes	characteristic
251	C₂×C₄	C₂×#106	1	1	yes	characteristic
252	C₂×C₄	C₂×#106	1	1	yes	characteristic
253	C₂×C₄	C₂×#106	1	1	yes	characteristic
254	C₂×C₄	C₂×#106	1	1	yes	characteristic
255	C₂×C₄	C₂×#106	1	1	yes	characteristic
256	C₂×C₄	C₂×#106	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#525	1	2	yes	—
259	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
260	C₂×C₄	#575	1	2	yes	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	8	4	no	—
268	C₈	C₄×#41	1	2	yes	—
269	C₈	#533	1	2	yes	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₂×#365	1	2	yes	—
278	C₂²	#2155	1	2	yes	—
279	C₂²	C₂²×#106	1	1	yes	characteristic
280	C₂²	#3101	1	2	yes	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	8	8	no	—
290	C₄	C₂×#365	1	2	yes	—
291	C₄	C₂×#365	1	2	yes	—
292	C₄	#2296	1	1	yes	characteristic
293	C₄	#2911	1	1	yes	characteristic
294	C₄	#3103	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#21359	1	1	yes	characteristic
299	C₂	C₂²×#365	1	1	yes	characteristic
300	C₂	C₂×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13185	C₂	semidirect
6. C₄×#365	C₂	semidirect
11. #2163	C₄	semidirect
12. #2163	C₄	semidirect
13. #2162	C₄	semidirect
14. #2162	C₄	semidirect
67. C₄⋊C₁₆	C₂×C₄	semidirect
68. C₄⋊C₁₆	C₂×C₄	semidirect
71. #346	C₂×C₄	semidirect
72. C₄×C₁₆	C₂×C₄	semidirect
246. C₂×C₄	#365	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#20826	2
C₂×C₄×#106	2
#13185	2
#12929	2
C₄×#365	2