Subgroups — #494192

Order 512.

585 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#20826	C₂	1	1	yes	maximal, characteristic
3	#20816	C₂	1	1	yes	maximal, characteristic
4	#12903	C₂	1	2	yes	maximal, complemented
5	#12886	C₂	1	2	yes	maximal, complemented
6	#12831	C₂	1	4	yes	maximal
7	C₄×#365	C₂	1	4	yes	maximal, complemented
8	C₂²×M₃₂	C₂²	1	2	yes	—
9	#2920	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2285	—	2	1	no	—
12	C₂×#365	C₂²	1	8	yes	—
13	C₄×M₃₂	C₄	1	2	yes	complemented
14	C₄×M₃₂	C₄	1	2	yes	complemented
15	C₄×M₃₂	C₂²	1	1	yes	characteristic
16	C₄×M₃₂	C₂²	1	1	yes	characteristic
17	C₄×M₃₂	C₂²	1	2	yes	—
18	C₂×#346	C₄	1	2	yes	complemented
19	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
20	#1848	C₂²	1	2	yes	—
21	#1844	C₂²	1	2	yes	—
22	#1837	—	2	1	no	—
23	#1834	—	2	1	no	—
24	#1810	—	2	1	no	—
25	#1810	C₂²	1	2	yes	—
26	#1807	—	2	1	no	—
27	#1807	C₂²	1	2	yes	—
28	C₄×#108	—	2	1	no	—
29	C₄×#108	C₂²	1	2	yes	—
30	C₄×#106	—	2	1	no	—
31	C₄×#106	C₂²	1	2	yes	—
32	#1383	C₂²	1	8	yes	—
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	—	4	1	no	—
36	C₂²×C₄²	D₄	1	1	yes	characteristic
37	C₂×M₃₂	—	2	8	no	—
38	C₂×M₃₂	C₂×C₄	1	4	yes	—
39	C₂×M₃₂	C₂³	1	2	yes	—
40	C₂×M₃₂	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂×C₄	1	4	yes	—
42	#432	—	4	1	no	—
43	#432	—	4	1	no	—
44	#431	—	4	1	no	—
45	#431	—	4	1	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	C₂³	1	4	yes	—
48	C₂×#106	—	2	2	no	—
49	C₂×#106	C₂³	1	4	yes	—
50	#423	—	2	2	no	—
51	#407	—	2	2	no	—
52	C₄×M₁₆	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#384	—	2	2	no	—
66	#382	—	4	1	no	—
67	#380	—	2	2	no	—
68	#380	—	4	1	no	—
69	C₄×C₄⋊C₄	—	2	2	no	—
70	C₄×C₄⋊C₄	—	4	1	no	—
71	#365	—	2	16	no	—
72	#346	C₂×C₄	1	2	yes	—
73	#346	C₂×C₄	1	4	yes	—
74	#346	C₂×C₄	1	2	yes	—
75	C₄×C₁₆	C₂×C₄	1	2	yes	—
76	C₄×C₁₆	C₂×C₄	1	4	yes	—
77	C₄×C₁₆	C₂×C₄	1	2	yes	—
78	#341	—	2	2	no	—
79	#341	—	2	2	no	—
80	#340	—	2	2	no	—
81	#340	—	2	2	no	—
82	C₂³×C₄	SD₁₆	1	1	yes	characteristic
83	C₂³×C₄	SD₁₆	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×M₁₆	—	2	4	no	—
86	C₂²×C₈	—	2	4	no	—
87	C₂²×C₈	—	2	4	no	—
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂⁴	1	1	yes	characteristic
100	#117	—	8	4	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₂²⋊C₄	—	4	2	no	—
105	C₂×C₄²	—	4	2	no	—
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄²	C₄⋊C₄	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	SD₁₆	1	1	yes	characteristic
114	C₂×C₄²	SD₁₆	1	1	yes	characteristic
115	C₂×C₄²	SD₁₆	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
118	M₃₂	—	2	4	no	—
119	M₃₂	—	2	8	no	—
120	M₃₂	#41	1	8	yes	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	C₂²×C₄	1	4	yes	—
123	C₂×C₁₆	C₂²×C₄	1	4	yes	—
124	#108	—	2	8	no	—
125	#108	—	4	4	no	—
126	#106	—	2	8	no	—
127	#106	—	4	4	no	—
128	C₄⋊C₈	—	4	2	no	—
129	C₄⋊C₈	—	4	2	no	—
130	C₂²⋊C₈	—	4	2	no	—
131	C₂²⋊C₈	—	4	2	no	—
132	#97	—	4	2	no	—
133	C₄×C₈	—	4	2	no	—
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
147	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
148	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
149	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
150	#95	—	4	2	no	—
151	#95	—	4	2	no	—
152	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	4	4	no	—
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	#106	1	1	yes	characteristic
164	C₂²×C₄	#106	1	1	yes	characteristic
165	C₂²×C₄	#106	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄⋊C₄	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	C₂×SD₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
176	M₁₆	—	4	4	no	—
177	M₁₆	—	4	4	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	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	4	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	#135	1	4	yes	—
196	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	4	yes	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	8	4	no	—
202	C₄⋊C₄	—	8	4	no	—
203	C₂²⋊C₄	—	8	4	no	—
204	C₄²	—	8	4	no	—
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	#106	1	1	yes	characteristic
210	C₄²	#106	1	1	yes	characteristic
211	C₄²	#106	1	1	yes	characteristic
212	C₄²	#106	1	1	yes	characteristic
213	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₄²	C₂×SD₁₆	1	1	yes	characteristic
218	C₄²	C₂×SD₁₆	1	1	yes	characteristic
219	C₄²	C₂×SD₁₆	1	1	yes	characteristic
220	C₄²	C₂×SD₁₆	1	1	yes	characteristic
221	C₁₆	—	2	4	no	—
222	C₁₆	#117	1	8	yes	complemented
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	C₂×#106	1	1	yes	characteristic
227	C₂³	C₂×#106	1	1	yes	characteristic
228	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	#365	1	4	yes	complemented
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	C₂×#106	1	1	yes	characteristic
258	C₂×C₄	C₂×#106	1	1	yes	characteristic
259	C₂×C₄	C₂×#106	1	1	yes	characteristic
260	C₂×C₄	C₂×#106	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
262	C₂×C₄	#532	1	2	yes	—
263	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
264	C₂×C₄	C₂×#135	1	2	yes	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	8	4	no	—
270	C₈	#427	1	2	yes	—
271	C₈	#427	1	2	yes	—
272	C₈	C₂×#117	1	2	yes	—
273	C₈	#533	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	4	4	no	—
282	C₂²	C₂×#365	1	2	yes	—
283	C₂²	C₂²×#106	1	1	yes	characteristic
284	C₂²	#3167	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	16	4	no	—
293	C₄	C₂×#365	1	2	yes	—
294	C₄	C₂×#365	1	2	yes	—
295	C₄	#2155	1	2	yes	—
296	C₄	#2285	1	1	yes	characteristic
297	C₄	C₂×#427	1	1	yes	characteristic
298	C₄	#3167	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	4	1	no	—
302	C₂	#20816	1	1	yes	characteristic
303	C₂	C₂²×#365	1	1	yes	characteristic
304	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. #12903	C₂	semidirect
5. #12886	C₂	semidirect
7. C₄×#365	C₂	semidirect
13. C₄×M₃₂	C₄	semidirect
14. C₄×M₃₂	C₄	semidirect
18. C₂×#346	C₄	semidirect
19. C₂×C₄×C₁₆	C₄	semidirect
222. C₁₆	#117	semidirect
250. 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
C₂×C₄×M₃₂	2
#20826	2
#20816	2
#12903	2
#12886	2
#12831	2
C₄×#365	2