Subgroups — #494193

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 303 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	#20818	C₂	1	1	yes	maximal, characteristic
4	#12904	C₂	1	2	yes	maximal, complemented
5	#12887	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	#2283	—	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	#1846	C₂²	1	2	yes	—
22	#1837	—	2	1	no	—
23	#1835	—	2	1	no	—
24	#1810	—	2	1	no	—
25	#1810	C₂²	1	2	yes	—
26	#1808	—	2	1	no	—
27	#1807	C₂²	1	2	yes	—
28	C₄×#108	—	2	1	no	—
29	C₄×#108	C₂²	1	2	yes	—
30	C₄×#107	—	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	C₂²×C₄²	D₄	1	1	yes	characteristic
36	C₂×M₃₂	—	2	8	no	—
37	C₂×M₃₂	C₂×C₄	1	4	yes	—
38	C₂×M₃₂	C₂³	1	2	yes	—
39	C₂×M₃₂	C₂³	1	2	yes	—
40	C₂²×C₁₆	C₂×C₄	1	4	yes	—
41	#432	—	4	1	no	—
42	#432	—	4	1	no	—
43	#431	—	4	1	no	—
44	#431	—	4	1	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	C₂³	1	4	yes	—
47	C₂×#107	—	2	2	no	—
48	C₂×#106	C₂³	1	4	yes	—
49	#423	—	2	2	no	—
50	#407	—	2	2	no	—
51	C₄×M₁₆	—	2	2	no	—
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	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	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	#383	—	4	1	no	—
65	#382	—	2	2	no	—
66	#381	—	4	1	no	—
67	#380	—	2	2	no	—
68	C₄×C₄⋊C₄	—	2	2	no	—
69	C₄×C₂²⋊C₄	—	4	1	no	—
70	#376	—	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	C₂×C₄⋊C₄	—	2	4	no	—
101	C₂×C₂²⋊C₄	—	4	2	no	—
102	C₂×C₄²	—	4	2	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	4	no	—
116	M₃₂	—	2	8	no	—
117	M₃₂	#41	1	8	yes	—
118	C₂×C₁₆	—	2	8	no	—
119	C₂×C₁₆	C₂²×C₄	1	4	yes	—
120	C₂×C₁₆	C₂²×C₄	1	4	yes	—
121	#108	—	2	8	no	—
122	#108	—	4	4	no	—
123	#107	—	4	4	no	—
124	#106	—	2	8	no	—
125	C₄⋊C₈	—	4	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₂²⋊C₈	—	4	2	no	—
128	C₂²⋊C₈	—	4	2	no	—
129	#97	—	4	2	no	—
130	C₄×C₈	—	4	2	no	—
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	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	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
146	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
147	#95	—	4	2	no	—
148	#95	—	4	2	no	—
149	#95	—	4	2	no	—
150	#95	—	4	2	no	—
151	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	4	no	—
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	#106	1	1	yes	characteristic
164	C₂²×C₄	#106	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₄⋊C₄	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	C₂×SD₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
175	M₁₆	—	4	4	no	—
176	M₁₆	—	4	4	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	—	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	—	4	4	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	#135	1	4	yes	—
195	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
197	C₂×C₈	C₂×#41	1	4	yes	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₂²⋊C₄	—	8	4	no	—
201	C₄²	—	8	4	no	—
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	#106	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×C₄⋊C₄	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	C₄²	C₂×SD₁₆	1	1	yes	characteristic
218	C₁₆	—	2	4	no	—
219	C₁₆	#117	1	8	yes	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₂×#106	1	1	yes	characteristic
224	C₂³	C₂×#106	1	1	yes	characteristic
225	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	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	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
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	C₂×#106	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#532	1	2	yes	—
260	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
261	C₂×C₄	C₂×#135	1	2	yes	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	8	4	no	—
267	C₈	#427	1	2	yes	—
268	C₈	#427	1	2	yes	—
269	C₈	C₂×#117	1	2	yes	—
270	C₈	#533	1	2	yes	—
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	—	2	1	no	—
278	C₂²	—	4	4	no	—
279	C₂²	#2155	1	2	yes	—
280	C₂²	C₂²×#106	1	1	yes	characteristic
281	C₂²	#3167	1	2	yes	—
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	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	16	4	no	—
292	C₄	#2155	1	2	yes	—
293	C₄	#2285	1	1	yes	characteristic
294	C₄	C₂×#427	1	1	yes	characteristic
295	C₄	#3167	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	4	1	no	—
299	C₂	#20816	1	1	yes	characteristic
300	C₂	C₂×#2155	1	1	yes	characteristic
301	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. #12904	C₂	semidirect
5. #12887	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

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
#20818	2
#12904	2
#12887	2
#12831	2
C₄×#365	2