Subgroups — #127370

Order 512.

461 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2162	C₂	1	1	yes	maximal, characteristic
2	#20826	C₂	1	1	yes	maximal, characteristic
3	#20816	C₂	1	1	yes	maximal, characteristic
4	#7232	C₂	1	4	yes	maximal
5	#2920	—	2	1	no	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#2285	—	2	1	no	—
8	#2162	—	2	1	no	—
9	#2162	—	2	1	no	—
10	#2162	—	2	1	no	—
11	#2162	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
13	C₂×C₂²⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
15	#1848	—	2	1	no	—
16	#1844	—	2	1	no	—
17	#1837	C₄	1	2	yes	complemented
18	#1834	C₄	1	2	yes	—
19	#1810	—	2	1	no	—
20	#1810	C₂²	1	2	yes	—
21	#1807	—	2	1	no	—
22	#1807	C₂²	1	2	yes	—
23	C₄×#108	—	2	1	no	—
24	C₄×#108	C₄	1	2	yes	complemented
25	C₄×#106	—	2	1	no	—
26	C₄×#106	C₄	1	2	yes	—
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	4	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂×C₄	1	4	yes	—
32	#432	—	4	1	no	—
33	#432	—	4	1	no	—
34	#431	—	4	1	no	—
35	#431	—	4	1	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#106	—	2	2	no	—
39	C₂×#106	—	2	2	no	—
40	#423	—	2	2	no	—
41	#407	—	2	2	no	—
42	C₄×M₁₆	—	2	2	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
54	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
55	#384	—	2	2	no	—
56	#382	—	4	1	no	—
57	#380	—	2	2	no	—
58	#380	—	4	1	no	—
59	C₄×C₄⋊C₄	—	2	2	no	—
60	C₄×C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₂²⋊C₁₆	—	2	2	no	—
64	C₂²⋊C₁₆	—	2	2	no	—
65	C₄×C₁₆	—	2	2	no	—
66	C₄×C₁₆	—	2	2	no	—
67	#341	—	2	2	no	—
68	#341	C₂×C₄	1	4	yes	—
69	#340	—	2	2	no	—
70	#340	C₂×C₄	1	4	yes	—
71	C₂³×C₄	SD₁₆	1	1	yes	characteristic
72	C₂³×C₄	SD₁₆	1	1	yes	characteristic
73	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×M₁₆	—	2	4	no	—
75	C₂²×C₈	—	4	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
89	#117	—	8	4	no	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₂²⋊C₄	—	4	2	no	—
94	C₂×C₄²	—	4	2	no	—
95	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	D₈	1	1	yes	characteristic
102	C₂×C₄²	D₈	1	1	yes	characteristic
103	C₂×C₄²	SD₁₆	1	1	yes	characteristic
104	C₂×C₄²	SD₁₆	1	1	yes	characteristic
105	C₂×C₄²	Q₁₆	1	1	yes	characteristic
106	C₂×C₄²	Q₁₆	1	1	yes	characteristic
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	—	4	4	no	—
109	#108	—	4	4	no	—
110	#108	—	4	4	no	—
111	#106	—	4	4	no	—
112	#106	—	4	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₂²⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	2	no	—
117	#97	—	4	2	no	—
118	C₄×C₈	—	4	2	no	—
119	C₄×C₈	C₄²	1	1	yes	characteristic
120	C₄×C₈	C₄²	1	1	yes	characteristic
121	C₄×C₈	C₄²	1	1	yes	characteristic
122	C₄×C₈	C₄²	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
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	#95	—	4	2	no	—
136	#95	—	4	2	no	—
137	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	#102	1	1	yes	characteristic
144	C₂²×C₄	#102	1	1	yes	characteristic
145	C₂²×C₄	#102	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
147	C₂²×C₄	#103	1	1	yes	characteristic
148	C₂²×C₄	#103	1	1	yes	characteristic
149	C₂²×C₄	#103	1	1	yes	characteristic
150	C₂²×C₄	#103	1	1	yes	characteristic
151	C₂²×C₄	#106	1	1	yes	characteristic
152	C₂²×C₄	#106	1	1	yes	characteristic
153	C₂²×C₄	#106	1	1	yes	characteristic
154	C₂²×C₄	#106	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
161	M₁₆	—	4	4	no	—
162	M₁₆	—	4	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	#95	1	1	yes	characteristic
177	C₂×C₈	#95	1	1	yes	characteristic
178	C₂×C₈	#95	1	1	yes	characteristic
179	C₂×C₈	#95	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	8	4	no	—
187	C₄⋊C₄	—	8	4	no	—
188	C₂²⋊C₄	—	8	4	no	—
189	C₄²	—	8	4	no	—
190	C₄²	#95	1	1	yes	characteristic
191	C₄²	#95	1	1	yes	characteristic
192	C₄²	#95	1	1	yes	characteristic
193	C₄²	#95	1	1	yes	characteristic
194	C₄²	#102	1	1	yes	characteristic
195	C₄²	#102	1	1	yes	characteristic
196	C₄²	#102	1	1	yes	characteristic
197	C₄²	#102	1	1	yes	characteristic
198	C₄²	#103	1	1	yes	characteristic
199	C₄²	#103	1	1	yes	characteristic
200	C₄²	#103	1	1	yes	characteristic
201	C₄²	#103	1	1	yes	characteristic
202	C₄²	#107	1	1	yes	characteristic
203	C₄²	#107	1	1	yes	characteristic
204	C₄²	#107	1	1	yes	characteristic
205	C₄²	#107	1	1	yes	characteristic
206	C₁₆	—	4	4	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	4	1	no	—
210	C₂³	C₂×#102	1	1	yes	characteristic
211	C₂³	C₂×#103	1	1	yes	characteristic
212	C₂³	C₂×#106	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	#340	1	1	yes	characteristic
236	C₂×C₄	#340	1	1	yes	characteristic
237	C₂×C₄	#340	1	1	yes	characteristic
238	C₂×C₄	#340	1	1	yes	characteristic
239	C₂×C₄	#340	1	1	yes	characteristic
240	C₂×C₄	#340	1	1	yes	characteristic
241	C₂×C₄	#340	1	1	yes	characteristic
242	C₂×C₄	#340	1	1	yes	characteristic
243	C₂×C₄	#359	1	4	yes	—
244	C₂×C₄	#365	1	4	yes	—
245	C₂×C₄	C₂×#95	1	1	yes	characteristic
246	C₂×C₄	C₂×#102	1	1	yes	characteristic
247	C₂×C₄	C₂×#103	1	1	yes	characteristic
248	C₂×C₄	C₂×#107	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	8	4	no	—
256	C₈	H(3,ℤ/4)	1	2	yes	—
257	C₈	#410	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	4	4	no	—
266	C₂²	C₂×#340	1	1	yes	characteristic
267	C₂²	C₂×#359	1	2	yes	—
268	C₂²	C₂×#365	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	16	4	no	—
277	C₄	#1382	1	2	yes	—
278	C₄	#1382	1	2	yes	—
279	C₄	#1383	1	2	yes	—
280	C₄	#1383	1	2	yes	—
281	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
282	C₄	#1740	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	1	no	—
286	C₂	#7981	1	1	yes	characteristic
287	C₂	C₂×#1382	1	1	yes	characteristic
288	C₂	C₂×#1383	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. C₂×C₄⋊C₁₆	C₄	semidirect
14. C₂×C₄×C₁₆	C₄	semidirect
17. #1837	C₄	semidirect
24. C₄×#108	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2162	2
#20826	2
#20816	2
#7232	2