Subgroups — #127967

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 311 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	#20817	C₂	1	1	yes	maximal, characteristic
3	C₂×#1567	C₂	1	1	yes	maximal, characteristic
4	#7296	C₂	1	4	yes	maximal, complemented
5	C₂×#423	—	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	C₄	1	2	yes	complemented
11	#2162	C₂²	1	2	yes	—
12	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
13	C₂×C₂²⋊C₁₆	—	2	1	no	—
14	C₂×C₄×C₁₆	—	2	1	no	—
15	#1845	—	2	1	no	—
16	#1834	C₄	1	2	yes	—
17	#1808	—	2	1	no	—
18	#1808	C₄	1	2	yes	—
19	C₄×#107	—	2	1	no	—
20	C₄×#107	C₂²	1	2	yes	—
21	#1567	—	2	1	no	—
22	#1567	—	2	1	no	—
23	#1567	C₄	1	2	yes	complemented
24	#1567	C₂²	1	2	yes	—
25	C₂×#335	—	2	1	no	—
26	C₂×#335	C₄	1	2	yes	complemented
27	C₂²×M₁₆	—	4	1	no	—
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂×#117	—	4	1	no	—
31	C₂²×C₄²	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	#423	—	2	2	no	—
36	#423	—	2	2	no	—
37	#423	—	4	1	no	—
38	#423	—	4	1	no	—
39	C₂×C₄⋊C₈	—	2	2	no	—
40	C₂×C₄⋊C₈	—	4	1	no	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#384	—	2	2	no	—
54	#382	—	4	1	no	—
55	#380	—	2	2	no	—
56	#380	—	4	1	no	—
57	C₄×C₄⋊C₄	—	2	2	no	—
58	C₄×C₄⋊C₄	—	4	1	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
61	C₂²⋊C₁₆	—	2	2	no	—
62	C₂²⋊C₁₆	—	2	2	no	—
63	C₄×C₁₆	—	2	2	no	—
64	C₄×C₁₆	—	2	2	no	—
65	#340	—	2	2	no	—
66	#340	C₂×C₄	1	4	yes	—
67	#335	—	2	2	no	—
68	#335	—	2	2	no	—
69	#335	—	2	2	no	—
70	#335	C₂×C₄	1	4	yes	—
71	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₄	D₈	1	1	yes	characteristic
73	C₂³×C₄	SD₁₆	1	1	yes	characteristic
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	8	2	no	—
77	C₂×M₁₆	—	8	2	no	—
78	C₂²×C₈	—	4	2	no	—
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	—	4	2	no	—
81	C₂²×C₈	C₄²	1	1	yes	characteristic
82	C₂²×C₈	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	#117	—	8	2	no	—
94	#117	—	8	2	no	—
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₂²⋊C₄	—	4	2	no	—
99	C₂×C₄²	—	4	2	no	—
100	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	D₈	1	1	yes	characteristic
106	C₂×C₄²	SD₁₆	1	1	yes	characteristic
107	C₂×C₄²	SD₁₆	1	1	yes	characteristic
108	C₂×C₄²	SD₁₆	1	1	yes	characteristic
109	C₂×C₄²	Q₁₆	1	1	yes	characteristic
110	C₂×C₄²	Q₁₆	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	4	4	no	—
114	#107	—	4	4	no	—
115	#107	—	4	4	no	—
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	4	2	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₄⋊C₈	—	4	2	no	—
120	C₄×C₈	C₄²	1	1	yes	characteristic
121	C₄×C₈	C₄²	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊C₄	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	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
136	#95	—	4	2	no	—
137	#95	—	4	2	no	—
138	C₂⁴	#102	1	1	yes	characteristic
139	C₂²×C₄	—	4	1	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	#95	1	1	yes	characteristic
144	C₂²×C₄	#95	1	1	yes	characteristic
145	C₂²×C₄	#102	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
147	C₂²×C₄	#102	1	1	yes	characteristic
148	C₂²×C₄	#102	1	1	yes	characteristic
149	C₂²×C₄	#102	1	1	yes	characteristic
150	C₂²×C₄	#103	1	1	yes	characteristic
151	C₂²×C₄	#103	1	1	yes	characteristic
152	C₂²×C₄	#103	1	1	yes	characteristic
153	C₂²×C₄	#103	1	1	yes	characteristic
154	C₂²×C₄	#106	1	1	yes	characteristic
155	C₂²×C₄	#106	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	#107	1	1	yes	characteristic
158	C₂²×C₄	D₁₆	1	2	yes	—
159	C₂²×C₄	SD₃₂	1	2	yes	—
160	C₂²×C₄	SD₃₂	1	2	yes	—
161	C₂²×C₄	Q₃₂	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
165	M₁₆	—	8	4	no	—
166	M₁₆	—	8	2	no	—
167	M₁₆	—	8	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	8	4	no	—
182	C₂×C₈	#95	1	1	yes	characteristic
183	C₂×C₈	#95	1	1	yes	characteristic
184	C₂×C₈	#95	1	1	yes	characteristic
185	C₂×C₈	#95	1	1	yes	characteristic
186	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊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	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	8	4	no	—
193	C₄⋊C₄	—	8	4	no	—
194	C₂²⋊C₄	—	8	4	no	—
195	C₄²	—	8	2	no	—
196	C₄²	—	8	2	no	—
197	C₄²	#95	1	1	yes	characteristic
198	C₄²	#95	1	1	yes	characteristic
199	C₄²	#102	1	1	yes	characteristic
200	C₄²	#102	1	1	yes	characteristic
201	C₄²	#103	1	1	yes	characteristic
202	C₄²	#103	1	1	yes	characteristic
203	C₄²	#103	1	1	yes	characteristic
204	C₄²	#103	1	1	yes	characteristic
205	C₄²	#106	1	1	yes	characteristic
206	C₄²	#106	1	1	yes	characteristic
207	C₄²	#107	1	1	yes	characteristic
208	C₄²	#107	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₄²	C₂×SD₁₆	1	1	yes	characteristic
212	C₄²	C₂×Q₁₆	1	1	yes	characteristic
213	C₁₆	—	4	4	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	1	no	—
216	C₂³	#340	1	1	yes	characteristic
217	C₂³	#340	1	1	yes	characteristic
218	C₂³	#357	1	2	yes	—
219	C₂³	#358	1	2	yes	—
220	C₂³	C₂×#102	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	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	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	4	no	—
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	#340	1	1	yes	characteristic
244	C₂×C₄	#340	1	1	yes	characteristic
245	C₂×C₄	#357	1	2	yes	—
246	C₂×C₄	#357	1	2	yes	—
247	C₂×C₄	#357	1	2	yes	—
248	C₂×C₄	#358	1	2	yes	—
249	C₂×C₄	#358	1	2	yes	—
250	C₂×C₄	#358	1	2	yes	—
251	C₂×C₄	#366	1	2	yes	—
252	C₂×C₄	#366	1	2	yes	—
253	C₂×C₄	#367	1	2	yes	—
254	C₂×C₄	#367	1	2	yes	—
255	C₂×C₄	C₂×#95	1	1	yes	characteristic
256	C₂×C₄	C₂×#102	1	1	yes	characteristic
257	C₂×C₄	C₂×#103	1	1	yes	characteristic
258	C₂×C₄	C₂×#103	1	1	yes	characteristic
259	C₂×C₄	C₂×#106	1	1	yes	characteristic
260	C₂×C₄	C₂×#107	1	1	yes	characteristic
261	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
262	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
263	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	8	4	no	—
272	C₈	#344	1	2	yes	—
273	C₈	#421	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	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#1380	1	2	yes	—
283	C₂²	#1380	1	2	yes	—
284	C₂²	C₂×#340	1	1	yes	characteristic
285	C₂²	C₂×#357	1	1	yes	characteristic
286	C₂²	C₂×#358	1	1	yes	characteristic
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	—	2	1	no	—
293	C₄	—	16	4	no	—
294	C₄	#1380	1	2	yes	—
295	C₄	#1380	1	2	yes	—
296	C₄	#1389	1	2	yes	—
297	C₄	#1733	1	1	yes	characteristic
298	C₄	C₂×#344	1	1	yes	characteristic
299	C₄	C₂×#357	1	1	yes	characteristic
300	C₄	C₂×#358	1	1	yes	characteristic
301	C₄	C₂×#366	1	1	yes	characteristic
302	C₄	C₂×#367	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	#8095	1	1	yes	characteristic
308	C₂	C₂×#1380	1	1	yes	characteristic
309	C₂	C₂×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7296	C₂	semidirect
10. #2162	C₄	semidirect
12. C₂×C₄⋊C₁₆	C₄	semidirect
23. #1567	C₄	semidirect
26. C₂×#335	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
#20817	2
C₂×#1567	2
#7296	2