Subgroups — #126825

Order 512.

540 conjugacy classes of subgroups.

The subgroup lattice has 325 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	#33418	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#7169	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	—	2	1	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
9	#2163	—	2	1	no	—
10	#2163	C₄	1	2	yes	complemented
11	#2163	C₄	1	2	yes	complemented
12	#2163	C₄	1	2	yes	complemented
13	#2162	—	2	1	no	—
14	#2162	—	2	1	no	—
15	#2162	—	2	1	no	—
16	#2162	C₄	1	2	yes	complemented
17	#2150	—	2	1	no	—
18	#2150	C₂²	1	2	yes	—
19	#2112	—	2	1	no	—
20	#2112	—	2	1	no	—
21	C₄×M₃₂	—	2	1	no	—
22	C₄×M₃₂	C₂²	1	2	yes	—
23	C₈×C₄⋊C₄	—	2	1	no	—
24	C₈×C₄⋊C₄	—	2	1	no	—
25	C₈×C₄⋊C₄	C₄	1	2	yes	—
26	C₈×C₄⋊C₄	C₂²	1	2	yes	—
27	C₂×#336	—	2	1	no	—
28	C₂³×C₈	—	2	1	no	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₄⋊C₄	—	2	1	no	—
32	C₂²×C₄²	—	2	1	no	—
33	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	C₈	1	2	yes	—
40	C₂×C₄⋊C₈	C₈	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₈	—	2	2	no	—
43	C₂×C₄×C₈	—	2	2	no	—
44	C₂×C₄×C₈	—	4	4	no	—
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	C₂×C₄	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	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
55	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₄⋊C₄	—	2	1	no	—
58	C₄×C₄⋊C₄	—	2	1	no	—
59	C₄×C₄⋊C₄	C₈	1	2	yes	—
60	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₂×#95	C₈	1	2	yes	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₄⋊C₁₆	—	2	2	no	—
65	C₄⋊C₁₆	C₂×C₄	1	4	yes	complemented
66	C₂²⋊C₁₆	—	2	2	no	—
67	C₂²⋊C₁₆	—	2	2	no	—
68	C₂²⋊C₁₆	—	2	2	no	—
69	C₂²⋊C₁₆	—	2	2	no	—
70	#346	—	2	2	no	—
71	#346	C₂×C₄	1	4	yes	complemented
72	C₄×C₁₆	—	2	2	no	—
73	C₄×C₁₆	—	2	2	no	—
74	#336	—	2	2	no	—
75	#336	—	2	2	no	—
76	C₂³×C₄	—	2	2	no	—
77	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
79	C₂³×C₄	M₁₆	1	1	yes	characteristic
80	C₂²×C₈	—	2	2	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	4	1	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²⋊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	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄²	—	2	2	no	—
105	C₂×C₄²	—	2	2	no	—
106	C₂×C₄²	—	4	2	no	—
107	C₂×C₄²	—	4	2	no	—
108	C₂×C₄²	C₄²	1	1	yes	characteristic
109	C₂×C₄²	C₄²	1	1	yes	characteristic
110	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
114	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
115	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
116	C₂×C₄²	M₁₆	1	1	yes	characteristic
117	C₂×C₄²	M₁₆	1	1	yes	characteristic
118	C₂×C₄²	M₁₆	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	M₃₂	—	4	2	no	—
121	M₃₂	—	4	2	no	—
122	M₃₂	—	4	2	no	—
123	M₃₂	—	4	2	no	—
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	—	2	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	C₂×C₈	1	4	yes	—
130	C₄×C₈	—	4	4	no	—
131	C₄×C₈	—	4	4	no	—
132	C₄×C₈	C₄²	1	1	yes	characteristic
133	C₄×C₈	C₄²	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊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	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	C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₄⋊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×D₄	1	1	yes	characteristic
147	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
148	#95	—	2	2	no	—
149	#95	C₂×C₈	1	4	yes	—
150	C₂⁴	C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	—	4	4	no	—
161	C₂²×C₄	#95	1	1	yes	characteristic
162	C₂²×C₄	#95	1	1	yes	characteristic
163	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
164	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
165	C₂²×C₄	#97	1	1	yes	characteristic
166	C₂²×C₄	#97	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	#95	1	1	yes	characteristic
199	C₂×C₈	#95	1	1	yes	characteristic
200	C₂×C₈	#95	1	1	yes	characteristic
201	C₂×C₈	#95	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₈	#137	1	2	yes	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄²	—	4	4	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	#95	1	1	yes	characteristic
214	C₄²	#95	1	1	yes	characteristic
215	C₄²	C₄×C₈	1	1	yes	characteristic
216	C₄²	C₄×C₈	1	1	yes	characteristic
217	C₄²	#97	1	1	yes	characteristic
218	C₄²	#97	1	1	yes	characteristic
219	C₄²	C₂²⋊C₈	1	1	yes	characteristic
220	C₄²	C₂²⋊C₈	1	1	yes	characteristic
221	C₄²	C₄⋊C₈	1	1	yes	characteristic
222	C₄²	C₄⋊C₈	1	1	yes	characteristic
223	C₄²	C₄⋊C₈	1	1	yes	characteristic
224	C₄²	C₄⋊C₈	1	1	yes	characteristic
225	C₄²	C₂×C₄²	1	1	yes	characteristic
226	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
227	C₄²	C₂²×C₈	1	1	yes	characteristic
228	C₄²	C₂×M₁₆	1	1	yes	characteristic
229	C₁₆	—	4	4	no	—
230	C₁₆	—	4	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	#336	1	1	yes	characteristic
236	C₂³	#336	1	1	yes	characteristic
237	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	#336	1	1	yes	characteristic
264	C₂×C₄	#336	1	1	yes	characteristic
265	C₂×C₄	#336	1	1	yes	characteristic
266	C₂×C₄	#336	1	1	yes	characteristic
267	C₂×C₄	#336	1	1	yes	characteristic
268	C₂×C₄	#336	1	1	yes	characteristic
269	C₂×C₄	#364	1	4	yes	—
270	C₂×C₄	C₂×#95	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#97	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
274	C₂×C₄	#418	1	1	yes	characteristic
275	C₂×C₄	#419	1	1	yes	characteristic
276	C₂×C₄	#421	1	1	yes	characteristic
277	C₂×C₄	#421	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	4	4	no	—
287	C₈	#417	1	2	yes	—
288	C₈	#428	1	2	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	#1376	1	2	yes	—
298	C₂²	#1378	1	2	yes	—
299	C₂²	#1478	1	1	yes	characteristic
300	C₂²	#1479	1	1	yes	characteristic
301	C₂²	C₂×#336	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	8	no	—
311	C₄	#1376	1	2	yes	—
312	C₄	#1478	1	1	yes	characteristic
313	C₄	#1479	1	1	yes	characteristic
314	C₄	#1733	1	1	yes	characteristic
315	C₄	#1736	1	1	yes	characteristic
316	C₄	C₂×#364	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	#7892	1	1	yes	characteristic
322	C₂	C₂×#1376	1	1	yes	characteristic
323	C₂	C₂×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #7169	C₂	semidirect
10. #2163	C₄	semidirect
11. #2163	C₄	semidirect
12. #2163	C₄	semidirect
16. #2162	C₄	semidirect
65. C₄⋊C₁₆	C₂×C₄	semidirect
71. #346	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂×C₈×C₄⋊C₄	2
#7169	2