Subgroups — #493201

Order 512.

633 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	#33418	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#20473	C₂	1	1	yes	maximal, characteristic
4	#12666	C₂	1	4	yes	maximal, complemented
5	#12616	C₂	1	4	yes	maximal
6	#12594	C₂	1	2	yes	maximal, complemented
7	C₄×#347	C₂	1	2	yes	maximal, complemented
8	C₂²×M₃₂	C₂²	1	2	yes	—
9	C₂×#431	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×#117	C₄	1	1	yes	characteristic
12	#2163	C₄	1	2	yes	complemented
13	#2163	C₄	1	2	yes	complemented
14	#2162	C₄	1	2	yes	complemented
15	#2162	C₄	1	2	yes	complemented
16	#2150	C₂²	1	2	yes	—
17	#2112	C₂²	1	2	yes	—
18	C₂×#347	C₂²	1	8	yes	—
19	C₄×M₃₂	C₄	1	2	yes	complemented
20	C₄×M₃₂	C₄	1	2	yes	complemented
21	C₄×M₃₂	C₂²	1	1	yes	characteristic
22	C₄×M₃₂	C₂²	1	2	yes	—
23	C₄×M₃₂	C₂²	1	1	yes	characteristic
24	C₄×M₃₂	C₂²	1	2	yes	—
25	C₂×#346	C₄	1	2	yes	complemented
26	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
27	#1770	C₄	1	2	yes	—
28	#1770	C₂²	1	2	yes	—
29	#1752	C₄	1	2	yes	—
30	#1752	C₂²	1	2	yes	—
31	#1744	C₄	1	2	yes	—
32	C₄×#97	C₂²	1	2	yes	—
33	#1375	C₂²	1	8	yes	—
34	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₈	C₂³	1	1	yes	characteristic
36	C₂×#117	—	2	1	no	—
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	8	no	—
39	C₂×M₃₂	C₂×C₄	1	4	yes	—
40	C₂×M₃₂	C₂³	1	2	yes	—
41	C₂×M₃₂	C₂³	1	4	yes	—
42	C₂×M₃₂	C₂³	1	2	yes	—
43	C₂²×C₁₆	C₂×C₄	1	4	yes	—
44	#431	—	2	1	no	—
45	#431	—	2	1	no	—
46	#431	—	2	1	no	—
47	#431	—	2	1	no	—
48	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×#97	C₂×C₄	1	4	yes	—
51	C₂×#97	C₂³	1	4	yes	—
52	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₄⋊C₄	—	2	1	no	—
66	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×C₂²⋊C₄	—	2	1	no	—
68	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	#376	—	2	1	no	—
70	C₄³	C₂×C₄	1	2	yes	—
71	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
72	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
73	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
74	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
75	#347	—	2	16	no	—
76	#346	C₂×C₄	1	2	yes	—
77	#346	C₂×C₄	1	4	yes	—
78	#346	C₂×C₄	1	2	yes	—
79	#346	C₂×C₄	1	4	yes	—
80	C₄×C₁₆	C₂×C₄	1	4	yes	—
81	C₄×C₁₆	C₂×C₄	1	2	yes	—
82	C₄×C₁₆	C₂×C₄	1	2	yes	—
83	C₄×C₁₆	C₂×C₄	1	4	yes	—
84	#336	C₂×C₄	1	4	yes	—
85	#336	C₂×C₄	1	4	yes	—
86	C₂³×C₄	M₁₆	1	1	yes	characteristic
87	C₂³×C₄	M₁₆	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	C₄²	1	1	yes	characteristic
92	C₂²×C₈	C₄²	1	1	yes	characteristic
93	C₂²×C₈	C₄²	1	1	yes	characteristic
94	C₂²×C₈	C₄²	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₄	1	4	yes	—
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂⁴	1	1	yes	characteristic
104	#117	—	4	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₂²⋊C₄	—	2	2	no	—
107	C₂×C₄²	—	2	2	no	—
108	C₂×C₄²	—	2	2	no	—
109	C₂×C₄²	C₄²	1	1	yes	characteristic
110	C₂×C₄²	C₄²	1	1	yes	characteristic
111	C₂×C₄²	C₄²	1	1	yes	characteristic
112	C₂×C₄²	C₄²	1	1	yes	characteristic
113	C₂×C₄²	M₁₆	1	1	yes	characteristic
114	C₂×C₄²	M₁₆	1	1	yes	characteristic
115	C₂×C₄²	M₁₆	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	4	yes	—
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	M₃₂	—	2	8	no	—
123	M₃₂	—	2	4	no	—
124	M₃₂	—	2	8	no	—
125	M₃₂	#41	1	8	yes	—
126	C₂×C₁₆	—	2	8	no	—
127	C₂×C₁₆	C₂²×C₄	1	4	yes	—
128	C₂×C₁₆	C₂²×C₄	1	8	yes	—
129	C₂×C₁₆	C₂²×C₄	1	4	yes	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	#97	—	2	8	no	—
135	#97	—	2	8	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	C₄²	1	1	yes	characteristic
139	C₄×C₈	C₄²	1	1	yes	characteristic
140	C₄×C₈	C₄²	1	1	yes	characteristic
141	C₄×C₈	C₄²	1	1	yes	characteristic
142	C₄×C₈	C₄²	1	1	yes	characteristic
143	C₄×C₈	C₄²	1	1	yes	characteristic
144	C₄×C₈	C₄²	1	1	yes	characteristic
145	C₄×C₈	C₄²	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	#95	—	2	2	no	—
155	#95	—	2	2	no	—
156	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	#97	1	1	yes	characteristic
163	C₂²×C₄	#97	1	1	yes	characteristic
164	C₂²×C₄	#97	1	1	yes	characteristic
165	C₂²×C₄	#97	1	1	yes	characteristic
166	C₂²×C₄	#97	1	1	yes	characteristic
167	C₂²×C₄	#97	1	1	yes	characteristic
168	C₂²×C₄	#97	1	1	yes	characteristic
169	C₂²×C₄	#97	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
172	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
177	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
178	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
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	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
199	C₂×C₈	#131	1	4	yes	—
200	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₈	C₂×#41	1	4	yes	—
203	C₄⋊C₄	—	4	4	no	—
204	C₂²⋊C₄	—	4	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	4	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	#97	1	1	yes	characteristic
210	C₄²	#97	1	1	yes	characteristic
211	C₄²	#97	1	1	yes	characteristic
212	C₄²	#97	1	1	yes	characteristic
213	C₄²	#97	1	1	yes	characteristic
214	C₄²	#97	1	1	yes	characteristic
215	C₄²	#97	1	1	yes	characteristic
216	C₄²	#97	1	1	yes	characteristic
217	C₄²	C₂×C₄²	1	1	yes	characteristic
218	C₄²	C₂×C₄²	1	1	yes	characteristic
219	C₄²	C₂×C₄²	1	1	yes	characteristic
220	C₄²	C₂×C₄²	1	1	yes	characteristic
221	C₄²	C₂×M₁₆	1	1	yes	characteristic
222	C₄²	C₂×M₁₆	1	1	yes	characteristic
223	C₄²	C₂×M₁₆	1	1	yes	characteristic
224	C₄²	C₂×M₁₆	1	1	yes	characteristic
225	C₁₆	—	2	4	no	—
226	C₁₆	—	2	8	no	—
227	C₁₆	#117	1	8	yes	complemented
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₂×#97	1	1	yes	characteristic
232	C₂³	C₂×#97	1	1	yes	characteristic
233	C₂³	C₂²×M₁₆	1	1	yes	characteristic
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	—	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	2	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	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	#347	1	4	yes	complemented
256	C₂×C₄	C₂×#97	1	1	yes	characteristic
257	C₂×C₄	C₂×#97	1	1	yes	characteristic
258	C₂×C₄	C₂×#97	1	1	yes	characteristic
259	C₂×C₄	C₂×#97	1	1	yes	characteristic
260	C₂×C₄	C₂×#97	1	1	yes	characteristic
261	C₂×C₄	C₂×#97	1	1	yes	characteristic
262	C₂×C₄	C₂×#97	1	1	yes	characteristic
263	C₂×C₄	C₂×#97	1	1	yes	characteristic
264	C₂×C₄	C₂×#97	1	1	yes	characteristic
265	C₂×C₄	C₂×#97	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
267	C₂×C₄	C₄×#41	1	2	yes	—
268	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×#131	1	2	yes	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	4	no	—
275	C₈	#405	1	2	yes	—
276	C₈	#405	1	2	yes	—
277	C₈	C₂×#117	1	2	yes	—
278	C₈	C₄×#41	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×#347	1	2	yes	—
288	C₂²	C₂²×#97	1	1	yes	characteristic
289	C₂²	#2965	1	2	yes	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂×#347	1	2	yes	—
299	C₄	C₂×#347	1	2	yes	—
300	C₄	#2110	1	2	yes	—
301	C₄	C₄×#117	1	1	yes	characteristic
302	C₄	C₂×#405	1	1	yes	characteristic
303	C₄	#2965	1	2	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	4	1	no	—
307	C₂	#20473	1	1	yes	characteristic
308	C₂	C₂²×#347	1	1	yes	characteristic
309	C₂	C₂×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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. #12666	C₂	semidirect
6. #12594	C₂	semidirect
7. C₄×#347	C₂	semidirect
12. #2163	C₄	semidirect
13. #2163	C₄	semidirect
14. #2162	C₄	semidirect
15. #2162	C₄	semidirect
19. C₄×M₃₂	C₄	semidirect
20. C₄×M₃₂	C₄	semidirect
25. C₂×#346	C₄	semidirect
26. C₂×C₄×C₁₆	C₄	semidirect
227. C₁₆	#117	semidirect
255. C₂×C₄	#347	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×C₄×M₃₂	2
#20473	2
#12666	2
#12616	2
#12594	2
C₄×#347	2