Subgroups — #512752

Order 512.

441 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2258	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#33432	C₂	1	1	yes	maximal, characteristic
4	#13646	C₂	1	2	yes	maximal, complemented
5	#13645	C₂	1	2	yes	maximal, complemented
6	#13645	C₂	1	2	yes	maximal, complemented
7	D₄×C₃₂	C₂	1	2	yes	maximal, complemented
8	#13628	C₂	1	1	yes	maximal, characteristic, complemented
9	#13554	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#13549	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#504	C₂²	1	1	yes	characteristic
13	C₂³×C₁₆	C₂²	1	1	yes	characteristic
14	#2929	C₄	1	1	yes	characteristic
15	#2258	—	2	4	no	—
16	C₂×M₆₄	—	2	4	no	—
17	C₂×M₆₄	C₂²	1	1	yes	characteristic
18	C₂×M₆₄	C₂²	1	2	yes	—
19	C₂×M₆₄	C₂²	1	1	yes	characteristic
20	C₂×M₆₄	C₂²	1	2	yes	complemented
21	C₂²×C₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₃₂	C₂²	1	2	yes	—
23	C₂²×C₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₃₂	C₂²	1	2	yes	complemented
25	#2168	C₄	1	2	yes	—
26	#2168	C₄	1	2	yes	—
27	D₄×C₁₆	C₂²	1	2	yes	—
28	D₄×C₁₆	C₂²	1	2	yes	—
29	#2151	C₄	1	1	yes	characteristic
30	#2113	C₄	1	1	yes	characteristic
31	#2112	C₄	1	1	yes	characteristic
32	#2108	C₂²	1	1	yes	characteristic
33	C₄⋊C₃₂	C₂²	1	2	yes	complemented
34	C₄⋊C₃₂	C₂²	1	2	yes	complemented
35	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
36	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
37	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
38	C₂²⋊C₃₂	C₂²	1	1	yes	characteristic
39	C₂²⋊C₃₂	C₂²	1	2	yes	complemented
40	C₂²⋊C₃₂	C₂²	1	2	yes	complemented
41	#1397	C₂²	1	2	yes	complemented
42	C₄×C₃₂	C₂²	1	2	yes	complemented
43	C₂×#131	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
45	#525	C₈	1	1	yes	characteristic
46	#504	—	2	2	no	—
47	#504	—	2	2	no	—
48	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
49	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
50	C₂×M₃₂	C₂×C₄	1	2	yes	—
51	C₂²×C₁₆	—	2	2	no	—
52	C₂²×C₁₆	—	2	2	no	—
53	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₁₆	C₂³	1	1	yes	characteristic
57	C₂²×C₁₆	C₂³	1	1	yes	characteristic
58	C₂²×C₁₆	C₂³	1	2	yes	—
59	C₂²×C₁₆	C₂³	1	1	yes	characteristic
60	#435	C₈	1	2	yes	—
61	#435	C₈	1	2	yes	—
62	C₈×D₄	C₂×C₄	1	2	yes	—
63	C₈×D₄	C₂×C₄	1	2	yes	—
64	#424	C₈	1	1	yes	characteristic
65	#408	C₈	1	1	yes	characteristic
66	#407	C₈	1	1	yes	characteristic
67	#405	C₂×C₄	1	1	yes	characteristic
68	M₆₄	—	2	2	no	—
69	M₆₄	—	2	4	no	—
70	M₆₄	—	2	4	no	—
71	M₆₄	D₄	1	4	yes	complemented
72	C₂×C₃₂	—	2	2	no	—
73	C₂×C₃₂	—	2	2	no	—
74	C₂×C₃₂	—	2	4	no	—
75	C₂×C₃₂	—	2	4	no	—
76	C₂×C₃₂	C₂³	1	1	yes	characteristic
77	C₂×C₃₂	C₂³	1	1	yes	characteristic
78	C₂×C₃₂	C₂³	1	1	yes	characteristic
79	C₂×C₃₂	C₂³	1	1	yes	characteristic
80	C₂×C₃₂	C₂³	1	2	yes	—
81	C₂×C₃₂	C₂³	1	2	yes	—
82	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
83	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
84	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
85	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
86	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
87	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
88	#346	C₂×C₄	1	2	yes	—
89	C₄×C₁₆	C₂³	1	2	yes	—
90	C₂×#41	C₂×C₈	1	1	yes	characteristic
91	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
92	#131	—	2	2	no	—
93	#131	—	2	2	no	—
94	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
95	C₂×M₁₆	C₂×C₈	1	2	yes	—
96	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	2	yes	—
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	#123	C₁₆	1	2	yes	—
107	C₂²⋊Q₈	C₁₆	1	1	yes	characteristic
108	C₂²⋊Q₈	C₁₆	1	1	yes	characteristic
109	#121	C₁₆	1	1	yes	characteristic
110	#121	C₁₆	1	1	yes	characteristic
111	#120	C₁₆	1	2	yes	—
112	C₄×D₄	C₂×C₈	1	2	yes	—
113	C₄×D₄	C₂×C₈	1	2	yes	—
114	#117	C₂×C₈	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	M₃₂	—	2	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	C₂²×C₄	1	2	yes	—
128	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₆	C₂²×C₄	1	2	yes	—
131	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₁₆	C₂×D₄	1	2	yes	—
133	C₂×C₁₆	#41	1	2	yes	—
134	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
135	C₄⋊C₈	C₂×C₈	1	2	yes	—
136	C₄⋊C₈	C₂×C₈	1	2	yes	—
137	C₂²⋊C₈	C₂×C₈	1	2	yes	—
138	C₂²⋊C₈	C₂×C₈	1	2	yes	—
139	C₂²⋊C₈	C₂×C₈	1	2	yes	—
140	C₂²⋊C₈	C₂×C₈	1	2	yes	—
141	#97	C₂×C₈	1	2	yes	—
142	C₄×C₈	C₂²×C₄	1	2	yes	—
143	C₃₂	—	2	1	no	—
144	C₃₂	—	2	1	no	—
145	C₃₂	—	2	2	no	—
146	C₃₂	—	2	1	no	—
147	C₃₂	—	2	1	no	—
148	C₃₂	C₂×D₄	1	4	yes	—
149	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
153	C₂×D₄	C₂×C₁₆	1	2	yes	—
154	C₂×D₄	C₂×C₁₆	1	2	yes	—
155	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
156	C₂×D₄	C₂×C₁₆	1	2	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₈	1	2	yes	—
165	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	M₁₆	—	2	2	no	—
168	M₁₆	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₄×D₄	1	2	yes	—
179	C₂×C₈	C₂²×C₈	1	2	yes	—
180	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₈	1	2	yes	—
183	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₈	#131	1	2	yes	—
185	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
186	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
187	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
188	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
189	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
190	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
191	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
192	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
193	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
194	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
195	C₄²	C₂²×C₈	1	2	yes	—
196	C₁₆	—	2	1	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	—	2	2	no	—
200	C₁₆	C₄×D₄	1	2	yes	—
201	C₁₆	C₂²×D₄	1	1	yes	characteristic
202	C₁₆	C₂×#41	1	1	yes	characteristic
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	4	1	no	—
206	C₂³	C₂²×C₁₆	1	1	yes	characteristic
207	C₂³	C₂²×C₁₆	1	2	yes	—
208	C₂³	C₂²×C₁₆	1	1	yes	characteristic
209	C₂³	C₂²×C₁₆	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	D₄	—	2	4	no	—
212	D₄	—	2	4	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	C₈×D₄	1	2	yes	—
225	C₂×C₄	C₂²×C₁₆	1	2	yes	—
226	C₂×C₄	C₂²×C₁₆	1	2	yes	—
227	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₁₆	1	2	yes	—
230	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
231	C₂×C₄	#504	1	2	yes	—
232	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	C₈×D₄	1	2	yes	—
238	C₈	C₂×C₄×D₄	1	1	yes	characteristic
239	C₈	C₂×#131	1	1	yes	characteristic
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	D₄×C₁₆	1	2	yes	—
252	C₂²	#2258	1	2	yes	complemented
253	C₂²	C₂³×C₁₆	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	D₄×C₁₆	1	2	yes	—
260	C₄	C₂×C₈×D₄	1	1	yes	characteristic
261	C₄	C₂×#504	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	—	4	2	no	—
265	C₂	—	4	1	no	—
266	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic
267	C₂	C₂×#2258	1	1	yes	characteristic
268	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#525	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2258	C₂	semidirect
2. C₂²×M₆₄	C₂	semidirect
4. #13646	C₂	semidirect
5. #13645	C₂	semidirect
6. #13645	C₂	semidirect
7. D₄×C₃₂	C₂	semidirect
8. #13628	C₂	semidirect
9. #13554	C₂	semidirect
10. C₂×C₂²⋊C₃₂	C₂	semidirect
11. #13549	C₂	semidirect
20. C₂×M₆₄	C₂²	semidirect
24. C₂²×C₃₂	C₂²	semidirect
33. C₄⋊C₃₂	C₂²	semidirect
34. C₄⋊C₃₂	C₂²	semidirect
39. C₂²⋊C₃₂	C₂²	semidirect
40. C₂²⋊C₃₂	C₂²	semidirect
41. #1397	C₂²	semidirect
42. C₄×C₃₂	C₂²	semidirect
71. M₆₄	D₄	semidirect
252. C₂²	#2258	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2258	2
C₂²×M₆₄	2
#33432	2
#13646	2
#13645	2
D₄×C₃₂	2
#13628	2
#13554	2
C₂×C₂²⋊C₃₂	2
#13549	2