Subgroups — #128651

Order 512.

289 conjugacy classes of subgroups.

The subgroup lattice has 247 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33480	C₂	1	1	yes	maximal, characteristic
2	#12592	C₂	1	1	yes	maximal, characteristic, complemented
3	#11618	C₂	1	1	yes	maximal, characteristic
4	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#8247	C₂	1	1	yes	maximal, characteristic
6	#7165	C₂	1	1	yes	maximal, characteristic, complemented
7	#7150	C₂	1	1	yes	maximal, characteristic, complemented
8	#2966	C₂²	1	1	yes	characteristic
9	#2183	C₄	1	1	yes	characteristic, complemented
10	Q₈×C₁₆	C₄	1	1	yes	characteristic, complemented
11	#2169	C₄	1	1	yes	characteristic, complemented
12	#2168	—	2	1	no	—
13	D₄×C₁₆	—	2	1	no	—
14	D₄×C₁₆	C₄	1	1	yes	characteristic, complemented
15	#2164	—	2	1	no	—
16	#2163	C₂²	1	1	yes	characteristic
17	#2162	C₂²	1	1	yes	characteristic
18	#2108	—	2	1	no	—
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	#1726	C₂²	1	1	yes	characteristic
22	#1524	C₄	1	1	yes	characteristic
23	#1500	C₄	1	1	yes	characteristic
24	C₈×M₁₆	C₂²	1	1	yes	characteristic
25	#1374	—	2	1	no	—
26	#1374	—	2	1	no	—
27	#1338	C₄	1	2	yes	complemented
28	#1330	C₄	1	2	yes	complemented
29	#1312	—	2	1	no	—
30	C₈×C₁₆	—	2	1	no	—
31	#1302	C₄	1	1	yes	characteristic
32	#1300	C₄	1	1	yes	characteristic
33	#529	C₂×C₄	1	1	yes	characteristic
34	C₂×M₃₂	—	2	1	no	—
35	C₂²×C₁₆	—	2	1	no	—
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₁₆	D₄	1	1	yes	characteristic
39	#446	C₂×C₄	1	1	yes	characteristic
40	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
41	#436	C₂×C₄	1	1	yes	characteristic
42	#435	—	2	1	no	—
43	C₈×D₄	—	2	1	no	—
44	C₈×D₄	C₂×C₄	1	1	yes	characteristic
45	#433	—	2	1	no	—
46	#432	C₂×C₄	1	1	yes	characteristic
47	#431	C₂×C₄	1	1	yes	characteristic
48	#405	—	2	1	no	—
49	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×#97	—	2	1	no	—
51	C₂×#97	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	—	2	1	no	—
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#376	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₁₆	—	2	1	no	—
56	C₄⋊C₁₆	—	2	1	no	—
57	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
58	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₁₆	—	2	1	no	—
61	C₂²⋊C₁₆	—	2	1	no	—
62	C₂²⋊C₁₆	—	2	1	no	—
63	C₂²⋊C₁₆	—	2	1	no	—
64	#346	—	2	1	no	—
65	#346	—	2	1	no	—
66	#346	—	2	1	no	—
67	#346	C₂×C₄	1	2	yes	—
68	C₄×C₁₆	—	2	1	no	—
69	C₄×C₁₆	—	2	1	no	—
70	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
71	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
72	C₄×C₁₆	D₄	1	1	yes	characteristic
73	C₄×C₁₆	D₄	1	1	yes	characteristic
74	#330	C₈	1	2	yes	—
75	#329	C₈	1	2	yes	—
76	#328	C₈	1	2	yes	—
77	#327	C₈	1	2	yes	—
78	#322	—	2	1	no	—
79	C₈²	—	2	1	no	—
80	C₂×M₁₆	—	2	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₈	#41	1	1	yes	characteristic
86	#126	—	2	1	no	—
87	#125	C₂×C₈	1	1	yes	characteristic
88	#124	C₂×C₈	1	1	yes	characteristic
89	#123	—	2	1	no	—
90	C₂²⋊Q₈	C₂×C₈	1	1	yes	characteristic
91	#121	C₂×C₈	1	1	yes	characteristic
92	C₄×Q₈	C₄²	1	1	yes	characteristic
93	C₄×D₄	—	2	1	no	—
94	C₄×D₄	C₄²	1	1	yes	characteristic
95	#117	—	2	1	no	—
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	M₃₂	—	4	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	4	1	no	—
104	C₂×C₁₆	—	4	1	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₂×C₁₆	—	4	1	no	—
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	C₄²	1	1	yes	characteristic
116	C₄⋊C₈	C₄²	1	1	yes	characteristic
117	C₂²⋊C₈	—	2	1	no	—
118	C₂²⋊C₈	—	2	1	no	—
119	C₂²⋊C₈	—	2	1	no	—
120	C₂²⋊C₈	C₂×C₈	1	2	yes	—
121	#97	—	2	1	no	—
122	#97	—	2	1	no	—
123	#97	—	2	1	no	—
124	#97	—	4	2	no	—
125	#97	C₂×C₈	1	2	yes	—
126	#97	C₂×C₈	1	2	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	—	4	1	no	—
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	#41	1	1	yes	characteristic
136	#95	—	2	1	no	—
137	#95	C₂×C₈	1	2	yes	—
138	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
139	C₂×D₄	—	2	1	no	—
140	C₂×D₄	C₄×C₈	1	1	yes	characteristic
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
145	C₂²×C₄	#131	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₈	#117	1	1	yes	characteristic
162	C₂×C₈	C₄×D₄	1	1	yes	characteristic
163	C₂×C₈	C₄×D₄	1	1	yes	characteristic
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
169	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	4	1	no	—
178	C₄²	C₂×C₄²	1	1	yes	characteristic
179	C₄²	C₄×D₄	1	1	yes	characteristic
180	C₄²	C₂²×C₈	1	1	yes	characteristic
181	C₄²	#131	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	1	no	—
184	C₁₆	—	2	1	no	—
185	C₁₆	—	2	1	no	—
186	C₁₆	—	4	2	no	—
187	C₁₆	—	4	1	no	—
188	C₁₆	—	4	1	no	—
189	C₁₆	—	4	2	no	—
190	C₂³	—	2	1	no	—
191	C₂³	C₈×D₄	1	1	yes	characteristic
192	Q₈	—	4	1	no	—
193	D₄	—	4	1	no	—
194	D₄	—	4	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
209	C₂×C₄	#405	1	1	yes	characteristic
210	C₂×C₄	C₈×D₄	1	1	yes	characteristic
211	C₂×C₄	C₈×D₄	1	1	yes	characteristic
212	C₂×C₄	C₈×D₄	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	4	2	no	—
217	C₈	—	4	2	no	—
218	C₈	—	4	1	no	—
219	C₈	—	4	1	no	—
220	C₈	#421	1	1	yes	characteristic
221	C₈	#421	1	1	yes	characteristic
222	C₈	#444	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²	#2170	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	8	2	no	—
239	C₄	#1759	1	1	yes	characteristic
240	C₄	#1765	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	8	1	no	—
243	C₂	#8191	1	1	yes	characteristic
244	C₂	#12613	1	1	yes	characteristic
245	C₂	#12614	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	C₈×C₂²⋊C₄
prime radical	#121	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
2. #12592	C₂	semidirect
4. C₁₆×M₁₆	C₂	semidirect
6. #7165	C₂	semidirect
7. #7150	C₂	semidirect
9. #2183	C₄	semidirect
10. Q₈×C₁₆	C₄	semidirect
11. #2169	C₄	semidirect
14. D₄×C₁₆	C₄	semidirect
27. #1338	C₄	semidirect
28. #1330	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33480	2
#12592	2
#11618	2
C₁₆×M₁₆	2
#8247	2
#7165	2
#7150	2