Subgroups — C₁₆×#102 #128658

Order 512.

408 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#102	C₂	1	1	yes	maximal, characteristic
6	#7167	C₂	1	1	yes	maximal, characteristic, complemented
7	#7151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₆	—	2	1	no	—
9	C₂×C₈×D₄	C₂²	1	1	yes	characteristic
10	D₄×C₁₆	—	2	2	no	—
11	D₄×C₁₆	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₁₆	—	2	1	no	—
15	C₂×C₄×C₁₆	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄×#102	C₄	1	1	yes	characteristic
20	C₂×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₂×C₈²	C₂²	1	1	yes	characteristic
22	#1374	—	2	1	no	—
23	D₄⋊C₁₆	C₄	1	4	yes	complemented
24	C₈×C₁₆	—	2	2	no	—
25	#1279	C₄	1	1	yes	characteristic
26	#1277	C₄	1	1	yes	characteristic
27	C₂³×C₈	—	2	1	no	—
28	C₂×C₄×D₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	—	2	1	no	—
32	C₂²×C₁₆	—	2	4	no	—
33	C₂²×C₁₆	—	4	2	no	—
34	C₂²×C₁₆	—	4	2	no	—
35	C₂²×C₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₁₆	D₄	1	1	yes	characteristic
37	C₈×D₄	—	2	1	no	—
38	C₈×D₄	—	2	1	no	—
39	C₈×D₄	C₂×C₄	1	2	yes	—
40	C₈×D₄	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×#102	C₈	1	2	yes	—
44	C₂×C₂²⋊C₈	—	2	1	no	—
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	1	no	—
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
53	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
54	C₂²⋊C₁₆	—	2	4	no	—
55	C₄×C₁₆	—	2	1	no	—
56	C₄×C₁₆	—	4	2	no	—
57	C₄×C₁₆	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	C₂×C₄	1	2	yes	—
59	C₄×C₁₆	D₄	1	2	yes	—
60	#340	C₈	1	2	yes	—
61	#336	—	2	1	no	—
62	D₄⋊C₈	C₈	1	4	yes	—
63	C₈²	—	2	1	no	—
64	C₈²	—	2	1	no	—
65	C₂²×D₄	C₂×C₈	1	1	yes	characteristic
66	C₂³×C₄	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	—	2	2	no	—
72	C₂²×C₈	—	4	1	no	—
73	C₂²×C₈	—	4	1	no	—
74	C₂²×C₈	—	4	1	no	—
75	C₂²×C₈	—	4	1	no	—
76	C₂²×C₈	C₂×C₈	1	2	yes	—
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	#41	1	1	yes	characteristic
79	C₄×D₄	—	2	1	no	—
80	C₄×D₄	—	2	1	no	—
81	C₄×D₄	C₄²	1	2	yes	—
82	C₄×D₄	C₂×C₈	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
85	C₂×C₂²⋊C₄	—	2	1	no	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	—	4	1	no	—
97	C₂×C₁₆	—	4	1	no	—
98	C₂×C₁₆	—	4	1	no	—
99	C₂×C₁₆	—	4	8	no	—
100	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
101	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
102	C₂×C₁₆	D₈	1	2	yes	—
103	C₂×C₁₆	SD₁₆	1	2	yes	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	C₄²	1	2	yes	—
107	C₄⋊C₈	C₂×C₈	1	2	yes	—
108	#102	C₁₆	1	8	yes	direct product factor
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	2	2	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	C₂×C₈	1	2	yes	—
117	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×C₈	C₂×D₄	1	1	yes	characteristic
120	C₄×C₈	#41	1	1	yes	characteristic
121	#95	—	2	1	no	—
122	C₂⁴	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	C₄×C₈	1	2	yes	—
126	C₂×D₄	C₂×C₁₆	1	4	yes	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	1	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₄	#131	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₂×C₁₆	1	4	yes	complemented
155	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₈	#117	1	1	yes	characteristic
157	C₂×C₈	C₄×D₄	1	1	yes	characteristic
158	C₂×C₈	C₄×D₄	1	1	yes	characteristic
159	C₂×C₈	C₂×D₈	1	1	yes	characteristic
160	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₈	#135	1	1	yes	characteristic
162	C₂×C₈	#135	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	C₄×C₈	1	2	yes	—
166	C₄⋊C₄	C₂×C₁₆	1	4	yes	complemented
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×C₄²	1	1	yes	characteristic
172	C₄²	C₄×D₄	1	1	yes	characteristic
173	C₄²	C₂²×C₈	1	1	yes	characteristic
174	C₄²	#131	1	1	yes	characteristic
175	C₁₆	—	2	4	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	4	2	no	—
178	C₁₆	—	4	2	no	—
179	C₁₆	#102	1	4	yes	direct product factor
180	C₂³	—	2	4	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	4	1	no	—
183	C₂³	—	4	1	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₈×D₄	1	1	yes	characteristic
186	D₄	—	2	2	no	—
187	D₄	—	2	4	no	—
188	D₄	C₄×C₁₆	1	4	yes	complemented
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
202	C₂×C₄	#405	1	1	yes	characteristic
203	C₂×C₄	C₈×D₄	1	1	yes	characteristic
204	C₂×C₄	C₈×D₄	1	1	yes	characteristic
205	C₂×C₄	C₈×D₄	1	1	yes	characteristic
206	C₂×C₄	C₄×D₈	1	1	yes	characteristic
207	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
208	C₂×C₄	#443	1	1	yes	characteristic
209	C₂×C₄	#443	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₁₆	1	2	yes	—
211	C₂×C₄	#504	1	2	yes	—
212	C₈	—	2	4	no	—
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	C₂×#102	1	1	yes	characteristic
219	C₈	#416	1	1	yes	characteristic
220	C₈	C₄×D₈	1	1	yes	characteristic
221	C₈	C₄×SD₁₆	1	1	yes	characteristic
222	C₂²	—	2	4	no	—
223	C₂²	—	4	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₈×D₈	1	1	yes	characteristic
228	C₂²	C₈×SD₁₆	1	1	yes	characteristic
229	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²	D₄×C₁₆	1	2	yes	—
231	C₂²	#2170	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₈×D₈	1	1	yes	characteristic
237	C₄	C₈×SD₁₆	1	1	yes	characteristic
238	C₄	C₄×#102	1	1	yes	characteristic
239	C₄	#1764	1	1	yes	characteristic
240	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
241	C₄	#2108	1	1	yes	characteristic
242	C₄	D₄×C₁₆	1	2	yes	—
243	C₂	—	4	4	no	—
244	C₂	C₈×#102	1	1	yes	characteristic
245	C₂	C₁₆×D₈	1	2	yes	—
246	C₂	C₁₆×SD₁₆	1	2	yes	—
247	C₂	C₁₆×C₂²⋊C₄	1	1	yes	characteristic
248	C₂	#12614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ 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. C₁₆×C₄⋊C₄	C₂	semidirect
4. C₂×C₈×C₁₆	C₂	semidirect
6. #7167	C₂	semidirect
7. #7151	C₂	semidirect
11. D₄×C₁₆	C₄	semidirect
23. D₄⋊C₁₆	C₄	semidirect
108. #102	C₁₆	direct (N × H)
154. C₂×C₈	C₂×C₁₆	semidirect
166. C₄⋊C₄	C₂×C₁₆	semidirect
179. C₁₆	#102	direct (N × H)
188. D₄	C₄×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₁₆	2
C₁₆×C₄⋊C₄	2
C₂×D₄⋊C₁₆	2
C₂×C₈×C₁₆	2
C₈×#102	2
#7167	2
#7151	2

Subgroups — C16×#102 #128658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#102 #128658