Subgroups — Q₁₆×M₃₂ #483539

Order 512.

445 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	Q₈×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#12054	C₂	1	2	yes	maximal, complemented
4	#12053	C₂	1	4	yes	maximal, complemented
5	C₁₆×Q₁₆	C₂	1	2	yes	maximal, complemented
6	#12004	C₂	1	1	yes	maximal, characteristic
7	#11607	C₂	1	2	yes	maximal
8	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₁₆	C₄	1	1	yes	characteristic
11	#2183	—	2	4	no	—
12	#2183	C₂²	1	4	yes	—
13	Q₈×C₁₆	C₂²	1	4	yes	—
14	#2150	—	2	2	no	—
15	#2150	C₂²	1	2	yes	—
16	C₄×M₃₂	—	2	2	no	—
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₄×M₃₂	C₂²	1	1	yes	characteristic
19	C₈×Q₁₆	C₄	1	2	yes	—
20	C₈×Q₁₆	C₄	1	1	yes	characteristic
21	C₈×Q₁₆	C₄	1	1	yes	characteristic
22	C₈×Q₁₆	C₂²	1	1	yes	characteristic
23	C₈×Q₁₆	C₂²	1	2	yes	—
24	C₈×Q₁₆	C₂²	1	1	yes	characteristic
25	C₂×#335	C₄	1	1	yes	characteristic
26	C₂×Q₈⋊C₈	C₄	1	2	yes	—
27	C₂×C₈²	C₂²	1	1	yes	characteristic
28	#1370	C₂²	1	2	yes	—
29	#1370	C₂²	1	2	yes	—
30	Q₈⋊C₁₆	C₂²	1	4	yes	—
31	Q₈⋊C₁₆	C₂²	1	4	yes	—
32	#1311	C₂²	1	2	yes	—
33	C₈×C₁₆	C₂²	1	2	yes	—
34	C₂²×Q₁₆	C₈	1	1	yes	characteristic
35	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	2	no	—
38	C₂×M₃₂	D₄	1	1	yes	characteristic
39	C₂×M₃₂	D₄	1	1	yes	characteristic
40	C₈×Q₈	—	2	2	no	—
41	C₈×Q₈	—	2	2	no	—
42	C₈×Q₈	C₂×C₄	1	2	yes	—
43	C₈×Q₈	C₂×C₄	1	2	yes	—
44	C₈×Q₈	C₂³	1	2	yes	—
45	C₈×Q₈	C₂³	1	2	yes	—
46	C₄×Q₁₆	C₈	1	2	yes	—
47	C₄×Q₁₆	C₈	1	1	yes	characteristic
48	C₄×Q₁₆	C₈	1	1	yes	characteristic
49	C₄×Q₁₆	C₂×C₄	1	2	yes	—
50	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
52	C₂×#107	C₈	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	2	2	no	—
54	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×#103	C₈	1	2	yes	—
56	C₂×C₄×C₈	—	2	2	no	—
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₄⋊C₁₆	—	2	4	no	—
61	C₄⋊C₁₆	—	2	4	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₄⋊C₁₆	C₂³	1	4	yes	—
64	#346	—	2	4	no	—
65	#346	—	2	1	no	—
66	#346	D₄	1	2	yes	—
67	C₄×C₁₆	—	2	4	no	—
68	C₄×C₁₆	C₂³	1	2	yes	—
69	C₄×C₁₆	C₂³	1	2	yes	—
70	#335	C₂×C₄	1	1	yes	characteristic
71	#335	C₂×C₄	1	2	yes	—
72	#335	C₂×C₄	1	1	yes	characteristic
73	Q₈⋊C₈	C₂×C₄	1	2	yes	—
74	Q₈⋊C₈	C₂×C₄	1	2	yes	—
75	Q₈⋊C₈	C₂×C₄	1	2	yes	—
76	Q₈⋊C₈	C₂×C₄	1	2	yes	—
77	C₈²	C₂×C₄	1	1	yes	characteristic
78	C₈²	C₂×C₄	1	1	yes	characteristic
79	C₈²	C₂³	1	1	yes	characteristic
80	C₈²	C₂³	1	1	yes	characteristic
81	C₂²×Q₈	C₂×C₈	1	2	yes	—
82	C₂×Q₁₆	—	2	4	no	—
83	C₂×Q₁₆	C₂×C₈	1	2	yes	—
84	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
85	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₈	—	2	2	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	#41	1	1	yes	characteristic
92	C₄×Q₈	—	2	2	no	—
93	C₄×Q₈	—	2	2	no	—
94	C₄×Q₈	C₂×C₈	1	2	yes	—
95	C₄×Q₈	C₂×C₈	1	2	yes	—
96	C₄×Q₈	C₂²×C₄	1	2	yes	—
97	C₄×Q₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₂×C₄²	—	2	2	no	—
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	M₃₂	—	2	4	no	—
103	M₃₂	—	2	2	no	—
104	M₃₂	—	4	4	no	—
105	M₃₂	Q₁₆	1	4	yes	direct product factor
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×C₁₆	C₂×D₄	1	2	yes	—
110	#107	C₂×C₈	1	2	yes	—
111	#107	C₂×C₈	1	1	yes	characteristic
112	#107	C₂×C₈	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	C₂²×C₄	1	2	yes	—
119	C₄⋊C₈	C₂²×C₄	1	2	yes	—
120	#103	C₂×C₈	1	2	yes	—
121	#103	C₂×C₈	1	2	yes	—
122	#103	C₂×C₈	1	2	yes	—
123	#103	C₂×C₈	1	2	yes	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₂×C₈	1	1	yes	characteristic
130	C₄×C₈	C₂×C₈	1	1	yes	characteristic
131	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
137	C₄×C₈	#41	1	1	yes	characteristic
138	C₄×C₈	C₂⁴	1	1	yes	characteristic
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	C₂²×C₈	1	2	yes	—
143	C₂×Q₈	C₂²×C₈	1	2	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
148	C₂²×C₄	#131	1	1	yes	characteristic
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	2	2	no	—
151	Q₁₆	M₃₂	1	4	yes	direct product factor
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	C₄×D₄	1	1	yes	characteristic
162	C₂×C₈	C₄×D₄	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
167	C₂×C₈	C₂×Q₁₆	1	2	yes	—
168	C₂×C₈	#135	1	2	yes	—
169	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₈	C₂×#41	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂²×C₈	1	2	yes	—
177	C₄⋊C₄	C₂²×C₈	1	2	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₄×D₄	1	1	yes	characteristic
183	C₄²	C₂²×C₈	1	1	yes	characteristic
184	C₄²	#131	1	1	yes	characteristic
185	C₄²	C₂³×C₄	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₁₆	—	2	2	no	—
188	C₁₆	—	4	4	no	—
189	C₁₆	C₂×Q₁₆	1	4	yes	complemented
190	C₂³	C₈×D₄	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	4	no	—
195	Q₈	C₂×M₃₂	1	4	yes	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	C₈×D₄	1	1	yes	characteristic
206	C₂×C₄	C₈×D₄	1	1	yes	characteristic
207	C₂×C₄	C₈×D₄	1	1	yes	characteristic
208	C₂×C₄	C₄×Q₁₆	1	2	yes	—
209	C₂×C₄	#443	1	2	yes	—
210	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
212	C₂×C₄	C₂×#131	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	4	1	no	—
220	C₈	—	4	2	no	—
221	C₈	—	4	2	no	—
222	C₈	C₄×Q₁₆	1	2	yes	—
223	C₈	C₂×M₃₂	1	2	yes	—
224	C₈	C₂²×Q₁₆	1	1	yes	characteristic
225	C₈	C₂×#135	1	1	yes	characteristic
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	C₈×Q₁₆	1	2	yes	—
229	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	4	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	C₈×Q₁₆	1	2	yes	—
238	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
239	C₄	C₂×#443	1	1	yes	characteristic
240	C₄	C₂²×M₃₂	1	1	yes	characteristic
241	C₄	#3407	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	C₂×C₈×Q₁₆	1	1	yes	characteristic
244	C₂	D₄×M₃₂	1	1	yes	characteristic
245	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12054	C₂	semidirect
4. #12053	C₂	semidirect
5. C₁₆×Q₁₆	C₂	semidirect
105. M₃₂	Q₁₆	direct (N × H)
151. Q₁₆	M₃₂	direct (N × H)
189. C₁₆	C₂×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
C₂×C₈×Q₁₆	2
#12054	2
#12053	2
C₁₆×Q₁₆	2
#12004	2
#11607	2
C₈×M₃₂	2

Subgroups — Q16×M32 #483539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×M₃₂ #483539