Subgroups — D₈×Q₁₆×C₃₁ #21476733

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal
2	D₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1674	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1669	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1649	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₁	—	2	2	no	—
10	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#500	—	2	2	no	—
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	2	yes	—
14	C₃₁×#494	C₂²	1	2	yes	—
15	C₃₁×#474	—	2	4	no	—
16	C₃₁×#474	C₂²	1	4	yes	—
17	C₃₁×#467	—	2	2	no	—
18	C₃₁×#462	C₂²	1	4	yes	—
19	C₃₁×#451	—	2	4	no	—
20	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×D₈×C₃₁	—	2	2	no	—
23	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#335	C₂²	1	1	yes	characteristic
26	C₃₁×#335	C₂²	1	1	yes	characteristic
27	C₃₁×#332	C₂²	1	4	yes	—
28	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	4	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	—	2	2	no	—
35	C₂×Q₁₆×C₃₁	—	4	8	no	—
36	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
37	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	—	2	4	no	—
40	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×C₂²⋊Q₈	—	4	4	no	—
43	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₄×D₄×C₃₁	—	2	4	no	—
45	C₄×D₄×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#107	—	2	2	no	—
47	C₃₁×#107	—	2	2	no	—
48	C₃₁×#107	—	2	4	no	—
49	C₃₁×#107	—	4	2	no	—
50	C₃₁×#107	D₄	1	1	yes	characteristic
51	C₃₁×#107	D₄	1	1	yes	characteristic
52	C₃₁×C₄⋊C₈	—	2	2	no	—
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₃₁×#103	—	2	4	no	—
56	C₃₁×#103	D₄	1	2	yes	—
57	C₃₁×#102	—	2	4	no	—
58	C₃₁×#102	D₄	1	2	yes	—
59	C₃₁×C₂²⋊C₈	—	2	2	no	—
60	C₄×C₈×C₃₁	—	2	2	no	—
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₃₁	—	2	4	no	—
65	C₂×Q₈×C₃₁	—	4	4	no	—
66	C₂×Q₈×C₃₁	—	4	8	no	—
67	C₂×Q₈×C₃₁	—	8	4	no	—
68	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	4	no	—
72	Q₁₆×C₃₁	—	4	8	no	—
73	Q₁₆×C₃₁	—	4	4	no	—
74	Q₁₆×C₃₁	—	8	2	no	—
75	Q₁₆×C₃₁	D₈	1	4	yes	direct product factor
76	D₈×C₃₁	Q₁₆	1	4	yes	direct product factor
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	4	no	—
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	4	4	no	—
89	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
95	D₈×Q₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	2	no	—
97	Q₈×C₃₁	—	2	2	no	—
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	4	4	no	—
100	Q₈×C₃₁	—	8	8	no	—
101	Q₈×C₃₁	C₂×D₈	1	4	yes	—
102	D₄×C₃₁	—	2	2	no	—
103	D₄×C₃₁	C₂×Q₁₆	1	4	yes	complemented
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	8	8	no	—
111	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₈×C₃₁	—	2	2	no	—
115	C₈×C₃₁	—	2	1	no	—
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	4	2	no	—
118	C₈×C₃₁	—	4	2	no	—
119	C₈×C₃₁	C₂×D₈	1	2	yes	—
120	C₈×C₃₁	C₂×Q₁₆	1	2	yes	complemented
121	Q₈×D₈	C₂×C₃₁	1	2	yes	—
122	D₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
123	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1674	C₂×C₃₁	1	2	yes	complemented
125	#1669	C₂×C₃₁	1	2	yes	complemented
126	#1649	C₂×C₃₁	1	4	yes	complemented
127	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	4	2	no	—
131	C₂²×C₃₁	—	4	2	no	—
132	C₂²×C₃₁	D₄²	1	1	yes	characteristic
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	4	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	8	4	no	—
138	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₃₁	#576	1	1	yes	characteristic
141	C₄×C₃₁	#578	1	1	yes	characteristic
142	C₂²×Q₁₆	—	2	2	no	—
143	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
144	#500	—	2	2	no	—
145	#500	C₂²×C₃₁	1	2	yes	—
146	#500	C₂²×C₃₁	1	2	yes	—
147	#494	C₂²×C₃₁	1	2	yes	—
148	#474	—	2	4	no	—
149	#474	C₂²×C₃₁	1	4	yes	—
150	#467	—	2	2	no	—
151	#462	C₂²×C₃₁	1	4	yes	—
152	#451	—	2	4	no	—
153	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
154	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
155	C₄×D₈	—	2	2	no	—
156	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₈×D₄	C₂²×C₃₁	1	2	yes	—
158	#335	C₂²×C₃₁	1	1	yes	characteristic
159	#335	C₂²×C₃₁	1	1	yes	characteristic
160	#332	C₂²×C₃₁	1	4	yes	—
161	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
162	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
163	C₈²	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×C₃₁	—	4	4	no	—
165	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
166	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
167	C₂×C₃₁	#3293	1	1	yes	characteristic
168	C₂²×Q₈	—	2	4	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	4	8	no	—
173	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
174	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
178	C₂²⋊Q₈	—	2	4	no	—
179	C₂²⋊Q₈	—	4	4	no	—
180	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	C₂³×C₃₁	1	2	yes	—
183	#107	—	2	2	no	—
184	#107	—	2	2	no	—
185	#107	—	2	4	no	—
186	#107	—	4	2	no	—
187	#107	D₄×C₃₁	1	1	yes	characteristic
188	#107	D₄×C₃₁	1	1	yes	characteristic
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
191	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
192	#103	—	2	4	no	—
193	#103	D₄×C₃₁	1	2	yes	—
194	#102	—	2	4	no	—
195	#102	D₄×C₃₁	1	2	yes	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
201	C₃₁	D₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	8	no	—
205	C₂×Q₈	—	8	4	no	—
206	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
207	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	4	no	—
210	Q₁₆	—	4	8	no	—
211	Q₁₆	—	4	4	no	—
212	Q₁₆	—	8	2	no	—
213	Q₁₆	D₈×C₃₁	1	4	yes	direct product factor
214	D₈	Q₁₆×C₃₁	1	4	yes	direct product factor
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
228	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	4	4	no	—
236	Q₈	—	4	4	no	—
237	Q₈	—	8	8	no	—
238	Q₈	C₂×D₈×C₃₁	1	4	yes	—
239	D₄	—	2	2	no	—
240	D₄	C₂×Q₁₆×C₃₁	1	4	yes	complemented
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	8	no	—
248	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	C₂×D₈×C₃₁	1	2	yes	—
257	C₈	C₂×Q₁₆×C₃₁	1	2	yes	complemented
258	C₂²	—	2	4	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	D₄²×C₃₁	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	8	4	no	—
267	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
268	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
269	C₄	C₃₁×#576	1	1	yes	characteristic
270	C₄	C₃₁×#578	1	1	yes	characteristic
271	C₂	—	4	4	no	—
272	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
273	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
274	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#1713	C₂	semidirect
4. C₃₁×#1674	C₂	semidirect
5. C₃₁×#1669	C₂	semidirect
6. C₃₁×#1649	C₂	semidirect
7. C₈×Q₁₆×C₃₁	C₂	semidirect
75. Q₁₆×C₃₁	D₈	direct (N × H)
76. D₈×C₃₁	Q₁₆	direct (N × H)
95. D₈×Q₁₆	C₃₁	direct (N × H)
103. D₄×C₃₁	C₂×Q₁₆	semidirect
120. C₈×C₃₁	C₂×Q₁₆	semidirect
122. D₄×Q₁₆	C₂×C₃₁	semidirect
123. #1713	C₂×C₃₁	semidirect
124. #1674	C₂×C₃₁	semidirect
125. #1669	C₂×C₃₁	semidirect
126. #1649	C₂×C₃₁	semidirect
127. C₈×Q₁₆	C₂×C₃₁	semidirect
201. C₃₁	D₈×Q₁₆	direct (N × H)
213. Q₁₆	D₈×C₃₁	direct (N × H)
214. D₈	Q₁₆×C₃₁	direct (N × H)
240. D₄	C₂×Q₁₆×C₃₁	semidirect
257. C₈	C₂×Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₈×C₃₁	2
D₄×Q₁₆×C₃₁	2
C₃₁×#1713	2
C₃₁×#1674	2
C₃₁×#1669	2
C₃₁×#1649	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
D₈×Q₁₆	31

Subgroups — D8×Q16×C31 #21476733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×Q₁₆×C₃₁ #21476733