Subgroups — C₂×C₈×D₈×C₃₁ #21474601

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₈×D₈×C₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	—	2	2	no	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
10	C₄×D₈×C₃₁	C₄	1	4	yes	—
11	C₄×D₈×C₃₁	C₂²	1	4	yes	—
12	C₈×D₄×C₃₁	—	2	4	no	—
13	C₈×D₄×C₃₁	C₂²	1	8	yes	complemented
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₄	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#335	C₂²	1	4	yes	complemented
22	C₃₁×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₈²×C₃₁	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	C₈	1	8	yes	direct product factor
27	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	—	4	8	no	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	—	2	2	no	—
38	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	C₂×C₄	1	4	yes	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
46	C₃₁×#102	C₂×C₄	1	8	yes	—
47	C₃₁×C₂²⋊C₈	—	2	8	no	—
48	C₄×C₈×C₃₁	—	2	1	no	—
49	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₃₁	D₄	1	2	yes	—
51	C₄×C₈×C₃₁	C₂³	1	2	yes	—
52	C₄×C₈×C₃₁	C₂³	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂⁴×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	4	no	—
57	C₂×D₄×C₃₁	C₂×C₈	1	8	yes	complemented
58	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
67	D₈×C₃₁	C₂×C₈	1	16	yes	direct product factor
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	—	4	16	no	—
75	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
76	C₂×C₈×C₃₁	D₈	1	4	yes	direct product factor
77	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₃₁	—	2	4	no	—
84	C₂²⋊C₄×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₁	#41	1	1	yes	characteristic
88	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×D₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	4	no	—
91	C₂³×C₃₁	—	2	8	no	—
92	C₂³×C₃₁	—	4	4	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
95	D₄×C₃₁	—	2	4	no	—
96	D₄×C₃₁	—	2	8	no	—
97	D₄×C₃₁	C₂²×C₈	1	8	yes	complemented
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	8	no	—
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₃₁	#131	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₃₁	#135	1	2	yes	—
111	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
114	C₈×C₃₁	—	2	4	no	—
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	—	2	2	no	—
117	C₈×C₃₁	—	4	4	no	—
118	C₈×C₃₁	C₂²×C₈	1	4	yes	complemented
119	C₈×C₃₁	C₂×D₈	1	4	yes	direct product factor
120	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
121	C₂×C₈×D₄	C₂×C₃₁	1	2	yes	complemented
122	C₈×D₈	C₂×C₃₁	1	8	yes	direct product factor
123	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
125	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂²×C₃₁	—	2	8	no	—
127	C₂²×C₃₁	—	4	16	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	4	4	no	—
130	C₂²×C₃₁	C₈×D₄	1	2	yes	—
131	C₂²×C₃₁	C₄×D₈	1	2	yes	—
132	C₂²×C₃₁	#443	1	2	yes	—
133	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	4	4	no	—
137	C₄×C₃₁	C₈×D₄	1	2	yes	—
138	C₄×C₃₁	C₄×D₈	1	2	yes	—
139	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
141	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
143	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
144	C₂³×C₈	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
146	C₄×D₈	C₄×C₃₁	1	4	yes	—
147	C₄×D₈	C₂²×C₃₁	1	4	yes	—
148	C₈×D₄	—	2	4	no	—
149	C₈×D₄	C₂²×C₃₁	1	8	yes	complemented
150	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
152	C₂×#102	C₄×C₃₁	1	2	yes	—
153	C₂×C₂²⋊C₈	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	#335	C₂²×C₃₁	1	4	yes	complemented
158	D₄⋊C₈	C₂²×C₃₁	1	8	yes	complemented
159	C₈²	C₂²×C₃₁	1	4	yes	complemented
160	C₂×C₃₁	—	4	8	no	—
161	C₂×C₃₁	C₈×D₈	1	4	yes	direct product factor
162	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
163	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
165	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂×D₈	C₈×C₃₁	1	8	yes	direct product factor
168	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	8	no	—
172	C₂²×C₈	—	4	8	no	—
173	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
180	C₄×D₄	C₂³×C₃₁	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
184	#107	C₂×C₄×C₃₁	1	4	yes	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
187	#102	C₂×C₄×C₃₁	1	8	yes	—
188	C₂²⋊C₈	—	2	8	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
191	C₄×C₈	D₄×C₃₁	1	2	yes	—
192	C₄×C₈	C₂³×C₃₁	1	2	yes	—
193	C₄×C₈	C₂³×C₃₁	1	2	yes	—
194	C₄×C₈	C₂³×C₃₁	1	2	yes	—
195	C₃₁	C₂×C₈×D₈	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	C₂×C₈×C₃₁	1	8	yes	complemented
200	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
209	D₈	C₂×C₈×C₃₁	1	16	yes	direct product factor
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	16	no	—
217	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
218	C₂×C₈	D₈×C₃₁	1	4	yes	direct product factor
219	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₄²	#41×C₃₁	1	1	yes	characteristic
230	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₂³	—	2	8	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
236	D₄	—	2	8	no	—
237	D₄	—	2	4	no	—
238	D₄	C₂²×C₈×C₃₁	1	8	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
249	C₂×C₄	C₃₁×#131	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
251	C₂×C₄	C₃₁×#135	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	C₂²×C₈×C₃₁	1	4	yes	complemented
260	C₈	C₂×D₈×C₃₁	1	4	yes	direct product factor
261	C₂²	—	2	8	no	—
262	C₂²	—	4	16	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	4	4	no	—
265	C₂²	C₈×D₄×C₃₁	1	2	yes	—
266	C₂²	C₄×D₈×C₃₁	1	2	yes	—
267	C₂²	C₃₁×#443	1	2	yes	—
268	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	C₈×D₄×C₃₁	1	2	yes	—
273	C₄	C₄×D₈×C₃₁	1	2	yes	—
274	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
275	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
276	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
277	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
278	C₂	—	4	8	no	—
279	C₂	C₈×D₈×C₃₁	1	4	yes	direct product factor
280	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
281	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
282	C₂	C₂×C₃₁×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₃₁	D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂²×D₈×C₃₁	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₈×D₈×C₃₁	C₂	direct (N × H)
4. C₂×C₃₁×#335	C₂	semidirect
5. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
6. C₂×C₈²×C₃₁	C₂	semidirect
13. C₈×D₄×C₃₁	C₂²	semidirect
21. C₃₁×#335	C₂²	semidirect
22. C₃₁×D₄⋊C₈	C₂²	semidirect
23. C₈²×C₃₁	C₂²	semidirect
26. C₂×D₈×C₃₁	C₈	direct (N × H)
57. C₂×D₄×C₃₁	C₂×C₈	semidirect
67. D₈×C₃₁	C₂×C₈	direct (N × H)
75. C₂×C₈×C₃₁	C₂×C₈	semidirect
76. C₂×C₈×C₃₁	D₈	direct (N × H)
89. C₂×C₈×D₈	C₃₁	direct (N × H)
97. D₄×C₃₁	C₂²×C₈	semidirect
118. C₈×C₃₁	C₂²×C₈	semidirect
119. C₈×C₃₁	C₂×D₈	direct (N × H)
121. C₂×C₈×D₄	C₂×C₃₁	semidirect
122. C₈×D₈	C₂×C₃₁	direct (N × H)
123. C₂×#335	C₂×C₃₁	semidirect
124. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
125. C₂×C₈²	C₂×C₃₁	semidirect
149. C₈×D₄	C₂²×C₃₁	semidirect
157. #335	C₂²×C₃₁	semidirect
158. D₄⋊C₈	C₂²×C₃₁	semidirect
159. C₈²	C₂²×C₃₁	semidirect
161. C₂×C₃₁	C₈×D₈	direct (N × H)
167. C₂×D₈	C₈×C₃₁	direct (N × H)
195. C₃₁	C₂×C₈×D₈	direct (N × H)
199. C₂×D₄	C₂×C₈×C₃₁	semidirect
209. D₈	C₂×C₈×C₃₁	direct (N × H)
217. C₂×C₈	C₂×C₈×C₃₁	semidirect
218. C₂×C₈	D₈×C₃₁	direct (N × H)
238. D₄	C₂²×C₈×C₃₁	semidirect
259. C₈	C₂²×C₈×C₃₁	semidirect
260. C₈	C₂×D₈×C₃₁	direct (N × H)
279. C₂	C₈×D₈×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₈×D₈×C₃₁	2
C₂×C₃₁×#335	2
C₂×C₃₁×D₄⋊C₈	2
C₂×C₈²×C₃₁	2
C₂×C₈×D₈	31

Subgroups — C2×C8×D8×C31 #21474601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₈×C₃₁ #21474601