Subgroups — C₂×Q₈×D₈×C₃₁ #21518110

Order 7936.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
2	Q₈×D₈×C₃₁	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#500	C₂	1	3	yes	maximal, complemented
4	C₂×C₃₁×#474	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₃₁	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₃₁	—	2	2	no	—
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	D₄×Q₈×C₃₁	—	2	8	no	—
10	D₄×Q₈×C₃₁	C₂²	1	16	yes	complemented
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	6	yes	—
12	C₂×C₃₁×C₂²⋊Q₈	—	2	6	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
15	C₃₁×#500	C₂²	1	24	yes	complemented
16	C₃₁×#474	C₂²	1	48	yes	complemented
17	C₈×Q₈×C₃₁	C₂²	1	8	yes	complemented
18	C₄×D₈×C₃₁	C₂²	1	24	yes	—
19	C₂×C₃₁×#107	C₂²	1	3	yes	—
20	C₂×C₃₁×#107	C₂²	1	6	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
22	C₂×C₃₁×#102	C₂²	1	6	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
24	C₂²×Q₈×C₃₁	—	2	16	no	—
25	C₂²×Q₈×C₃₁	—	2	6	no	—
26	C₂²×Q₈×C₃₁	—	4	16	no	—
27	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₁	—	2	6	no	—
30	C₂×D₈×C₃₁	Q₈	1	8	yes	direct product factor
31	C₂×D₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	—	2	12	no	—
35	C₃₁×C₄⋊Q₈	C₂³	1	24	yes	—
36	C₃₁×C₂²⋊Q₈	—	2	48	no	—
37	C₄×Q₈×C₃₁	D₄	1	4	yes	—
38	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
39	C₄×D₄×C₃₁	—	2	12	no	—
40	C₄×D₄×C₃₁	C₂³	1	24	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
42	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
45	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
46	C₃₁×#107	C₂³	1	12	yes	—
47	C₃₁×#107	C₂³	1	24	yes	—
48	C₃₁×C₄⋊C₈	C₂³	1	12	yes	—
49	C₃₁×#102	C₂³	1	24	yes	—
50	C₄×C₈×C₃₁	C₂³	1	12	yes	—
51	C₂⁴×C₃₁	—	2	2	no	—
52	C₂×Q₈×C₃₁	—	2	24	no	—
53	C₂×Q₈×C₃₁	—	4	64	no	—
54	C₂×Q₈×C₃₁	—	4	24	no	—
55	C₂×Q₈×C₃₁	D₈	1	8	yes	direct product factor
56	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	4	no	—
59	C₂×D₄×C₃₁	C₂×Q₈	1	8	yes	complemented
60	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₃₁	—	2	6	no	—
62	C₂²×C₄×C₃₁	—	2	24	no	—
63	C₂²×C₄×C₃₁	—	4	24	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
67	D₈×C₃₁	C₂×Q₈	1	16	yes	direct product factor
68	C₂×C₈×C₃₁	—	2	6	no	—
69	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
70	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
71	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
72	C₄⋊C₄×C₃₁	—	2	24	no	—
73	C₄⋊C₄×C₃₁	—	2	6	no	—
74	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
75	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
76	C₂²⋊C₄×C₃₁	—	2	24	no	—
77	C₄²×C₃₁	C₂×D₄	1	6	yes	—
78	C₄²×C₃₁	C₂⁴	1	6	yes	—
79	C₂×Q₈×D₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	2	8	no	—
81	C₂³×C₃₁	—	2	4	no	—
82	C₂³×C₃₁	—	4	4	no	—
83	C₂³×C₃₁	—	4	4	no	—
84	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₃₁	—	4	48	no	—
86	Q₈×C₃₁	C₂×D₈	1	16	yes	direct product factor
87	D₄×C₃₁	—	2	8	no	—
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	C₂²×Q₈	1	8	yes	complemented
90	C₂×C₄×C₃₁	—	2	6	no	—
91	C₂×C₄×C₃₁	—	2	12	no	—
92	C₂×C₄×C₃₁	—	4	12	no	—
93	C₂×C₄×C₃₁	—	4	48	no	—
94	C₂×C₄×C₃₁	C₂×D₈	1	12	yes	—
95	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
98	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
100	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
101	C₈×C₃₁	—	2	6	no	—
102	C₈×C₃₁	C₂²×Q₈	1	4	yes	complemented
103	C₂×D₄×Q₈	C₂×C₃₁	1	2	yes	complemented
104	Q₈×D₈	C₂×C₃₁	1	16	yes	direct product factor
105	C₂×#500	C₂×C₃₁	1	3	yes	complemented
106	C₂×#474	C₂×C₃₁	1	6	yes	complemented
107	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×C₄×D₈	C₂×C₃₁	1	3	yes	—
109	C₂²×C₃₁	—	2	8	no	—
110	C₂²×C₃₁	—	4	16	no	—
111	C₂²×C₃₁	—	4	4	no	—
112	C₂²×C₃₁	—	4	4	no	—
113	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
114	C₂²×C₃₁	C₂²×D₈	1	2	yes	—
115	C₂²×C₃₁	#578	1	2	yes	—
116	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₃₁	—	2	6	no	—
118	C₄×C₃₁	—	4	12	no	—
119	C₄×C₃₁	D₄×Q₈	1	2	yes	—
120	C₄×C₃₁	C₂²×D₈	1	12	yes	—
121	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
122	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
123	C₂³×Q₈	—	2	2	no	—
124	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
125	D₄×Q₈	—	2	8	no	—
126	D₄×Q₈	C₂²×C₃₁	1	16	yes	complemented
127	C₂×C₄⋊Q₈	C₂²×C₃₁	1	6	yes	—
128	C₂×C₂²⋊Q₈	—	2	6	no	—
129	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
131	#500	C₂²×C₃₁	1	24	yes	complemented
132	#474	C₂²×C₃₁	1	48	yes	complemented
133	C₈×Q₈	C₂²×C₃₁	1	8	yes	complemented
134	C₄×D₈	C₂²×C₃₁	1	24	yes	—
135	C₂×#107	C₂²×C₃₁	1	3	yes	—
136	C₂×#107	C₂²×C₃₁	1	6	yes	—
137	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
138	C₂×#102	C₂²×C₃₁	1	6	yes	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
140	C₂×C₃₁	—	4	8	no	—
141	C₂×C₃₁	Q₈×D₈	1	4	yes	direct product factor
142	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
143	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
144	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
145	C₂²×Q₈	—	2	16	no	—
146	C₂²×Q₈	—	2	6	no	—
147	C₂²×Q₈	—	4	16	no	—
148	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
149	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
150	C₂³×C₄	—	2	6	no	—
151	C₂×D₈	Q₈×C₃₁	1	8	yes	direct product factor
152	C₂×D₈	C₂³×C₃₁	1	4	yes	—
153	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
154	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	12	no	—
156	C₄⋊Q₈	C₂³×C₃₁	1	24	yes	—
157	C₂²⋊Q₈	—	2	48	no	—
158	C₄×Q₈	D₄×C₃₁	1	4	yes	—
159	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
160	C₄×D₄	—	2	12	no	—
161	C₄×D₄	C₂³×C₃₁	1	24	yes	—
162	C₂×C₄⋊C₄	—	2	6	no	—
163	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
164	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
165	C₂×C₂²⋊C₄	—	2	6	no	—
166	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
167	#107	C₂³×C₃₁	1	12	yes	—
168	#107	C₂³×C₃₁	1	24	yes	—
169	C₄⋊C₈	C₂³×C₃₁	1	12	yes	—
170	#102	C₂³×C₃₁	1	24	yes	—
171	C₄×C₈	C₂³×C₃₁	1	12	yes	—
172	C₃₁	C₂×Q₈×D₈	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	—	2	2	no	—
174	C₂×Q₈	—	2	24	no	—
175	C₂×Q₈	—	4	64	no	—
176	C₂×Q₈	—	4	24	no	—
177	C₂×Q₈	D₈×C₃₁	1	8	yes	direct product factor
178	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	C₂×Q₈×C₃₁	1	8	yes	complemented
182	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	—	2	24	no	—
185	C₂²×C₄	—	4	24	no	—
186	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
187	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
188	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
189	D₈	C₂×Q₈×C₃₁	1	16	yes	direct product factor
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
192	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
193	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
194	C₄⋊C₄	—	2	24	no	—
195	C₄⋊C₄	—	2	6	no	—
196	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
197	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
198	C₂²⋊C₄	—	2	24	no	—
199	C₄²	C₂×D₄×C₃₁	1	6	yes	—
200	C₄²	C₂⁴×C₃₁	1	6	yes	—
201	C₂³	—	2	8	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
206	Q₈	—	4	48	no	—
207	Q₈	C₂×D₈×C₃₁	1	16	yes	direct product factor
208	D₄	—	2	8	no	—
209	D₄	—	2	4	no	—
210	D₄	C₂²×Q₈×C₃₁	1	8	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	4	12	no	—
214	C₂×C₄	—	4	48	no	—
215	C₂×C₄	C₂×D₈×C₃₁	1	12	yes	—
216	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
217	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
219	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
220	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
221	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂²×Q₈×C₃₁	1	4	yes	complemented
224	C₂²	—	2	8	no	—
225	C₂²	—	4	16	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
229	C₂²	C₂²×D₈×C₃₁	1	2	yes	—
230	C₂²	C₃₁×#578	1	2	yes	—
231	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	4	12	no	—
234	C₄	D₄×Q₈×C₃₁	1	2	yes	—
235	C₄	C₂²×D₈×C₃₁	1	12	yes	—
236	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
237	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	8	no	—
239	C₂	Q₈×D₈×C₃₁	1	4	yes	direct product factor
240	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
241	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
242	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈×C₃₁	C₂	semidirect
2. Q₈×D₈×C₃₁	C₂	direct (N × H)
3. C₂×C₃₁×#500	C₂	semidirect
4. C₂×C₃₁×#474	C₂	semidirect
5. C₂×C₈×Q₈×C₃₁	C₂	semidirect
10. D₄×Q₈×C₃₁	C₂²	semidirect
15. C₃₁×#500	C₂²	semidirect
16. C₃₁×#474	C₂²	semidirect
17. C₈×Q₈×C₃₁	C₂²	semidirect
30. C₂×D₈×C₃₁	Q₈	direct (N × H)
55. C₂×Q₈×C₃₁	D₈	direct (N × H)
59. C₂×D₄×C₃₁	C₂×Q₈	semidirect
67. D₈×C₃₁	C₂×Q₈	direct (N × H)
69. C₂×C₈×C₃₁	C₂×Q₈	semidirect
79. C₂×Q₈×D₈	C₃₁	direct (N × H)
86. Q₈×C₃₁	C₂×D₈	direct (N × H)
89. D₄×C₃₁	C₂²×Q₈	semidirect
102. C₈×C₃₁	C₂²×Q₈	semidirect
103. C₂×D₄×Q₈	C₂×C₃₁	semidirect
104. Q₈×D₈	C₂×C₃₁	direct (N × H)
105. C₂×#500	C₂×C₃₁	semidirect
106. C₂×#474	C₂×C₃₁	semidirect
107. C₂×C₈×Q₈	C₂×C₃₁	semidirect
126. D₄×Q₈	C₂²×C₃₁	semidirect
131. #500	C₂²×C₃₁	semidirect
132. #474	C₂²×C₃₁	semidirect
133. C₈×Q₈	C₂²×C₃₁	semidirect
141. C₂×C₃₁	Q₈×D₈	direct (N × H)
151. C₂×D₈	Q₈×C₃₁	direct (N × H)
172. C₃₁	C₂×Q₈×D₈	direct (N × H)
177. C₂×Q₈	D₈×C₃₁	direct (N × H)
181. C₂×D₄	C₂×Q₈×C₃₁	semidirect
189. D₈	C₂×Q₈×C₃₁	direct (N × H)
191. C₂×C₈	C₂×Q₈×C₃₁	semidirect
207. Q₈	C₂×D₈×C₃₁	direct (N × H)
210. D₄	C₂²×Q₈×C₃₁	semidirect
223. C₈	C₂²×Q₈×C₃₁	semidirect
239. C₂	Q₈×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×D₄×Q₈×C₃₁	2
Q₈×D₈×C₃₁	2
C₂×C₃₁×#500	2
C₂×C₃₁×#474	2
C₂×C₈×Q₈×C₃₁	2
C₂×C₄×D₈×C₃₁	2
C₂×Q₈×D₈	31

Subgroups — C2×Q8×D8×C31 #21518110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₈×C₃₁ #21518110