Subgroups — C₂×D₄×Q₁₆×C₃₁ #21517805

Order 7936.

2742 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	2	yes	maximal
3	D₄×Q₁₆×C₃₁	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#467	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#462	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#451	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₃₁	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₃₁	—	2	16	no	—
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₃₁	C₂²	1	16	yes	complemented
17	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
18	D₄×Q₈×C₃₁	—	2	8	no	—
19	D₄×Q₈×C₃₁	C₂²	1	16	yes	—
20	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	—	2	4	no	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₃₁	—	2	2	no	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#494	C₂²	1	8	yes	complemented
28	C₃₁×#467	C₂²	1	16	yes	complemented
29	C₃₁×#462	C₂²	1	16	yes	complemented
30	C₃₁×#451	C₂²	1	32	yes	complemented
31	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
32	C₈×D₄×C₃₁	C₂²	1	8	yes	—
33	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#103	C₂²	1	2	yes	—
36	C₂×C₃₁×#103	C₂²	1	4	yes	—
37	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	—	2	4	no	—
40	C₂²×Q₈×C₃₁	—	2	8	no	—
41	C₂²×Q₈×C₃₁	—	2	16	no	—
42	C₂²×Q₈×C₃₁	—	4	8	no	—
43	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	16	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	4	no	—
48	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₃₁	—	2	16	no	—
50	C₂×Q₁₆×C₃₁	—	2	64	no	—
51	C₂×Q₁₆×C₃₁	—	2	8	no	—
52	C₂×Q₁₆×C₃₁	—	4	4	no	—
53	C₂×Q₁₆×C₃₁	D₄	1	8	yes	direct product factor
54	C₂×Q₁₆×C₃₁	C₂³	1	16	yes	—
55	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
56	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
57	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂²×C₈×C₃₁	—	2	8	no	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	—	2	16	no	—
64	C₃₁×C₄⋊Q₈	—	2	4	no	—
65	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
66	C₃₁×C₂²⋊Q₈	—	2	8	no	—
67	C₃₁×C₂²⋊Q₈	—	2	32	no	—
68	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
69	C₄×Q₈×C₃₁	—	2	4	no	—
70	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
71	C₄×D₄×C₃₁	—	2	16	no	—
72	C₄×D₄×C₃₁	D₄	1	4	yes	—
73	C₄×D₄×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#107	C₂³	1	4	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
86	C₃₁×#103	C₂³	1	8	yes	—
87	C₃₁×#103	C₂³	1	16	yes	—
88	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
89	C₄×C₈×C₃₁	C₂³	1	4	yes	—
90	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	16	no	—
93	C₂×Q₈×C₃₁	—	2	4	no	—
94	C₂×Q₈×C₃₁	—	2	4	no	—
95	C₂×Q₈×C₃₁	—	2	32	no	—
96	C₂×Q₈×C₃₁	—	2	8	no	—
97	C₂×Q₈×C₃₁	—	2	8	no	—
98	C₂×Q₈×C₃₁	—	4	16	no	—
99	C₂×Q₈×C₃₁	—	4	16	no	—
100	C₂×Q₈×C₃₁	—	4	32	no	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
102	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
103	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
104	C₂×D₄×C₃₁	Q₁₆	1	8	yes	direct product factor
105	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	16	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	4	16	no	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	Q₁₆×C₃₁	—	2	32	no	—
120	Q₁₆×C₃₁	—	2	16	no	—
121	Q₁₆×C₃₁	—	4	8	no	—
122	Q₁₆×C₃₁	C₂×D₄	1	16	yes	direct product factor
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	16	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
128	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
129	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
130	C₄⋊C₄×C₃₁	—	2	8	no	—
131	C₄⋊C₄×C₃₁	—	2	8	no	—
132	C₄⋊C₄×C₃₁	—	2	8	no	—
133	C₄⋊C₄×C₃₁	—	2	16	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
136	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
137	C₂²⋊C₄×C₃₁	—	2	16	no	—
138	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
139	C₄²×C₃₁	—	2	8	no	—
140	C₄²×C₃₁	C₂×D₄	1	2	yes	—
141	C₄²×C₃₁	C₂⁴	1	2	yes	—
142	C₂×D₄×Q₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	4	no	—
144	C₂³×C₃₁	—	2	4	no	—
145	C₂³×C₃₁	C₂×Q₁₆	1	8	yes	complemented
146	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
147	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₃₁	—	2	8	no	—
149	Q₈×C₃₁	—	2	4	no	—
150	Q₈×C₃₁	—	2	8	no	—
151	Q₈×C₃₁	—	2	8	no	—
152	Q₈×C₃₁	—	4	32	no	—
153	Q₈×C₃₁	—	4	8	no	—
154	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
155	D₄×C₃₁	C₂×Q₁₆	1	16	yes	direct product factor
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	8	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	8	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	8	no	—
165	C₂×C₄×C₃₁	—	4	32	no	—
166	C₂×C₄×C₃₁	—	4	8	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	complemented
169	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
170	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
175	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
176	C₈×C₃₁	—	2	4	no	—
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	C₂²×D₄	1	4	yes	—
179	C₂³×Q₁₆	C₂×C₃₁	1	2	yes	complemented
180	C₂×D₄×Q₈	C₂×C₃₁	1	2	yes	—
181	D₄×Q₁₆	C₂×C₃₁	1	16	yes	direct product factor
182	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂×#467	C₂×C₃₁	1	2	yes	complemented
184	C₂×#462	C₂×C₃₁	1	2	yes	complemented
185	C₂×#451	C₂×C₃₁	1	4	yes	complemented
186	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
188	C₂²×C₃₁	—	2	4	no	—
189	C₂²×C₃₁	—	2	16	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	D₄²	1	2	yes	—
192	C₂²×C₃₁	C₂²×Q₁₆	1	2	yes	—
193	C₂²×C₃₁	C₂²×Q₁₆	1	8	yes	complemented
194	C₂²×C₃₁	#578	1	2	yes	—
195	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
196	C₄×C₃₁	—	2	4	no	—
197	C₄×C₃₁	—	2	8	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	4	8	no	—
200	C₄×C₃₁	—	4	4	no	—
201	C₄×C₃₁	D₄²	1	2	yes	—
202	C₄×C₃₁	C₂²×Q₁₆	1	4	yes	complemented
203	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂³×Q₈	C₂²×C₃₁	1	4	yes	—
206	C₂²×Q₁₆	—	2	16	no	—
207	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
208	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
209	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
210	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	C₂²×Q₁₆	C₂²×C₃₁	1	16	yes	complemented
212	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
213	D₄×Q₈	—	2	8	no	—
214	D₄×Q₈	C₂²×C₃₁	1	16	yes	—
215	C₂×C₄⋊Q₈	—	2	2	no	—
216	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₂²⋊Q₈	—	2	4	no	—
218	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
219	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
220	C₂×C₄×D₄	—	2	2	no	—
221	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
222	#494	C₂²×C₃₁	1	8	yes	complemented
223	#467	C₂²×C₃₁	1	16	yes	complemented
224	#462	C₂²×C₃₁	1	16	yes	complemented
225	#451	C₂²×C₃₁	1	32	yes	complemented
226	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
227	C₈×D₄	C₂²×C₃₁	1	8	yes	—
228	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#103	C₂²×C₃₁	1	2	yes	—
231	C₂×#103	C₂²×C₃₁	1	4	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₃₁	—	2	8	no	—
235	C₂×C₃₁	D₄×Q₁₆	1	4	yes	direct product factor
236	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
237	C₂×C₃₁	C₂³×Q₁₆	1	1	yes	characteristic
238	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
239	C₂²×Q₈	—	2	4	no	—
240	C₂²×Q₈	—	2	8	no	—
241	C₂²×Q₈	—	2	16	no	—
242	C₂²×Q₈	—	4	8	no	—
243	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
244	C₂²×Q₈	C₂³×C₃₁	1	16	yes	—
245	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
246	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
247	C₂³×C₄	—	2	4	no	—
248	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
249	C₂×Q₁₆	—	2	16	no	—
250	C₂×Q₁₆	—	2	64	no	—
251	C₂×Q₁₆	—	2	8	no	—
252	C₂×Q₁₆	—	4	4	no	—
253	C₂×Q₁₆	D₄×C₃₁	1	8	yes	direct product factor
254	C₂×Q₁₆	C₂³×C₃₁	1	16	yes	—
255	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
256	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
257	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
258	C₂²×C₈	—	2	8	no	—
259	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₄⋊Q₈	—	2	16	no	—
264	C₄⋊Q₈	—	2	4	no	—
265	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
266	C₂²⋊Q₈	—	2	8	no	—
267	C₂²⋊Q₈	—	2	32	no	—
268	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
269	C₄×Q₈	—	2	4	no	—
270	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
271	C₄×D₄	—	2	16	no	—
272	C₄×D₄	D₄×C₃₁	1	4	yes	—
273	C₄×D₄	C₂³×C₃₁	1	4	yes	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	4	no	—
278	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	4	no	—
281	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	#107	C₂³×C₃₁	1	4	yes	—
285	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
286	#103	C₂³×C₃₁	1	8	yes	—
287	#103	C₂³×C₃₁	1	16	yes	—
288	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
289	C₄×C₈	C₂³×C₃₁	1	4	yes	—
290	C₃₁	C₂×D₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
291	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	16	no	—
294	C₂×Q₈	—	2	4	no	—
295	C₂×Q₈	—	2	4	no	—
296	C₂×Q₈	—	2	32	no	—
297	C₂×Q₈	—	2	8	no	—
298	C₂×Q₈	—	2	8	no	—
299	C₂×Q₈	—	4	16	no	—
300	C₂×Q₈	—	4	16	no	—
301	C₂×Q₈	—	4	32	no	—
302	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
303	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
304	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
305	C₂×D₄	Q₁₆×C₃₁	1	8	yes	direct product factor
306	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	16	no	—
313	C₂²×C₄	—	4	16	no	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
317	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
320	Q₁₆	—	2	32	no	—
321	Q₁₆	—	2	16	no	—
322	Q₁₆	—	4	8	no	—
323	Q₁₆	C₂×D₄×C₃₁	1	16	yes	direct product factor
324	C₂×C₈	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	16	no	—
327	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
328	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
329	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
330	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	16	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
337	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
338	C₂²⋊C₄	—	2	16	no	—
339	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
340	C₄²	—	2	8	no	—
341	C₄²	C₂×D₄×C₃₁	1	2	yes	—
342	C₄²	C₂⁴×C₃₁	1	2	yes	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	4	no	—
345	C₂³	C₂×Q₁₆×C₃₁	1	8	yes	complemented
346	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
347	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
348	Q₈	—	2	8	no	—
349	Q₈	—	2	4	no	—
350	Q₈	—	2	8	no	—
351	Q₈	—	2	8	no	—
352	Q₈	—	4	32	no	—
353	Q₈	—	4	8	no	—
354	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
355	D₄	C₂×Q₁₆×C₃₁	1	16	yes	direct product factor
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	4	32	no	—
366	C₂×C₄	—	4	8	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	complemented
369	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
370	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
371	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
373	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
374	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
375	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	—	2	2	no	—
378	C₈	C₂²×D₄×C₃₁	1	4	yes	—
379	C₂²	—	2	4	no	—
380	C₂²	—	2	16	no	—
381	C₂²	—	2	4	no	—
382	C₂²	D₄²×C₃₁	1	2	yes	—
383	C₂²	C₂²×Q₁₆×C₃₁	1	2	yes	—
384	C₂²	C₂²×Q₁₆×C₃₁	1	8	yes	complemented
385	C₂²	C₃₁×#578	1	2	yes	—
386	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
387	C₄	—	2	4	no	—
388	C₄	—	2	8	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	8	no	—
391	C₄	—	4	4	no	—
392	C₄	D₄²×C₃₁	1	2	yes	—
393	C₄	C₂²×Q₁₆×C₃₁	1	4	yes	complemented
394	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
395	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
396	C₂	—	2	8	no	—
397	C₂	D₄×Q₁₆×C₃₁	1	4	yes	direct product factor
398	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
399	C₂	C₂³×Q₁₆×C₃₁	1	1	yes	characteristic
400	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×D₄×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₃₁	C₂	semidirect
3. D₄×Q₁₆×C₃₁	C₂	direct (N × H)
4. C₂×C₃₁×#494	C₂	semidirect
5. C₂×C₃₁×#467	C₂	semidirect
6. C₂×C₃₁×#462	C₂	semidirect
7. C₂×C₃₁×#451	C₂	semidirect
8. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
16. C₂²×Q₁₆×C₃₁	C₂²	semidirect
27. C₃₁×#494	C₂²	semidirect
28. C₃₁×#467	C₂²	semidirect
29. C₃₁×#462	C₂²	semidirect
30. C₃₁×#451	C₂²	semidirect
31. C₄×Q₁₆×C₃₁	C₂²	semidirect
53. C₂×Q₁₆×C₃₁	D₄	direct (N × H)
104. C₂×D₄×C₃₁	Q₁₆	direct (N × H)
122. Q₁₆×C₃₁	C₂×D₄	direct (N × H)
142. C₂×D₄×Q₁₆	C₃₁	direct (N × H)
145. C₂³×C₃₁	C₂×Q₁₆	semidirect
155. D₄×C₃₁	C₂×Q₁₆	direct (N × H)
168. C₂×C₄×C₃₁	C₂×Q₁₆	semidirect
179. C₂³×Q₁₆	C₂×C₃₁	semidirect
181. D₄×Q₁₆	C₂×C₃₁	direct (N × H)
182. C₂×#494	C₂×C₃₁	semidirect
183. C₂×#467	C₂×C₃₁	semidirect
184. C₂×#462	C₂×C₃₁	semidirect
185. C₂×#451	C₂×C₃₁	semidirect
186. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
193. C₂²×C₃₁	C₂²×Q₁₆	semidirect
202. C₄×C₃₁	C₂²×Q₁₆	semidirect
211. C₂²×Q₁₆	C₂²×C₃₁	semidirect
222. #494	C₂²×C₃₁	semidirect
223. #467	C₂²×C₃₁	semidirect
224. #462	C₂²×C₃₁	semidirect
225. #451	C₂²×C₃₁	semidirect
226. C₄×Q₁₆	C₂²×C₃₁	semidirect
235. C₂×C₃₁	D₄×Q₁₆	direct (N × H)
253. C₂×Q₁₆	D₄×C₃₁	direct (N × H)
290. C₃₁	C₂×D₄×Q₁₆	direct (N × H)
305. C₂×D₄	Q₁₆×C₃₁	direct (N × H)
323. Q₁₆	C₂×D₄×C₃₁	direct (N × H)
345. C₂³	C₂×Q₁₆×C₃₁	semidirect
355. D₄	C₂×Q₁₆×C₃₁	direct (N × H)
368. C₂×C₄	C₂×Q₁₆×C₃₁	semidirect
384. C₂²	C₂²×Q₁₆×C₃₁	semidirect
393. C₄	C₂²×Q₁₆×C₃₁	semidirect
397. C₂	D₄×Q₁₆×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
C₂³×Q₁₆×C₃₁	2
C₂×D₄×Q₈×C₃₁	2
D₄×Q₁₆×C₃₁	2
C₂×C₃₁×#494	2
C₂×C₃₁×#467	2
C₂×C₃₁×#462	2
C₂×C₃₁×#451	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₂×D₄×Q₁₆	31

Subgroups — C2×D4×Q16×C31 #21517805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Q₁₆×C₃₁ #21517805