Subgroups — C₄×D₄×Q₈×C₃₁ #21490920

Order 7936.

3148 conjugacy classes of subgroups.

The subgroup lattice has 456 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	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2365	C₂	1	3	yes	maximal, complemented
4	C₃₁×#2360	C₂	1	6	yes	maximal, complemented
5	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₄⋊C₄×C₃₁	C₂	1	3	yes	maximal
7	Q₈×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
9	C₄×C₃₁×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
10	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₃₁	C₂	1	3	yes	maximal
12	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
13	D₄×Q₈×C₃₁	C₄	1	16	yes	direct product factor
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	—
16	C₂×C₄×Q₈×C₃₁	—	2	16	no	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	6	yes	—
28	C₃₁×#391	C₂²	1	6	yes	—
29	C₃₁×#391	C₂²	1	6	yes	—
30	C₃₁×#389	C₂²	1	6	yes	—
31	C₃₁×#389	C₂²	1	6	yes	—
32	C₃₁×#389	C₂²	1	6	yes	—
33	C₃₁×#387	C₂²	1	12	yes	—
34	C₃₁×#385	C₂²	1	12	yes	—
35	C₃₁×#384	C₂²	1	3	yes	—
36	C₃₁×#380	C₂²	1	6	yes	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
42	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
44	C₄×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
45	C₄×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
46	C₄×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
47	C₄³×C₃₁	C₂²	1	3	yes	—
48	C₂²×Q₈×C₃₁	—	2	8	no	—
49	C₂²×Q₈×C₃₁	—	2	8	no	—
50	C₂²×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
51	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
52	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	24	yes	complemented
58	C₃₁×C₂²⋊Q₈	C₂×C₄	1	48	yes	complemented
59	C₄×Q₈×C₃₁	—	2	8	no	—
60	C₄×Q₈×C₃₁	—	2	4	no	—
61	C₄×Q₈×C₃₁	—	2	24	no	—
62	C₄×Q₈×C₃₁	—	2	24	no	—
63	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
64	C₄×Q₈×C₃₁	D₄	1	8	yes	direct product factor
65	C₄×D₄×C₃₁	—	2	12	no	—
66	C₄×D₄×C₃₁	C₂×C₄	1	24	yes	—
67	C₄×D₄×C₃₁	Q₈	1	8	yes	direct product factor
68	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₃₁	—	2	24	no	—
83	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
91	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#95	C₂³	1	6	yes	—
94	C₃₁×#95	C₂³	1	6	yes	—
95	C₃₁×#95	C₂³	1	6	yes	—
96	C₃₁×#95	C₂³	1	6	yes	—
97	C₃₁×#95	C₂³	1	12	yes	—
98	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
99	C₂×Q₈×C₃₁	—	2	64	no	—
100	C₂×Q₈×C₃₁	—	2	12	no	—
101	C₂×Q₈×C₃₁	—	2	12	no	—
102	C₂×Q₈×C₃₁	C₂²×C₄	1	24	yes	—
103	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
105	C₂×Q₈×C₃₁	#41	1	4	yes	—
106	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
108	C₂×D₄×C₃₁	#41	1	4	yes	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	12	no	—
111	C₂²×C₄×C₃₁	—	2	12	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	12	no	—
114	C₂²×C₄×C₃₁	—	2	12	no	—
115	C₂²×C₄×C₃₁	—	4	6	no	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	24	yes	—
117	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	complemented
118	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	—	2	24	no	—
129	C₄⋊C₄×C₃₁	—	2	12	no	—
130	C₄⋊C₄×C₃₁	—	2	6	no	—
131	C₄⋊C₄×C₃₁	—	2	6	no	—
132	C₄⋊C₄×C₃₁	—	4	6	no	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	12	yes	—
134	C₄⋊C₄×C₃₁	C₂²×C₄	1	12	yes	—
135	C₄⋊C₄×C₃₁	C₂²×C₄	1	24	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	12	yes	—
137	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	complemented
138	C₂²⋊C₄×C₃₁	—	2	12	no	—
139	C₂²⋊C₄×C₃₁	C₂²×C₄	1	24	yes	—
140	C₂²⋊C₄×C₃₁	C₂×Q₈	1	8	yes	complemented
141	C₄²×C₃₁	—	2	12	no	—
142	C₄²×C₃₁	—	2	12	no	—
143	C₄²×C₃₁	—	2	12	no	—
144	C₄²×C₃₁	—	2	6	no	—
145	C₄²×C₃₁	—	2	6	no	—
146	C₄²×C₃₁	—	2	6	no	—
147	C₄²×C₃₁	—	4	3	no	—
148	C₄²×C₃₁	C₂²×C₄	1	12	yes	—
149	C₄²×C₃₁	C₂×D₄	1	12	yes	—
150	C₄²×C₃₁	C₂×Q₈	1	4	yes	complemented
151	C₄×D₄×Q₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
157	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
158	C₂³×C₃₁	C₂×#41	1	4	yes	—
159	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	48	no	—
161	Q₈×C₃₁	C₄×D₄	1	16	yes	direct product factor
162	D₄×C₃₁	C₄×Q₈	1	16	yes	direct product factor
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	6	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	3	no	—
167	C₂×C₄×C₃₁	—	2	6	no	—
168	C₂×C₄×C₃₁	—	2	3	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	3	no	—
172	C₂×C₄×C₃₁	—	2	3	no	—
173	C₂×C₄×C₃₁	—	2	48	no	—
174	C₂×C₄×C₃₁	—	2	16	no	—
175	C₂×C₄×C₃₁	—	2	6	no	—
176	C₂×C₄×C₃₁	—	2	3	no	—
177	C₂×C₄×C₃₁	—	2	3	no	—
178	C₂×C₄×C₃₁	—	2	6	no	—
179	C₂×C₄×C₃₁	—	2	3	no	—
180	C₂×C₄×C₃₁	—	2	6	no	—
181	C₂×C₄×C₃₁	—	4	24	no	—
182	C₂×C₄×C₃₁	C₂³×C₄	1	6	yes	—
183	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
184	C₂×C₄×C₃₁	C₂³×C₄	1	6	yes	—
185	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
186	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
187	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
188	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
190	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
191	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
192	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
193	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
194	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
195	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
196	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
197	C₂²×C₄×Q₈	C₂×C₃₁	1	2	yes	complemented
198	#2365	C₂×C₃₁	1	3	yes	complemented
199	#2360	C₂×C₃₁	1	6	yes	complemented
200	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
201	D₄×C₄⋊C₄	C₂×C₃₁	1	3	yes	—
202	Q₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
203	C₄×C₄⋊Q₈	C₂×C₃₁	1	3	yes	complemented
204	C₄×C₂²⋊Q₈	C₂×C₃₁	1	6	yes	complemented
205	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₄²×D₄	C₂×C₃₁	1	3	yes	—
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	2	4	no	—
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	C₂×C₄×Q₈	1	8	yes	complemented
214	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
215	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
217	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
218	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
219	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂²×C₃₁	#585	1	1	yes	characteristic
221	C₄×C₃₁	—	2	12	no	—
222	C₄×C₃₁	—	2	6	no	—
223	C₄×C₃₁	—	2	4	no	—
224	C₄×C₃₁	—	2	6	no	—
225	C₄×C₃₁	—	2	2	no	—
226	C₄×C₃₁	—	4	6	no	—
227	C₄×C₃₁	—	4	3	no	—
228	C₄×C₃₁	C₂×C₄×D₄	1	12	yes	—
229	C₄×C₃₁	C₂×C₄×Q₈	1	4	yes	complemented
230	C₄×C₃₁	D₄×Q₈	1	4	yes	direct product factor
231	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
232	D₄×Q₈	C₄×C₃₁	1	16	yes	direct product factor
233	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
234	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	—
235	C₂×C₄×Q₈	—	2	16	no	—
236	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
237	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
239	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
240	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
243	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
244	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
246	C₂²×C₄²	C₂²×C₃₁	1	6	yes	—
247	#391	C₂²×C₃₁	1	6	yes	—
248	#391	C₂²×C₃₁	1	6	yes	—
249	#389	C₂²×C₃₁	1	6	yes	—
250	#389	C₂²×C₃₁	1	6	yes	—
251	#389	C₂²×C₃₁	1	6	yes	—
252	#387	C₂²×C₃₁	1	12	yes	—
253	#385	C₂²×C₃₁	1	12	yes	—
254	#384	C₂²×C₃₁	1	3	yes	—
255	#380	C₂²×C₃₁	1	6	yes	—
256	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
257	C₄×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
258	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
259	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
260	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
262	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
263	C₄×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
264	C₄×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
265	C₄×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
266	C₄³	C₂²×C₃₁	1	3	yes	—
267	C₂×C₃₁	—	2	8	no	—
268	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
269	C₂×C₃₁	C₂²×C₄×Q₈	1	1	yes	characteristic
270	C₂×C₃₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic
271	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
272	C₂×C₃₁	D₄×#41	1	1	yes	characteristic
273	C₂×C₃₁	Q₈×#41	1	1	yes	characteristic
274	C₂×C₃₁	#3482	1	1	yes	characteristic
275	C₂²×Q₈	—	2	8	no	—
276	C₂²×Q₈	—	2	8	no	—
277	C₂²×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
278	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
279	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
283	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
284	C₄⋊Q₈	C₂×C₄×C₃₁	1	24	yes	complemented
285	C₂²⋊Q₈	C₂×C₄×C₃₁	1	48	yes	complemented
286	C₄×Q₈	—	2	8	no	—
287	C₄×Q₈	—	2	4	no	—
288	C₄×Q₈	—	2	24	no	—
289	C₄×Q₈	—	2	24	no	—
290	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
291	C₄×Q₈	D₄×C₃₁	1	8	yes	direct product factor
292	C₄×D₄	—	2	12	no	—
293	C₄×D₄	C₂×C₄×C₃₁	1	24	yes	—
294	C₄×D₄	Q₈×C₃₁	1	8	yes	direct product factor
295	C₂×C₄⋊C₄	—	2	24	no	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄²	—	2	24	no	—
310	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
311	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
312	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
313	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
314	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
315	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
316	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
317	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
318	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#95	C₂³×C₃₁	1	6	yes	—
321	#95	C₂³×C₃₁	1	6	yes	—
322	#95	C₂³×C₃₁	1	6	yes	—
323	#95	C₂³×C₃₁	1	6	yes	—
324	#95	C₂³×C₃₁	1	12	yes	—
325	C₃₁	C₄×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
327	C₂×Q₈	—	2	64	no	—
328	C₂×Q₈	—	2	12	no	—
329	C₂×Q₈	—	2	12	no	—
330	C₂×Q₈	C₂²×C₄×C₃₁	1	24	yes	—
331	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
332	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
333	C₂×Q₈	#41×C₃₁	1	4	yes	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
336	C₂×D₄	#41×C₃₁	1	4	yes	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	12	no	—
339	C₂²×C₄	—	2	12	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	12	no	—
342	C₂²×C₄	—	2	12	no	—
343	C₂²×C₄	—	4	6	no	—
344	C₂²×C₄	C₂²×C₄×C₃₁	1	24	yes	—
345	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	complemented
346	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
347	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
348	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
351	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
356	C₄⋊C₄	—	2	24	no	—
357	C₄⋊C₄	—	2	12	no	—
358	C₄⋊C₄	—	2	6	no	—
359	C₄⋊C₄	—	2	6	no	—
360	C₄⋊C₄	—	4	6	no	—
361	C₄⋊C₄	C₂²×C₄×C₃₁	1	12	yes	—
362	C₄⋊C₄	C₂²×C₄×C₃₁	1	12	yes	—
363	C₄⋊C₄	C₂²×C₄×C₃₁	1	24	yes	—
364	C₄⋊C₄	C₂×D₄×C₃₁	1	12	yes	—
365	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	complemented
366	C₂²⋊C₄	—	2	12	no	—
367	C₂²⋊C₄	C₂²×C₄×C₃₁	1	24	yes	—
368	C₂²⋊C₄	C₂×Q₈×C₃₁	1	8	yes	complemented
369	C₄²	—	2	12	no	—
370	C₄²	—	2	12	no	—
371	C₄²	—	2	12	no	—
372	C₄²	—	2	6	no	—
373	C₄²	—	2	6	no	—
374	C₄²	—	2	6	no	—
375	C₄²	—	4	3	no	—
376	C₄²	C₂²×C₄×C₃₁	1	12	yes	—
377	C₄²	C₂×D₄×C₃₁	1	12	yes	—
378	C₄²	C₂×Q₈×C₃₁	1	4	yes	complemented
379	C₂³	—	2	2	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
384	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
385	C₂³	C₂×#41×C₃₁	1	4	yes	—
386	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
387	Q₈	—	2	48	no	—
388	Q₈	C₄×D₄×C₃₁	1	16	yes	direct product factor
389	D₄	C₄×Q₈×C₃₁	1	16	yes	direct product factor
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	6	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	3	no	—
394	C₂×C₄	—	2	6	no	—
395	C₂×C₄	—	2	3	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	3	no	—
399	C₂×C₄	—	2	3	no	—
400	C₂×C₄	—	2	48	no	—
401	C₂×C₄	—	2	16	no	—
402	C₂×C₄	—	2	6	no	—
403	C₂×C₄	—	2	3	no	—
404	C₂×C₄	—	2	3	no	—
405	C₂×C₄	—	2	6	no	—
406	C₂×C₄	—	2	3	no	—
407	C₂×C₄	—	2	6	no	—
408	C₂×C₄	—	4	24	no	—
409	C₂×C₄	C₂³×C₄×C₃₁	1	6	yes	—
410	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
411	C₂×C₄	C₂³×C₄×C₃₁	1	6	yes	—
412	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
414	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
415	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
416	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
417	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
418	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
419	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
420	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
421	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
422	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	2	4	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	2	4	no	—
429	C₂²	C₂×C₄×Q₈×C₃₁	1	8	yes	complemented
430	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
431	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
432	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
433	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
434	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
435	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
436	C₂²	C₃₁×#585	1	1	yes	characteristic
437	C₄	—	2	12	no	—
438	C₄	—	2	6	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	6	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	6	no	—
443	C₄	—	4	3	no	—
444	C₄	C₂×C₄×D₄×C₃₁	1	12	yes	—
445	C₄	C₂×C₄×Q₈×C₃₁	1	4	yes	complemented
446	C₄	D₄×Q₈×C₃₁	1	4	yes	direct product factor
447	C₂	—	2	8	no	—
448	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
449	C₂	C₂²×C₄×Q₈×C₃₁	1	1	yes	characteristic
450	C₂	C₄×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
451	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
452	C₂	D₄×#41×C₃₁	1	1	yes	characteristic
453	C₂	Q₈×#41×C₃₁	1	1	yes	characteristic
454	C₂	C₃₁×#3482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#2365	C₂	semidirect
4. C₃₁×#2360	C₂	semidirect
5. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
7. Q₈×C₂²⋊C₄×C₃₁	C₂	semidirect
8. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
9. C₄×C₃₁×C₂²⋊Q₈	C₂	semidirect
10. C₄²×Q₈×C₃₁	C₂	semidirect
13. D₄×Q₈×C₃₁	C₄	direct (N × H)
50. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
57. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
58. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
63. C₄×Q₈×C₃₁	C₂×C₄	semidirect
64. C₄×Q₈×C₃₁	D₄	direct (N × H)
67. C₄×D₄×C₃₁	Q₈	direct (N × H)
117. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
137. C₄⋊C₄×C₃₁	C₂×Q₈	semidirect
140. C₂²⋊C₄×C₃₁	C₂×Q₈	semidirect
150. C₄²×C₃₁	C₂×Q₈	semidirect
151. C₄×D₄×Q₈	C₃₁	direct (N × H)
161. Q₈×C₃₁	C₄×D₄	direct (N × H)
162. D₄×C₃₁	C₄×Q₈	direct (N × H)
197. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
198. #2365	C₂×C₃₁	semidirect
199. #2360	C₂×C₃₁	semidirect
200. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
202. Q₈×C₂²⋊C₄	C₂×C₃₁	semidirect
203. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
204. C₄×C₂²⋊Q₈	C₂×C₃₁	semidirect
205. C₄²×Q₈	C₂×C₃₁	semidirect
213. C₂²×C₃₁	C₂×C₄×Q₈	semidirect
229. C₄×C₃₁	C₂×C₄×Q₈	semidirect
230. C₄×C₃₁	D₄×Q₈	direct (N × H)
232. D₄×Q₈	C₄×C₃₁	direct (N × H)
277. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
284. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
285. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
290. C₄×Q₈	C₂×C₄×C₃₁	semidirect
291. C₄×Q₈	D₄×C₃₁	direct (N × H)
294. C₄×D₄	Q₈×C₃₁	direct (N × H)
325. C₃₁	C₄×D₄×Q₈	direct (N × H)
345. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
365. C₄⋊C₄	C₂×Q₈×C₃₁	semidirect
368. C₂²⋊C₄	C₂×Q₈×C₃₁	semidirect
378. C₄²	C₂×Q₈×C₃₁	semidirect
388. Q₈	C₄×D₄×C₃₁	direct (N × H)
389. D₄	C₄×Q₈×C₃₁	direct (N × H)
429. C₂²	C₂×C₄×Q₈×C₃₁	semidirect
445. C₄	C₂×C₄×Q₈×C₃₁	semidirect
446. C₄	D₄×Q₈×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
C₂²×C₄×Q₈×C₃₁	2
C₃₁×#2365	2
C₃₁×#2360	2
Q₈×C₄⋊C₄×C₃₁	2
D₄×C₄⋊C₄×C₃₁	2
Q₈×C₂²⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×C₂²⋊Q₈	2
C₄²×Q₈×C₃₁	2
C₄²×D₄×C₃₁	2
C₄×D₄×Q₈	31

Subgroups — C4×D4×Q8×C31 #21490920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×Q₈×C₃₁ #21490920