Subgroups — C₂²×C₃₁×#468 #21517197

Order 7936.

3222 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#468	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₃₁	—	2	1	no	—
11	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	—	2	8	no	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	12	yes	complemented
15	C₂²×M₁₆×C₃₁	—	2	8	no	—
16	C₂²×M₁₆×C₃₁	C₂²	1	12	yes	complemented
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊Q₈	—	2	6	no	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	complemented
21	C₂×C₃₁×#121	—	2	6	no	—
22	C₂×C₃₁×#121	C₂²	1	12	yes	—
23	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
27	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
28	C₃₁×#468	C₂²	1	64	yes	direct product factor
29	C₂×C₃₁×#107	C₂²	1	12	yes	complemented
30	C₂×C₃₁×#103	C₂²	1	12	yes	complemented
31	C₂×C₃₁×#102	C₂²	1	12	yes	complemented
32	C₂⁵×C₃₁	—	2	1	no	—
33	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×Q₈×C₃₁	—	2	3	no	—
35	C₂²×Q₈×C₃₁	—	2	4	no	—
36	C₂²×Q₈×C₃₁	—	2	4	no	—
37	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
38	C₂²×D₄×C₃₁	—	2	12	no	—
39	C₂²×D₄×C₃₁	—	2	4	no	—
40	C₂²×D₄×C₃₁	—	2	3	no	—
41	C₂²×D₄×C₃₁	—	2	4	no	—
42	C₂²×D₄×C₃₁	—	4	8	no	—
43	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
44	C₂³×C₄×C₃₁	—	2	4	no	—
45	C₂³×C₄×C₃₁	—	2	4	no	—
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	D₄	1	6	yes	—
50	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₁	—	2	48	no	—
54	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	complemented
55	C₂×M₁₆×C₃₁	—	2	48	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	16	yes	complemented
57	C₂²×C₈×C₃₁	—	2	4	no	—
58	C₂²×C₈×C₃₁	D₄	1	8	yes	complemented
59	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
61	C₃₁×C₂²⋊Q₈	—	2	24	no	—
62	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	complemented
63	C₃₁×#121	—	2	24	no	—
64	C₃₁×#121	C₂³	1	16	yes	—
65	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
72	C₃₁×#107	C₂³	1	16	yes	complemented
73	C₃₁×#103	C₂³	1	16	yes	complemented
74	C₃₁×#102	C₂³	1	16	yes	complemented
75	C₂⁴×C₃₁	—	2	4	no	—
76	C₂⁴×C₃₁	—	2	4	no	—
77	C₂⁴×C₃₁	—	2	4	no	—
78	C₂⁴×C₃₁	—	2	6	no	—
79	C₂⁴×C₃₁	—	4	4	no	—
80	C₂⁴×C₃₁	—	4	4	no	—
81	C₂⁴×C₃₁	—	4	4	no	—
82	C₂⁴×C₃₁	C₂×D₄	1	6	yes	—
83	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×Q₈×C₃₁	—	2	6	no	—
85	C₂×Q₈×C₃₁	—	2	12	no	—
86	C₂×Q₈×C₃₁	—	2	12	no	—
87	C₂×Q₈×C₃₁	—	4	12	no	—
88	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
89	C₂×D₄×C₃₁	—	2	16	no	—
90	C₂×D₄×C₃₁	—	2	12	no	—
91	C₂×D₄×C₃₁	—	2	6	no	—
92	C₂×D₄×C₃₁	—	2	12	no	—
93	C₂×D₄×C₃₁	—	4	48	no	—
94	C₂×D₄×C₃₁	—	4	12	no	—
95	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₃₁	—	2	4	no	—
97	C₂²×C₄×C₃₁	—	2	24	no	—
98	C₂²×C₄×C₃₁	—	2	12	no	—
99	C₂²×C₄×C₃₁	—	2	12	no	—
100	C₂²×C₄×C₃₁	—	2	6	no	—
101	C₂²×C₄×C₃₁	—	2	6	no	—
102	C₂²×C₄×C₃₁	—	2	6	no	—
103	C₂²×C₄×C₃₁	—	4	4	no	—
104	C₂²×C₄×C₃₁	—	4	4	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₃₁	#41	1	4	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
113	SD₁₆×C₃₁	—	2	32	no	—
114	M₁₆×C₃₁	—	2	32	no	—
115	C₂×C₈×C₃₁	—	2	12	no	—
116	C₂×C₈×C₃₁	C₂×D₄	1	24	yes	complemented
117	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
118	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
119	C₄⋊C₄×C₃₁	—	2	16	no	—
120	C₄⋊C₄×C₃₁	—	2	6	no	—
121	C₄⋊C₄×C₃₁	—	2	6	no	—
122	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
123	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
124	C₂²⋊C₄×C₃₁	—	2	16	no	—
125	C₂²⋊C₄×C₃₁	—	2	16	no	—
126	C₂²×#468	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	2	12	no	—
128	C₂³×C₃₁	—	2	12	no	—
129	C₂³×C₃₁	—	2	12	no	—
130	C₂³×C₃₁	—	2	4	no	—
131	C₂³×C₃₁	—	4	16	no	—
132	C₂³×C₃₁	—	4	12	no	—
133	C₂³×C₃₁	—	4	12	no	—
134	C₂³×C₃₁	—	4	12	no	—
135	C₂³×C₃₁	—	8	16	no	—
136	C₂³×C₃₁	#121	1	4	yes	—
137	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
138	C₂³×C₃₁	#137	1	4	yes	—
139	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
140	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
141	Q₈×C₃₁	—	2	4	no	—
142	Q₈×C₃₁	—	2	4	no	—
143	Q₈×C₃₁	—	4	12	no	—
144	D₄×C₃₁	—	2	4	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	4	32	no	—
147	D₄×C₃₁	—	4	12	no	—
148	C₂×C₄×C₃₁	—	2	12	no	—
149	C₂×C₄×C₃₁	—	2	24	no	—
150	C₂×C₄×C₃₁	—	2	4	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	4	12	no	—
156	C₂×C₄×C₃₁	—	4	12	no	—
157	C₂×C₄×C₃₁	—	4	12	no	—
158	C₂×C₄×C₃₁	#121	1	12	yes	—
159	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
160	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
161	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
162	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
163	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
164	C₈×C₃₁	—	2	4	no	—
165	C₈×C₃₁	C₂²×D₄	1	8	yes	complemented
166	C₂³×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
167	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
168	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
170	C₂×#468	C₂×C₃₁	1	24	yes	direct product factor
171	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
172	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
173	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
174	C₂²×C₃₁	—	2	4	no	—
175	C₂²×C₃₁	—	2	4	no	—
176	C₂²×C₃₁	—	2	4	no	—
177	C₂²×C₃₁	—	4	24	no	—
178	C₂²×C₃₁	—	4	4	no	—
179	C₂²×C₃₁	—	4	4	no	—
180	C₂²×C₃₁	—	4	4	no	—
181	C₂²×C₃₁	—	8	24	no	—
182	C₂²×C₃₁	#468	1	16	yes	direct product factor
183	C₂²×C₃₁	C₂×#121	1	6	yes	—
184	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
185	C₂²×C₃₁	C₂×#137	1	6	yes	—
186	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
187	C₄×C₃₁	—	2	4	no	—
188	C₄×C₃₁	—	4	4	no	—
189	C₄×C₃₁	—	4	4	no	—
190	C₄×C₃₁	—	4	4	no	—
191	C₄×C₃₁	C₂×#121	1	6	yes	—
192	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
193	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
194	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
195	C₂³×D₄	—	2	1	no	—
196	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
197	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
198	C₂²×SD₁₆	—	2	8	no	—
199	C₂²×SD₁₆	C₂²×C₃₁	1	12	yes	complemented
200	C₂²×M₁₆	—	2	8	no	—
201	C₂²×M₁₆	C₂²×C₃₁	1	12	yes	complemented
202	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
203	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₂²⋊Q₈	—	2	6	no	—
205	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	12	yes	complemented
206	C₂×#121	—	2	6	no	—
207	C₂×#121	C₂²×C₃₁	1	12	yes	—
208	C₂²×C₄⋊C₄	—	2	1	no	—
209	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
210	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
211	C₂²×C₂²⋊C₄	—	2	1	no	—
212	C₂²×C₂²⋊C₄	—	2	1	no	—
213	#468	C₂²×C₃₁	1	64	yes	direct product factor
214	C₂×#107	C₂²×C₃₁	1	12	yes	complemented
215	C₂×#103	C₂²×C₃₁	1	12	yes	complemented
216	C₂×#102	C₂²×C₃₁	1	12	yes	complemented
217	C₂×C₃₁	—	4	4	no	—
218	C₂×C₃₁	—	8	4	no	—
219	C₂×C₃₁	C₂×#468	1	12	yes	direct product factor
220	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
221	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
223	C₂⁵	—	2	1	no	—
224	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
225	C₂²×Q₈	—	2	3	no	—
226	C₂²×Q₈	—	2	4	no	—
227	C₂²×Q₈	—	2	4	no	—
228	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
229	C₂²×D₄	—	2	12	no	—
230	C₂²×D₄	—	2	4	no	—
231	C₂²×D₄	—	2	3	no	—
232	C₂²×D₄	—	2	4	no	—
233	C₂²×D₄	—	4	8	no	—
234	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
235	C₂³×C₄	—	2	4	no	—
236	C₂³×C₄	—	2	4	no	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
241	C₂³×C₄	D₄×C₃₁	1	6	yes	—
242	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
243	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
244	C₂×SD₁₆	—	2	48	no	—
245	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	complemented
246	C₂×M₁₆	—	2	48	no	—
247	C₂×M₁₆	C₂³×C₃₁	1	16	yes	complemented
248	C₂²×C₈	—	2	4	no	—
249	C₂²×C₈	D₄×C₃₁	1	8	yes	complemented
250	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
252	C₂²⋊Q₈	—	2	24	no	—
253	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	complemented
254	#121	—	2	24	no	—
255	#121	C₂³×C₃₁	1	16	yes	—
256	C₂×C₄⋊C₄	—	2	12	no	—
257	C₂×C₄⋊C₄	—	2	3	no	—
258	C₂×C₄⋊C₄	—	2	3	no	—
259	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
260	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
261	C₂×C₂²⋊C₄	—	2	12	no	—
262	C₂×C₂²⋊C₄	—	2	12	no	—
263	#107	C₂³×C₃₁	1	16	yes	complemented
264	#103	C₂³×C₃₁	1	16	yes	complemented
265	#102	C₂³×C₃₁	1	16	yes	complemented
266	C₃₁	C₂²×#468	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁴	—	2	4	no	—
268	C₂⁴	—	2	4	no	—
269	C₂⁴	—	2	4	no	—
270	C₂⁴	—	2	6	no	—
271	C₂⁴	—	4	4	no	—
272	C₂⁴	—	4	4	no	—
273	C₂⁴	—	4	4	no	—
274	C₂⁴	C₂×D₄×C₃₁	1	6	yes	—
275	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×Q₈	—	2	6	no	—
277	C₂×Q₈	—	2	12	no	—
278	C₂×Q₈	—	2	12	no	—
279	C₂×Q₈	—	4	12	no	—
280	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
281	C₂×D₄	—	2	16	no	—
282	C₂×D₄	—	2	12	no	—
283	C₂×D₄	—	2	6	no	—
284	C₂×D₄	—	2	12	no	—
285	C₂×D₄	—	4	48	no	—
286	C₂×D₄	—	4	12	no	—
287	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	24	no	—
290	C₂²×C₄	—	2	12	no	—
291	C₂²×C₄	—	2	12	no	—
292	C₂²×C₄	—	2	6	no	—
293	C₂²×C₄	—	2	6	no	—
294	C₂²×C₄	—	2	6	no	—
295	C₂²×C₄	—	4	4	no	—
296	C₂²×C₄	—	4	4	no	—
297	C₂²×C₄	—	4	4	no	—
298	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
299	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
300	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
301	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
302	C₂²×C₄	#41×C₃₁	1	4	yes	—
303	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
304	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
305	SD₁₆	—	2	32	no	—
306	M₁₆	—	2	32	no	—
307	C₂×C₈	—	2	12	no	—
308	C₂×C₈	C₂×D₄×C₃₁	1	24	yes	complemented
309	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
310	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
311	C₄⋊C₄	—	2	16	no	—
312	C₄⋊C₄	—	2	6	no	—
313	C₄⋊C₄	—	2	6	no	—
314	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
315	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
316	C₂²⋊C₄	—	2	16	no	—
317	C₂²⋊C₄	—	2	16	no	—
318	C₂³	—	2	12	no	—
319	C₂³	—	2	12	no	—
320	C₂³	—	2	12	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	4	16	no	—
323	C₂³	—	4	12	no	—
324	C₂³	—	4	12	no	—
325	C₂³	—	4	12	no	—
326	C₂³	—	8	16	no	—
327	C₂³	C₃₁×#121	1	4	yes	—
328	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
329	C₂³	C₃₁×#137	1	4	yes	—
330	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
331	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
332	Q₈	—	2	4	no	—
333	Q₈	—	2	4	no	—
334	Q₈	—	4	12	no	—
335	D₄	—	2	4	no	—
336	D₄	—	2	4	no	—
337	D₄	—	4	32	no	—
338	D₄	—	4	12	no	—
339	C₂×C₄	—	2	12	no	—
340	C₂×C₄	—	2	24	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	12	no	—
347	C₂×C₄	—	4	12	no	—
348	C₂×C₄	—	4	12	no	—
349	C₂×C₄	C₃₁×#121	1	12	yes	—
350	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
351	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
352	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
353	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
354	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
355	C₈	—	2	4	no	—
356	C₈	C₂²×D₄×C₃₁	1	8	yes	complemented
357	C₂²	—	2	4	no	—
358	C₂²	—	2	4	no	—
359	C₂²	—	2	4	no	—
360	C₂²	—	4	24	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	8	24	no	—
365	C₂²	C₃₁×#468	1	16	yes	direct product factor
366	C₂²	C₂×C₃₁×#121	1	6	yes	—
367	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
368	C₂²	C₂×C₃₁×#137	1	6	yes	—
369	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
370	C₄	—	2	4	no	—
371	C₄	—	4	4	no	—
372	C₄	—	4	4	no	—
373	C₄	—	4	4	no	—
374	C₄	C₂×C₃₁×#121	1	6	yes	—
375	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
376	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
377	C₂	—	4	4	no	—
378	C₂	—	8	4	no	—
379	C₂	C₂×C₃₁×#468	1	12	yes	direct product factor
380	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
381	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
382	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₃₁	C₂	semidirect
2. C₂³×M₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
5. C₂×C₃₁×#468	C₂	direct (N × H)
6. C₂²×C₃₁×#107	C₂	semidirect
7. C₂²×C₃₁×#103	C₂	semidirect
8. C₂²×C₃₁×#102	C₂	semidirect
14. C₂²×SD₁₆×C₃₁	C₂²	semidirect
16. C₂²×M₁₆×C₃₁	C₂²	semidirect
20. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
28. C₃₁×#468	C₂²	direct (N × H)
29. C₂×C₃₁×#107	C₂²	semidirect
30. C₂×C₃₁×#103	C₂²	semidirect
31. C₂×C₃₁×#102	C₂²	semidirect
54. C₂×SD₁₆×C₃₁	C₂³	semidirect
56. C₂×M₁₆×C₃₁	C₂³	semidirect
58. C₂²×C₈×C₃₁	D₄	semidirect
62. C₃₁×C₂²⋊Q₈	C₂³	semidirect
72. C₃₁×#107	C₂³	semidirect
73. C₃₁×#103	C₂³	semidirect
74. C₃₁×#102	C₂³	semidirect
116. C₂×C₈×C₃₁	C₂×D₄	semidirect
126. C₂²×#468	C₃₁	direct (N × H)
165. C₈×C₃₁	C₂²×D₄	semidirect
166. C₂³×SD₁₆	C₂×C₃₁	semidirect
167. C₂³×M₁₆	C₂×C₃₁	semidirect
168. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
170. C₂×#468	C₂×C₃₁	direct (N × H)
171. C₂²×#107	C₂×C₃₁	semidirect
172. C₂²×#103	C₂×C₃₁	semidirect
173. C₂²×#102	C₂×C₃₁	semidirect
182. C₂²×C₃₁	#468	direct (N × H)
199. C₂²×SD₁₆	C₂²×C₃₁	semidirect
201. C₂²×M₁₆	C₂²×C₃₁	semidirect
205. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
213. #468	C₂²×C₃₁	direct (N × H)
214. C₂×#107	C₂²×C₃₁	semidirect
215. C₂×#103	C₂²×C₃₁	semidirect
216. C₂×#102	C₂²×C₃₁	semidirect
219. C₂×C₃₁	C₂×#468	direct (N × H)
245. C₂×SD₁₆	C₂³×C₃₁	semidirect
247. C₂×M₁₆	C₂³×C₃₁	semidirect
249. C₂²×C₈	D₄×C₃₁	semidirect
253. C₂²⋊Q₈	C₂³×C₃₁	semidirect
263. #107	C₂³×C₃₁	semidirect
264. #103	C₂³×C₃₁	semidirect
265. #102	C₂³×C₃₁	semidirect
266. C₃₁	C₂²×#468	direct (N × H)
308. C₂×C₈	C₂×D₄×C₃₁	semidirect
356. C₈	C₂²×D₄×C₃₁	semidirect
365. C₂²	C₃₁×#468	direct (N × H)
379. C₂	C₂×C₃₁×#468	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×SD₁₆×C₃₁	2
C₂³×M₁₆×C₃₁	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₂×C₃₁×#468	2
C₂²×C₃₁×#107	2
C₂²×C₃₁×#103	2
C₂²×C₃₁×#102	2
C₂²×#468	31

Subgroups — C22×C31×#468 #21517197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#468 #21517197