Subgroups — Q₈×C₃₁×#135 #21518116

Order 7936.

1786 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3383	C₂	1	3	yes	maximal, complemented
3	Q₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3381	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3380	C₂	1	3	yes	maximal, complemented
6	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
7	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3160	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3076	C₂	1	6	yes	maximal, complemented
10	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#135	C₂	1	3	yes	maximal
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	Q₈²×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#557	C₂²	1	6	yes	—
15	C₃₁×#554	—	2	6	no	—
16	C₃₁×#554	C₂²	1	6	yes	—
17	D₄×Q₈×C₃₁	—	2	2	no	—
18	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#533	—	2	6	no	—
20	C₃₁×#533	C₂²	1	6	yes	—
21	C₄×#41×C₃₁	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#500	C₂²	1	3	yes	—
25	C₃₁×#500	C₂²	1	3	yes	—
26	C₃₁×#499	C₂²	1	6	yes	—
27	C₃₁×#499	C₂²	1	6	yes	—
28	C₃₁×#498	C₂²	1	6	yes	—
29	C₃₁×#479	C₂²	1	12	yes	—
30	C₃₁×#478	C₂²	1	12	yes	—
31	C₃₁×#477	C₂²	1	6	yes	—
32	C₃₁×#476	C₂²	1	6	yes	—
33	C₃₁×#475	C₂²	1	6	yes	—
34	C₃₁×#474	C₂²	1	6	yes	—
35	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₃₁	C₂²	1	3	yes	—
38	C₈×Q₈×C₃₁	C₂²	1	3	yes	—
39	C₄×Q₁₆×C₃₁	C₂²	1	3	yes	—
40	C₄×Q₁₆×C₃₁	C₂²	1	3	yes	—
41	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
42	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
43	C₄×D₈×C₃₁	C₂²	1	3	yes	—
44	C₄×D₈×C₃₁	C₂²	1	3	yes	—
45	C₃₁×#427	C₂²	1	6	yes	—
46	C₃₁×#427	C₂²	1	3	yes	—
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
48	C₃₁×#416	C₂²	1	6	yes	—
49	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
50	C₂×#41×C₃₁	C₂³	1	2	yes	—
51	C₂²×Q₈×C₃₁	—	2	2	no	—
52	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×#135	Q₈	1	8	yes	direct product factor
54	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	—	2	6	no	—
60	C₃₁×C₄⋊Q₈	—	2	6	no	—
61	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
62	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
63	C₃₁×#125	—	2	12	no	—
64	C₃₁×#125	C₂³	1	12	yes	—
65	C₃₁×C₂²⋊Q₈	—	2	6	no	—
66	C₃₁×C₂²⋊Q₈	—	2	6	no	—
67	C₃₁×C₂²⋊Q₈	—	2	6	no	—
68	C₃₁×C₂²⋊Q₈	—	2	6	no	—
69	C₃₁×C₂²⋊Q₈	—	2	12	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	6	no	—
72	C₄×Q₈×C₃₁	—	2	6	no	—
73	C₄×Q₈×C₃₁	—	2	6	no	—
74	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
75	C₄×Q₈×C₃₁	D₄	1	3	yes	—
76	C₄×Q₈×C₃₁	C₂³	1	6	yes	—
77	C₄×Q₈×C₃₁	C₂³	1	6	yes	—
78	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
80	C₄×D₄×C₃₁	—	2	6	no	—
81	C₄×D₄×C₃₁	—	2	6	no	—
82	C₄×D₄×C₃₁	C₂³	1	6	yes	—
83	C₄×D₄×C₃₁	C₂³	1	6	yes	—
84	C₃₁×#117	—	2	6	no	—
85	C₃₁×#117	C₂³	1	6	yes	—
86	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
87	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
88	C₂×C₄²×C₃₁	—	2	6	no	—
89	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
90	C₃₁×#107	C₂³	1	3	yes	—
91	C₃₁×#107	C₂³	1	3	yes	—
92	C₃₁×#107	C₂³	1	6	yes	—
93	C₃₁×#107	C₂³	1	6	yes	—
94	C₃₁×#106	C₂³	1	6	yes	—
95	C₃₁×#106	C₂³	1	12	yes	—
96	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
99	C₃₁×#103	C₂³	1	6	yes	—
100	C₃₁×#103	C₂³	1	6	yes	—
101	C₃₁×#102	C₂³	1	6	yes	—
102	C₃₁×#102	C₂³	1	6	yes	—
103	C₄×C₈×C₃₁	C₂³	1	3	yes	—
104	C₄×C₈×C₃₁	C₂³	1	3	yes	—
105	C₄×C₈×C₃₁	C₂³	1	3	yes	—
106	C₄×C₈×C₃₁	C₂³	1	3	yes	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	C₂×Q₈	1	8	yes	complemented
109	C₂×Q₈×C₃₁	—	2	4	no	—
110	C₂×Q₈×C₃₁	—	2	6	no	—
111	C₂×Q₈×C₃₁	—	2	6	no	—
112	C₂×Q₈×C₃₁	—	4	8	no	—
113	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
114	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	6	no	—
120	C₂²×C₄×C₃₁	—	2	6	no	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
125	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
126	D₈×C₃₁	C₂×Q₈	1	4	yes	complemented
127	C₂×C₈×C₃₁	—	2	6	no	—
128	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
129	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
130	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
131	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₄⋊C₄×C₃₁	—	2	12	no	—
134	C₄⋊C₄×C₃₁	—	2	6	no	—
135	C₄⋊C₄×C₃₁	—	2	6	no	—
136	C₄⋊C₄×C₃₁	—	2	6	no	—
137	C₄⋊C₄×C₃₁	—	2	6	no	—
138	C₄⋊C₄×C₃₁	—	2	12	no	—
139	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
142	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
143	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
144	C₂²⋊C₄×C₃₁	—	2	6	no	—
145	C₂²⋊C₄×C₃₁	—	2	6	no	—
146	C₂²⋊C₄×C₃₁	—	2	6	no	—
147	C₄²×C₃₁	—	2	6	no	—
148	C₄²×C₃₁	—	2	6	no	—
149	C₄²×C₃₁	—	2	6	no	—
150	C₄²×C₃₁	C₂×D₄	1	3	yes	—
151	C₄²×C₃₁	C₂×D₄	1	3	yes	—
152	C₄²×C₃₁	C₂⁴	1	3	yes	—
153	C₄²×C₃₁	C₂⁴	1	3	yes	—
154	Q₈×#135	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	6	no	—
158	Q₈×C₃₁	—	2	2	no	—
159	Q₈×C₃₁	—	4	12	no	—
160	Q₈×C₃₁	—	8	6	no	—
161	Q₈×C₃₁	#135	1	4	yes	direct product factor
162	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
163	D₄×C₃₁	—	2	8	no	—
164	D₄×C₃₁	—	2	2	no	—
165	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	6	no	—
169	C₂×C₄×C₃₁	—	2	6	no	—
170	C₂×C₄×C₃₁	—	2	6	no	—
171	C₂×C₄×C₃₁	—	4	3	no	—
172	C₂×C₄×C₃₁	—	4	12	no	—
173	C₂×C₄×C₃₁	—	8	6	no	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
175	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
180	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
181	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
182	C₈×C₃₁	—	2	3	no	—
183	C₈×C₃₁	—	2	3	no	—
184	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
185	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
186	Q₈×#41	C₂×C₃₁	1	2	yes	complemented
187	#3383	C₂×C₃₁	1	3	yes	complemented
188	Q₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3381	C₂×C₃₁	1	6	yes	complemented
190	#3380	C₂×C₃₁	1	3	yes	complemented
191	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
192	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#3160	C₂×C₃₁	1	3	yes	complemented
194	#3076	C₂×C₃₁	1	6	yes	complemented
195	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₄×#135	C₂×C₃₁	1	3	yes	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	4	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
203	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₃₁	—	2	3	no	—
205	C₄×C₃₁	—	2	3	no	—
206	C₄×C₃₁	—	2	4	no	—
207	C₄×C₃₁	—	2	3	no	—
208	C₄×C₃₁	—	4	6	no	—
209	C₄×C₃₁	—	4	6	no	—
210	C₄×C₃₁	D₄×Q₈	1	2	yes	—
211	C₄×C₃₁	C₂×#135	1	6	yes	—
212	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
213	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
215	Q₈²	C₂²×C₃₁	1	2	yes	—
216	#557	C₂²×C₃₁	1	6	yes	—
217	#554	—	2	6	no	—
218	#554	C₂²×C₃₁	1	6	yes	—
219	D₄×Q₈	—	2	2	no	—
220	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
221	#533	—	2	6	no	—
222	#533	C₂²×C₃₁	1	6	yes	—
223	C₄×#41	C₂²×C₃₁	1	6	yes	—
224	C₂×C₄×Q₈	—	2	2	no	—
225	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	#500	C₂²×C₃₁	1	3	yes	—
227	#500	C₂²×C₃₁	1	3	yes	—
228	#499	C₂²×C₃₁	1	6	yes	—
229	#499	C₂²×C₃₁	1	6	yes	—
230	#498	C₂²×C₃₁	1	6	yes	—
231	#479	C₂²×C₃₁	1	12	yes	—
232	#478	C₂²×C₃₁	1	12	yes	—
233	#477	C₂²×C₃₁	1	6	yes	—
234	#476	C₂²×C₃₁	1	6	yes	—
235	#475	C₂²×C₃₁	1	6	yes	—
236	#474	C₂²×C₃₁	1	6	yes	—
237	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₈×Q₈	C₂²×C₃₁	1	3	yes	—
240	C₈×Q₈	C₂²×C₃₁	1	3	yes	—
241	C₄×Q₁₆	C₂²×C₃₁	1	3	yes	—
242	C₄×Q₁₆	C₂²×C₃₁	1	3	yes	—
243	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
244	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
245	C₄×D₈	C₂²×C₃₁	1	3	yes	—
246	C₄×D₈	C₂²×C₃₁	1	3	yes	—
247	#427	C₂²×C₃₁	1	6	yes	—
248	#427	C₂²×C₃₁	1	3	yes	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
250	#416	C₂²×C₃₁	1	6	yes	—
251	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
252	C₂×C₃₁	—	2	2	no	—
253	C₂×C₃₁	—	4	4	no	—
254	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
255	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
256	C₂×C₃₁	#3584	1	1	yes	characteristic
257	C₂×#41	C₂³×C₃₁	1	2	yes	—
258	C₂²×Q₈	—	2	2	no	—
259	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
260	#135	Q₈×C₃₁	1	8	yes	direct product factor
261	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
263	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
265	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₄⋊Q₈	—	2	6	no	—
267	C₄⋊Q₈	—	2	6	no	—
268	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
269	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
270	#125	—	2	12	no	—
271	#125	C₂³×C₃₁	1	12	yes	—
272	C₂²⋊Q₈	—	2	6	no	—
273	C₂²⋊Q₈	—	2	6	no	—
274	C₂²⋊Q₈	—	2	6	no	—
275	C₂²⋊Q₈	—	2	6	no	—
276	C₂²⋊Q₈	—	2	12	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	—	2	6	no	—
279	C₄×Q₈	—	2	6	no	—
280	C₄×Q₈	—	2	6	no	—
281	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
282	C₄×Q₈	D₄×C₃₁	1	3	yes	—
283	C₄×Q₈	C₂³×C₃₁	1	6	yes	—
284	C₄×Q₈	C₂³×C₃₁	1	6	yes	—
285	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
287	C₄×D₄	—	2	6	no	—
288	C₄×D₄	—	2	6	no	—
289	C₄×D₄	C₂³×C₃₁	1	6	yes	—
290	C₄×D₄	C₂³×C₃₁	1	6	yes	—
291	#117	—	2	6	no	—
292	#117	C₂³×C₃₁	1	6	yes	—
293	C₂×C₄⋊C₄	—	2	6	no	—
294	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
295	C₂×C₄²	—	2	6	no	—
296	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
297	#107	C₂³×C₃₁	1	3	yes	—
298	#107	C₂³×C₃₁	1	3	yes	—
299	#107	C₂³×C₃₁	1	6	yes	—
300	#107	C₂³×C₃₁	1	6	yes	—
301	#106	C₂³×C₃₁	1	6	yes	—
302	#106	C₂³×C₃₁	1	12	yes	—
303	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
304	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
305	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
306	#103	C₂³×C₃₁	1	6	yes	—
307	#103	C₂³×C₃₁	1	6	yes	—
308	#102	C₂³×C₃₁	1	6	yes	—
309	#102	C₂³×C₃₁	1	6	yes	—
310	C₄×C₈	C₂³×C₃₁	1	3	yes	—
311	C₄×C₈	C₂³×C₃₁	1	3	yes	—
312	C₄×C₈	C₂³×C₃₁	1	3	yes	—
313	C₄×C₈	C₂³×C₃₁	1	3	yes	—
314	C₃₁	Q₈×#135	1	1	yes	characteristic, direct product factor, Sylow
315	#41	—	2	4	no	—
316	#41	C₂×Q₈×C₃₁	1	8	yes	complemented
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	6	no	—
319	C₂×Q₈	—	2	6	no	—
320	C₂×Q₈	—	4	8	no	—
321	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
322	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	6	no	—
328	C₂²×C₄	—	2	6	no	—
329	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
331	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
332	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
333	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
334	D₈	C₂×Q₈×C₃₁	1	4	yes	complemented
335	C₂×C₈	—	2	6	no	—
336	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
337	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
338	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
339	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
341	C₄⋊C₄	—	2	12	no	—
342	C₄⋊C₄	—	2	6	no	—
343	C₄⋊C₄	—	2	6	no	—
344	C₄⋊C₄	—	2	6	no	—
345	C₄⋊C₄	—	2	6	no	—
346	C₄⋊C₄	—	2	12	no	—
347	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
348	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
349	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
350	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
351	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
352	C₂²⋊C₄	—	2	6	no	—
353	C₂²⋊C₄	—	2	6	no	—
354	C₂²⋊C₄	—	2	6	no	—
355	C₄²	—	2	6	no	—
356	C₄²	—	2	6	no	—
357	C₄²	—	2	6	no	—
358	C₄²	C₂×D₄×C₃₁	1	3	yes	—
359	C₄²	C₂×D₄×C₃₁	1	3	yes	—
360	C₄²	C₂⁴×C₃₁	1	3	yes	—
361	C₄²	C₂⁴×C₃₁	1	3	yes	—
362	C₂³	—	2	2	no	—
363	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
364	Q₈	—	2	6	no	—
365	Q₈	—	2	2	no	—
366	Q₈	—	4	12	no	—
367	Q₈	—	8	6	no	—
368	Q₈	C₃₁×#135	1	4	yes	direct product factor
369	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
370	D₄	—	2	8	no	—
371	D₄	—	2	2	no	—
372	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	6	no	—
376	C₂×C₄	—	2	6	no	—
377	C₂×C₄	—	2	6	no	—
378	C₂×C₄	—	4	3	no	—
379	C₂×C₄	—	4	12	no	—
380	C₂×C₄	—	8	6	no	—
381	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
382	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
383	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
384	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
386	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
387	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
388	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
389	C₈	—	2	3	no	—
390	C₈	—	2	3	no	—
391	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
392	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	2	no	—
398	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
399	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
400	C₄	—	2	4	no	—
401	C₄	—	2	3	no	—
402	C₄	—	2	3	no	—
403	C₄	—	2	3	no	—
404	C₄	—	4	6	no	—
405	C₄	—	4	6	no	—
406	C₄	D₄×Q₈×C₃₁	1	2	yes	—
407	C₄	C₂×C₃₁×#135	1	6	yes	—
408	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
409	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
410	C₂	—	2	2	no	—
411	C₂	—	4	4	no	—
412	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
413	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
414	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₃₁	C₂	semidirect
2. C₃₁×#3383	C₂	semidirect
3. Q₈×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#3381	C₂	semidirect
5. C₃₁×#3380	C₂	semidirect
6. Q₈×SD₁₆×C₃₁	C₂	semidirect
7. Q₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#3160	C₂	semidirect
9. C₃₁×#3076	C₂	semidirect
10. C₂×C₈×Q₈×C₃₁	C₂	semidirect
53. C₃₁×#135	Q₈	direct (N × H)
108. #41×C₃₁	C₂×Q₈	semidirect
124. Q₁₆×C₃₁	C₂×Q₈	semidirect
125. SD₁₆×C₃₁	C₂×Q₈	semidirect
126. D₈×C₃₁	C₂×Q₈	semidirect
128. C₂×C₈×C₃₁	C₂×Q₈	semidirect
154. Q₈×#135	C₃₁	direct (N × H)
161. Q₈×C₃₁	#135	direct (N × H)
186. Q₈×#41	C₂×C₃₁	semidirect
187. #3383	C₂×C₃₁	semidirect
188. Q₈×Q₁₆	C₂×C₃₁	semidirect
189. #3381	C₂×C₃₁	semidirect
190. #3380	C₂×C₃₁	semidirect
191. Q₈×SD₁₆	C₂×C₃₁	semidirect
192. Q₈×D₈	C₂×C₃₁	semidirect
193. #3160	C₂×C₃₁	semidirect
194. #3076	C₂×C₃₁	semidirect
195. C₂×C₈×Q₈	C₂×C₃₁	semidirect
260. #135	Q₈×C₃₁	direct (N × H)
314. C₃₁	Q₈×#135	direct (N × H)
316. #41	C₂×Q₈×C₃₁	semidirect
332. Q₁₆	C₂×Q₈×C₃₁	semidirect
333. SD₁₆	C₂×Q₈×C₃₁	semidirect
334. D₈	C₂×Q₈×C₃₁	semidirect
336. C₂×C₈	C₂×Q₈×C₃₁	semidirect
368. Q₈	C₃₁×#135	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41×C₃₁	2
C₃₁×#3383	2
Q₈×Q₁₆×C₃₁	2
C₃₁×#3381	2
C₃₁×#3380	2
Q₈×SD₁₆×C₃₁	2
Q₈×D₈×C₃₁	2
C₃₁×#3160	2
C₃₁×#3076	2
C₂×C₈×Q₈×C₃₁	2
C₄×C₃₁×#135	2
Q₈×#135	31

Subgroups — Q8×C31×#135 #21518116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₁×#135 #21518116