Subgroups — D₃₁×#552 #16859214

Order 3968.

1124 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521741	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521740	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521739	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521738	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521737	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521736	C₂	1	2	yes	maximal, complemented
7	#14521735	C₂	1	2	yes	maximal, complemented
8	#14521734	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521733	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521732	C₂	1	2	yes	maximal, complemented
11	#14521731	C₂	1	2	yes	maximal, complemented
12	#126×D₃₁	C₂	1	2	yes	maximal, complemented
13	#125×D₃₁	C₂	1	2	yes	maximal, complemented
14	#125×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#123×D₃₁	C₂	1	2	yes	maximal
16	C₂²⋊Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²⋊Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
19	C₄×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic
20	#117×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
21	#117×D₃₁	C₂	1	2	yes	maximal, complemented
22	C₂×C₄⋊C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#552	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
25	C₂×D₄×D₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
27	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#126	C₂²	1	2	yes	complemented
30	C₃₁×#125	C₂²	1	2	yes	complemented
31	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#123	C₂²	1	2	yes	—
33	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
35	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#117	C₂²	1	2	yes	complemented
38	C₃₁×#117	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
40	#11758523	C₂²	1	1	yes	characteristic, complemented
41	#11758523	C₂²	1	1	yes	characteristic, complemented
42	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
43	#11758512	C₂²	1	2	yes	—
44	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
45	#11758506	C₂²	1	2	yes	—
46	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
48	#11758503	C₂²	1	1	yes	characteristic, complemented
49	#11758503	C₂²	1	2	yes	complemented
50	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic, complemented
51	#11758501	C₂²	1	1	yes	characteristic, complemented
52	#11758501	C₂²	1	1	yes	characteristic, complemented
53	C₂×Dic₃₁⋊C₄	C₂²	1	2	yes	complemented
54	#11758475	C₂²	1	2	yes	complemented
55	#11758475	C₂²	1	2	yes	complemented
56	#11758475	C₂²	1	2	yes	complemented
57	#11758474	C₂²	1	1	yes	characteristic, complemented
58	#11758474	C₂²	1	1	yes	characteristic, complemented
59	#11758473	C₂²	1	2	yes	complemented
60	#11758473	C₂²	1	2	yes	complemented
61	#11758471	C₂²	1	2	yes	—
62	#11758471	C₂²	1	2	yes	—
63	#11758470	C₂²	1	1	yes	characteristic
64	#11758469	C₂²	1	2	yes	complemented
65	#11758469	C₂²	1	1	yes	characteristic, complemented
66	#11758469	C₂²	1	2	yes	complemented
67	#11758469	C₂²	1	1	yes	characteristic, complemented
68	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
69	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
70	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
71	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
72	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
73	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
74	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
75	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
76	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
77	C₄⋊C₄×D₃₁	C₂²	1	2	yes	—
78	C₄⋊C₄×D₃₁	C₂²	1	2	yes	complemented
79	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic, complemented
80	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic, complemented
81	#11758467	C₂²	1	1	yes	characteristic, complemented
82	#11758467	C₂²	1	2	yes	complemented
83	#11758467	C₂²	1	2	yes	complemented
84	#11758467	C₂²	1	1	yes	characteristic, complemented
85	#11758466	C₂²	1	2	yes	complemented
86	#11758466	C₂²	1	2	yes	complemented
87	#11758466	C₂²	1	2	yes	complemented
88	#11758466	C₂²	1	2	yes	complemented
89	#11758466	C₂²	1	2	yes	complemented
90	#11758466	C₂²	1	2	yes	complemented
91	#11758464	C₂²	1	2	yes	complemented
92	#11758464	C₂²	1	1	yes	characteristic, complemented
93	#11758463	C₂²	1	2	yes	—
94	#11758460	C₂²	1	2	yes	—
95	#11758460	C₂²	1	2	yes	—
96	(C₂²⋊D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
97	(C₂²⋊D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
98	C₂²⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
99	C₂²⋊C₄×D₃₁	C₂²	1	2	yes	—
100	C₂²⋊C₄×D₃₁	C₂²	1	2	yes	—
101	C₂²⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
102	C₂²⋊C₄×D₃₁	C₂²	1	2	yes	—
103	#11758456	C₂²	1	2	yes	complemented
104	#11758456	C₂²	1	2	yes	complemented
105	#11758456	C₂²	1	2	yes	complemented
106	#11758455	C₂²	1	2	yes	complemented
107	#11758455	C₂²	1	2	yes	complemented
108	#11758454	C₂²	1	2	yes	complemented
109	#11758454	C₂²	1	2	yes	complemented
110	#11758454	C₂²	1	2	yes	complemented
111	#11758453	C₂²	1	2	yes	complemented
112	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
113	#11758449	C₂²	1	2	yes	complemented
114	#11758449	C₂²	1	2	yes	complemented
115	#11758449	C₂²	1	2	yes	complemented
116	C₄²×D₃₁	C₂²	1	2	yes	—
117	C₄²×D₃₁	C₂²	1	2	yes	—
118	C₄²×D₃₁	C₂²	1	1	yes	characteristic
119	#11758447	C₂²	1	2	yes	complemented
120	#11758447	C₂²	1	1	yes	characteristic, complemented
121	C₄×Dic₆₂	C₂²	1	2	yes	complemented
122	C₄×Dic₆₂	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₃₁	C₂³	1	1	yes	characteristic
124	C₂³×D₃₁	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊D₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₂²⋊D₃₁	C₂³	1	1	yes	characteristic
132	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
133	C₂²×Dic₃₁	C₂³	1	2	yes	—
134	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
135	Q₈×D₃₁	—	2	4	no	—
136	D₄×D₃₁	—	2	4	no	—
137	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
138	C₂×C₄×D₃₁	—	2	8	no	—
139	C₂×C₄×D₃₁	—	2	4	no	—
140	C₂×C₄×D₃₁	—	2	4	no	—
141	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
142	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₃₁	C₂³	1	2	yes	—
144	C₂×C₄×D₃₁	C₂³	1	2	yes	—
145	C₂×C₄×D₃₁	C₂³	1	2	yes	—
146	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₃₁	C₂³	1	2	yes	—
149	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₃₁	C₂³	1	2	yes	—
153	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₃₁	C₂³	1	2	yes	—
155	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
156	C₂×Dic₆₂	C₂³	1	2	yes	—
157	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
158	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
160	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
161	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
162	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
163	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
169	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
170	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
171	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
173	C₄²×C₃₁	C₂³	1	1	yes	characteristic
174	C₄²×C₃₁	C₂³	1	2	yes	—
175	C₄²×C₃₁	C₂³	1	2	yes	—
176	C₂²⋊Dic₃₁	C₂³	1	1	yes	characteristic
177	C₂²⋊Dic₃₁	C₂³	1	2	yes	—
178	C₂²⋊Dic₃₁	C₂³	1	2	yes	—
179	C₂²⋊Dic₃₁	C₂³	1	2	yes	—
180	C₂²⋊Dic₃₁	C₂³	1	1	yes	characteristic
181	(C₂×D₃₁)⋊C₄	C₂³	1	1	yes	characteristic
182	(C₂×D₃₁)⋊C₄	C₂³	1	1	yes	characteristic
183	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
184	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
186	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
187	(C₂×D₃₁)⋊C₄	C₂³	1	1	yes	characteristic
188	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
189	(C₂×D₃₁)⋊C₄	C₂³	1	1	yes	characteristic
190	(C₂×D₃₁)⋊C₄	C₂³	1	2	yes	—
191	C₄⋊Dic₃₁	C₂³	1	2	yes	—
192	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
193	C₄⋊Dic₃₁	C₂³	1	2	yes	—
194	C₄⋊Dic₃₁	C₂³	1	2	yes	—
195	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
196	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
197	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
198	C₄⋊Dic₃₁	C₂³	1	2	yes	—
199	C₄⋊Dic₃₁	C₂³	1	2	yes	—
200	C₄⋊Dic₃₁	C₂³	1	2	yes	—
201	Dic₃₁⋊C₄	C₂³	1	2	yes	—
202	Dic₃₁⋊C₄	C₂³	1	2	yes	—
203	Dic₃₁⋊C₄	C₂³	1	2	yes	—
204	Dic₃₁⋊C₄	C₂³	1	1	yes	characteristic
205	Dic₃₁⋊C₄	C₂³	1	2	yes	—
206	Dic₃₁⋊C₄	C₂³	1	2	yes	—
207	Dic₃₁⋊C₄	C₂³	1	2	yes	—
208	Dic₃₁⋊C₄	C₂³	1	2	yes	—
209	Dic₃₁⋊C₄	C₂³	1	2	yes	—
210	Dic₃₁⋊C₄	C₂³	1	2	yes	—
211	Dic₃₁⋊C₄	C₂³	1	2	yes	—
212	Dic₃₁⋊C₄	C₂³	1	2	yes	—
213	Dic₃₁⋊C₄	C₂³	1	2	yes	—
214	Dic₃₁⋊C₄	C₂³	1	2	yes	—
215	Dic₃₁⋊C₄	C₂³	1	2	yes	—
216	Dic₃₁⋊C₄	C₂³	1	1	yes	characteristic
217	Dic₃₁⋊C₄	C₂³	1	2	yes	—
218	C₄×Dic₃₁	C₂³	1	2	yes	—
219	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
220	C₄×Dic₃₁	C₂³	1	2	yes	—
221	C₄×Dic₃₁	C₂³	1	2	yes	—
222	C₄×Dic₃₁	C₂³	1	2	yes	—
223	C₄×Dic₃₁	C₂³	1	2	yes	—
224	C₄×Dic₃₁	C₂³	1	2	yes	—
225	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
226	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
227	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
228	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
229	C₂²×D₃₁	—	2	2	no	—
230	C₂²×D₃₁	—	2	2	no	—
231	C₂²×D₃₁	—	2	2	no	—
232	C₂²×D₃₁	—	2	2	no	—
233	C₂²×D₃₁	—	2	2	no	—
234	C₂²×D₃₁	#41	1	4	yes	—
235	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
236	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
237	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
238	Q₈×C₃₁	—	2	2	no	—
239	D₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
246	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
248	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
249	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
250	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
252	C₂²⋊D₃₁	—	2	2	no	—
253	C₂²⋊D₃₁	—	2	2	no	—
254	C₂×Dic₃₁	—	2	4	no	—
255	C₂×Dic₃₁	—	2	2	no	—
256	C₂×Dic₃₁	—	2	2	no	—
257	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
258	C₂×Dic₃₁	C₂⁴	1	2	yes	—
259	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
260	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
261	C₂×Dic₃₁	C₂⁴	1	2	yes	—
262	C₂×Dic₃₁	C₂⁴	1	2	yes	—
263	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₃₁	C₂⁴	1	2	yes	—
265	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
266	D₁₂₄	—	2	2	no	—
267	C₄×D₃₁	—	2	4	no	—
268	C₄×D₃₁	—	2	4	no	—
269	C₄×D₃₁	—	2	4	no	—
270	C₄×D₃₁	—	2	4	no	—
271	C₄×D₃₁	—	2	2	no	—
272	C₄×D₃₁	—	2	2	no	—
273	C₄×D₃₁	—	2	2	no	—
274	C₄×D₃₁	—	2	4	no	—
275	C₄×D₃₁	—	2	2	no	—
276	C₄×D₃₁	—	2	4	no	—
277	C₄×D₃₁	—	2	2	no	—
278	C₄×D₃₁	—	2	2	no	—
279	C₄×D₃₁	—	2	2	no	—
280	C₄×D₃₁	—	2	2	no	—
281	C₄×D₃₁	#41	1	4	yes	—
282	Dic₆₂	—	2	2	no	—
283	Dic₆₂	—	2	4	no	—
284	C₂×#552	—	31	1	no	maximal, Sylow
285	C₂²×C₃₁	—	2	1	no	—
286	C₂²×C₃₁	—	2	1	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	C₂×#41	1	2	yes	—
291	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
292	C₂×D₃₁	—	2	1	no	—
293	C₂×D₃₁	—	2	1	no	—
294	C₂×D₃₁	—	2	8	no	—
295	C₂×D₃₁	—	2	1	no	—
296	C₂×D₃₁	—	2	1	no	—
297	C₂×D₃₁	—	2	1	no	—
298	C₂×D₃₁	—	4	4	no	—
299	C₂×D₃₁	C₂×#41	1	2	yes	—
300	C₂×D₃₁	C₂×#41	1	2	yes	—
301	C₂×D₃₁	C₂×#41	1	2	yes	—
302	C₂×D₃₁	2₋¹⁺⁴	1	2	yes	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	C₂×#41	1	2	yes	—
312	Dic₃₁	—	2	2	no	—
313	Dic₃₁	—	2	2	no	—
314	Dic₃₁	—	2	2	no	—
315	Dic₃₁	—	2	1	no	—
316	Dic₃₁	—	2	1	no	—
317	Dic₃₁	—	2	1	no	—
318	Dic₃₁	—	2	1	no	—
319	Dic₃₁	—	2	2	no	—
320	Dic₃₁	C₂×#41	1	2	yes	—
321	#552	—	31	2	no	—
322	#552	—	31	2	no	—
323	#552	—	31	1	no	—
324	#552	—	31	2	no	—
325	#552	—	31	1	no	—
326	#552	—	31	1	no	—
327	#552	—	31	1	no	—
328	#552	—	31	1	no	—
329	#552	—	31	2	no	—
330	#552	—	31	1	no	—
331	#552	—	31	1	no	—
332	#552	D₃₁	1	1	yes	characteristic, direct product factor
333	C₂×#126	—	31	2	no	—
334	C₂×#125	—	31	2	no	—
335	C₂×#125	—	31	1	no	—
336	C₂×#123	—	31	2	no	—
337	C₂×C₂²⋊Q₈	—	31	1	no	—
338	C₂×C₂²⋊Q₈	—	31	1	no	—
339	C₂×C₄×Q₈	—	31	1	no	—
340	C₂×C₄×D₄	—	31	1	no	—
341	C₂×#117	—	31	1	no	—
342	C₂×#117	—	31	2	no	—
343	C₂²×C₄⋊C₄	—	31	1	no	—
344	C₂×C₃₁	—	2	2	no	—
345	C₂×C₃₁	—	4	1	no	—
346	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
347	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
348	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
349	D₃₁	—	2	2	no	—
350	D₃₁	—	4	1	no	—
351	D₃₁	#552	1	4	yes	direct product factor
352	C₂²×Q₈	—	31	1	no	—
353	C₂²×D₄	—	31	1	no	—
354	C₂³×C₄	—	31	2	no	—
355	C₂³×C₄	—	31	1	no	—
356	C₂³×C₄	—	31	1	no	—
357	#126	—	31	2	no	—
358	#126	—	31	2	no	—
359	#126	—	31	2	no	—
360	#126	—	31	2	no	—
361	#126	—	31	2	no	—
362	#126	—	31	2	no	—
363	#126	—	31	2	no	—
364	#126	C₂×D₃₁	1	2	yes	complemented
365	#125	—	31	2	no	—
366	#125	—	31	2	no	—
367	#125	—	31	2	no	—
368	#125	—	31	2	no	—
369	#125	—	31	2	no	—
370	#125	—	31	2	no	—
371	#125	—	31	2	no	—
372	#125	—	31	2	no	—
373	#125	—	31	1	no	—
374	#125	—	31	1	no	—
375	#125	—	31	2	no	—
376	#125	—	31	1	no	—
377	#125	C₂×D₃₁	1	2	yes	complemented
378	#125	C₂×D₃₁	1	1	yes	characteristic, complemented
379	#123	—	31	2	no	—
380	#123	—	31	2	no	—
381	#123	—	31	2	no	—
382	#123	—	31	2	no	—
383	#123	—	31	2	no	—
384	#123	—	31	2	no	—
385	#123	—	31	2	no	—
386	#123	C₂×D₃₁	1	2	yes	—
387	C₂²⋊Q₈	—	31	1	no	—
388	C₂²⋊Q₈	—	31	1	no	—
389	C₂²⋊Q₈	—	31	1	no	—
390	C₂²⋊Q₈	—	31	1	no	—
391	C₂²⋊Q₈	—	31	2	no	—
392	C₂²⋊Q₈	—	31	2	no	—
393	C₂²⋊Q₈	—	31	1	no	—
394	C₂²⋊Q₈	—	31	1	no	—
395	C₂²⋊Q₈	—	31	2	no	—
396	C₂²⋊Q₈	—	31	2	no	—
397	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
398	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
399	C₄×Q₈	—	31	1	no	—
400	C₄×Q₈	—	31	2	no	—
401	C₄×Q₈	—	31	1	no	—
402	C₄×Q₈	—	31	1	no	—
403	C₄×Q₈	—	31	2	no	—
404	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
405	C₄×D₄	—	31	1	no	—
406	C₄×D₄	—	31	1	no	—
407	C₄×D₄	—	31	1	no	—
408	C₄×D₄	—	31	1	no	—
409	C₄×D₄	—	31	1	no	—
410	C₄×D₄	—	31	1	no	—
411	C₄×D₄	—	31	1	no	—
412	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
413	#117	—	31	1	no	—
414	#117	—	31	2	no	—
415	#117	—	31	1	no	—
416	#117	—	31	1	no	—
417	#117	—	31	2	no	—
418	#117	—	31	2	no	—
419	#117	—	31	2	no	—
420	#117	—	31	2	no	—
421	#117	—	31	2	no	—
422	#117	—	31	2	no	—
423	#117	—	31	2	no	—
424	#117	—	31	2	no	—
425	#117	C₂×D₃₁	1	1	yes	characteristic, complemented
426	#117	C₂×D₃₁	1	2	yes	complemented
427	C₂×C₄⋊C₄	—	31	2	no	—
428	C₂×C₄⋊C₄	—	31	1	no	—
429	C₂×C₄⋊C₄	—	31	1	no	—
430	C₂×C₄⋊C₄	—	31	1	no	—
431	C₂×C₄⋊C₄	—	31	2	no	—
432	C₂×C₄⋊C₄	—	31	2	no	—
433	C₂×C₄⋊C₄	—	31	1	no	—
434	C₂×C₄⋊C₄	—	31	1	no	—
435	C₂×C₄⋊C₄	—	31	1	no	—
436	C₂×C₄⋊C₄	—	31	2	no	—
437	C₂×C₄⋊C₄	—	31	2	no	—
438	C₂×C₄⋊C₄	—	31	1	no	—
439	C₂×C₄⋊C₄	—	31	2	no	—
440	C₂×C₄⋊C₄	—	31	2	no	—
441	C₂×C₄⋊C₄	—	31	2	no	—
442	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic, complemented
443	C₂×C₂²⋊C₄	—	31	1	no	—
444	C₂×C₂²⋊C₄	—	31	2	no	—
445	C₂×C₂²⋊C₄	—	31	2	no	—
446	C₂×C₂²⋊C₄	—	31	1	no	—
447	C₂×C₂²⋊C₄	—	31	2	no	—
448	C₂×C₄²	—	31	2	no	—
449	C₂×C₄²	—	31	2	no	—
450	C₂×C₄²	—	31	1	no	—
451	C₃₁	C₂×#552	1	1	yes	characteristic, complemented, Sylow
452	C₂⁴	—	31	1	no	—
453	C₂⁴	—	31	1	no	—
454	C₂×Q₈	—	31	2	no	—
455	C₂×Q₈	—	31	1	no	—
456	C₂×Q₈	—	62	4	no	—
457	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
458	C₂×D₄	—	31	1	no	—
459	C₂×D₄	—	31	1	no	—
460	C₂×D₄	—	31	1	no	—
461	C₂×D₄	—	62	4	no	—
462	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
463	C₂²×C₄	—	31	1	no	—
464	C₂²×C₄	—	31	2	no	—
465	C₂²×C₄	—	31	1	no	—
466	C₂²×C₄	—	31	1	no	—
467	C₂²×C₄	—	31	1	no	—
468	C₂²×C₄	—	31	1	no	—
469	C₂²×C₄	—	31	2	no	—
470	C₂²×C₄	—	31	2	no	—
471	C₂²×C₄	—	31	2	no	—
472	C₂²×C₄	—	31	1	no	—
473	C₂²×C₄	—	31	1	no	—
474	C₂²×C₄	—	31	2	no	—
475	C₂²×C₄	—	31	2	no	—
476	C₂²×C₄	—	31	2	no	—
477	C₂²×C₄	—	31	1	no	—
478	C₂²×C₄	—	31	1	no	—
479	C₂²×C₄	—	31	1	no	—
480	C₂²×C₄	—	31	1	no	—
481	C₂²×C₄	—	62	4	no	—
482	C₂²×C₄	—	62	4	no	—
483	C₂²×C₄	—	62	8	no	—
484	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
485	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
487	C₄⋊C₄	—	31	2	no	—
488	C₄⋊C₄	—	31	2	no	—
489	C₄⋊C₄	—	31	2	no	—
490	C₄⋊C₄	—	31	2	no	—
491	C₄⋊C₄	—	31	2	no	—
492	C₄⋊C₄	—	31	2	no	—
493	C₄⋊C₄	—	31	2	no	—
494	C₄⋊C₄	—	31	2	no	—
495	C₄⋊C₄	—	31	2	no	—
496	C₄⋊C₄	—	31	1	no	—
497	C₄⋊C₄	—	31	1	no	—
498	C₄⋊C₄	—	31	2	no	—
499	C₄⋊C₄	—	31	2	no	—
500	C₄⋊C₄	—	31	2	no	—
501	C₄⋊C₄	—	31	2	no	—
502	C₄⋊C₄	—	31	2	no	—
503	C₄⋊C₄	—	31	2	no	—
504	C₄⋊C₄	—	31	1	no	—
505	C₄⋊C₄	—	31	1	no	—
506	C₄⋊C₄	—	31	2	no	—
507	C₄⋊C₄	—	31	2	no	—
508	C₄⋊C₄	—	31	1	no	—
509	C₄⋊C₄	—	31	2	no	—
510	C₄⋊C₄	—	31	1	no	—
511	C₄⋊C₄	—	31	2	no	—
512	C₄⋊C₄	—	31	2	no	—
513	C₄⋊C₄	—	31	2	no	—
514	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
517	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
518	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
520	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
522	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
523	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
524	C₂²⋊C₄	—	31	1	no	—
525	C₂²⋊C₄	—	31	2	no	—
526	C₂²⋊C₄	—	31	2	no	—
527	C₂²⋊C₄	—	31	1	no	—
528	C₂²⋊C₄	—	31	2	no	—
529	C₂²⋊C₄	—	31	1	no	—
530	C₂²⋊C₄	—	31	1	no	—
531	C₂²⋊C₄	—	31	2	no	—
532	C₂²⋊C₄	—	31	2	no	—
533	C₂²⋊C₄	—	31	2	no	—
534	C₂²⋊C₄	—	31	2	no	—
535	C₂²⋊C₄	—	31	1	no	—
536	C₂²⋊C₄	—	31	1	no	—
537	C₂²⋊C₄	—	31	2	no	—
538	C₂²⋊C₄	—	31	2	no	—
539	C₂²⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
540	C₂²⋊C₄	C₂²×D₃₁	1	2	yes	—
541	C₂²⋊C₄	C₂²×D₃₁	1	2	yes	—
542	C₂²⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×D₃₁	1	2	yes	—
544	C₄²	—	31	2	no	—
545	C₄²	—	31	2	no	—
546	C₄²	—	31	2	no	—
547	C₄²	—	31	2	no	—
548	C₄²	—	31	2	no	—
549	C₄²	—	31	2	no	—
550	C₄²	—	31	1	no	—
551	C₄²	—	31	1	no	—
552	C₄²	—	31	1	no	—
553	C₄²	C₂²×D₃₁	1	2	yes	—
554	C₄²	C₂²×D₃₁	1	2	yes	—
555	C₄²	C₂²×D₃₁	1	1	yes	characteristic
556	C₂³	—	31	1	no	—
557	C₂³	—	31	4	no	—
558	C₂³	—	31	1	no	—
559	C₂³	—	31	1	no	—
560	C₂³	—	62	2	no	—
561	C₂³	—	62	2	no	—
562	C₂³	—	62	2	no	—
563	C₂³	—	62	2	no	—
564	C₂³	—	62	2	no	—
565	C₂³	C₂³×D₃₁	1	1	yes	characteristic
566	C₂³	C₂³×D₃₁	1	1	yes	characteristic
567	Q₈	—	2	2	no	—
568	Q₈	—	62	4	no	—
569	Q₈	—	62	2	no	—
570	D₄	—	2	2	no	—
571	D₄	—	62	2	no	—
572	D₄	—	62	2	no	—
573	D₄	—	62	2	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	31	1	no	—
578	C₂×C₄	—	31	4	no	—
579	C₂×C₄	—	31	1	no	—
580	C₂×C₄	—	31	1	no	—
581	C₂×C₄	—	31	2	no	—
582	C₂×C₄	—	31	2	no	—
583	C₂×C₄	—	31	2	no	—
584	C₂×C₄	—	31	1	no	—
585	C₂×C₄	—	31	1	no	—
586	C₂×C₄	—	31	2	no	—
587	C₂×C₄	—	62	2	no	—
588	C₂×C₄	—	62	2	no	—
589	C₂×C₄	—	62	2	no	—
590	C₂×C₄	—	62	4	no	—
591	C₂×C₄	—	62	2	no	—
592	C₂×C₄	—	62	2	no	—
593	C₂×C₄	—	62	4	no	—
594	C₂×C₄	—	62	4	no	—
595	C₂×C₄	—	62	4	no	—
596	C₂×C₄	—	62	4	no	—
597	C₂×C₄	—	62	2	no	—
598	C₂×C₄	—	62	4	no	—
599	C₂×C₄	—	62	2	no	—
600	C₂×C₄	—	62	2	no	—
601	C₂×C₄	—	62	4	no	—
602	C₂×C₄	—	62	2	no	—
603	C₂×C₄	—	62	2	no	—
604	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
605	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂³×D₃₁	1	2	yes	—
607	C₂×C₄	C₂³×D₃₁	1	2	yes	—
608	C₂×C₄	C₂³×D₃₁	1	2	yes	—
609	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
611	C₂×C₄	C₂³×D₃₁	1	2	yes	—
612	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	31	2	no	—
619	C₂²	—	31	2	no	—
620	C₂²	—	31	2	no	—
621	C₂²	—	31	2	no	—
622	C₂²	—	62	8	no	—
623	C₂²	—	62	1	no	—
624	C₂²	—	62	1	no	—
625	C₂²	—	62	1	no	—
626	C₂²	—	62	1	no	—
627	C₂²	—	62	1	no	—
628	C₂²	—	124	4	no	—
629	C₂²	#41×D₃₁	1	2	yes	—
630	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	31	2	no	—
640	C₄	—	62	2	no	—
641	C₄	—	62	2	no	—
642	C₄	—	62	1	no	—
643	C₄	—	62	1	no	—
644	C₄	—	62	1	no	—
645	C₄	—	62	2	no	—
646	C₄	—	62	1	no	—
647	C₄	—	62	2	no	—
648	C₄	#41×D₃₁	1	2	yes	—
649	C₂	—	2	2	no	—
650	C₂	—	4	1	no	—
651	C₂	—	31	4	no	—
652	C₂	—	62	2	no	—
653	C₂	—	124	1	no	—
654	C₂	C₂×#41×D₃₁	1	1	yes	characteristic
655	C₂	C₂×#41×D₃₁	1	1	yes	characteristic
656	C₂	2₋¹⁺⁴×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁵
Fitting	C₃₁×#552	C₂
prime radical	C₂×D₄×D₃₁	C₂²
hypercenter	#552	D₃₁
hypocenter	C₃₁	C₂×#552
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #552

Complements

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

N	H ≅ G/N	type
1. #14521741	C₂	semidirect
2. #14521740	C₂	semidirect
3. #14521739	C₂	semidirect
4. #14521738	C₂	semidirect
5. #14521737	C₂	semidirect
6. #14521736	C₂	semidirect
7. #14521735	C₂	semidirect
8. #14521734	C₂	semidirect
9. #14521733	C₂	semidirect
10. #14521732	C₂	semidirect
11. #14521731	C₂	semidirect
12. #126×D₃₁	C₂	semidirect
13. #125×D₃₁	C₂	semidirect
14. #125×D₃₁	C₂	semidirect
16. C₂²⋊Q₈×D₃₁	C₂	semidirect
17. C₂²⋊Q₈×D₃₁	C₂	semidirect
18. C₄×Q₈×D₃₁	C₂	semidirect
20. #117×D₃₁	C₂	semidirect
21. #117×D₃₁	C₂	semidirect
22. C₂×C₄⋊C₄×D₃₁	C₂	semidirect
23. C₃₁×#552	C₂	semidirect
29. C₃₁×#126	C₂²	semidirect
30. C₃₁×#125	C₂²	semidirect
31. C₃₁×#125	C₂²	semidirect
33. C₃₁×C₂²⋊Q₈	C₂²	semidirect
34. C₃₁×C₂²⋊Q₈	C₂²	semidirect
35. C₄×Q₈×C₃₁	C₂²	semidirect
37. C₃₁×#117	C₂²	semidirect
38. C₃₁×#117	C₂²	semidirect
39. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
40. #11758523	C₂²	semidirect
41. #11758523	C₂²	semidirect
42. Q₈×Dic₃₁	C₂²	semidirect
48. #11758503	C₂²	semidirect
49. #11758503	C₂²	semidirect
50. C₂×C₄⋊Dic₃₁	C₂²	semidirect
51. #11758501	C₂²	semidirect
52. #11758501	C₂²	semidirect
53. C₂×Dic₃₁⋊C₄	C₂²	semidirect
54. #11758475	C₂²	semidirect
55. #11758475	C₂²	semidirect
56. #11758475	C₂²	semidirect
57. #11758474	C₂²	semidirect
58. #11758474	C₂²	semidirect
59. #11758473	C₂²	semidirect
60. #11758473	C₂²	semidirect
64. #11758469	C₂²	semidirect
65. #11758469	C₂²	semidirect
66. #11758469	C₂²	semidirect
67. #11758469	C₂²	semidirect
78. C₄⋊C₄×D₃₁	C₂²	semidirect
79. C₄⋊C₄×D₃₁	C₂²	semidirect
80. C₄⋊C₄×D₃₁	C₂²	semidirect
81. #11758467	C₂²	semidirect
82. #11758467	C₂²	semidirect
83. #11758467	C₂²	semidirect
84. #11758467	C₂²	semidirect
85. #11758466	C₂²	semidirect
86. #11758466	C₂²	semidirect
87. #11758466	C₂²	semidirect
88. #11758466	C₂²	semidirect
89. #11758466	C₂²	semidirect
90. #11758466	C₂²	semidirect
91. #11758464	C₂²	semidirect
92. #11758464	C₂²	semidirect
103. #11758456	C₂²	semidirect
104. #11758456	C₂²	semidirect
105. #11758456	C₂²	semidirect
106. #11758455	C₂²	semidirect
107. #11758455	C₂²	semidirect
108. #11758454	C₂²	semidirect
109. #11758454	C₂²	semidirect
110. #11758454	C₂²	semidirect
111. #11758453	C₂²	semidirect
113. #11758449	C₂²	semidirect
114. #11758449	C₂²	semidirect
115. #11758449	C₂²	semidirect
119. #11758447	C₂²	semidirect
120. #11758447	C₂²	semidirect
121. C₄×Dic₆₂	C₂²	semidirect
122. C₄×Dic₆₂	C₂²	semidirect
332. #552	D₃₁	direct (N × H)
351. D₃₁	#552	direct (N × H)
364. #126	C₂×D₃₁	semidirect
377. #125	C₂×D₃₁	semidirect
378. #125	C₂×D₃₁	semidirect
397. C₂²⋊Q₈	C₂×D₃₁	semidirect
398. C₂²⋊Q₈	C₂×D₃₁	semidirect
404. C₄×Q₈	C₂×D₃₁	semidirect
425. #117	C₂×D₃₁	semidirect
426. #117	C₂×D₃₁	semidirect
442. C₂×C₄⋊C₄	C₂×D₃₁	semidirect
451. C₃₁	C₂×#552	semidirect

Maximal subgroups

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

subgroup	index
#14521741	2
#14521740	2
#14521739	2
#14521738	2
#14521737	2
#14521736	2
#14521735	2
#14521734	2
#14521733	2
#14521732	2
#14521731	2
#126×D₃₁	2
#125×D₃₁	2
#123×D₃₁	2
C₂²⋊Q₈×D₃₁	2
C₄×Q₈×D₃₁	2
C₄×D₄×D₃₁	2
#117×D₃₁	2
C₂×C₄⋊C₄×D₃₁	2
C₃₁×#552	2
C₂×#552	31

Subgroups — D31×#552 #16859214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#552 #16859214