Subgroups — C₂²×C₃₁×#552 #21519463

Order 7936.

7934 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#552	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×#126	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#125	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃₁×#123	C₂	1	2	yes	maximal
6	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₃₁×#117	C₂	1	2	yes	maximal, complemented
12	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#552	C₂²	1	256	yes	direct product factor
19	C₂×C₃₁×#126	C₂²	1	48	yes	complemented
20	C₂×C₃₁×#125	C₂²	1	48	yes	complemented
21	C₂×C₃₁×#125	C₂²	1	24	yes	complemented
22	C₂×C₃₁×#123	C₂²	1	48	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
25	C₂×C₄×Q₈×C₃₁	C₂²	1	24	yes	complemented
26	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	—
27	C₂×C₃₁×#117	C₂²	1	24	yes	complemented
28	C₂×C₃₁×#117	C₂²	1	48	yes	complemented
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₂²×C₄⋊C₄×C₃₁	C₂²	1	24	yes	complemented
40	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
43	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
45	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
46	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
47	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
48	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	—	2	8	no	—
51	C₂²×Q₈×C₃₁	C₂³	1	12	yes	—
52	C₂²×D₄×C₃₁	—	2	8	no	—
53	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
54	C₂³×C₄×C₃₁	—	2	16	no	—
55	C₂³×C₄×C₃₁	—	2	8	no	—
56	C₂³×C₄×C₃₁	—	2	8	no	—
57	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#126	C₂³	1	128	yes	complemented
70	C₃₁×#125	C₂³	1	128	yes	complemented
71	C₃₁×#125	C₂³	1	64	yes	complemented
72	C₃₁×#123	C₂³	1	128	yes	—
73	C₃₁×C₂²⋊Q₈	C₂³	1	64	yes	complemented
74	C₃₁×C₂²⋊Q₈	C₂³	1	64	yes	complemented
75	C₄×Q₈×C₃₁	C₂³	1	64	yes	complemented
76	C₄×D₄×C₃₁	C₂³	1	64	yes	—
77	C₃₁×#117	C₂³	1	64	yes	complemented
78	C₃₁×#117	C₂³	1	128	yes	complemented
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	64	yes	complemented
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	24	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	24	yes	—
97	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
98	C₂⁴×C₃₁	—	2	4	no	—
99	C₂⁴×C₃₁	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	4	no	—
102	C₂⁴×C₃₁	—	2	4	no	—
103	C₂⁴×C₃₁	#41	1	8	yes	—
104	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×Q₈×C₃₁	—	2	48	no	—
108	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	—
109	C₂×D₄×C₃₁	—	2	48	no	—
110	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
111	C₂²×C₄×C₃₁	—	2	96	no	—
112	C₂²×C₄×C₃₁	—	2	8	no	—
113	C₂²×C₄×C₃₁	—	2	8	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	4	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	2	48	no	—
121	C₂²×C₄×C₃₁	—	2	48	no	—
122	C₂²×C₄×C₃₁	#41	1	8	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
135	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
136	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
137	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
138	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
139	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
140	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
141	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
142	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
143	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
144	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
145	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
146	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
147	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
148	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
149	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
150	C₄²×C₃₁	C₂⁴	1	32	yes	—
151	C₄²×C₃₁	C₂⁴	1	32	yes	—
152	C₄²×C₃₁	C₂⁴	1	16	yes	—
153	C₂²×#552	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	32	no	—
155	C₂³×C₃₁	—	2	12	no	—
156	C₂³×C₃₁	—	2	12	no	—
157	C₂³×C₃₁	—	2	12	no	—
158	C₂³×C₃₁	—	2	12	no	—
159	C₂³×C₃₁	—	2	12	no	—
160	C₂³×C₃₁	—	4	16	no	—
161	C₂³×C₃₁	C₂×#41	1	24	yes	—
162	C₂³×C₃₁	C₂×#41	1	4	yes	—
163	C₂³×C₃₁	C₂×#41	1	4	yes	—
164	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
165	C₂³×C₃₁	C₂⁵	1	4	yes	—
166	C₂³×C₃₁	C₂⁵	1	4	yes	—
167	C₂³×C₃₁	C₂⁵	1	3	yes	—
168	Q₈×C₃₁	—	2	32	no	—
169	D₄×C₃₁	—	2	32	no	—
170	C₂×C₄×C₃₁	—	2	64	no	—
171	C₂×C₄×C₃₁	—	2	24	no	—
172	C₂×C₄×C₃₁	—	2	24	no	—
173	C₂×C₄×C₃₁	—	2	12	no	—
174	C₂×C₄×C₃₁	—	2	12	no	—
175	C₂×C₄×C₃₁	—	2	12	no	—
176	C₂×C₄×C₃₁	—	2	24	no	—
177	C₂×C₄×C₃₁	—	2	12	no	—
178	C₂×C₄×C₃₁	—	2	24	no	—
179	C₂×C₄×C₃₁	—	2	32	no	—
180	C₂×C₄×C₃₁	—	2	32	no	—
181	C₂×C₄×C₃₁	C₂×#41	1	24	yes	—
182	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
183	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
184	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
185	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
186	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
187	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
188	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
189	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
190	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
191	C₂×#552	C₂×C₃₁	1	48	yes	direct product factor
192	C₂²×#126	C₂×C₃₁	1	2	yes	complemented
193	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×#125	C₂×C₃₁	1	2	yes	complemented
195	C₂²×#123	C₂×C₃₁	1	2	yes	—
196	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
200	C₂²×#117	C₂×C₃₁	1	2	yes	complemented
201	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	2	48	no	—
206	C₂²×C₃₁	—	2	4	no	—
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	2	4	no	—
209	C₂²×C₃₁	—	4	24	no	—
210	C₂²×C₃₁	#552	1	16	yes	direct product factor
211	C₂²×C₃₁	C₂²×#41	1	8	yes	—
212	C₂²×C₃₁	C₂²×#41	1	6	yes	—
213	C₂²×C₃₁	C₂²×#41	1	6	yes	—
214	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	6	yes	—
215	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
216	C₄×C₃₁	—	2	8	no	—
217	C₄×C₃₁	—	2	8	no	—
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	—	2	4	no	—
221	C₄×C₃₁	—	2	8	no	—
222	C₄×C₃₁	—	2	4	no	—
223	C₄×C₃₁	—	2	8	no	—
224	C₄×C₃₁	C₂²×#41	1	8	yes	—
225	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
228	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
229	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
230	#552	C₂²×C₃₁	1	256	yes	direct product factor
231	C₂×#126	C₂²×C₃₁	1	48	yes	complemented
232	C₂×#125	C₂²×C₃₁	1	48	yes	complemented
233	C₂×#125	C₂²×C₃₁	1	24	yes	complemented
234	C₂×#123	C₂²×C₃₁	1	48	yes	—
235	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
236	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
237	C₂×C₄×Q₈	C₂²×C₃₁	1	24	yes	complemented
238	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	—
239	C₂×#117	C₂²×C₃₁	1	24	yes	complemented
240	C₂×#117	C₂²×C₃₁	1	48	yes	complemented
241	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
250	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₄⋊C₄	C₂²×C₃₁	1	24	yes	complemented
252	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
253	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
254	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
255	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
256	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
257	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
258	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
259	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₃₁	—	2	8	no	—
261	C₂×C₃₁	—	4	4	no	—
262	C₂×C₃₁	C₂×#552	1	12	yes	direct product factor
263	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
264	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
265	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
266	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
267	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×Q₈	—	2	8	no	—
269	C₂²×Q₈	C₂³×C₃₁	1	12	yes	—
270	C₂²×D₄	—	2	8	no	—
271	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
272	C₂³×C₄	—	2	16	no	—
273	C₂³×C₄	—	2	8	no	—
274	C₂³×C₄	—	2	8	no	—
275	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
276	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
277	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
280	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
284	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
285	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
286	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
287	#126	C₂³×C₃₁	1	128	yes	complemented
288	#125	C₂³×C₃₁	1	128	yes	complemented
289	#125	C₂³×C₃₁	1	64	yes	complemented
290	#123	C₂³×C₃₁	1	128	yes	—
291	C₂²⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
292	C₂²⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
293	C₄×Q₈	C₂³×C₃₁	1	64	yes	complemented
294	C₄×D₄	C₂³×C₃₁	1	64	yes	—
295	#117	C₂³×C₃₁	1	64	yes	complemented
296	#117	C₂³×C₃₁	1	128	yes	complemented
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	64	yes	complemented
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
313	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
314	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
315	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
316	C₃₁	C₂²×#552	1	1	yes	characteristic, direct product factor, Sylow
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	4	no	—
320	C₂⁴	—	2	4	no	—
321	C₂⁴	—	2	4	no	—
322	C₂⁴	#41×C₃₁	1	8	yes	—
323	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
324	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
325	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
326	C₂×Q₈	—	2	48	no	—
327	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	—
328	C₂×D₄	—	2	48	no	—
329	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
330	C₂²×C₄	—	2	96	no	—
331	C₂²×C₄	—	2	8	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	8	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	48	no	—
340	C₂²×C₄	—	2	48	no	—
341	C₂²×C₄	#41×C₃₁	1	8	yes	—
342	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
343	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
344	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
345	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
346	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
347	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
348	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
351	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
352	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
354	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
355	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
356	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
357	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
358	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
359	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
360	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
361	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
362	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
363	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
364	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
365	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
366	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
367	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
368	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
369	C₄²	C₂⁴×C₃₁	1	32	yes	—
370	C₄²	C₂⁴×C₃₁	1	32	yes	—
371	C₄²	C₂⁴×C₃₁	1	16	yes	—
372	C₂³	—	2	32	no	—
373	C₂³	—	2	12	no	—
374	C₂³	—	2	12	no	—
375	C₂³	—	2	12	no	—
376	C₂³	—	2	12	no	—
377	C₂³	—	2	12	no	—
378	C₂³	—	4	16	no	—
379	C₂³	C₂×#41×C₃₁	1	24	yes	—
380	C₂³	C₂×#41×C₃₁	1	4	yes	—
381	C₂³	C₂×#41×C₃₁	1	4	yes	—
382	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
383	C₂³	C₂⁵×C₃₁	1	4	yes	—
384	C₂³	C₂⁵×C₃₁	1	4	yes	—
385	C₂³	C₂⁵×C₃₁	1	3	yes	—
386	Q₈	—	2	32	no	—
387	D₄	—	2	32	no	—
388	C₂×C₄	—	2	64	no	—
389	C₂×C₄	—	2	24	no	—
390	C₂×C₄	—	2	24	no	—
391	C₂×C₄	—	2	12	no	—
392	C₂×C₄	—	2	12	no	—
393	C₂×C₄	—	2	12	no	—
394	C₂×C₄	—	2	24	no	—
395	C₂×C₄	—	2	12	no	—
396	C₂×C₄	—	2	24	no	—
397	C₂×C₄	—	2	32	no	—
398	C₂×C₄	—	2	32	no	—
399	C₂×C₄	C₂×#41×C₃₁	1	24	yes	—
400	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
401	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
402	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
403	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
404	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
405	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
406	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
407	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
408	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	2	48	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	4	24	no	—
416	C₂²	C₃₁×#552	1	16	yes	direct product factor
417	C₂²	C₂²×#41×C₃₁	1	8	yes	—
418	C₂²	C₂²×#41×C₃₁	1	6	yes	—
419	C₂²	C₂²×#41×C₃₁	1	6	yes	—
420	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	6	yes	—
421	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
422	C₄	—	2	8	no	—
423	C₄	—	2	8	no	—
424	C₄	—	2	4	no	—
425	C₄	—	2	4	no	—
426	C₄	—	2	4	no	—
427	C₄	—	2	8	no	—
428	C₄	—	2	4	no	—
429	C₄	—	2	8	no	—
430	C₄	C₂²×#41×C₃₁	1	8	yes	—
431	C₂	—	2	8	no	—
432	C₂	—	4	4	no	—
433	C₂	C₂×C₃₁×#552	1	12	yes	direct product factor
434	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
435	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
436	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#552	C₂	direct (N × H)
2. C₂²×C₃₁×#126	C₂	semidirect
3. C₂²×C₃₁×#125	C₂	semidirect
4. C₂²×C₃₁×#125	C₂	semidirect
6. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
7. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
8. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
10. C₂²×C₃₁×#117	C₂	semidirect
11. C₂²×C₃₁×#117	C₂	semidirect
12. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
18. C₃₁×#552	C₂²	direct (N × H)
19. C₂×C₃₁×#126	C₂²	semidirect
20. C₂×C₃₁×#125	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
23. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
24. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
25. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
27. C₂×C₃₁×#117	C₂²	semidirect
28. C₂×C₃₁×#117	C₂²	semidirect
39. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
69. C₃₁×#126	C₂³	semidirect
70. C₃₁×#125	C₂³	semidirect
71. C₃₁×#125	C₂³	semidirect
73. C₃₁×C₂²⋊Q₈	C₂³	semidirect
74. C₃₁×C₂²⋊Q₈	C₂³	semidirect
75. C₄×Q₈×C₃₁	C₂³	semidirect
77. C₃₁×#117	C₂³	semidirect
78. C₃₁×#117	C₂³	semidirect
89. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
153. C₂²×#552	C₃₁	direct (N × H)
191. C₂×#552	C₂×C₃₁	direct (N × H)
192. C₂²×#126	C₂×C₃₁	semidirect
193. C₂²×#125	C₂×C₃₁	semidirect
194. C₂²×#125	C₂×C₃₁	semidirect
196. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
197. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
198. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
200. C₂²×#117	C₂×C₃₁	semidirect
201. C₂²×#117	C₂×C₃₁	semidirect
202. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
210. C₂²×C₃₁	#552	direct (N × H)
230. #552	C₂²×C₃₁	direct (N × H)
231. C₂×#126	C₂²×C₃₁	semidirect
232. C₂×#125	C₂²×C₃₁	semidirect
233. C₂×#125	C₂²×C₃₁	semidirect
235. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
236. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
237. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
239. C₂×#117	C₂²×C₃₁	semidirect
240. C₂×#117	C₂²×C₃₁	semidirect
251. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
262. C₂×C₃₁	C₂×#552	direct (N × H)
287. #126	C₂³×C₃₁	semidirect
288. #125	C₂³×C₃₁	semidirect
289. #125	C₂³×C₃₁	semidirect
291. C₂²⋊Q₈	C₂³×C₃₁	semidirect
292. C₂²⋊Q₈	C₂³×C₃₁	semidirect
293. C₄×Q₈	C₂³×C₃₁	semidirect
295. #117	C₂³×C₃₁	semidirect
296. #117	C₂³×C₃₁	semidirect
307. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
316. C₃₁	C₂²×#552	direct (N × H)
416. C₂²	C₃₁×#552	direct (N × H)
433. C₂	C₂×C₃₁×#552	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×C₃₁×#552	2
C₂²×C₃₁×#126	2
C₂²×C₃₁×#125	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₂²×C₃₁×#117	2
C₂³×C₄⋊C₄×C₃₁	2
C₂²×#552	31

Subgroups — C22×C31×#552 #21519463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#552 #21519463