Subgroups — C₂×C₃×D₄×M₁₆ #10678745

Order 768.

3218 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×M₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄×M₁₆	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#436	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#435	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×C₈×D₄	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#407	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₃×D₄	C₄	1	1	yes	characteristic
12	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
13	C₂²×C₃×M₁₆	—	2	16	no	—
14	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
16	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×M₁₆	C₂²	1	4	yes	complemented
18	C₂²×C₃×M₁₆	C₂²	1	16	yes	complemented
19	C₂³×C₃×C₈	C₂²	1	4	yes	complemented
20	C₂×C₃×C₄×D₄	C₄	1	2	yes	—
21	C₂×C₃×C₄×D₄	C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×D₄	C₄	1	8	yes	—
23	C₂×C₃×C₄×D₄	C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄×D₄	C₂²	1	2	yes	—
26	C₂×C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄×D₄	C₂²	1	8	yes	—
28	C₂²×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
29	C₂²×C₃×C₂²⋊C₄	C₄	1	2	yes	—
30	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×#436	C₂²	1	16	yes	complemented
32	C₃×#435	C₂²	1	32	yes	complemented
33	C₃×C₈×D₄	C₂²	1	16	yes	complemented
34	C₃×#423	C₂²	1	8	yes	complemented
35	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃×#407	C₂²	1	16	yes	complemented
38	C₂×C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
39	C₂×C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
40	C₃×C₄×M₁₆	C₂²	1	8	yes	complemented
41	C₂×C₃×#97	C₂²	1	2	yes	complemented
42	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
43	C₂³×M₁₆	C₂×C₃	1	2	yes	complemented
44	C₂²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
45	D₄×M₁₆	C₂×C₃	1	16	yes	direct product factor
46	C₂×#436	C₂×C₃	1	2	yes	complemented
47	C₂×#435	C₂×C₃	1	4	yes	complemented
48	C₂×C₈×D₄	C₂×C₃	1	2	yes	complemented
49	C₂×#423	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#407	C₂×C₃	1	2	yes	complemented
51	C₂×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂⁵×C₃	C₂×C₄	1	2	yes	—
53	C₂²×C₃×D₄	—	2	8	no	—
54	C₂²×C₃×D₄	C₂×C₄	1	2	yes	—
55	C₂²×C₃×D₄	C₂×C₄	1	8	yes	—
56	C₂²×C₃×D₄	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×D₄	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₃×C₄	—	2	8	no	—
59	C₂³×C₃×C₄	—	2	8	no	—
60	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
61	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
62	C₂³×C₃×C₄	C₂×C₄	1	8	yes	—
63	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
64	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₃×C₄	C₂³	1	8	yes	—
67	C₂³×C₃×C₄	C₂³	1	2	yes	—
68	C₂³×C₃×C₄	C₂³	1	2	yes	—
69	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂×C₃×M₁₆	—	2	8	no	—
71	C₂×C₃×M₁₆	—	2	4	no	—
72	C₂×C₃×M₁₆	—	2	64	no	—
73	C₂×C₃×M₁₆	—	2	16	no	—
74	C₂×C₃×M₁₆	D₄	1	8	yes	direct product factor
75	C₂×C₃×M₁₆	C₂³	1	8	yes	—
76	C₂×C₃×M₁₆	C₂³	1	4	yes	—
77	C₂×C₃×M₁₆	C₂³	1	4	yes	—
78	C₂×C₃×M₁₆	C₂³	1	16	yes	complemented
79	C₂²×C₃×C₈	—	2	16	no	—
80	C₂²×C₃×C₈	C₂³	1	4	yes	—
81	C₂²×C₃×C₈	C₂³	1	2	yes	—
82	C₂²×C₃×C₈	C₂³	1	2	yes	—
83	C₂²×C₃×C₈	C₂³	1	16	yes	complemented
84	C₃×C₄×D₄	C₂×C₄	1	8	yes	—
85	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
86	C₃×C₄×D₄	C₂×C₄	1	16	yes	—
87	C₃×C₄×D₄	C₂×C₄	1	8	yes	—
88	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
89	C₃×C₄×D₄	C₂×C₄	1	8	yes	—
90	C₃×C₄×D₄	C₂³	1	4	yes	—
91	C₃×C₄×D₄	C₂³	1	8	yes	—
92	C₃×C₄×D₄	C₂³	1	4	yes	—
93	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
94	C₂×C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
95	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
98	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
99	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
100	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
101	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	16	yes	—
102	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
103	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
104	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
105	C₂×C₃×C₄²	C₂³	1	4	yes	—
106	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
107	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
108	C₃×C₄⋊C₈	C₂³	1	8	yes	complemented
109	C₃×C₄⋊C₈	C₂³	1	8	yes	complemented
110	C₃×C₂²⋊C₈	C₂³	1	16	yes	complemented
111	C₃×C₂²⋊C₈	C₂³	1	16	yes	complemented
112	C₃×#97	C₂³	1	8	yes	complemented
113	C₃×C₄×C₈	C₂³	1	8	yes	complemented
114	C₂³×D₄	C₃×C₄	1	1	yes	characteristic
115	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
116	C₂²×M₁₆	—	2	16	no	—
117	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
118	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
119	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
120	C₂²×M₁₆	C₂²×C₃	1	4	yes	complemented
121	C₂²×M₁₆	C₂²×C₃	1	16	yes	complemented
122	C₂³×C₈	C₂²×C₃	1	4	yes	complemented
123	C₂×C₄×D₄	C₃×C₄	1	2	yes	—
124	C₂×C₄×D₄	C₃×C₄	1	1	yes	characteristic
125	C₂×C₄×D₄	C₃×C₄	1	8	yes	—
126	C₂×C₄×D₄	C₃×C₄	1	1	yes	characteristic
127	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
128	C₂×C₄×D₄	C₂²×C₃	1	2	yes	—
129	C₂×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
130	C₂×C₄×D₄	C₂²×C₃	1	8	yes	—
131	C₂²×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
133	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
134	#436	C₂²×C₃	1	16	yes	complemented
135	#435	C₂²×C₃	1	32	yes	complemented
136	C₈×D₄	C₂²×C₃	1	16	yes	complemented
137	#423	C₂²×C₃	1	8	yes	complemented
138	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
139	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
140	#407	C₂²×C₃	1	16	yes	complemented
141	C₂×C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
142	C₂×C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
143	C₄×M₁₆	C₂²×C₃	1	8	yes	complemented
144	C₂×#97	C₂²×C₃	1	2	yes	complemented
145	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
146	C₂⁴×C₃	—	2	8	no	—
147	C₂⁴×C₃	—	2	8	no	—
148	C₂⁴×C₃	—	4	4	no	—
149	C₂⁴×C₃	C₂²×C₄	1	2	yes	—
150	C₂⁴×C₃	C₂²×C₄	1	8	yes	—
151	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
152	C₂⁴×C₃	C₂²×C₄	1	2	yes	—
153	C₂×C₃×D₄	—	2	8	no	—
154	C₂×C₃×D₄	—	2	32	no	—
155	C₂×C₃×D₄	—	2	4	no	—
156	C₂×C₃×D₄	M₁₆	1	8	yes	direct product factor
157	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
158	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
159	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
160	C₂²×C₃×C₄	—	2	8	no	—
161	C₂²×C₃×C₄	—	2	8	no	—
162	C₂²×C₃×C₄	—	2	32	no	—
163	C₂²×C₃×C₄	—	2	8	no	—
164	C₂²×C₃×C₄	—	2	4	no	—
165	C₂²×C₃×C₄	—	2	4	no	—
166	C₂²×C₃×C₄	—	2	16	no	—
167	C₂²×C₃×C₄	—	2	4	no	—
168	C₂²×C₃×C₄	—	2	4	no	—
169	C₂²×C₃×C₄	—	2	4	no	—
170	C₂²×C₃×C₄	—	2	16	no	—
171	C₂²×C₃×C₄	—	4	2	no	—
172	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
176	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
177	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
178	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
179	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
180	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
181	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
182	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
183	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
184	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
185	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
186	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
187	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
188	C₂²×C₃×C₄	#41	1	4	yes	—
189	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
190	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
191	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
192	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
193	C₃×M₁₆	—	2	16	no	—
194	C₃×M₁₆	—	2	8	no	—
195	C₃×M₁₆	—	2	32	no	—
196	C₃×M₁₆	C₂×D₄	1	16	yes	direct product factor
197	C₂×C₃×C₈	—	2	8	no	—
198	C₂×C₃×C₈	—	2	4	no	—
199	C₂×C₃×C₈	—	2	32	no	—
200	C₂×C₃×C₈	C₂×D₄	1	8	yes	complemented
201	C₂×C₃×C₈	C₂⁴	1	4	yes	—
202	C₂×C₃×C₈	C₂⁴	1	8	yes	—
203	C₂×C₃×C₈	C₂⁴	1	4	yes	—
204	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
205	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
206	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
207	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
208	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
209	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
210	C₃×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
211	C₃×C₄²	C₂²×C₄	1	4	yes	—
212	C₃×C₄²	C₂²×C₄	1	2	yes	—
213	C₃×C₄²	C₂²×C₄	1	4	yes	—
214	C₃×C₄²	C₂²×C₄	1	2	yes	—
215	C₃×C₄²	C₂⁴	1	2	yes	—
216	C₃×C₄²	C₂⁴	1	2	yes	—
217	C₂⁵	C₂×C₃×C₄	1	2	yes	—
218	C₂²×D₄	—	2	8	no	—
219	C₂²×D₄	C₂×C₃×C₄	1	2	yes	—
220	C₂²×D₄	C₂×C₃×C₄	1	8	yes	—
221	C₂²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
222	C₂²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
223	C₂³×C₄	—	2	8	no	—
224	C₂³×C₄	—	2	8	no	—
225	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
226	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
227	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
228	C₂³×C₄	C₂×C₃×C₄	1	8	yes	—
229	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
230	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₃	1	2	yes	—
232	C₂³×C₄	C₂³×C₃	1	2	yes	—
233	C₂³×C₄	C₂³×C₃	1	8	yes	—
234	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
235	C₂×M₁₆	—	2	8	no	—
236	C₂×M₁₆	—	2	4	no	—
237	C₂×M₁₆	—	2	64	no	—
238	C₂×M₁₆	—	2	16	no	—
239	C₂×M₁₆	C₃×D₄	1	8	yes	direct product factor
240	C₂×M₁₆	C₂³×C₃	1	8	yes	—
241	C₂×M₁₆	C₂³×C₃	1	4	yes	—
242	C₂×M₁₆	C₂³×C₃	1	4	yes	—
243	C₂×M₁₆	C₂³×C₃	1	16	yes	complemented
244	C₂²×C₈	—	2	16	no	—
245	C₂²×C₈	C₂³×C₃	1	4	yes	—
246	C₂²×C₈	C₂³×C₃	1	2	yes	—
247	C₂²×C₈	C₂³×C₃	1	2	yes	—
248	C₂²×C₈	C₂³×C₃	1	16	yes	complemented
249	C₄×D₄	C₂×C₃×C₄	1	8	yes	—
250	C₄×D₄	C₂×C₃×C₄	1	4	yes	—
251	C₄×D₄	C₂×C₃×C₄	1	16	yes	—
252	C₄×D₄	C₂×C₃×C₄	1	8	yes	—
253	C₄×D₄	C₂×C₃×C₄	1	4	yes	—
254	C₄×D₄	C₂×C₃×C₄	1	8	yes	—
255	C₄×D₄	C₂³×C₃	1	4	yes	—
256	C₄×D₄	C₂³×C₃	1	8	yes	—
257	C₄×D₄	C₂³×C₃	1	4	yes	—
258	C₂×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
260	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
263	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
265	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
266	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
267	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
268	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
269	C₂×C₄²	C₂×C₃×C₄	1	4	yes	—
270	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₃	1	4	yes	—
273	C₄⋊C₈	C₂³×C₃	1	8	yes	complemented
274	C₄⋊C₈	C₂³×C₃	1	8	yes	complemented
275	C₂²⋊C₈	C₂³×C₃	1	16	yes	complemented
276	C₂²⋊C₈	C₂³×C₃	1	16	yes	complemented
277	#97	C₂³×C₃	1	8	yes	complemented
278	C₄×C₈	C₂³×C₃	1	8	yes	complemented
279	C₂³×C₃	—	2	4	no	—
280	C₂³×C₃	—	2	2	no	—
281	C₂³×C₃	—	2	2	no	—
282	C₂³×C₃	—	2	4	no	—
283	C₂³×C₃	—	2	16	no	—
284	C₂³×C₃	—	2	4	no	—
285	C₂³×C₃	—	2	4	no	—
286	C₂³×C₃	—	2	16	no	—
287	C₂³×C₃	—	2	4	no	—
288	C₂³×C₃	—	4	8	no	—
289	C₂³×C₃	—	4	32	no	—
290	C₂³×C₃	C₄×D₄	1	4	yes	—
291	C₂³×C₃	C₂×M₁₆	1	8	yes	complemented
292	C₂³×C₃	C₂³×C₄	1	4	yes	—
293	C₂³×C₃	C₂³×C₄	1	2	yes	—
294	C₂³×C₃	C₂³×C₄	1	4	yes	—
295	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
296	C₃×D₄	—	2	16	no	—
297	C₃×D₄	—	2	8	no	—
298	C₃×D₄	C₂×M₁₆	1	16	yes	direct product factor
299	C₂×C₃×C₄	—	2	16	no	—
300	C₂×C₃×C₄	—	2	4	no	—
301	C₂×C₃×C₄	—	2	4	no	—
302	C₂×C₃×C₄	—	2	2	no	—
303	C₂×C₃×C₄	—	2	2	no	—
304	C₂×C₃×C₄	—	2	2	no	—
305	C₂×C₃×C₄	—	2	8	no	—
306	C₂×C₃×C₄	—	2	16	no	—
307	C₂×C₃×C₄	—	2	16	no	—
308	C₂×C₃×C₄	—	2	8	no	—
309	C₂×C₃×C₄	—	2	8	no	—
310	C₂×C₃×C₄	—	2	4	no	—
311	C₂×C₃×C₄	—	2	4	no	—
312	C₂×C₃×C₄	—	2	4	no	—
313	C₂×C₃×C₄	—	2	4	no	—
314	C₂×C₃×C₄	—	4	4	no	—
315	C₂×C₃×C₄	C₄×D₄	1	8	yes	—
316	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
317	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	complemented
318	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
319	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
320	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
321	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
322	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
323	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
324	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
325	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
326	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
327	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
328	C₂×C₃×C₄	C₂×#41	1	2	yes	—
329	C₂×C₃×C₄	C₂×#41	1	4	yes	—
330	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
331	C₃×C₈	—	2	8	no	—
332	C₃×C₈	—	2	4	no	—
333	C₃×C₈	C₂²×D₄	1	8	yes	complemented
334	C₂⁴	—	2	8	no	—
335	C₂⁴	—	2	8	no	—
336	C₂⁴	—	4	4	no	—
337	C₂⁴	C₂²×C₃×C₄	1	2	yes	—
338	C₂⁴	C₂²×C₃×C₄	1	8	yes	—
339	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
340	C₂⁴	C₂²×C₃×C₄	1	2	yes	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	2	32	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	C₃×M₁₆	1	8	yes	direct product factor
345	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
346	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
347	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	32	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	16	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	16	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
364	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
365	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
366	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
367	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
368	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
369	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
370	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
371	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
372	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
373	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
375	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
376	C₂²×C₄	C₃×#41	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
379	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
381	M₁₆	—	2	16	no	—
382	M₁₆	—	2	8	no	—
383	M₁₆	—	2	32	no	—
384	M₁₆	C₂×C₃×D₄	1	16	yes	direct product factor
385	C₂×C₈	—	2	8	no	—
386	C₂×C₈	—	2	4	no	—
387	C₂×C₈	—	2	32	no	—
388	C₂×C₈	C₂×C₃×D₄	1	8	yes	complemented
389	C₂×C₈	C₂⁴×C₃	1	4	yes	—
390	C₂×C₈	C₂⁴×C₃	1	8	yes	—
391	C₂×C₈	C₂⁴×C₃	1	4	yes	—
392	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
393	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
394	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
395	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
396	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
397	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
398	C₂²⋊C₄	C₂²×C₃×C₄	1	8	yes	—
399	C₄²	C₂²×C₃×C₄	1	2	yes	—
400	C₄²	C₂²×C₃×C₄	1	4	yes	—
401	C₄²	C₂²×C₃×C₄	1	2	yes	—
402	C₄²	C₂²×C₃×C₄	1	4	yes	—
403	C₄²	C₂⁴×C₃	1	2	yes	—
404	C₄²	C₂⁴×C₃	1	2	yes	—
405	C₂²×C₃	—	2	4	no	—
406	C₂²×C₃	—	2	8	no	—
407	C₂²×C₃	—	2	16	no	—
408	C₂²×C₃	—	2	2	no	—
409	C₂²×C₃	—	2	2	no	—
410	C₂²×C₃	—	2	4	no	—
411	C₂²×C₃	—	2	4	no	—
412	C₂²×C₃	—	2	4	no	—
413	C₂²×C₃	—	2	4	no	—
414	C₂²×C₃	—	4	32	no	—
415	C₂²×C₃	—	4	8	no	—
416	C₂²×C₃	C₂×C₄×D₄	1	4	yes	—
417	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
418	C₂²×C₃	C₂²×M₁₆	1	2	yes	—
419	C₂²×C₃	C₂²×M₁₆	1	8	yes	complemented
420	C₂²×C₃	#568	1	2	yes	—
421	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
422	C₃×C₄	—	2	4	no	—
423	C₃×C₄	—	2	4	no	—
424	C₃×C₄	—	2	2	no	—
425	C₃×C₄	—	2	2	no	—
426	C₃×C₄	—	2	2	no	—
427	C₃×C₄	C₂×C₄×D₄	1	4	yes	—
428	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
429	C₃×C₄	C₂²×M₁₆	1	4	yes	complemented
430	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
431	C₃×C₄	C₂²×#41	1	1	yes	characteristic
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	2	16	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	16	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	32	no	—
443	C₂³	C₃×C₄×D₄	1	4	yes	—
444	C₂³	C₂×C₃×M₁₆	1	8	yes	complemented
445	C₂³	C₂³×C₃×C₄	1	4	yes	—
446	C₂³	C₂³×C₃×C₄	1	4	yes	—
447	C₂³	C₂³×C₃×C₄	1	2	yes	—
448	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
449	D₄	—	2	16	no	—
450	D₄	—	2	8	no	—
451	D₄	C₂×C₃×M₁₆	1	16	yes	direct product factor
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	16	no	—
465	C₂×C₄	—	2	16	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	C₃×C₄×D₄	1	8	yes	—
469	C₂×C₄	C₃×C₄×D₄	1	4	yes	—
470	C₂×C₄	C₂×C₃×M₁₆	1	4	yes	complemented
471	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
472	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
473	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
474	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
475	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
476	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
477	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
478	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
479	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
480	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
481	C₂×C₄	C₂×C₃×#41	1	2	yes	—
482	C₂×C₄	C₂×C₃×#41	1	4	yes	—
483	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
484	C₈	—	2	8	no	—
485	C₈	—	2	4	no	—
486	C₈	C₂²×C₃×D₄	1	8	yes	complemented
487	C₂×C₃	—	2	4	no	—
488	C₂×C₃	—	2	8	no	—
489	C₂×C₃	—	4	4	no	—
490	C₂×C₃	D₄×M₁₆	1	4	yes	direct product factor
491	C₂×C₃	C₂²×C₄×D₄	1	1	yes	characteristic
492	C₂×C₃	C₂³×M₁₆	1	1	yes	characteristic
493	C₂×C₃	C₂×#568	1	1	yes	characteristic
494	C₂²	—	2	4	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	16	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	4	32	no	—
504	C₂²	—	4	8	no	—
505	C₂²	C₂×C₃×C₄×D₄	1	4	yes	—
506	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
507	C₂²	C₂²×C₃×M₁₆	1	2	yes	—
508	C₂²	C₂²×C₃×M₁₆	1	8	yes	complemented
509	C₂²	C₃×#568	1	2	yes	—
510	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
511	C₄	—	2	4	no	—
512	C₄	—	2	4	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
517	C₄	C₂×C₃×C₄×D₄	1	4	yes	—
518	C₄	C₂²×C₃×M₁₆	1	4	yes	complemented
519	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
520	C₄	C₂²×C₃×#41	1	1	yes	characteristic
521	C₃	C₂×D₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
522	C₂	—	2	4	no	—
523	C₂	—	2	8	no	—
524	C₂	—	4	4	no	—
525	C₂	C₃×D₄×M₁₆	1	4	yes	direct product factor
526	C₂	C₂²×C₃×C₄×D₄	1	1	yes	characteristic
527	C₂	C₂³×C₃×M₁₆	1	1	yes	characteristic
528	C₂	C₂×C₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂³×C₃×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×M₁₆	C₂	semidirect
3. C₃×D₄×M₁₆	C₂	direct (N × H)
4. C₂×C₃×#436	C₂	semidirect
5. C₂×C₃×#435	C₂	semidirect
6. C₂×C₃×C₈×D₄	C₂	semidirect
7. C₂×C₃×#423	C₂	semidirect
8. C₂×C₃×#407	C₂	semidirect
9. C₂×C₃×C₄×M₁₆	C₂	semidirect
10. C₂×D₄×M₁₆	C₃	direct (N × H)
17. C₂²×C₃×M₁₆	C₂²	semidirect
18. C₂²×C₃×M₁₆	C₂²	semidirect
19. C₂³×C₃×C₈	C₂²	semidirect
31. C₃×#436	C₂²	semidirect
32. C₃×#435	C₂²	semidirect
33. C₃×C₈×D₄	C₂²	semidirect
34. C₃×#423	C₂²	semidirect
35. C₂×C₃×C₄⋊C₈	C₂²	semidirect
36. C₂×C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#407	C₂²	semidirect
38. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
39. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
40. C₃×C₄×M₁₆	C₂²	semidirect
41. C₂×C₃×#97	C₂²	semidirect
42. C₂×C₃×C₄×C₈	C₂²	semidirect
43. C₂³×M₁₆	C₂×C₃	semidirect
45. D₄×M₁₆	C₂×C₃	direct (N × H)
46. C₂×#436	C₂×C₃	semidirect
47. C₂×#435	C₂×C₃	semidirect
48. C₂×C₈×D₄	C₂×C₃	semidirect
49. C₂×#423	C₂×C₃	semidirect
50. C₂×#407	C₂×C₃	semidirect
51. C₂×C₄×M₁₆	C₂×C₃	semidirect
74. C₂×C₃×M₁₆	D₄	direct (N × H)
78. C₂×C₃×M₁₆	C₂³	semidirect
83. C₂²×C₃×C₈	C₂³	semidirect
108. C₃×C₄⋊C₈	C₂³	semidirect
109. C₃×C₄⋊C₈	C₂³	semidirect
110. C₃×C₂²⋊C₈	C₂³	semidirect
111. C₃×C₂²⋊C₈	C₂³	semidirect
112. C₃×#97	C₂³	semidirect
113. C₃×C₄×C₈	C₂³	semidirect
120. C₂²×M₁₆	C₂²×C₃	semidirect
121. C₂²×M₁₆	C₂²×C₃	semidirect
122. C₂³×C₈	C₂²×C₃	semidirect
134. #436	C₂²×C₃	semidirect
135. #435	C₂²×C₃	semidirect
136. C₈×D₄	C₂²×C₃	semidirect
137. #423	C₂²×C₃	semidirect
138. C₂×C₄⋊C₈	C₂²×C₃	semidirect
139. C₂×C₄⋊C₈	C₂²×C₃	semidirect
140. #407	C₂²×C₃	semidirect
141. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
142. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
143. C₄×M₁₆	C₂²×C₃	semidirect
144. C₂×#97	C₂²×C₃	semidirect
145. C₂×C₄×C₈	C₂²×C₃	semidirect
156. C₂×C₃×D₄	M₁₆	direct (N × H)
196. C₃×M₁₆	C₂×D₄	direct (N × H)
200. C₂×C₃×C₈	C₂×D₄	semidirect
239. C₂×M₁₆	C₃×D₄	direct (N × H)
243. C₂×M₁₆	C₂³×C₃	semidirect
248. C₂²×C₈	C₂³×C₃	semidirect
273. C₄⋊C₈	C₂³×C₃	semidirect
274. C₄⋊C₈	C₂³×C₃	semidirect
275. C₂²⋊C₈	C₂³×C₃	semidirect
276. C₂²⋊C₈	C₂³×C₃	semidirect
277. #97	C₂³×C₃	semidirect
278. C₄×C₈	C₂³×C₃	semidirect
291. C₂³×C₃	C₂×M₁₆	semidirect
298. C₃×D₄	C₂×M₁₆	direct (N × H)
317. C₂×C₃×C₄	C₂×M₁₆	semidirect
333. C₃×C₈	C₂²×D₄	semidirect
344. C₂×D₄	C₃×M₁₆	direct (N × H)
384. M₁₆	C₂×C₃×D₄	direct (N × H)
388. C₂×C₈	C₂×C₃×D₄	semidirect
419. C₂²×C₃	C₂²×M₁₆	semidirect
429. C₃×C₄	C₂²×M₁₆	semidirect
444. C₂³	C₂×C₃×M₁₆	semidirect
451. D₄	C₂×C₃×M₁₆	direct (N × H)
470. C₂×C₄	C₂×C₃×M₁₆	semidirect
486. C₈	C₂²×C₃×D₄	semidirect
490. C₂×C₃	D₄×M₁₆	direct (N × H)
508. C₂²	C₂²×C₃×M₁₆	semidirect
518. C₄	C₂²×C₃×M₁₆	semidirect
521. C₃	C₂×D₄×M₁₆	direct (N × H)
525. C₂	C₃×D₄×M₁₆	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×C₃×M₁₆	2
C₂²×C₃×C₄×D₄	2
C₃×D₄×M₁₆	2
C₂×C₃×#436	2
C₂×C₃×#435	2
C₂×C₃×C₈×D₄	2
C₂×C₃×#423	2
C₂×C₃×#407	2
C₂×C₃×C₄×M₁₆	2
C₂×D₄×M₁₆	3

Subgroups — C2×C3×D4×M16 #10678745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₄×M₁₆ #10678745