Subgroups — C₂⁴×#529 #72295499

Order 1024.

553749 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#529	C₂	1	240	yes	maximal, direct product factor
2	C₂⁴×#126	C₂	1	2	yes	maximal, complemented
3	C₂⁴×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×#123	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×#121	C₂	1	1	yes	maximal, characteristic
8	C₂⁴×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×#117	C₂	1	2	yes	maximal, complemented
12	C₂⁵×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×Q₈	C₂²	1	1	yes	characteristic
14	C₂⁵×D₄	C₂²	1	2	yes	—
15	C₂⁵×D₄	C₂²	1	1	yes	characteristic
16	C₂⁶×C₄	C₂²	1	1	yes	characteristic
17	C₂⁶×C₄	C₂²	1	1	yes	characteristic
18	C₂⁶×C₄	C₂²	1	1	yes	characteristic
19	C₂⁶×C₄	C₂²	1	2	yes	—
20	C₂²×#529	C₂²	1	8960	yes	direct product factor
21	C₂³×#126	C₂²	1	240	yes	complemented
22	C₂³×#125	C₂²	1	120	yes	complemented
23	C₂³×#124	C₂²	1	120	yes	complemented
24	C₂³×#123	C₂²	1	240	yes	complemented
25	C₂³×C₂²⋊Q₈	C₂²	1	120	yes	complemented
26	C₂³×#121	C₂²	1	120	yes	—
27	C₂³×C₄×Q₈	C₂²	1	120	yes	complemented
28	C₂³×C₄×D₄	C₂²	1	240	yes	complemented
29	C₂³×C₄×D₄	C₂²	1	120	yes	complemented
30	C₂³×#117	C₂²	1	240	yes	complemented
31	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂⁴×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂⁴×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
39	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
43	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
44	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
45	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
46	C₂⁴×C₄²	C₂²	1	2	yes	—
47	C₂⁴×C₄²	C₂²	1	120	yes	complemented
48	C₂⁷	C₂³	1	2	yes	—
49	C₂⁷	C₂³	1	1	yes	characteristic
50	C₂⁴×Q₈	—	2	32	no	—
51	C₂⁴×Q₈	C₂³	1	60	yes	—
52	C₂⁴×D₄	—	2	64	no	—
53	C₂⁴×D₄	—	2	32	no	—
54	C₂⁴×D₄	C₂³	1	120	yes	—
55	C₂⁴×D₄	C₂³	1	60	yes	—
56	C₂⁵×C₄	—	2	32	no	—
57	C₂⁵×C₄	—	2	32	no	—
58	C₂⁵×C₄	—	2	64	no	—
59	C₂⁵×C₄	—	2	32	no	—
60	C₂⁵×C₄	C₂³	1	1	yes	characteristic
61	C₂⁵×C₄	C₂³	1	1	yes	characteristic
62	C₂⁵×C₄	C₂³	1	1	yes	characteristic
63	C₂⁵×C₄	C₂³	1	1	yes	characteristic
64	C₂⁵×C₄	C₂³	1	60	yes	—
65	C₂⁵×C₄	C₂³	1	60	yes	—
66	C₂⁵×C₄	C₂³	1	60	yes	—
67	C₂⁵×C₄	C₂³	1	2	yes	—
68	C₂⁵×C₄	C₂³	1	120	yes	—
69	C₂⁵×C₄	C₂³	1	2	yes	—
70	C₂⁵×C₄	C₂³	1	2	yes	—
71	C₂⁵×C₄	C₂³	1	1	yes	characteristic
72	C₂⁵×C₄	C₂³	1	1	yes	characteristic
73	C₂×#529	C₂³	1	61440	yes	direct product factor
74	C₂²×#126	C₂³	1	4480	yes	complemented
75	C₂²×#125	C₂³	1	2240	yes	complemented
76	C₂²×#124	C₂³	1	2240	yes	complemented
77	C₂²×#123	C₂³	1	4480	yes	complemented
78	C₂²×C₂²⋊Q₈	C₂³	1	2240	yes	complemented
79	C₂²×#121	C₂³	1	2240	yes	—
80	C₂²×C₄×Q₈	C₂³	1	2240	yes	complemented
81	C₂²×C₄×D₄	C₂³	1	4480	yes	complemented
82	C₂²×C₄×D₄	C₂³	1	2240	yes	complemented
83	C₂²×#117	C₂³	1	4480	yes	complemented
84	C₂³×C₄⋊C₄	C₂³	1	120	yes	—
85	C₂³×C₄⋊C₄	C₂³	1	120	yes	—
86	C₂³×C₄⋊C₄	C₂³	1	60	yes	—
87	C₂³×C₄⋊C₄	C₂³	1	120	yes	—
88	C₂³×C₄⋊C₄	C₂³	1	60	yes	—
89	C₂³×C₄⋊C₄	C₂³	1	120	yes	—
90	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
91	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
92	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
93	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
94	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
95	C₂³×C₄²	C₂³	1	60	yes	—
96	C₂³×C₄²	C₂³	1	60	yes	—
97	C₂³×C₄²	C₂³	1	60	yes	—
98	C₂³×C₄²	C₂³	1	60	yes	—
99	C₂³×C₄²	C₂³	1	120	yes	—
100	C₂³×C₄²	C₂³	1	2240	yes	complemented
101	C₂⁶	—	2	16	no	—
102	C₂⁶	—	2	16	no	—
103	C₂⁶	—	2	32	no	—
104	C₂⁶	—	2	32	no	—
105	C₂⁶	—	2	32	no	—
106	C₂⁶	#41	1	32	yes	—
107	C₂⁶	C₂⁴	1	60	yes	—
108	C₂⁶	C₂⁴	1	30	yes	—
109	C₂⁶	C₂⁴	1	1	yes	characteristic
110	C₂³×Q₈	—	2	960	no	—
111	C₂³×Q₈	C₂⁴	1	560	yes	—
112	C₂³×D₄	—	2	1920	no	—
113	C₂³×D₄	—	2	960	no	—
114	C₂³×D₄	C₂⁴	1	1120	yes	—
115	C₂³×D₄	C₂⁴	1	560	yes	—
116	C₂⁴×C₄	—	2	960	no	—
117	C₂⁴×C₄	—	2	16	no	—
118	C₂⁴×C₄	—	2	16	no	—
119	C₂⁴×C₄	—	2	16	no	—
120	C₂⁴×C₄	—	2	16	no	—
121	C₂⁴×C₄	—	2	32	no	—
122	C₂⁴×C₄	—	2	32	no	—
123	C₂⁴×C₄	—	2	32	no	—
124	C₂⁴×C₄	—	2	960	no	—
125	C₂⁴×C₄	—	2	1920	no	—
126	C₂⁴×C₄	—	2	960	no	—
127	C₂⁴×C₄	#41	1	32	yes	—
128	C₂⁴×C₄	#41	1	32	yes	—
129	C₂⁴×C₄	C₂⁴	1	30	yes	—
130	C₂⁴×C₄	C₂⁴	1	30	yes	—
131	C₂⁴×C₄	C₂⁴	1	30	yes	—
132	C₂⁴×C₄	C₂⁴	1	30	yes	—
133	C₂⁴×C₄	C₂⁴	1	560	yes	—
134	C₂⁴×C₄	C₂⁴	1	560	yes	—
135	C₂⁴×C₄	C₂⁴	1	560	yes	—
136	C₂⁴×C₄	C₂⁴	1	60	yes	—
137	C₂⁴×C₄	C₂⁴	1	1120	yes	—
138	C₂⁴×C₄	C₂⁴	1	60	yes	—
139	C₂⁴×C₄	C₂⁴	1	60	yes	—
140	C₂⁴×C₄	C₂⁴	1	30	yes	—
141	C₂⁴×C₄	C₂⁴	1	30	yes	—
142	#529	C₂⁴	1	65536	yes	direct product factor
143	C₂×#126	C₂⁴	1	15360	yes	complemented
144	C₂×#125	C₂⁴	1	7680	yes	complemented
145	C₂×#124	C₂⁴	1	7680	yes	complemented
146	C₂×#123	C₂⁴	1	15360	yes	complemented
147	C₂×C₂²⋊Q₈	C₂⁴	1	7680	yes	complemented
148	C₂×#121	C₂⁴	1	7680	yes	—
149	C₂×C₄×Q₈	C₂⁴	1	7680	yes	complemented
150	C₂×C₄×D₄	C₂⁴	1	15360	yes	complemented
151	C₂×C₄×D₄	C₂⁴	1	7680	yes	complemented
152	C₂×#117	C₂⁴	1	15360	yes	complemented
153	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	—
154	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	—
155	C₂²×C₄⋊C₄	C₂⁴	1	560	yes	—
156	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	—
157	C₂²×C₄⋊C₄	C₂⁴	1	560	yes	—
158	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	—
159	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
160	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
161	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
162	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
163	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
164	C₂²×C₄²	C₂⁴	1	560	yes	—
165	C₂²×C₄²	C₂⁴	1	560	yes	—
166	C₂²×C₄²	C₂⁴	1	560	yes	—
167	C₂²×C₄²	C₂⁴	1	560	yes	—
168	C₂²×C₄²	C₂⁴	1	1120	yes	—
169	C₂²×C₄²	C₂⁴	1	7680	yes	complemented
170	C₂⁵	—	2	512	no	—
171	C₂⁵	—	2	240	no	—
172	C₂⁵	—	2	240	no	—
173	C₂⁵	—	2	480	no	—
174	C₂⁵	—	2	480	no	—
175	C₂⁵	—	2	480	no	—
176	C₂⁵	—	4	512	no	—
177	C₂⁵	C₂×#41	1	480	yes	—
178	C₂⁵	C₂×#41	1	16	yes	—
179	C₂⁵	C₂×#41	1	16	yes	—
180	C₂⁵	C₂×#41	1	16	yes	—
181	C₂⁵	C₂⁵	1	280	yes	—
182	C₂⁵	C₂⁵	1	140	yes	—
183	C₂⁵	C₂⁵	1	15	yes	—
184	C₂²×Q₈	—	2	4480	no	—
185	C₂²×Q₈	C₂⁵	1	960	yes	—
186	C₂²×D₄	—	2	8960	no	—
187	C₂²×D₄	—	2	4480	no	—
188	C₂²×D₄	C₂⁵	1	1920	yes	—
189	C₂²×D₄	C₂⁵	1	960	yes	—
190	C₂³×C₄	—	2	4480	no	—
191	C₂³×C₄	—	2	240	no	—
192	C₂³×C₄	—	2	240	no	—
193	C₂³×C₄	—	2	240	no	—
194	C₂³×C₄	—	2	240	no	—
195	C₂³×C₄	—	2	480	no	—
196	C₂³×C₄	—	2	480	no	—
197	C₂³×C₄	—	2	480	no	—
198	C₂³×C₄	—	2	4480	no	—
199	C₂³×C₄	—	2	8960	no	—
200	C₂³×C₄	—	2	4480	no	—
201	C₂³×C₄	C₂×#41	1	480	yes	—
202	C₂³×C₄	C₂×#41	1	480	yes	—
203	C₂³×C₄	C₂⁵	1	280	yes	—
204	C₂³×C₄	C₂⁵	1	280	yes	—
205	C₂³×C₄	C₂⁵	1	140	yes	—
206	C₂³×C₄	C₂⁵	1	140	yes	—
207	C₂³×C₄	C₂⁵	1	140	yes	—
208	C₂³×C₄	C₂⁵	1	140	yes	—
209	C₂³×C₄	C₂⁵	1	280	yes	—
210	C₂³×C₄	C₂⁵	1	1920	yes	—
211	C₂³×C₄	C₂⁵	1	960	yes	—
212	C₂³×C₄	C₂⁵	1	960	yes	—
213	C₂³×C₄	C₂⁵	1	960	yes	—
214	C₂³×C₄	C₂⁵	1	140	yes	—
215	C₂³×C₄	C₂⁵	1	140	yes	—
216	#126	C₂⁵	1	8192	yes	complemented
217	#125	C₂⁵	1	4096	yes	complemented
218	#124	C₂⁵	1	4096	yes	complemented
219	#123	C₂⁵	1	8192	yes	complemented
220	C₂²⋊Q₈	C₂⁵	1	4096	yes	complemented
221	#121	C₂⁵	1	4096	yes	—
222	C₄×Q₈	C₂⁵	1	4096	yes	complemented
223	C₄×D₄	C₂⁵	1	4096	yes	complemented
224	C₄×D₄	C₂⁵	1	8192	yes	complemented
225	#117	C₂⁵	1	8192	yes	complemented
226	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	—
227	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	—
228	C₂×C₄⋊C₄	C₂⁵	1	960	yes	—
229	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	—
230	C₂×C₄⋊C₄	C₂⁵	1	960	yes	—
231	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	—
232	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
233	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
234	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
235	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
236	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
237	C₂×C₄²	C₂⁵	1	960	yes	—
238	C₂×C₄²	C₂⁵	1	1920	yes	—
239	C₂×C₄²	C₂⁵	1	960	yes	—
240	C₂×C₄²	C₂⁵	1	960	yes	—
241	C₂×C₄²	C₂⁵	1	960	yes	—
242	C₂×C₄²	C₂⁵	1	4096	yes	complemented
243	C₂⁴	—	2	560	no	—
244	C₂⁴	—	2	560	no	—
245	C₂⁴	—	2	3840	no	—
246	C₂⁴	—	2	1120	no	—
247	C₂⁴	—	2	1120	no	—
248	C₂⁴	—	2	1120	no	—
249	C₂⁴	—	4	3840	no	—
250	C₂⁴	#529	1	256	yes	direct product factor
251	C₂⁴	C₂²×#41	1	1120	yes	—
252	C₂⁴	C₂²×#41	1	120	yes	—
253	C₂⁴	C₂²×#41	1	120	yes	—
254	C₂⁴	C₂²×#41	1	120	yes	—
255	C₂⁴	C₂⁶	1	240	yes	—
256	C₂⁴	C₂⁶	1	120	yes	—
257	C₂⁴	C₂⁶	1	35	yes	—
258	C₂×Q₈	—	2	3840	no	—
259	C₂×Q₈	C₂⁶	1	256	yes	—
260	C₂×D₄	—	2	7680	no	—
261	C₂×D₄	—	2	3840	no	—
262	C₂×D₄	C₂⁶	1	512	yes	—
263	C₂×D₄	C₂⁶	1	256	yes	—
264	C₂²×C₄	—	2	3840	no	—
265	C₂²×C₄	—	2	560	no	—
266	C₂²×C₄	—	2	560	no	—
267	C₂²×C₄	—	2	1120	no	—
268	C₂²×C₄	—	2	560	no	—
269	C₂²×C₄	—	2	560	no	—
270	C₂²×C₄	—	2	1120	no	—
271	C₂²×C₄	—	2	1120	no	—
272	C₂²×C₄	—	2	3840	no	—
273	C₂²×C₄	—	2	7680	no	—
274	C₂²×C₄	—	2	3840	no	—
275	C₂²×C₄	C₂²×#41	1	1120	yes	—
276	C₂²×C₄	C₂²×#41	1	1120	yes	—
277	C₂²×C₄	C₂⁶	1	120	yes	—
278	C₂²×C₄	C₂⁶	1	120	yes	—
279	C₂²×C₄	C₂⁶	1	120	yes	—
280	C₂²×C₄	C₂⁶	1	120	yes	—
281	C₂²×C₄	C₂⁶	1	240	yes	—
282	C₂²×C₄	C₂⁶	1	240	yes	—
283	C₂²×C₄	C₂⁶	1	256	yes	—
284	C₂²×C₄	C₂⁶	1	256	yes	—
285	C₂²×C₄	C₂⁶	1	256	yes	—
286	C₂²×C₄	C₂⁶	1	240	yes	—
287	C₂²×C₄	C₂⁶	1	512	yes	—
288	C₂²×C₄	C₂⁶	1	120	yes	—
289	C₂²×C₄	C₂⁶	1	120	yes	—
290	C₄⋊C₄	C₂⁶	1	512	yes	—
291	C₄⋊C₄	C₂⁶	1	512	yes	—
292	C₄⋊C₄	C₂⁶	1	256	yes	—
293	C₄⋊C₄	C₂⁶	1	512	yes	—
294	C₄⋊C₄	C₂⁶	1	256	yes	—
295	C₄⋊C₄	C₂⁶	1	512	yes	—
296	C₂²⋊C₄	C₂⁶	1	512	yes	—
297	C₂²⋊C₄	C₂⁶	1	512	yes	—
298	C₂²⋊C₄	C₂⁶	1	512	yes	—
299	C₂²⋊C₄	C₂⁶	1	512	yes	—
300	C₂²⋊C₄	C₂⁶	1	512	yes	—
301	C₄²	C₂⁶	1	256	yes	—
302	C₄²	C₂⁶	1	256	yes	—
303	C₄²	C₂⁶	1	256	yes	—
304	C₄²	C₂⁶	1	256	yes	—
305	C₄²	C₂⁶	1	512	yes	—
306	C₂³	—	2	240	no	—
307	C₂³	—	2	240	no	—
308	C₂³	—	2	4480	no	—
309	C₂³	—	2	480	no	—
310	C₂³	—	2	480	no	—
311	C₂³	—	2	480	no	—
312	C₂³	—	4	4480	no	—
313	C₂³	C₂×#529	1	960	yes	direct product factor
314	C₂³	C₂³×#41	1	480	yes	—
315	C₂³	C₂³×#41	1	140	yes	—
316	C₂³	C₂³×#41	1	140	yes	—
317	C₂³	C₂³×#41	1	140	yes	—
318	C₂³	C₂⁷	1	32	yes	—
319	C₂³	C₂⁷	1	16	yes	—
320	C₂³	C₂⁷	1	15	yes	—
321	Q₈	—	2	512	no	—
322	D₄	—	2	1024	no	—
323	D₄	—	2	512	no	—
324	C₂×C₄	—	2	512	no	—
325	C₂×C₄	—	2	240	no	—
326	C₂×C₄	—	2	240	no	—
327	C₂×C₄	—	2	240	no	—
328	C₂×C₄	—	2	240	no	—
329	C₂×C₄	—	2	480	no	—
330	C₂×C₄	—	2	480	no	—
331	C₂×C₄	—	2	480	no	—
332	C₂×C₄	—	2	512	no	—
333	C₂×C₄	—	2	1024	no	—
334	C₂×C₄	—	2	512	no	—
335	C₂×C₄	C₂³×#41	1	480	yes	—
336	C₂×C₄	C₂³×#41	1	480	yes	—
337	C₂×C₄	C₂⁷	1	16	yes	—
338	C₂×C₄	C₂⁷	1	16	yes	—
339	C₂×C₄	C₂⁷	1	16	yes	—
340	C₂×C₄	C₂⁷	1	16	yes	—
341	C₂×C₄	C₂⁷	1	32	yes	—
342	C₂×C₄	C₂⁷	1	32	yes	—
343	C₂×C₄	C₂⁷	1	32	yes	—
344	C₂×C₄	C₂⁷	1	16	yes	—
345	C₂×C₄	C₂⁷	1	16	yes	—
346	C₂²	—	2	960	no	—
347	C₂²	—	2	16	no	—
348	C₂²	—	2	16	no	—
349	C₂²	—	2	32	no	—
350	C₂²	—	2	32	no	—
351	C₂²	—	2	32	no	—
352	C₂²	—	4	960	no	—
353	C₂²	C₂²×#529	1	560	yes	direct product factor
354	C₂²	C₂⁴×#41	1	32	yes	—
355	C₂²	C₂⁴×#41	1	30	yes	—
356	C₂²	C₂⁴×#41	1	30	yes	—
357	C₂²	C₂⁴×#41	1	30	yes	—
358	C₂²	C₂⁸	1	1	yes	characteristic
359	C₄	—	2	16	no	—
360	C₄	—	2	16	no	—
361	C₄	—	2	16	no	—
362	C₄	—	2	16	no	—
363	C₄	—	2	32	no	—
364	C₄	—	2	32	no	—
365	C₄	—	2	32	no	—
366	C₄	C₂⁴×#41	1	32	yes	—
367	C₄	C₂⁴×#41	1	32	yes	—
368	C₂	—	2	32	no	—
369	C₂	—	4	32	no	—
370	C₂	C₂³×#529	1	60	yes	direct product factor
371	C₂	C₂⁵×#41	1	1	yes	characteristic
372	C₂	C₂⁵×#41	1	1	yes	characteristic
373	C₂	C₂⁵×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#529	C₂	direct (N × H)
2. C₂⁴×#126	C₂	semidirect
3. C₂⁴×#125	C₂	semidirect
4. C₂⁴×#124	C₂	semidirect
5. C₂⁴×#123	C₂	semidirect
6. C₂⁴×C₂²⋊Q₈	C₂	semidirect
8. C₂⁴×C₄×Q₈	C₂	semidirect
9. C₂⁴×C₄×D₄	C₂	semidirect
10. C₂⁴×C₄×D₄	C₂	semidirect
11. C₂⁴×#117	C₂	semidirect
12. C₂⁵×C₄²	C₂	semidirect
20. C₂²×#529	C₂²	direct (N × H)
21. C₂³×#126	C₂²	semidirect
22. C₂³×#125	C₂²	semidirect
23. C₂³×#124	C₂²	semidirect
24. C₂³×#123	C₂²	semidirect
25. C₂³×C₂²⋊Q₈	C₂²	semidirect
27. C₂³×C₄×Q₈	C₂²	semidirect
28. C₂³×C₄×D₄	C₂²	semidirect
29. C₂³×C₄×D₄	C₂²	semidirect
30. C₂³×#117	C₂²	semidirect
47. C₂⁴×C₄²	C₂²	semidirect
73. C₂×#529	C₂³	direct (N × H)
74. C₂²×#126	C₂³	semidirect
75. C₂²×#125	C₂³	semidirect
76. C₂²×#124	C₂³	semidirect
77. C₂²×#123	C₂³	semidirect
78. C₂²×C₂²⋊Q₈	C₂³	semidirect
80. C₂²×C₄×Q₈	C₂³	semidirect
81. C₂²×C₄×D₄	C₂³	semidirect
82. C₂²×C₄×D₄	C₂³	semidirect
83. C₂²×#117	C₂³	semidirect
100. C₂³×C₄²	C₂³	semidirect
142. #529	C₂⁴	direct (N × H)
143. C₂×#126	C₂⁴	semidirect
144. C₂×#125	C₂⁴	semidirect
145. C₂×#124	C₂⁴	semidirect
146. C₂×#123	C₂⁴	semidirect
147. C₂×C₂²⋊Q₈	C₂⁴	semidirect
149. C₂×C₄×Q₈	C₂⁴	semidirect
150. C₂×C₄×D₄	C₂⁴	semidirect
151. C₂×C₄×D₄	C₂⁴	semidirect
152. C₂×#117	C₂⁴	semidirect
169. C₂²×C₄²	C₂⁴	semidirect
216. #126	C₂⁵	semidirect
217. #125	C₂⁵	semidirect
218. #124	C₂⁵	semidirect
219. #123	C₂⁵	semidirect
220. C₂²⋊Q₈	C₂⁵	semidirect
222. C₄×Q₈	C₂⁵	semidirect
223. C₄×D₄	C₂⁵	semidirect
224. C₄×D₄	C₂⁵	semidirect
225. #117	C₂⁵	semidirect
242. C₂×C₄²	C₂⁵	semidirect
250. C₂⁴	#529	direct (N × H)
313. C₂³	C₂×#529	direct (N × H)
353. C₂²	C₂²×#529	direct (N × H)
370. C₂	C₂³×#529	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×#529	2
C₂⁴×#126	2
C₂⁴×#125	2
C₂⁴×#124	2
C₂⁴×#123	2
C₂⁴×C₂²⋊Q₈	2
C₂⁴×#121	2
C₂⁴×C₄×Q₈	2
C₂⁴×C₄×D₄	2
C₂⁴×#117	2
C₂⁵×C₄²	2

Subgroups — C24×#529 #72295499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#529 #72295499