Subgroups — C₂⁴×#538 #72295507

Order 1024.

634325 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×#538	C₂	1	240	yes	maximal, direct product factor
2	C₂⁴×#126	C₂	1	2	yes	maximal, complemented
3	C₂⁴×#124	C₂	1	2	yes	maximal, complemented
4	C₂⁴×#123	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×#121	C₂	1	2	yes	maximal, complemented
8	C₂⁴×#120	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂⁵×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁸	C₂²	1	1	yes	characteristic
12	C₂⁵×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂⁵×D₄	C₂²	1	2	yes	—
14	C₂⁵×D₄	C₂²	1	2	yes	—
15	C₂⁵×D₄	C₂²	1	2	yes	—
16	C₂⁵×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂⁶×C₄	C₂²	1	1	yes	characteristic
18	C₂⁶×C₄	C₂²	1	1	yes	characteristic
19	C₂⁶×C₄	C₂²	1	2	yes	complemented
20	C₂²×#538	C₂²	1	8960	yes	direct product factor
21	C₂³×#126	C₂²	1	240	yes	complemented
22	C₂³×#124	C₂²	1	240	yes	complemented
23	C₂³×#123	C₂²	1	240	yes	complemented
24	C₂³×C₂²⋊Q₈	C₂²	1	120	yes	complemented
25	C₂³×#121	C₂²	1	120	yes	complemented
26	C₂³×#121	C₂²	1	240	yes	complemented
27	C₂³×#120	C₂²	1	240	yes	complemented
28	C₂³×C₄×D₄	C₂²	1	240	yes	complemented
29	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	complemented
31	C₂⁴×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₂⁴×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
33	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
35	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
37	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂⁴×C₂²⋊C₄	C₂²	1	120	yes	complemented
39	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	complemented
40	C₂⁴×C₂²⋊C₄	C₂²	1	2	yes	complemented
41	C₂⁴×C₄²	C₂²	1	2	yes	complemented
42	C₂⁷	—	2	32	no	—
43	C₂⁷	—	4	16	no	—
44	C₂⁷	—	4	16	no	—
45	C₂⁷	C₂³	1	2	yes	—
46	C₂⁷	C₂³	1	2	yes	—
47	C₂⁷	C₂³	1	1	yes	characteristic
48	C₂⁷	C₂³	1	60	yes	—
49	C₂⁴×Q₈	—	4	16	no	—
50	C₂⁴×Q₈	C₂³	1	60	yes	complemented
51	C₂⁴×D₄	—	2	64	no	—
52	C₂⁴×D₄	—	4	32	no	—
53	C₂⁴×D₄	—	4	32	no	—
54	C₂⁴×D₄	—	4	16	no	—
55	C₂⁴×D₄	C₂³	1	120	yes	—
56	C₂⁴×D₄	C₂³	1	120	yes	—
57	C₂⁴×D₄	C₂³	1	120	yes	—
58	C₂⁴×D₄	C₂³	1	60	yes	complemented
59	C₂⁵×C₄	—	4	16	no	—
60	C₂⁵×C₄	—	4	16	no	—
61	C₂⁵×C₄	—	4	32	no	—
62	C₂⁵×C₄	C₂³	1	2	yes	—
63	C₂⁵×C₄	C₂³	1	1	yes	characteristic
64	C₂⁵×C₄	C₂³	1	1	yes	characteristic
65	C₂⁵×C₄	C₂³	1	2	yes	—
66	C₂⁵×C₄	C₂³	1	2	yes	—
67	C₂⁵×C₄	C₂³	1	2	yes	—
68	C₂⁵×C₄	C₂³	1	60	yes	—
69	C₂⁵×C₄	C₂³	1	60	yes	—
70	C₂⁵×C₄	C₂³	1	120	yes	complemented
71	C₂×#538	C₂³	1	61440	yes	direct product factor
72	C₂²×#126	C₂³	1	4480	yes	complemented
73	C₂²×#124	C₂³	1	4480	yes	complemented
74	C₂²×#123	C₂³	1	4480	yes	complemented
75	C₂²×C₂²⋊Q₈	C₂³	1	2240	yes	complemented
76	C₂²×#121	C₂³	1	4480	yes	complemented
77	C₂²×#121	C₂³	1	2240	yes	complemented
78	C₂²×#120	C₂³	1	4480	yes	complemented
79	C₂²×C₄×D₄	C₂³	1	4480	yes	complemented
80	C₂³×C₄⋊C₄	C₂³	1	120	yes	complemented
81	C₂³×C₄⋊C₄	C₂³	1	120	yes	complemented
82	C₂³×C₄⋊C₄	C₂³	1	60	yes	complemented
83	C₂³×C₄⋊C₄	C₂³	1	60	yes	complemented
84	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
85	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
86	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
87	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
88	C₂³×C₂²⋊C₄	C₂³	1	120	yes	—
89	C₂³×C₂²⋊C₄	C₂³	1	2240	yes	complemented
90	C₂³×C₂²⋊C₄	C₂³	1	120	yes	complemented
91	C₂³×C₂²⋊C₄	C₂³	1	120	yes	complemented
92	C₂³×C₄²	C₂³	1	120	yes	complemented
93	C₂⁶	—	2	16	no	—
94	C₂⁶	—	2	16	no	—
95	C₂⁶	—	2	32	no	—
96	C₂⁶	—	2	960	no	—
97	C₂⁶	—	2	32	no	—
98	C₂⁶	—	2	32	no	—
99	C₂⁶	—	2	32	no	—
100	C₂⁶	—	2	32	no	—
101	C₂⁶	—	2	32	no	—
102	C₂⁶	—	4	480	no	—
103	C₂⁶	—	4	480	no	—
104	C₂⁶	—	4	1024	no	—
105	C₂⁶	#41	1	32	yes	—
106	C₂⁶	C₂⁴	1	60	yes	—
107	C₂⁶	C₂⁴	1	60	yes	—
108	C₂⁶	C₂⁴	1	30	yes	—
109	C₂⁶	C₂⁴	1	560	yes	—
110	C₂⁶	C₂⁴	1	1	yes	characteristic
111	C₂³×Q₈	—	4	480	no	—
112	C₂³×Q₈	C₂⁴	1	560	yes	complemented
113	C₂³×D₄	—	2	1920	no	—
114	C₂³×D₄	—	4	960	no	—
115	C₂³×D₄	—	4	960	no	—
116	C₂³×D₄	—	4	480	no	—
117	C₂³×D₄	C₂⁴	1	1120	yes	—
118	C₂³×D₄	C₂⁴	1	1120	yes	—
119	C₂³×D₄	C₂⁴	1	1120	yes	—
120	C₂³×D₄	C₂⁴	1	560	yes	complemented
121	C₂⁴×C₄	—	2	32	no	—
122	C₂⁴×C₄	—	2	32	no	—
123	C₂⁴×C₄	—	2	16	no	—
124	C₂⁴×C₄	—	2	32	no	—
125	C₂⁴×C₄	—	2	16	no	—
126	C₂⁴×C₄	—	2	32	no	—
127	C₂⁴×C₄	—	4	480	no	—
128	C₂⁴×C₄	—	4	480	no	—
129	C₂⁴×C₄	—	4	960	no	—
130	C₂⁴×C₄	C₂⁴	1	60	yes	—
131	C₂⁴×C₄	C₂⁴	1	30	yes	—
132	C₂⁴×C₄	C₂⁴	1	30	yes	—
133	C₂⁴×C₄	C₂⁴	1	60	yes	—
134	C₂⁴×C₄	C₂⁴	1	60	yes	—
135	C₂⁴×C₄	C₂⁴	1	60	yes	—
136	C₂⁴×C₄	C₂⁴	1	560	yes	—
137	C₂⁴×C₄	C₂⁴	1	560	yes	—
138	C₂⁴×C₄	C₂⁴	1	1120	yes	complemented
139	#538	C₂⁴	1	65536	yes	direct product factor
140	C₂×#126	C₂⁴	1	15360	yes	complemented
141	C₂×#124	C₂⁴	1	15360	yes	complemented
142	C₂×#123	C₂⁴	1	15360	yes	complemented
143	C₂×C₂²⋊Q₈	C₂⁴	1	7680	yes	complemented
144	C₂×#121	C₂⁴	1	15360	yes	complemented
145	C₂×#121	C₂⁴	1	7680	yes	complemented
146	C₂×#120	C₂⁴	1	15360	yes	complemented
147	C₂×C₄×D₄	C₂⁴	1	15360	yes	complemented
148	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	complemented
149	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	complemented
150	C₂²×C₄⋊C₄	C₂⁴	1	560	yes	complemented
151	C₂²×C₄⋊C₄	C₂⁴	1	560	yes	complemented
152	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
153	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
154	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
155	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
156	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	—
157	C₂²×C₂²⋊C₄	C₂⁴	1	7680	yes	complemented
158	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	complemented
159	C₂²×C₂²⋊C₄	C₂⁴	1	1120	yes	complemented
160	C₂²×C₄²	C₂⁴	1	1120	yes	complemented
161	C₂⁵	—	2	512	no	—
162	C₂⁵	—	2	240	no	—
163	C₂⁵	—	2	240	no	—
164	C₂⁵	—	2	480	no	—
165	C₂⁵	—	2	4480	no	—
166	C₂⁵	—	2	480	no	—
167	C₂⁵	—	2	480	no	—
168	C₂⁵	—	2	480	no	—
169	C₂⁵	—	2	480	no	—
170	C₂⁵	—	2	480	no	—
171	C₂⁵	—	4	2240	no	—
172	C₂⁵	—	4	2240	no	—
173	C₂⁵	—	4	15360	no	—
174	C₂⁵	—	4	512	no	—
175	C₂⁵	—	4	512	no	—
176	C₂⁵	C₂×#41	1	480	yes	—
177	C₂⁵	C₂×#41	1	16	yes	—
178	C₂⁵	H(5,𝔽₂)	1	16	yes	—
179	C₂⁵	H(5,𝔽₂)	1	16	yes	—
180	C₂⁵	C₂⁵	1	280	yes	—
181	C₂⁵	C₂⁵	1	280	yes	—
182	C₂⁵	C₂⁵	1	140	yes	—
183	C₂⁵	C₂⁵	1	960	yes	—
184	C₂⁵	C₂⁵	1	15	yes	—
185	C₂²×Q₈	—	4	2240	no	—
186	C₂²×Q₈	C₂⁵	1	960	yes	complemented
187	C₂²×D₄	—	2	8960	no	—
188	C₂²×D₄	—	4	4480	no	—
189	C₂²×D₄	—	4	4480	no	—
190	C₂²×D₄	—	4	2240	no	—
191	C₂²×D₄	C₂⁵	1	1920	yes	—
192	C₂²×D₄	C₂⁵	1	1920	yes	—
193	C₂²×D₄	C₂⁵	1	1920	yes	—
194	C₂²×D₄	C₂⁵	1	960	yes	complemented
195	C₂³×C₄	—	2	480	no	—
196	C₂³×C₄	—	2	480	no	—
197	C₂³×C₄	—	2	240	no	—
198	C₂³×C₄	—	2	480	no	—
199	C₂³×C₄	—	2	240	no	—
200	C₂³×C₄	—	2	480	no	—
201	C₂³×C₄	—	4	2240	no	—
202	C₂³×C₄	—	4	2240	no	—
203	C₂³×C₄	—	4	4480	no	—
204	C₂³×C₄	C₂⁵	1	140	yes	—
205	C₂³×C₄	C₂⁵	1	280	yes	—
206	C₂³×C₄	C₂⁵	1	280	yes	—
207	C₂³×C₄	C₂⁵	1	280	yes	—
208	C₂³×C₄	C₂⁵	1	140	yes	—
209	C₂³×C₄	C₂⁵	1	280	yes	—
210	C₂³×C₄	C₂⁵	1	960	yes	—
211	C₂³×C₄	C₂⁵	1	960	yes	—
212	C₂³×C₄	C₂⁵	1	1920	yes	complemented
213	#126	C₂⁵	1	8192	yes	complemented
214	#124	C₂⁵	1	8192	yes	complemented
215	#123	C₂⁵	1	8192	yes	complemented
216	C₂²⋊Q₈	C₂⁵	1	4096	yes	complemented
217	#121	C₂⁵	1	8192	yes	complemented
218	#121	C₂⁵	1	4096	yes	complemented
219	#120	C₂⁵	1	8192	yes	complemented
220	C₄×D₄	C₂⁵	1	8192	yes	complemented
221	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	complemented
222	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	complemented
223	C₂×C₄⋊C₄	C₂⁵	1	960	yes	complemented
224	C₂×C₄⋊C₄	C₂⁵	1	960	yes	complemented
225	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
226	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
227	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
228	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
229	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	—
230	C₂×C₂²⋊C₄	C₂⁵	1	4096	yes	complemented
231	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	complemented
232	C₂×C₂²⋊C₄	C₂⁵	1	1920	yes	complemented
233	C₂×C₄²	C₂⁵	1	1920	yes	complemented
234	C₂⁴	—	2	560	no	—
235	C₂⁴	—	2	560	no	—
236	C₂⁴	—	2	3840	no	—
237	C₂⁴	—	2	1120	no	—
238	C₂⁴	—	2	3840	no	—
239	C₂⁴	—	2	1120	no	—
240	C₂⁴	—	2	1120	no	—
241	C₂⁴	—	2	1120	no	—
242	C₂⁴	—	2	1120	no	—
243	C₂⁴	—	2	1120	no	—
244	C₂⁴	—	4	1920	no	—
245	C₂⁴	—	4	1920	no	—
246	C₂⁴	—	4	35840	no	—
247	C₂⁴	—	4	3840	no	—
248	C₂⁴	—	4	3840	no	—
249	C₂⁴	#538	1	256	yes	direct product factor
250	C₂⁴	C₂²×#41	1	1120	yes	—
251	C₂⁴	C₂²×#41	1	120	yes	—
252	C₂⁴	C₂×H(5,𝔽₂)	1	120	yes	—
253	C₂⁴	C₂×H(5,𝔽₂)	1	120	yes	—
254	C₂⁴	C₂⁶	1	240	yes	—
255	C₂⁴	C₂⁶	1	240	yes	—
256	C₂⁴	C₂⁶	1	120	yes	—
257	C₂⁴	C₂⁶	1	256	yes	—
258	C₂⁴	C₂⁶	1	35	yes	—
259	C₂×Q₈	—	4	1920	no	—
260	C₂×Q₈	C₂⁶	1	256	yes	complemented
261	C₂×D₄	—	2	7680	no	—
262	C₂×D₄	—	4	3840	no	—
263	C₂×D₄	—	4	3840	no	—
264	C₂×D₄	—	4	1920	no	—
265	C₂×D₄	C₂⁶	1	512	yes	—
266	C₂×D₄	C₂⁶	1	512	yes	—
267	C₂×D₄	C₂⁶	1	512	yes	—
268	C₂×D₄	C₂⁶	1	256	yes	complemented
269	C₂²×C₄	—	2	1120	no	—
270	C₂²×C₄	—	2	1120	no	—
271	C₂²×C₄	—	2	560	no	—
272	C₂²×C₄	—	2	1120	no	—
273	C₂²×C₄	—	2	560	no	—
274	C₂²×C₄	—	2	1120	no	—
275	C₂²×C₄	—	4	1920	no	—
276	C₂²×C₄	—	4	1920	no	—
277	C₂²×C₄	—	4	3840	no	—
278	C₂²×C₄	C₂⁶	1	240	yes	—
279	C₂²×C₄	C₂⁶	1	240	yes	—
280	C₂²×C₄	C₂⁶	1	240	yes	—
281	C₂²×C₄	C₂⁶	1	120	yes	—
282	C₂²×C₄	C₂⁶	1	120	yes	—
283	C₂²×C₄	C₂⁶	1	240	yes	—
284	C₂²×C₄	C₂⁶	1	256	yes	—
285	C₂²×C₄	C₂⁶	1	256	yes	—
286	C₂²×C₄	C₂⁶	1	512	yes	complemented
287	C₄⋊C₄	C₂⁶	1	512	yes	complemented
288	C₄⋊C₄	C₂⁶	1	512	yes	complemented
289	C₄⋊C₄	C₂⁶	1	256	yes	complemented
290	C₄⋊C₄	C₂⁶	1	256	yes	complemented
291	C₂²⋊C₄	C₂⁶	1	512	yes	—
292	C₂²⋊C₄	C₂⁶	1	512	yes	—
293	C₂²⋊C₄	C₂⁶	1	512	yes	—
294	C₂²⋊C₄	C₂⁶	1	512	yes	—
295	C₂²⋊C₄	C₂⁶	1	512	yes	—
296	C₂²⋊C₄	C₂⁶	1	512	yes	complemented
297	C₂²⋊C₄	C₂⁶	1	512	yes	complemented
298	C₄²	C₂⁶	1	512	yes	complemented
299	C₂³	—	2	240	no	—
300	C₂³	—	2	240	no	—
301	C₂³	—	2	4480	no	—
302	C₂³	—	2	480	no	—
303	C₂³	—	2	512	no	—
304	C₂³	—	2	480	no	—
305	C₂³	—	2	480	no	—
306	C₂³	—	2	480	no	—
307	C₂³	—	2	480	no	—
308	C₂³	—	2	480	no	—
309	C₂³	—	4	256	no	—
310	C₂³	—	4	256	no	—
311	C₂³	—	4	15360	no	—
312	C₂³	—	4	4480	no	—
313	C₂³	—	4	4480	no	—
314	C₂³	C₂×#538	1	960	yes	direct product factor
315	C₂³	C₂³×#41	1	480	yes	—
316	C₂³	C₂³×#41	1	140	yes	—
317	C₂³	C₂²×H(5,𝔽₂)	1	140	yes	—
318	C₂³	C₂²×H(5,𝔽₂)	1	140	yes	—
319	C₂³	C₂⁷	1	32	yes	—
320	C₂³	C₂⁷	1	32	yes	—
321	C₂³	C₂⁷	1	16	yes	—
322	C₂³	C₂⁷	1	15	yes	—
323	Q₈	—	4	256	no	—
324	D₄	—	2	1024	no	—
325	D₄	—	4	512	no	—
326	D₄	—	4	512	no	—
327	D₄	—	4	256	no	—
328	C₂×C₄	—	2	480	no	—
329	C₂×C₄	—	2	480	no	—
330	C₂×C₄	—	2	240	no	—
331	C₂×C₄	—	2	480	no	—
332	C₂×C₄	—	2	240	no	—
333	C₂×C₄	—	2	480	no	—
334	C₂×C₄	—	4	256	no	—
335	C₂×C₄	—	4	256	no	—
336	C₂×C₄	—	4	512	no	—
337	C₂×C₄	C₂⁷	1	32	yes	—
338	C₂×C₄	C₂⁷	1	32	yes	—
339	C₂×C₄	C₂⁷	1	32	yes	—
340	C₂×C₄	C₂⁷	1	16	yes	—
341	C₂×C₄	C₂⁷	1	16	yes	—
342	C₂×C₄	C₂⁷	1	32	yes	—
343	C₂²	—	2	960	no	—
344	C₂²	—	2	16	no	—
345	C₂²	—	2	16	no	—
346	C₂²	—	2	32	no	—
347	C₂²	—	2	32	no	—
348	C₂²	—	2	32	no	—
349	C₂²	—	2	32	no	—
350	C₂²	—	2	32	no	—
351	C₂²	—	2	32	no	—
352	C₂²	—	4	1024	no	—
353	C₂²	—	4	960	no	—
354	C₂²	—	4	960	no	—
355	C₂²	C₂²×#538	1	560	yes	direct product factor
356	C₂²	C₂⁴×#41	1	32	yes	—
357	C₂²	C₂⁴×#41	1	30	yes	—
358	C₂²	C₂³×H(5,𝔽₂)	1	30	yes	—
359	C₂²	C₂³×H(5,𝔽₂)	1	30	yes	—
360	C₂²	C₂⁸	1	1	yes	characteristic
361	C₄	—	2	32	no	—
362	C₄	—	2	32	no	—
363	C₄	—	2	16	no	—
364	C₄	—	2	32	no	—
365	C₄	—	2	16	no	—
366	C₄	—	2	32	no	—
367	C₂	—	2	32	no	—
368	C₂	—	4	32	no	—
369	C₂	—	4	32	no	—
370	C₂	C₂³×#538	1	60	yes	direct product factor
371	C₂	C₂⁵×#41	1	1	yes	characteristic
372	C₂	C₂⁴×H(5,𝔽₂)	1	1	yes	characteristic
373	C₂	C₂⁴×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#538	C₂	direct (N × H)
2. C₂⁴×#126	C₂	semidirect
3. C₂⁴×#124	C₂	semidirect
4. C₂⁴×#123	C₂	semidirect
5. C₂⁴×C₂²⋊Q₈	C₂	semidirect
6. C₂⁴×#121	C₂	semidirect
7. C₂⁴×#121	C₂	semidirect
8. C₂⁴×#120	C₂	semidirect
9. C₂⁴×C₄×D₄	C₂	semidirect
10. C₂⁵×C₂²⋊C₄	C₂	semidirect
12. C₂⁵×Q₈	C₂²	semidirect
16. C₂⁵×D₄	C₂²	semidirect
19. C₂⁶×C₄	C₂²	semidirect
20. C₂²×#538	C₂²	direct (N × H)
21. C₂³×#126	C₂²	semidirect
22. C₂³×#124	C₂²	semidirect
23. C₂³×#123	C₂²	semidirect
24. C₂³×C₂²⋊Q₈	C₂²	semidirect
25. C₂³×#121	C₂²	semidirect
26. C₂³×#121	C₂²	semidirect
27. C₂³×#120	C₂²	semidirect
28. C₂³×C₄×D₄	C₂²	semidirect
29. C₂⁴×C₄⋊C₄	C₂²	semidirect
30. C₂⁴×C₄⋊C₄	C₂²	semidirect
31. C₂⁴×C₄⋊C₄	C₂²	semidirect
32. C₂⁴×C₄⋊C₄	C₂²	semidirect
38. C₂⁴×C₂²⋊C₄	C₂²	semidirect
39. C₂⁴×C₂²⋊C₄	C₂²	semidirect
40. C₂⁴×C₂²⋊C₄	C₂²	semidirect
41. C₂⁴×C₄²	C₂²	semidirect
50. C₂⁴×Q₈	C₂³	semidirect
58. C₂⁴×D₄	C₂³	semidirect
70. C₂⁵×C₄	C₂³	semidirect
71. C₂×#538	C₂³	direct (N × H)
72. C₂²×#126	C₂³	semidirect
73. C₂²×#124	C₂³	semidirect
74. C₂²×#123	C₂³	semidirect
75. C₂²×C₂²⋊Q₈	C₂³	semidirect
76. C₂²×#121	C₂³	semidirect
77. C₂²×#121	C₂³	semidirect
78. C₂²×#120	C₂³	semidirect
79. C₂²×C₄×D₄	C₂³	semidirect
80. C₂³×C₄⋊C₄	C₂³	semidirect
81. C₂³×C₄⋊C₄	C₂³	semidirect
82. C₂³×C₄⋊C₄	C₂³	semidirect
83. C₂³×C₄⋊C₄	C₂³	semidirect
89. C₂³×C₂²⋊C₄	C₂³	semidirect
90. C₂³×C₂²⋊C₄	C₂³	semidirect
91. C₂³×C₂²⋊C₄	C₂³	semidirect
92. C₂³×C₄²	C₂³	semidirect
112. C₂³×Q₈	C₂⁴	semidirect
120. C₂³×D₄	C₂⁴	semidirect
138. C₂⁴×C₄	C₂⁴	semidirect
139. #538	C₂⁴	direct (N × H)
140. C₂×#126	C₂⁴	semidirect
141. C₂×#124	C₂⁴	semidirect
142. C₂×#123	C₂⁴	semidirect
143. C₂×C₂²⋊Q₈	C₂⁴	semidirect
144. C₂×#121	C₂⁴	semidirect
145. C₂×#121	C₂⁴	semidirect
146. C₂×#120	C₂⁴	semidirect
147. C₂×C₄×D₄	C₂⁴	semidirect
148. C₂²×C₄⋊C₄	C₂⁴	semidirect
149. C₂²×C₄⋊C₄	C₂⁴	semidirect
150. C₂²×C₄⋊C₄	C₂⁴	semidirect
151. C₂²×C₄⋊C₄	C₂⁴	semidirect
157. C₂²×C₂²⋊C₄	C₂⁴	semidirect
158. C₂²×C₂²⋊C₄	C₂⁴	semidirect
159. C₂²×C₂²⋊C₄	C₂⁴	semidirect
160. C₂²×C₄²	C₂⁴	semidirect
186. C₂²×Q₈	C₂⁵	semidirect
194. C₂²×D₄	C₂⁵	semidirect
212. C₂³×C₄	C₂⁵	semidirect
213. #126	C₂⁵	semidirect
214. #124	C₂⁵	semidirect
215. #123	C₂⁵	semidirect
216. C₂²⋊Q₈	C₂⁵	semidirect
217. #121	C₂⁵	semidirect
218. #121	C₂⁵	semidirect
219. #120	C₂⁵	semidirect
220. C₄×D₄	C₂⁵	semidirect
221. C₂×C₄⋊C₄	C₂⁵	semidirect
222. C₂×C₄⋊C₄	C₂⁵	semidirect
223. C₂×C₄⋊C₄	C₂⁵	semidirect
224. C₂×C₄⋊C₄	C₂⁵	semidirect
230. C₂×C₂²⋊C₄	C₂⁵	semidirect
231. C₂×C₂²⋊C₄	C₂⁵	semidirect
232. C₂×C₂²⋊C₄	C₂⁵	semidirect
233. C₂×C₄²	C₂⁵	semidirect
249. C₂⁴	#538	direct (N × H)
260. C₂×Q₈	C₂⁶	semidirect
268. C₂×D₄	C₂⁶	semidirect
286. C₂²×C₄	C₂⁶	semidirect
287. C₄⋊C₄	C₂⁶	semidirect
288. C₄⋊C₄	C₂⁶	semidirect
289. C₄⋊C₄	C₂⁶	semidirect
290. C₄⋊C₄	C₂⁶	semidirect
296. C₂²⋊C₄	C₂⁶	semidirect
297. C₂²⋊C₄	C₂⁶	semidirect
298. C₄²	C₂⁶	semidirect
314. C₂³	C₂×#538	direct (N × H)
355. C₂²	C₂²×#538	direct (N × H)
370. C₂	C₂³×#538	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×#538	2
C₂⁴×#126	2
C₂⁴×#124	2
C₂⁴×#123	2
C₂⁴×C₂²⋊Q₈	2
C₂⁴×#121	2
C₂⁴×#120	2
C₂⁴×C₄×D₄	2
C₂⁵×C₂²⋊C₄	2

Subgroups — C24×#538 #72295507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#538 #72295507