Subgroups — #72314507

Order 2048.

At least 9440 conjugacy classes of subgroups; only the 1524 normal subgroups are shown.

The subgroup lattice has 506 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#146484	C₂	1	2	yes	maximal, complemented
2	C₂×#147582	C₂	1	2	yes	maximal, complemented
3	#72260460	C₂	1	1	yes	maximal, characteristic, complemented
4	#72240229	C₂	1	2	yes	maximal, complemented
5	#72238533	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#147573	C₂	1	2	yes	maximal, complemented
7	C₂²×#12711	C₂	1	2	yes	maximal, complemented
8	C₂×#392132	C₂	1	2	yes	maximal, complemented
9	C₂×#7631286	C₂	1	1	yes	maximal, characteristic
10	C₂⁴×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
11	C₂²×#3484	C₄	1	1	yes	characteristic, complemented
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	C₂³×D₄²	C₄	1	1	yes	characteristic, complemented
14	C₂×#35527	C₄	1	1	yes	characteristic, complemented
15	C₂×#35527	C₂²	1	1	yes	characteristic
16	C₂×#35512	C₄	1	1	yes	characteristic, complemented
17	C₂×#35512	C₂²	1	1	yes	characteristic
18	C₂×#35250	C₄	1	1	yes	characteristic, complemented
19	C₂×#35250	C₂²	1	1	yes	characteristic
20	C₂×#34736	C₄	1	1	yes	characteristic, complemented
21	C₂×#34736	C₂²	1	1	yes	characteristic
22	C₂×#34256	C₄	1	1	yes	characteristic, complemented
23	C₂×#34256	C₂²	1	1	yes	characteristic
24	C₂×#34171	C₄	1	1	yes	characteristic, complemented
25	C₂²×#2134	C₂²	1	2	yes	complemented
26	C₂²×#2128	C₂²	1	2	yes	complemented
27	C₂²×#2120	C₂²	1	4	yes	complemented
28	C₂²×#1795	C₂²	1	4	yes	—
29	C₂²×#1795	C₂²	1	4	yes	complemented
30	C₂×#20591	C₂²	1	2	yes	complemented
31	C₂×#20584	C₂²	1	2	yes	complemented
32	C₂×#12587	C₂²	1	2	yes	complemented
33	C₂×#12583	C₂²	1	2	yes	complemented
34	C₂×#12573	C₂²	1	4	yes	complemented
35	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
36	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
37	C₂³×H(5,𝔽₂)	D₄	1	2	yes	—
38	C₂³×H(5,𝔽₂)	D₄	1	2	yes	—
39	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
40	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
41	C₂⁴×#41	C₂×C₄	1	1	yes	characteristic
42	C₂⁴×#41	D₄	1	2	yes	—
43	C₂⁴×#41	C₂³	1	1	yes	characteristic
44	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
45	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
46	C₂⁵×D₄	D₄	1	2	yes	—
47	C₂×#3484	D₄	1	8	yes	—
48	C₂²×#535	C₂×C₄	1	1	yes	characteristic
49	C₂²×#535	C₂×C₄	1	1	yes	characteristic
50	C₂²×#535	C₂×C₄	1	1	yes	characteristic
51	C₂²×#535	C₂³	1	1	yes	characteristic
52	C₂³×#127	C₂×C₄	1	1	yes	characteristic
53	C₂²×#525	C₂×C₄	1	1	yes	characteristic
54	C₂²×#525	C₂×C₄	1	1	yes	characteristic
55	C₂²×#525	C₂×C₄	1	1	yes	characteristic
56	C₂²×#525	C₂³	1	1	yes	characteristic
57	C₂³×#121	C₂×C₄	1	1	yes	characteristic
58	C₂³×#121	C₂×C₄	1	1	yes	characteristic
59	C₂²×#518	C₂×C₄	1	1	yes	characteristic
60	C₂²×#518	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×D₄	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×D₄	C₂×C₄	1	1	yes	characteristic
63	C₂²×#413	C₂³	1	4	yes	—
64	C₂²×#410	C₂³	1	4	yes	—
65	#35527	D₄	1	8	yes	—
66	#35512	D₄	1	8	yes	—
67	C₂×#2406	D₄	1	2	yes	—
68	C₂×#2404	D₄	1	2	yes	—
69	C₂×#2431	D₄	1	2	yes	—
70	C₂×#2431	D₄	1	2	yes	—
71	C₂×#2436	D₄	1	2	yes	—
72	C₂×#2433	D₄	1	2	yes	—
73	C₂×#2405	D₄	1	2	yes	—
74	C₂×#2403	D₄	1	2	yes	—
75	#34256	D₄	1	8	yes	—
76	C₂×#2311	C₂×C₄	1	1	yes	characteristic
77	C₂×#2311	C₂×C₄	1	1	yes	characteristic
78	C₂×#2311	C₂×C₄	1	1	yes	characteristic
79	C₂×#2311	C₂³	1	1	yes	characteristic
80	C₂×#2309	C₂×C₄	1	1	yes	characteristic
81	C₂×#2309	C₂×C₄	1	1	yes	characteristic
82	C₂×#2297	C₂×C₄	1	1	yes	characteristic
83	C₂×#2297	C₂×C₄	1	1	yes	characteristic
84	C₂×#2297	C₂×C₄	1	1	yes	characteristic
85	C₂×#2297	C₂³	1	1	yes	characteristic
86	C₂×#2280	C₂×C₄	1	1	yes	characteristic
87	C₂×#2280	C₂×C₄	1	1	yes	characteristic
88	C₂×#1795	D₄	1	16	yes	complemented
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
90	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
91	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
92	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
93	C₂²×H(5,𝔽₂)	C₂×D₄	1	8	yes	—
94	C₂²×H(5,𝔽₂)	#41	1	8	yes	—
95	C₂³×#41	C₂²⋊C₄	1	2	yes	—
96	C₂³×#41	C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×#41	C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×#41	C₂²⋊C₄	1	2	yes	—
99	C₂³×#41	C₂²×C₄	1	1	yes	characteristic
100	C₂³×#41	C₂²×C₄	1	1	yes	characteristic
101	C₂³×#41	C₂²×C₄	1	1	yes	characteristic
102	C₂³×#41	C₂×D₄	1	2	yes	—
103	C₂³×#41	C₂×D₄	1	2	yes	—
104	C₂³×#41	C₂×D₄	1	2	yes	—
105	C₂³×#41	C₂×D₄	1	1	yes	characteristic
106	C₂³×#41	C₂×D₄	1	2	yes	—
107	C₂³×#41	C₂×D₄	1	1	yes	characteristic
108	C₂³×#41	C₂⁴	1	1	yes	characteristic
109	C₂⁴×Q₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂⁴×Q₈	C₂×D₄	1	1	yes	characteristic
111	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
112	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
113	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
114	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
115	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
116	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
117	C₂⁴×D₄	C₂²⋊C₄	1	4	yes	complemented
118	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
119	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
120	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
121	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×D₄	C₂×D₄	1	2	yes	—
123	C₂⁴×D₄	C₂×D₄	1	2	yes	—
124	C₂⁴×D₄	C₂×D₄	1	2	yes	—
125	C₂⁵×C₄	C₂²×C₄	1	1	yes	characteristic
126	C₂⁵×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂⁵×C₄	C₂×D₄	1	2	yes	—
128	C₂×#535	C₂²⋊C₄	1	4	yes	complemented
129	C₂×#535	C₂²⋊C₄	1	4	yes	complemented
130	C₂×#535	C₂×D₄	1	4	yes	—
131	C₂×#535	C₂×D₄	1	4	yes	—
132	C₂²×#127	C₂²⋊C₄	1	2	yes	—
133	C₂²×#127	C₂²⋊C₄	1	4	yes	complemented
134	C₂²×#127	C₂×D₄	1	2	yes	—
135	C₂²×#124	C₂²⋊C₄	1	2	yes	—
136	C₂²×#124	C₂×D₄	1	2	yes	—
137	C₂×#525	C₂²⋊C₄	1	4	yes	complemented
138	C₂×#525	C₂²⋊C₄	1	4	yes	complemented
139	C₂×#525	C₂×D₄	1	4	yes	—
140	C₂×#525	C₂×D₄	1	4	yes	—
141	C₂²×#123	C₂²⋊C₄	1	2	yes	—
142	C₂²×#123	C₂×D₄	1	2	yes	—
143	C₂²×#121	C₂²⋊C₄	1	2	yes	—
144	C₂²×#121	C₂²⋊C₄	1	2	yes	—
145	C₂²×#121	C₂²⋊C₄	1	4	yes	complemented
146	C₂²×#120	C₂²⋊C₄	1	2	yes	—
147	C₂²×#120	C₂×D₄	1	2	yes	—
148	C₂×#518	C₂²⋊C₄	1	4	yes	complemented
149	C₂×#518	C₂²⋊C₄	1	4	yes	complemented
150	C₂²×C₄×D₄	C₂²⋊C₄	1	4	yes	complemented
151	C₂²×#117	C₂²×C₄	1	1	yes	characteristic
152	C₂²×#117	C₂²×C₄	1	1	yes	characteristic
153	C₂²×#117	C₂²×C₄	1	1	yes	characteristic
154	C₂²×#117	C₂²×C₄	1	1	yes	characteristic
155	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
156	C₂³×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
157	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
158	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
159	C₂³×C₄²	C₂²×C₄	1	1	yes	characteristic
160	C₂×#413	C₂×D₄	1	8	yes	complemented
161	C₂×#410	C₂×D₄	1	8	yes	complemented
162	#2311	C₂²⋊C₄	1	4	yes	complemented
163	#2311	C₂²⋊C₄	1	4	yes	complemented
164	#2311	C₂×D₄	1	4	yes	—
165	#2311	C₂×D₄	1	4	yes	—
166	#2309	C₂²⋊C₄	1	4	yes	complemented
167	#2309	C₂²⋊C₄	1	4	yes	complemented
168	#2297	C₂²⋊C₄	1	4	yes	complemented
169	#2297	C₂²⋊C₄	1	4	yes	complemented
170	#2297	C₂×D₄	1	4	yes	—
171	#2297	C₂×D₄	1	4	yes	—
172	C₂×#391	C₂²⋊C₄	1	2	yes	—
173	C₂×#391	C₂×D₄	1	2	yes	—
174	C₂×#390	C₂²⋊C₄	1	2	yes	—
175	C₂×#390	C₂×D₄	1	2	yes	—
176	C₂×#386	C₂²⋊C₄	1	2	yes	—
177	C₂×#386	C₂²⋊C₄	1	2	yes	—
178	C₂×#386	C₂×D₄	1	2	yes	—
179	C₂×#386	C₂×D₄	1	2	yes	—
180	C₂×#381	C₂²⋊C₄	1	2	yes	—
181	C₂×#379	C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
184	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁶	#120	1	2	yes	—
187	C₂×H(5,𝔽₂)	#120	1	8	yes	—
188	C₂×H(5,𝔽₂)	#121	1	16	yes	—
189	C₂×H(5,𝔽₂)	#121	1	16	yes	—
190	C₂×H(5,𝔽₂)	#123	1	8	yes	—
191	C₂×H(5,𝔽₂)	#124	1	8	yes	—
192	C₂×H(5,𝔽₂)	#127	1	8	yes	—
193	C₂²×#41	#99	1	8	yes	—
194	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
198	C₂²×#41	C₄×D₄	1	8	yes	—
199	C₂²×#41	#120	1	8	yes	—
200	C₂²×#41	#120	1	8	yes	—
201	C₂²×#41	#120	1	4	yes	—
202	C₂²×#41	#121	1	8	yes	—
203	C₂²×#41	#123	1	4	yes	—
204	C₂²×#41	C₂²×D₄	1	2	yes	—
205	C₂²×#41	C₂²×D₄	1	2	yes	—
206	C₂²×#41	C₂²×D₄	1	2	yes	—
207	C₂²×#41	C₂²×D₄	1	2	yes	—
208	C₂²×#41	C₂²×D₄	1	4	yes	—
209	C₂²×#41	C₂×#41	1	4	yes	—
210	C₂²×#41	H(5,𝔽₂)	1	4	yes	—
211	C₂²×#41	H(5,𝔽₂)	1	4	yes	—
212	C₂³×Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂³×Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂³×Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×Q₈	#120	1	2	yes	—
216	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
217	C₂³×D₄	#99	1	8	yes	—
218	C₂³×D₄	#99	1	8	yes	—
219	C₂³×D₄	#99	1	8	yes	—
220	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂³×D₄	C₄×D₄	1	8	yes	—
234	C₂³×D₄	#120	1	2	yes	—
235	C₂³×D₄	#120	1	8	yes	—
236	C₂³×D₄	#120	1	8	yes	—
237	C₂³×D₄	#120	1	4	yes	—
238	C₂³×D₄	#120	1	2	yes	—
239	C₂³×D₄	#120	1	2	yes	—
240	C₂³×D₄	#121	1	8	yes	—
241	C₂³×D₄	#123	1	4	yes	—
242	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
243	C₂³×D₄	C₂²×D₄	1	2	yes	—
244	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂⁴×C₄	#120	1	2	yes	—
256	C₂⁴×C₄	#120	1	2	yes	—
257	C₂⁴×C₄	#120	1	2	yes	—
258	C₂⁴×C₄	C₂³×C₄	1	1	yes	characteristic
259	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
260	C₂×#127	#120	1	8	yes	—
261	C₂×#124	#120	1	8	yes	—
262	C₂×#123	#120	1	8	yes	—
263	C₂×#120	#120	1	8	yes	—
264	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
269	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
271	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
272	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
273	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
276	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	—
279	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	—
280	C₂²×C₄²	C₂×C₂²⋊C₄	1	2	yes	—
281	#391	#120	1	8	yes	—
282	#390	#120	1	8	yes	—
283	#386	#120	1	8	yes	—
284	#386	#120	1	8	yes	—
285	C₂⁵	#379	1	1	yes	characteristic
286	C₂⁵	#379	1	1	yes	characteristic
287	C₂⁵	#379	1	2	yes	—
288	C₂⁵	#379	1	2	yes	—
289	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
290	C₂⁵	C₂×#120	1	2	yes	—
291	C₂×#41	#386	1	8	yes	—
292	C₂×#41	#390	1	8	yes	—
293	C₂×#41	#392	1	16	yes	—
294	C₂×#41	#394	1	16	yes	—
295	C₂×#41	C₂×#120	1	2	yes	—
296	C₂×#41	C₂×#121	1	4	yes	—
297	C₂×#41	C₂×#121	1	4	yes	—
298	C₂×#41	C₂×#123	1	2	yes	—
299	C₂×#41	C₂×#124	1	2	yes	—
300	C₂×#41	C₂×#127	1	2	yes	—
301	C₂×#41	#534	1	4	yes	—
302	C₂×#41	#534	1	2	yes	—
303	C₂×#41	#534	1	2	yes	—
304	C₂×#41	#535	1	2	yes	—
305	C₂×#41	#535	1	2	yes	—
306	C₂×#41	#535	1	4	yes	—
307	C₂×#41	D₄²	1	4	yes	—
308	C₂×#41	D₄²	1	4	yes	—
309	C₂×#41	D₄²	1	4	yes	—
310	C₂×#41	D₄²	1	4	yes	—
311	C₂×#41	#546	1	4	yes	—
312	C₂×#41	#546	1	4	yes	—
313	C₂×#41	#546	1	4	yes	—
314	C₂×#41	#546	1	4	yes	—
315	C₂²×Q₈	#379	1	2	yes	—
316	C₂²×Q₈	#379	1	2	yes	—
317	C₂²×Q₈	#386	1	4	yes	—
318	C₂²×Q₈	C₂×#99	1	4	yes	—
319	C₂²×Q₈	C₂×#120	1	2	yes	—
320	C₂²×Q₈	C₂×#120	1	2	yes	—
321	C₂²×Q₈	D₄²	1	4	yes	—
322	C₂²×Q₈	#546	1	4	yes	—
323	C₂²×D₄	#379	1	2	yes	—
324	C₂²×D₄	#379	1	4	yes	—
325	C₂²×D₄	#379	1	2	yes	—
326	C₂²×D₄	#379	1	4	yes	—
327	C₂²×D₄	#379	1	4	yes	—
328	C₂²×D₄	#386	1	8	yes	—
329	C₂²×D₄	#386	1	4	yes	—
330	C₂²×D₄	#386	1	4	yes	—
331	C₂²×D₄	#386	1	4	yes	—
332	C₂²×D₄	#390	1	8	yes	—
333	C₂²×D₄	#392	1	16	yes	—
334	C₂²×D₄	#394	1	16	yes	—
335	C₂²×D₄	C₂×#99	1	4	yes	—
336	C₂²×D₄	C₂×#99	1	4	yes	—
337	C₂²×D₄	C₂×#99	1	4	yes	—
338	C₂²×D₄	C₂×#99	1	4	yes	—
339	C₂²×D₄	C₂×#99	1	4	yes	—
340	C₂²×D₄	C₂×#99	1	4	yes	—
341	C₂²×D₄	C₂×#99	1	4	yes	—
342	C₂²×D₄	C₂×#120	1	4	yes	—
343	C₂²×D₄	C₂×#120	1	4	yes	—
344	C₂²×D₄	C₂×#120	1	2	yes	—
345	C₂²×D₄	C₂×#120	1	2	yes	—
346	C₂²×D₄	D₄²	1	4	yes	—
347	C₂²×D₄	D₄²	1	4	yes	—
348	C₂²×D₄	D₄²	1	4	yes	—
349	C₂²×D₄	#546	1	4	yes	—
350	C₂²×D₄	#546	1	4	yes	—
351	C₂²×D₄	#546	1	4	yes	—
352	C₂³×C₄	#379	1	1	yes	characteristic
353	C₂³×C₄	#379	1	2	yes	—
354	C₂³×C₄	#379	1	1	yes	characteristic
355	C₂³×C₄	#379	1	2	yes	—
356	C₂³×C₄	#379	1	2	yes	—
357	C₂³×C₄	#379	1	2	yes	—
358	C₂³×C₄	C₂×#99	1	4	yes	—
359	C₂³×C₄	C₂×#99	1	4	yes	—
360	C₂³×C₄	C₂×#99	1	4	yes	—
361	C₂³×C₄	C₂×#99	1	4	yes	—
362	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
363	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
364	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
365	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
366	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
367	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
368	C₂³×C₄	C₂×C₄×D₄	1	4	yes	—
369	C₂³×C₄	#518	1	4	yes	—
370	C₂³×C₄	C₂×#120	1	2	yes	—
371	C₂³×C₄	C₂×#120	1	4	yes	—
372	C₂³×C₄	C₂×#120	1	4	yes	—
373	C₂³×C₄	C₂×#120	1	2	yes	—
374	C₂³×C₄	C₂×#121	1	4	yes	—
375	C₂³×C₄	C₂×#123	1	2	yes	—
376	C₂³×C₄	#534	1	4	yes	—
377	C₂³×C₄	#534	1	2	yes	—
378	C₂³×C₄	#534	1	2	yes	—
379	C₂×C₄⋊C₄	#379	1	4	yes	—
380	C₂×C₂²⋊C₄	#379	1	4	yes	—
381	C₂×C₂²⋊C₄	#379	1	4	yes	—
382	C₂×C₄²	#379	1	4	yes	—
383	C₂×C₄²	#379	1	4	yes	—
384	C₂⁴	#1781	1	2	yes	—
385	C₂⁴	#1781	1	4	yes	—
386	C₂⁴	#1781	1	2	yes	—
387	C₂⁴	C₂×#379	1	1	yes	characteristic
388	C₂⁴	C₂×#379	1	1	yes	characteristic
389	C₂⁴	C₂×#379	1	1	yes	characteristic
390	C₂⁴	D₄×C₂²⋊C₄	1	4	yes	—
391	C₂⁴	#2403	1	2	yes	—
392	C₂⁴	#2405	1	2	yes	—
393	C₂⁴	#2410	1	4	yes	—
394	C₂⁴	#2846	1	4	yes	—
395	C₂×Q₈	#1438	1	4	yes	—
396	C₂×Q₈	#1781	1	4	yes	—
397	C₂×Q₈	#2403	1	2	yes	—
398	C₂×Q₈	#2405	1	2	yes	—
399	C₂×Q₈	#2408	1	4	yes	—
400	C₂×Q₈	#2410	1	4	yes	—
401	C₂×Q₈	#2433	1	2	yes	—
402	C₂×Q₈	#2435	1	2	yes	—
403	C₂×D₄	#1438	1	4	yes	—
404	C₂×D₄	#1438	1	8	yes	—
405	C₂×D₄	#1781	1	4	yes	—
406	C₂×D₄	#1781	1	8	yes	—
407	C₂×D₄	#1781	1	8	yes	—
408	C₂×D₄	#1781	1	8	yes	—
409	C₂×D₄	#1898	1	8	yes	—
410	C₂×D₄	#1898	1	8	yes	—
411	C₂×D₄	#1898	1	8	yes	—
412	C₂×D₄	#1898	1	8	yes	—
413	C₂×D₄	#2403	1	2	yes	—
414	C₂×D₄	#2403	1	4	yes	—
415	C₂×D₄	#2405	1	2	yes	—
416	C₂×D₄	#2405	1	4	yes	—
417	C₂×D₄	#2408	1	4	yes	—
418	C₂×D₄	#2408	1	4	yes	—
419	C₂×D₄	#2408	1	4	yes	—
420	C₂×D₄	#2410	1	4	yes	—
421	C₂×D₄	#2410	1	4	yes	—
422	C₂×D₄	#2410	1	4	yes	—
423	C₂×D₄	#2433	1	2	yes	—
424	C₂×D₄	#2433	1	4	yes	—
425	C₂×D₄	#2435	1	2	yes	—
426	C₂×D₄	#2435	1	4	yes	—
427	C₂²×C₄	#1781	1	4	yes	—
428	C₂²×C₄	#1781	1	2	yes	—
429	C₂²×C₄	#1781	1	2	yes	—
430	C₂²×C₄	C₂×#379	1	1	yes	characteristic
431	C₂²×C₄	C₂×#379	1	1	yes	characteristic
432	C₂²×C₄	C₂×#379	1	1	yes	characteristic
433	C₂²×C₄	C₂×#379	1	1	yes	characteristic
434	C₂²×C₄	C₂×#386	1	2	yes	—
435	C₂²×C₄	C₂×#386	1	2	yes	—
436	C₂²×C₄	C₂×#390	1	2	yes	—
437	C₂²×C₄	#2308	1	2	yes	—
438	C₂²×C₄	#2308	1	2	yes	—
439	C₂²×C₄	#2309	1	2	yes	—
440	C₂²×C₄	D₄×C₂²⋊C₄	1	4	yes	—
441	C₂²×C₄	D₄×C₂²⋊C₄	1	4	yes	—
442	C₂²×C₄	D₄×C₂²⋊C₄	1	4	yes	—
443	C₂²×C₄	C₂×#392	1	4	yes	—
444	C₂²×C₄	C₂×#394	1	4	yes	—
445	C₂²×C₄	#2403	1	4	yes	—
446	C₂²×C₄	#2403	1	2	yes	—
447	C₂²×C₄	#2405	1	4	yes	—
448	C₂²×C₄	#2405	1	2	yes	—
449	C₂²×C₄	#2410	1	4	yes	—
450	C₂²×C₄	#2410	1	4	yes	—
451	C₂²×C₄	#2410	1	4	yes	—
452	C₂²×C₄	#2613	1	4	yes	—
453	C₂²×C₄	#2613	1	4	yes	—
454	C₂²×C₄	#2846	1	4	yes	—
455	C₂²×C₄	C₂²×#99	1	2	yes	—
456	C₂²×C₄	C₂²×#99	1	2	yes	—
457	C₂²×C₄	C₂²×#99	1	2	yes	—
458	C₂²×C₄	C₂²×#99	1	2	yes	—
459	C₂³	#12711	1	4	yes	complemented
460	C₂³	#13867	1	2	yes	—
461	C₂³	#13867	1	2	yes	—
462	C₂³	#13902	1	2	yes	—
463	C₂³	#14679	1	1	yes	characteristic
464	C₂³	#14715	1	4	yes	—
465	C₂³	#14724	1	4	yes	—
466	C₂³	C₂×#1781	1	2	yes	—
467	C₂³	C₂×#1781	1	1	yes	characteristic
468	C₂³	C₂×#1781	1	1	yes	characteristic
469	D₄	#10965	1	16	yes	—
470	C₂×C₄	C₂×#1438	1	1	yes	characteristic
471	C₂×C₄	#13867	1	2	yes	—
472	C₂×C₄	#13867	1	2	yes	—
473	C₂×C₄	#13902	1	2	yes	—
474	C₂×C₄	#14335	1	1	yes	characteristic
475	C₂×C₄	#14715	1	4	yes	—
476	C₂×C₄	#14724	1	4	yes	—
477	C₂×C₄	C₂×#1781	1	1	yes	characteristic
478	C₂×C₄	C₂×#1781	1	1	yes	characteristic
479	C₂×C₄	C₂×#1781	1	2	yes	—
480	C₂×C₄	C₂×#1781	1	2	yes	—
481	C₂×C₄	C₂×#1781	1	2	yes	—
482	C₂×C₄	C₂×#1898	1	2	yes	—
483	C₂×C₄	C₂×#1898	1	2	yes	—
484	C₂×C₄	C₂×#1898	1	2	yes	—
485	C₂×C₄	C₂×#1898	1	2	yes	—
486	C₂×C₄	#21746	1	2	yes	—
487	C₂×C₄	#21746	1	2	yes	—
488	C₂×C₄	#21746	1	2	yes	—
489	C₂×C₄	#21746	1	2	yes	—
490	C₂²	#154094	1	1	yes	characteristic
491	C₂²	#359428	1	2	yes	—
492	C₂²	#359428	1	2	yes	—
493	C₂²	#392109	1	1	yes	characteristic
494	C₂²	#392109	1	1	yes	characteristic
495	C₂²	C₂×#12711	1	2	yes	—
496	C₂²	#493411	1	2	yes	—
497	C₄	C₂×#10965	1	2	yes	—
498	C₄	#359428	1	2	yes	—
499	C₄	#359428	1	2	yes	—
500	C₄	#382422	1	2	yes	—
501	C₂	#72214353	1	1	yes	characteristic
502	C₂	#72218375	1	1	yes	characteristic
503	C₂	#72238620	1	4	yes	—
504	C₂	#72314492	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×#1781
commutator	C₂⁴×C₄	C₂³×C₄
Frattini, cosocle	C₂³×#41	C₂⁴
prime radical	C₂×#7631286	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#146484	C₂	semidirect
2. C₂×#147582	C₂	semidirect
3. #72260460	C₂	semidirect
4. #72240229	C₂	semidirect
5. #72238533	C₂	semidirect
6. C₂×#147573	C₂	semidirect
7. C₂²×#12711	C₂	semidirect
8. C₂×#392132	C₂	semidirect
11. C₂²×#3484	C₄	semidirect
13. C₂³×D₄²	C₄	semidirect
14. C₂×#35527	C₄	semidirect
16. C₂×#35512	C₄	semidirect
18. C₂×#35250	C₄	semidirect
20. C₂×#34736	C₄	semidirect
22. C₂×#34256	C₄	semidirect
24. C₂×#34171	C₄	semidirect
25. C₂²×#2134	C₂²	semidirect
26. C₂²×#2128	C₂²	semidirect
27. C₂²×#2120	C₂²	semidirect
29. C₂²×#1795	C₂²	semidirect
30. C₂×#20591	C₂²	semidirect
31. C₂×#20584	C₂²	semidirect
32. C₂×#12587	C₂²	semidirect
33. C₂×#12583	C₂²	semidirect
34. C₂×#12573	C₂²	semidirect
88. C₂×#1795	D₄	semidirect
117. C₂⁴×D₄	C₂²⋊C₄	semidirect
128. C₂×#535	C₂²⋊C₄	semidirect
129. C₂×#535	C₂²⋊C₄	semidirect
133. C₂²×#127	C₂²⋊C₄	semidirect
137. C₂×#525	C₂²⋊C₄	semidirect
138. C₂×#525	C₂²⋊C₄	semidirect
145. C₂²×#121	C₂²⋊C₄	semidirect
148. C₂×#518	C₂²⋊C₄	semidirect
149. C₂×#518	C₂²⋊C₄	semidirect
150. C₂²×C₄×D₄	C₂²⋊C₄	semidirect
160. C₂×#413	C₂×D₄	semidirect
161. C₂×#410	C₂×D₄	semidirect
162. #2311	C₂²⋊C₄	semidirect
163. #2311	C₂²⋊C₄	semidirect
166. #2309	C₂²⋊C₄	semidirect
167. #2309	C₂²⋊C₄	semidirect
168. #2297	C₂²⋊C₄	semidirect
169. #2297	C₂²⋊C₄	semidirect
459. C₂³	#12711	semidirect

Maximal subgroups

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

subgroup	index
C₂×#146484	2
C₂×#147582	2
#72260460	2
#72240229	2
#72238533	2
C₂×#147573	2
C₂²×#12711	2
C₂×#392132	2
C₂×#7631286	2