Subgroups — #72252104

Order 2048.

At least 9160 conjugacy classes of subgroups; only the 1380 normal subgroups are shown.

The subgroup lattice has 659 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72250009	C₂	1	1	yes	maximal, characteristic, complemented
2	#72249967	C₂	1	1	yes	maximal, characteristic, complemented
3	#72238180	C₂	1	1	yes	maximal, characteristic, complemented
4	#72238135	C₂	1	2	yes	maximal, complemented
5	#72238128	C₂	1	2	yes	maximal, complemented
6	#72238122	C₂	1	1	yes	maximal, characteristic, complemented
7	#72236752	C₂	1	2	yes	maximal, complemented
8	#72236235	C₂	1	1	yes	maximal, characteristic
9	#72235750	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235087	C₂	1	2	yes	maximal, complemented
11	C₂×#185643	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#62773	C₂²	1	1	yes	characteristic
13	C₂³×D₄²	C₂²	1	1	yes	characteristic
14	C₂³×#418	C₂²	1	1	yes	characteristic, complemented
15	#7645254	C₄	1	4	yes	complemented
16	#7641850	C₂²	1	2	yes	—
17	#7631363	C₄	1	2	yes	complemented
18	#7631334	C₄	1	2	yes	complemented
19	C₂×#35222	C₂²	1	2	yes	—
20	#7625417	C₂²	1	1	yes	characteristic
21	#6374576	C₂²	1	2	yes	complemented
22	C₂×#20983	C₂²	1	2	yes	complemented
23	C₂×#20973	C₂²	1	1	yes	characteristic, complemented
24	C₂×#20912	C₂²	1	1	yes	characteristic, complemented
25	#6371361	C₂²	1	2	yes	complemented
26	C₂×#20898	C₂²	1	1	yes	characteristic
27	C₂×#20722	C₂²	1	1	yes	characteristic
28	C₂×#20598	C₂²	1	1	yes	characteristic, complemented
29	C₂×#20598	C₂²	1	2	yes	complemented
30	C₂×#17577	C₂²	1	2	yes	complemented
31	#6342640	C₂²	1	2	yes	complemented
32	C₂×#17362	C₂²	1	1	yes	characteristic
33	#514052	C₂²	1	1	yes	characteristic
34	C₂×#10717	C₂²	1	2	yes	complemented
35	C₂×#10716	C₂²	1	2	yes	complemented
36	#181125	C₂²	1	2	yes	complemented
37	#181125	C₂²	1	2	yes	complemented
38	C₂⁵×D₄	D₄	1	1	yes	characteristic
39	C₂⁵×D₄	C₂³	1	1	yes	characteristic
40	C₂⁴×M₁₆	C₂³	1	1	yes	characteristic
41	C₂×#3505	C₂×C₄	1	2	yes	—
42	C₂×#3490	D₄	1	2	yes	—
43	C₂×#3501	C₂×C₄	1	1	yes	characteristic
44	C₂×#3496	C₂×C₄	1	1	yes	characteristic
45	C₂×#3484	D₄	1	2	yes	—
46	C₂²×D₄²	C₂×C₄	1	2	yes	—
47	C₂²×D₄²	C₂×C₄	1	2	yes	—
48	C₂²×D₄²	D₄	1	2	yes	—
49	C₂²×D₄²	D₄	1	2	yes	—
50	C₂²×#535	D₄	1	4	yes	—
51	C₂×#3459	D₄	1	2	yes	—
52	C₂³×#127	D₄	1	1	yes	characteristic
53	C₂²×#525	C₂³	1	2	yes	—
54	C₂³×#121	D₄	1	1	yes	characteristic
55	C₂³×#121	C₂³	1	1	yes	characteristic
56	C₂×C₄×H(5,𝔽₂)	D₄	1	2	yes	—
57	C₂³×C₄×D₄	D₄	1	1	yes	characteristic
58	C₂³×C₄×D₄	C₂³	1	1	yes	characteristic
59	C₂³×#117	C₂³	1	1	yes	characteristic
60	C₂²×#407	C₂³	1	2	yes	—
61	C₂²×#407	C₂³	1	1	yes	characteristic
62	#37440	D₄	1	8	yes	—
63	#35076	D₄	1	8	yes	—
64	#35019	C₂×C₄	1	4	yes	—
65	#35001	C₂×C₄	1	4	yes	—
66	C₂×#2435	C₂×C₄	1	4	yes	—
67	C₂×#2435	D₄	1	4	yes	—
68	C₂×#2433	C₂×C₄	1	4	yes	—
69	C₂×#2433	D₄	1	4	yes	—
70	#34186	D₄	1	2	yes	—
71	#34185	D₄	1	2	yes	—
72	#34178	D₄	1	2	yes	—
73	#34176	D₄	1	2	yes	—
74	#34057	C₂×C₄	1	2	yes	—
75	C₂×#2309	D₄	1	4	yes	—
76	C₂×#2280	C₂³	1	2	yes	—
77	C₂×#2280	C₂³	1	1	yes	characteristic
78	C₂×#2280	C₂³	1	2	yes	—
79	C₄×#518	C₂×C₄	1	2	yes	—
80	C₂×#1869	D₄	1	4	yes	complemented
81	#20722	D₄	1	4	yes	complemented
82	C₂×#1459	D₄	1	4	yes	complemented
83	#13871	D₄	1	4	yes	complemented
84	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
85	C₂³×#41	C₂×D₄	1	2	yes	—
86	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
87	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
88	C₂⁴×D₄	C₂²×C₄	1	2	yes	—
89	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂×D₄	1	4	yes	—
92	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
94	C₂⁴×D₄	C₂×D₄	1	4	yes	—
95	C₂⁴×D₄	C₂×D₄	1	2	yes	—
96	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂⁵×C₄	C₂⁴	1	1	yes	characteristic
98	C₂⁴×C₈	C₂×D₄	1	2	yes	complemented
99	C₂×#543	C₂²⋊C₄	1	2	yes	—
100	C₂×#534	C₂²⋊C₄	1	4	yes	—
101	C₂×#534	C₂²⋊C₄	1	2	yes	—
102	C₂²×#127	C₂²⋊C₄	1	2	yes	—
103	C₂²×#127	C₂²⋊C₄	1	2	yes	—
104	C₂²×#127	C₂²⋊C₄	1	8	yes	complemented
105	C₂²×#124	C₂²⋊C₄	1	8	yes	complemented
106	C₂²×#123	C₂×D₄	1	4	yes	—
107	C₂²×C₂²⋊Q₈	C₂²×C₄	1	2	yes	—
108	C₂²×C₂²⋊Q₈	C₂×D₄	1	2	yes	—
109	C₂²×#121	C₂²⋊C₄	1	2	yes	—
110	C₂²×#121	C₂²⋊C₄	1	2	yes	—
111	C₂²×#121	C₂²×C₄	1	2	yes	—
112	C₂²×#121	C₂²×C₄	1	2	yes	—
113	C₂²×#121	C₂×D₄	1	2	yes	—
114	C₂²×#121	C₂×D₄	1	4	yes	—
115	C₂²×#121	C₂×D₄	1	4	yes	—
116	C₂²×#121	C₂×D₄	1	2	yes	—
117	C₂²×#120	C₂×D₄	1	4	yes	—
118	C₂×#518	C₂²⋊C₄	1	4	yes	—
119	C₂×#518	C₂²⋊C₄	1	2	yes	—
120	C₂²×C₄×D₄	C₂²⋊C₄	1	2	yes	—
121	C₂²×C₄×D₄	C₂²⋊C₄	1	2	yes	—
122	C₂²×C₄×D₄	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×D₄	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×D₄	C₂×D₄	1	4	yes	—
125	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×D₄	C₂×D₄	1	4	yes	—
128	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
133	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
134	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
135	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
136	C₂²×C₄×D₄	#41	1	2	yes	—
137	C₂²×C₄×D₄	#41	1	2	yes	—
138	C₂²×C₄×D₄	#41	1	2	yes	—
139	C₂²×C₄×D₄	#41	1	2	yes	—
140	C₂²×#117	C₂×D₄	1	4	yes	—
141	C₂²×#117	C₂×D₄	1	2	yes	—
142	C₂²×#117	C₂×D₄	1	2	yes	complemented
143	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
144	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
145	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
146	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
147	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₂²⋊C₈	C₂×D₄	1	4	yes	complemented
149	C₂²×C₂²⋊C₈	C₂×D₄	1	2	yes	complemented
150	#2591	C₂²⋊C₄	1	16	yes	complemented
151	#2539	C₂²⋊C₄	1	16	yes	complemented
152	#2313	C₂²⋊C₄	1	4	yes	—
153	#2312	C₂²⋊C₄	1	4	yes	—
154	C₂×#390	C₂²⋊C₄	1	8	yes	complemented
155	C₂×#390	C₂²⋊C₄	1	8	yes	complemented
156	#2280	C₂×D₄	1	4	yes	—
157	#2280	C₂×D₄	1	4	yes	—
158	#2280	C₂×D₄	1	4	yes	—
159	#2280	C₂×D₄	1	4	yes	—
160	#2280	C₂×D₄	1	4	yes	complemented
161	C₂×#380	C₂×D₄	1	2	yes	complemented
162	#2274	C₂²⋊C₄	1	4	yes	—
163	#2273	C₂²⋊C₄	1	4	yes	—
164	C₂×C₄×C₂²⋊C₄	C₂²×C₄	1	4	yes	—
165	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
166	C₂⁶	#120	1	1	yes	characteristic
167	C₂⁶	C₂²×D₄	1	1	yes	characteristic
168	C₂³×Q₈	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂³×D₄	C₂×C₂²⋊C₄	1	8	yes	—
171	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂³×D₄	C₂×C₂²⋊C₄	1	4	yes	—
176	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂³×D₄	#120	1	2	yes	—
178	C₂³×D₄	#120	1	2	yes	—
179	C₂³×D₄	#120	1	2	yes	—
180	C₂³×D₄	#120	1	8	yes	—
181	C₂³×D₄	#120	1	2	yes	—
182	C₂³×D₄	#120	1	2	yes	—
183	C₂³×D₄	#120	1	2	yes	—
184	C₂³×D₄	#120	1	4	yes	—
185	C₂³×D₄	#120	1	4	yes	—
186	C₂³×D₄	#120	1	1	yes	characteristic
187	C₂³×D₄	#120	1	1	yes	characteristic
188	C₂³×D₄	#120	1	2	yes	—
189	C₂³×D₄	#121	1	2	yes	—
190	C₂³×D₄	#121	1	2	yes	—
191	C₂³×D₄	#123	1	2	yes	—
192	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂⁴×C₄	#120	1	4	yes	—
195	C₂⁴×C₄	#120	1	1	yes	characteristic
196	C₂⁴×C₄	#120	1	1	yes	characteristic
197	C₂⁴×C₄	#120	1	2	yes	—
198	C₂⁴×C₄	#120	1	1	yes	characteristic
199	C₂⁴×C₄	#120	1	1	yes	characteristic
200	C₂⁴×C₄	#120	1	1	yes	characteristic
201	C₂⁴×C₄	#121	1	2	yes	—
202	C₂⁴×C₄	#121	1	2	yes	—
203	C₂⁴×C₄	#123	1	2	yes	—
204	C₂⁴×C₄	C₂³×C₄	1	1	yes	characteristic
205	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
206	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
207	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
208	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
209	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
210	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
211	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
212	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
213	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
214	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
216	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
217	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
218	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
219	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
220	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
221	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
222	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
223	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
224	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
225	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
226	C₂×#121	C₂×C₂²⋊C₄	1	8	yes	—
227	C₂×#121	#120	1	8	yes	—
228	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
229	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
230	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
231	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
232	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
233	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
234	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
235	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
236	C₂×C₄×D₄	#120	1	2	yes	—
237	C₂×C₄×D₄	#120	1	2	yes	—
238	C₂×C₄×D₄	#120	1	2	yes	—
239	C₂×C₄×D₄	#120	1	2	yes	—
240	C₂×C₄×D₄	#121	1	2	yes	—
241	C₂×C₄×D₄	#121	1	2	yes	—
242	C₂×C₄×D₄	#121	1	2	yes	—
243	C₂×C₄×D₄	#121	1	2	yes	—
244	C₂×C₄×D₄	#121	1	2	yes	—
245	C₂×C₄×D₄	#121	1	2	yes	—
246	C₂×C₄×D₄	#121	1	2	yes	—
247	C₂×C₄×D₄	#121	1	2	yes	—
248	C₂×C₄×D₄	#121	1	2	yes	—
249	C₂×C₄×D₄	#121	1	2	yes	—
250	C₂×C₄×D₄	#121	1	2	yes	—
251	C₂×C₄×D₄	#121	1	2	yes	—
252	C₂×C₄×D₄	#123	1	2	yes	—
253	C₂×C₄×D₄	#123	1	2	yes	—
254	C₂×C₄×D₄	#123	1	2	yes	—
255	C₂×C₄×D₄	#123	1	2	yes	—
256	C₂×C₄×D₄	#124	1	2	yes	—
257	C₂×C₄×D₄	#124	1	2	yes	—
258	C₂×C₄×D₄	#127	1	2	yes	—
259	C₂×C₄×D₄	#127	1	2	yes	—
260	C₂×#117	#120	1	8	yes	—
261	C₂×#117	#120	1	8	yes	—
262	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
263	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
264	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	#120	1	4	yes	—
269	C₂²×C₄⋊C₄	#120	1	2	yes	—
270	C₂²×C₄⋊C₄	#120	1	2	yes	—
271	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
273	C₂²×C₄⋊C₄	#121	1	2	yes	—
274	C₂²×C₄⋊C₄	#121	1	2	yes	—
275	C₂²×C₄⋊C₄	#124	1	2	yes	—
276	C₂²×C₄⋊C₄	#127	1	2	yes	—
277	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
278	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
279	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²×C₂²⋊C₄	#120	1	4	yes	—
284	C₂²×C₂²⋊C₄	#120	1	4	yes	—
285	C₂²×C₂²⋊C₄	#120	1	4	yes	—
286	C₂²×C₂²⋊C₄	#120	1	2	yes	—
287	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
288	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
289	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
290	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
291	C₂²×C₂²⋊C₄	#121	1	2	yes	—
292	C₂²×C₂²⋊C₄	#121	1	2	yes	—
293	C₂²×C₂²⋊C₄	#121	1	2	yes	—
294	C₂²×C₂²⋊C₄	#121	1	2	yes	—
295	C₂²×C₂²⋊C₄	#123	1	2	yes	—
296	C₂²×C₂²⋊C₄	#124	1	2	yes	—
297	C₂²×C₂²⋊C₄	#127	1	2	yes	—
298	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
299	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
300	C₂²×C₄²	C₂×C₂²⋊C₄	1	8	yes	—
301	C₂²×C₄²	#120	1	4	yes	—
302	C₂²×C₄²	#120	1	2	yes	—
303	C₂²×C₄²	#120	1	1	yes	characteristic
304	C₂²×C₄²	#120	1	1	yes	characteristic
305	C₂²×C₄²	#121	1	2	yes	—
306	C₂²×C₄²	#121	1	2	yes	—
307	C₂²×C₄²	#123	1	2	yes	—
308	C₄×C₂²⋊C₄	C₂×C₂²⋊C₄	1	8	yes	—
309	C₄×C₂²⋊C₄	C₂×C₂²⋊C₄	1	8	yes	—
310	C₄×C₂²⋊C₄	#120	1	8	yes	—
311	C₄×C₂²⋊C₄	#120	1	8	yes	—
312	C₂⁵	#379	1	1	yes	characteristic
313	C₂⁵	#379	1	4	yes	—
314	C₂⁵	#379	1	1	yes	characteristic
315	C₂⁵	#392	1	2	yes	—
316	C₂⁵	#394	1	2	yes	—
317	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
318	C₂⁵	C₂×#120	1	1	yes	characteristic
319	C₂⁵	C₂×#120	1	2	yes	—
320	C₂⁵	C₂×#120	1	2	yes	—
321	C₂⁵	C₂×#120	1	1	yes	characteristic
322	C₂⁵	C₂×#120	1	1	yes	characteristic
323	C₂⁵	C₂×#121	1	1	yes	characteristic
324	C₂⁵	C₂×#121	1	1	yes	characteristic
325	C₂⁵	C₂×#123	1	1	yes	characteristic
326	C₂⁵	#534	1	2	yes	—
327	C₂⁵	#534	1	2	yes	—
328	C₂²×D₄	#379	1	4	yes	—
329	C₂²×D₄	#379	1	4	yes	—
330	C₂²×D₄	#379	1	4	yes	—
331	C₂²×D₄	#386	1	2	yes	—
332	C₂²×D₄	#386	1	2	yes	—
333	C₂²×D₄	#386	1	2	yes	—
334	C₂²×D₄	#386	1	2	yes	—
335	C₂²×D₄	#392	1	2	yes	—
336	C₂²×D₄	#392	1	2	yes	—
337	C₂²×D₄	#394	1	2	yes	—
338	C₂²×D₄	#394	1	2	yes	—
339	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
340	C₂²×D₄	#458	1	4	yes	—
341	C₂³×C₄	#379	1	1	yes	characteristic
342	C₂³×C₄	#379	1	4	yes	—
343	C₂³×C₄	#379	1	4	yes	—
344	C₂³×C₄	#379	1	4	yes	—
345	C₂³×C₄	#379	1	1	yes	characteristic
346	C₂³×C₄	#379	1	1	yes	characteristic
347	C₂³×C₄	#379	1	1	yes	characteristic
348	C₂³×C₄	#379	1	1	yes	characteristic
349	C₂³×C₄	#379	1	1	yes	characteristic
350	C₂³×C₄	#386	1	2	yes	—
351	C₂³×C₄	#386	1	2	yes	—
352	C₂³×C₄	#386	1	2	yes	—
353	C₂³×C₄	#386	1	2	yes	—
354	C₂³×C₄	#392	1	2	yes	—
355	C₂³×C₄	#392	1	2	yes	—
356	C₂³×C₄	#392	1	2	yes	—
357	C₂³×C₄	#392	1	2	yes	—
358	C₂³×C₄	#392	1	2	yes	—
359	C₂³×C₄	#394	1	2	yes	—
360	C₂³×C₄	#394	1	2	yes	—
361	C₂³×C₄	#394	1	2	yes	—
362	C₂³×C₄	#394	1	2	yes	—
363	C₂³×C₄	#394	1	2	yes	—
364	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
365	C₂³×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
366	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
367	C₂³×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
368	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
369	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
370	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
371	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
372	C₂³×C₄	#518	1	2	yes	—
373	C₂³×C₄	#518	1	2	yes	—
374	C₂³×C₄	C₂×#120	1	1	yes	characteristic
375	C₂³×C₄	C₂×#120	1	2	yes	—
376	C₂³×C₄	C₂×#120	1	1	yes	characteristic
377	C₂³×C₄	C₂×#120	1	1	yes	characteristic
378	C₂³×C₄	C₂×#120	1	1	yes	characteristic
379	C₂³×C₄	C₂×#120	1	2	yes	—
380	C₂³×C₄	C₂×#120	1	2	yes	—
381	C₂³×C₄	C₂×#120	1	2	yes	—
382	C₂³×C₄	C₂×#120	1	1	yes	characteristic
383	C₂³×C₄	C₂×#120	1	1	yes	characteristic
384	C₂³×C₄	C₂×#120	1	2	yes	—
385	C₂³×C₄	C₂×#120	1	2	yes	—
386	C₂³×C₄	C₂×#120	1	2	yes	—
387	C₂³×C₄	C₂×#120	1	2	yes	—
388	C₂³×C₄	C₂×#120	1	1	yes	characteristic
389	C₂³×C₄	C₂×#120	1	1	yes	characteristic
390	C₂³×C₄	C₂×#120	1	1	yes	characteristic
391	C₂³×C₄	C₂×#121	1	1	yes	characteristic
392	C₂³×C₄	C₂×#121	1	1	yes	characteristic
393	C₂³×C₄	C₂×#121	1	1	yes	characteristic
394	C₂³×C₄	C₂×#121	1	1	yes	characteristic
395	C₂³×C₄	C₂×#121	1	1	yes	characteristic
396	C₂³×C₄	C₂×#121	1	1	yes	characteristic
397	C₂³×C₄	C₂×#121	1	1	yes	characteristic
398	C₂³×C₄	C₂×#121	1	1	yes	characteristic
399	C₂³×C₄	C₂×#121	1	1	yes	characteristic
400	C₂³×C₄	C₂×#121	1	1	yes	characteristic
401	C₂³×C₄	C₂×#123	1	1	yes	characteristic
402	C₂³×C₄	C₂×#123	1	1	yes	characteristic
403	C₂³×C₄	C₂×#123	1	1	yes	characteristic
404	C₂³×C₄	C₂×#124	1	1	yes	characteristic
405	C₂³×C₄	C₂×#124	1	1	yes	characteristic
406	C₂³×C₄	C₂×#127	1	1	yes	characteristic
407	C₂³×C₄	C₂×#127	1	1	yes	characteristic
408	C₂³×C₄	#534	1	1	yes	characteristic
409	C₂³×C₄	#534	1	1	yes	characteristic
410	C₂³×C₄	#534	1	1	yes	characteristic
411	C₂³×C₄	#534	1	1	yes	characteristic
412	C₂³×C₄	#534	1	2	yes	—
413	C₂³×C₄	#534	1	2	yes	—
414	C₂³×C₄	#534	1	2	yes	—
415	C₂³×C₄	#534	1	2	yes	—
416	C₂³×C₄	#535	1	2	yes	—
417	C₂³×C₄	#535	1	2	yes	—
418	C₂³×C₄	#535	1	2	yes	—
419	C₂³×C₄	#535	1	2	yes	—
420	C₂³×C₄	D₄²	1	2	yes	—
421	C₂³×C₄	D₄²	1	2	yes	—
422	C₂³×C₄	D₄²	1	2	yes	—
423	C₂³×C₄	D₄²	1	2	yes	—
424	C₂³×C₄	D₄²	1	2	yes	—
425	C₂³×C₄	D₄²	1	2	yes	—
426	C₂³×C₄	D₄²	1	2	yes	—
427	C₂³×C₄	D₄²	1	2	yes	—
428	C₂³×C₄	D₄²	1	2	yes	—
429	C₂³×C₄	D₄²	1	2	yes	—
430	C₂³×C₄	D₄²	1	2	yes	—
431	C₂³×C₄	D₄²	1	2	yes	—
432	C₂³×C₄	D₄²	1	2	yes	—
433	C₂³×C₄	D₄²	1	2	yes	—
434	C₂³×C₄	D₄²	1	2	yes	—
435	C₂³×C₄	D₄²	1	2	yes	—
436	C₂³×C₄	#546	1	2	yes	—
437	C₂³×C₄	#546	1	2	yes	—
438	C₂³×C₄	#546	1	2	yes	—
439	C₂³×C₄	#546	1	2	yes	—
440	C₂³×C₄	#546	1	2	yes	—
441	C₂³×C₄	#546	1	2	yes	—
442	C₂³×C₄	#546	1	2	yes	—
443	C₂³×C₄	#546	1	2	yes	—
444	C₂³×C₄	#546	1	2	yes	—
445	C₂³×C₄	#546	1	2	yes	—
446	C₂³×C₄	#546	1	2	yes	—
447	C₂³×C₄	#546	1	2	yes	—
448	C₂³×C₄	#546	1	2	yes	—
449	C₂³×C₄	#546	1	2	yes	—
450	C₂³×C₄	#546	1	2	yes	—
451	C₂³×C₄	#546	1	2	yes	—
452	C₂×C₄⋊C₄	#379	1	4	yes	—
453	C₂×C₄⋊C₄	#390	1	2	yes	—
454	C₂×C₄⋊C₄	#390	1	2	yes	—
455	C₂×C₄⋊C₄	#390	1	2	yes	—
456	C₂×C₄⋊C₄	#390	1	2	yes	—
457	C₂×C₄⋊C₄	#392	1	2	yes	—
458	C₂×C₄⋊C₄	#392	1	2	yes	—
459	C₂×C₄⋊C₄	#394	1	2	yes	—
460	C₂×C₄⋊C₄	#394	1	2	yes	—
461	C₂×C₂²⋊C₄	#386	1	2	yes	—
462	C₂×C₂²⋊C₄	#386	1	2	yes	—
463	C₂×C₂²⋊C₄	#386	1	2	yes	—
464	C₂×C₂²⋊C₄	#386	1	2	yes	—
465	C₂×C₂²⋊C₄	#390	1	2	yes	—
466	C₂×C₂²⋊C₄	#390	1	2	yes	—
467	C₂×C₂²⋊C₄	#390	1	2	yes	—
468	C₂×C₂²⋊C₄	#390	1	2	yes	—
469	C₂×C₂²⋊C₄	#392	1	2	yes	—
470	C₂×C₂²⋊C₄	#392	1	2	yes	—
471	C₂×C₂²⋊C₄	#392	1	2	yes	—
472	C₂×C₂²⋊C₄	#392	1	2	yes	—
473	C₂×C₂²⋊C₄	#394	1	2	yes	—
474	C₂×C₂²⋊C₄	#394	1	2	yes	—
475	C₂×C₂²⋊C₄	#394	1	2	yes	—
476	C₂×C₂²⋊C₄	#394	1	2	yes	—
477	C₂×C₄²	#379	1	16	yes	—
478	C₂×C₄²	#386	1	2	yes	—
479	C₂×C₄²	#386	1	2	yes	—
480	C₂×C₄²	#386	1	2	yes	—
481	C₂×C₄²	#386	1	2	yes	—
482	C₂×C₄²	#392	1	2	yes	—
483	C₂×C₄²	#392	1	2	yes	—
484	C₂×C₄²	#394	1	2	yes	—
485	C₂×C₄²	#394	1	2	yes	—
486	C₂⁴	#1438	1	2	yes	—
487	C₂⁴	#1438	1	2	yes	—
488	C₂⁴	C₂×#379	1	1	yes	characteristic
489	C₂⁴	C₂×#379	1	2	yes	—
490	C₂⁴	C₂×#386	1	1	yes	characteristic
491	C₂⁴	C₂×#386	1	1	yes	characteristic
492	C₂⁴	#2308	1	2	yes	—
493	C₂⁴	C₂×#392	1	1	yes	characteristic
494	C₂⁴	C₂×#392	1	1	yes	characteristic
495	C₂⁴	C₂×#394	1	1	yes	characteristic
496	C₂⁴	C₂×#394	1	1	yes	characteristic
497	C₂⁴	#2403	1	2	yes	—
498	C₂⁴	#2403	1	2	yes	—
499	C₂⁴	#2405	1	2	yes	—
500	C₂⁴	#2405	1	2	yes	—
501	C₂⁴	#2410	1	2	yes	—
502	C₂⁴	#2410	1	2	yes	—
503	C₂⁴	#2410	1	2	yes	—
504	C₂⁴	#2410	1	2	yes	—
505	C₂⁴	#2613	1	1	yes	characteristic
506	C₂⁴	#2613	1	1	yes	characteristic
507	C₂⁴	#2613	1	2	yes	—
508	C₂⁴	#2846	1	1	yes	characteristic
509	C₂⁴	#2846	1	2	yes	—
510	C₂⁴	#2846	1	1	yes	characteristic
511	C₂×D₄	#1790	1	8	yes	—
512	C₂×D₄	#1910	1	8	yes	—
513	C₂²×C₄	#1438	1	2	yes	—
514	C₂²×C₄	#1438	1	2	yes	—
515	C₂²×C₄	#1438	1	2	yes	—
516	C₂²×C₄	#1438	1	2	yes	—
517	C₂²×C₄	#1438	1	2	yes	—
518	C₂²×C₄	#1438	1	2	yes	—
519	C₂²×C₄	C₂×#379	1	1	yes	characteristic
520	C₂²×C₄	C₂×#379	1	2	yes	—
521	C₂²×C₄	C₂×#379	1	1	yes	characteristic
522	C₂²×C₄	C₂×#386	1	1	yes	characteristic
523	C₂²×C₄	C₂×#386	1	1	yes	characteristic
524	C₂²×C₄	C₂×#390	1	1	yes	characteristic
525	C₂²×C₄	C₂×#390	1	1	yes	characteristic
526	C₂²×C₄	#2308	1	1	yes	characteristic
527	C₂²×C₄	#2308	1	1	yes	characteristic
528	C₂²×C₄	#2309	1	2	yes	—
529	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
530	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
531	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
532	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
533	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
534	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
535	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
536	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
537	C₂²×C₄	C₂×#392	1	1	yes	characteristic
538	C₂²×C₄	C₂×#392	1	1	yes	characteristic
539	C₂²×C₄	C₂×#394	1	1	yes	characteristic
540	C₂²×C₄	C₂×#394	1	1	yes	characteristic
541	C₂²×C₄	#2403	1	1	yes	characteristic
542	C₂²×C₄	#2403	1	1	yes	characteristic
543	C₂²×C₄	#2403	1	1	yes	characteristic
544	C₂²×C₄	#2403	1	2	yes	—
545	C₂²×C₄	#2403	1	2	yes	—
546	C₂²×C₄	#2403	1	2	yes	—
547	C₂²×C₄	#2403	1	2	yes	—
548	C₂²×C₄	#2403	1	1	yes	characteristic
549	C₂²×C₄	#2405	1	1	yes	characteristic
550	C₂²×C₄	#2405	1	1	yes	characteristic
551	C₂²×C₄	#2405	1	1	yes	characteristic
552	C₂²×C₄	#2405	1	2	yes	—
553	C₂²×C₄	#2405	1	2	yes	—
554	C₂²×C₄	#2405	1	2	yes	—
555	C₂²×C₄	#2405	1	2	yes	—
556	C₂²×C₄	#2405	1	1	yes	characteristic
557	C₂²×C₄	#2408	1	2	yes	—
558	C₂²×C₄	#2408	1	2	yes	—
559	C₂²×C₄	#2408	1	2	yes	—
560	C₂²×C₄	#2408	1	2	yes	—
561	C₂²×C₄	#2408	1	2	yes	—
562	C₂²×C₄	#2408	1	2	yes	—
563	C₂²×C₄	#2408	1	2	yes	—
564	C₂²×C₄	#2408	1	2	yes	—
565	C₂²×C₄	#2410	1	1	yes	characteristic
566	C₂²×C₄	#2410	1	1	yes	characteristic
567	C₂²×C₄	#2410	1	1	yes	characteristic
568	C₂²×C₄	#2410	1	2	yes	—
569	C₂²×C₄	#2410	1	1	yes	characteristic
570	C₂²×C₄	#2410	1	1	yes	characteristic
571	C₂²×C₄	#2410	1	1	yes	characteristic
572	C₂²×C₄	#2410	1	2	yes	—
573	C₂²×C₄	#2410	1	2	yes	—
574	C₂²×C₄	#2410	1	2	yes	—
575	C₂²×C₄	#2410	1	2	yes	—
576	C₂²×C₄	#2410	1	2	yes	—
577	C₂²×C₄	#2410	1	2	yes	—
578	C₂²×C₄	#2410	1	2	yes	—
579	C₂²×C₄	#2410	1	1	yes	characteristic
580	C₂²×C₄	#2410	1	1	yes	characteristic
581	C₂²×C₄	#2433	1	2	yes	—
582	C₂²×C₄	#2433	1	2	yes	—
583	C₂²×C₄	#2433	1	2	yes	—
584	C₂²×C₄	#2433	1	2	yes	—
585	C₂²×C₄	#2435	1	2	yes	—
586	C₂²×C₄	#2435	1	2	yes	—
587	C₂²×C₄	#2435	1	2	yes	—
588	C₂²×C₄	#2435	1	2	yes	—
589	C₂²×C₄	#2613	1	1	yes	characteristic
590	C₂²×C₄	#2613	1	1	yes	characteristic
591	C₂²×C₄	#2613	1	2	yes	—
592	C₂²×C₄	#2846	1	1	yes	characteristic
593	C₂²×C₄	#2846	1	2	yes	—
594	C₂²×C₄	#2846	1	1	yes	characteristic
595	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
596	C₂²×C₄	C₂×#458	1	2	yes	—
597	C₂²×C₄	#3026	1	2	yes	—
598	C₂²×C₄	#3026	1	2	yes	—
599	C₂³	#10987	1	4	yes	complemented
600	C₂³	C₂×#1438	1	1	yes	characteristic
601	C₂³	#13867	1	2	yes	—
602	C₂³	#13867	1	2	yes	—
603	C₂³	#14335	1	1	yes	characteristic
604	C₂³	#14679	1	1	yes	characteristic
605	C₂³	#14715	1	2	yes	—
606	C₂³	#14715	1	1	yes	characteristic
607	C₂³	#14715	1	1	yes	characteristic
608	C₂³	#14724	1	2	yes	—
609	C₂³	#14724	1	1	yes	characteristic
610	C₂³	#14724	1	1	yes	characteristic
611	C₂³	C₂×#1910	1	4	yes	—
612	C₂³	#32898	1	2	yes	—
613	C₂³	#32900	1	2	yes	—
614	C₂×C₄	#10987	1	4	yes	complemented
615	C₂×C₄	#13867	1	1	yes	characteristic
616	C₂×C₄	#13867	1	1	yes	characteristic
617	C₂×C₄	#13867	1	1	yes	characteristic
618	C₂×C₄	#13867	1	1	yes	characteristic
619	C₂×C₄	#13902	1	2	yes	—
620	C₂×C₄	#13902	1	2	yes	—
621	C₂×C₄	#14715	1	1	yes	characteristic
622	C₂×C₄	#14715	1	2	yes	—
623	C₂×C₄	#14715	1	1	yes	characteristic
624	C₂×C₄	#14724	1	1	yes	characteristic
625	C₂×C₄	#14724	1	2	yes	—
626	C₂×C₄	#14724	1	1	yes	characteristic
627	C₂×C₄	C₂×#1790	1	2	yes	—
628	C₂×C₄	#20609	1	2	yes	—
629	C₂×C₄	#23900	1	1	yes	characteristic
630	C₂×C₄	#23901	1	1	yes	characteristic
631	C₂×C₄	#23903	1	1	yes	characteristic
632	C₂×C₄	#23903	1	1	yes	characteristic
633	C₂×C₄	#24056	1	2	yes	—
634	C₂×C₄	#24058	1	2	yes	—
635	C₂×C₄	#24404	1	2	yes	—
636	C₂×C₄	#24404	1	2	yes	—
637	C₂×C₄	#24405	1	2	yes	—
638	C₂×C₄	#24405	1	2	yes	—
639	C₂×C₄	#32898	1	2	yes	—
640	C₂×C₄	#32900	1	2	yes	—
641	C₂²	#154094	1	1	yes	characteristic
642	C₂²	C₂×#10987	1	2	yes	—
643	C₂²	#359513	1	4	yes	—
644	C₂²	#392125	1	2	yes	—
645	C₂²	#394066	1	1	yes	characteristic
646	C₂²	#394068	1	1	yes	characteristic
647	C₂²	#394086	1	2	yes	—
648	C₂²	#395418	1	2	yes	—
649	C₄	#359531	1	2	yes	—
650	C₄	#359552	1	1	yes	characteristic
651	C₄	#359554	1	1	yes	characteristic
652	C₄	#394066	1	1	yes	characteristic
653	C₄	#394068	1	1	yes	characteristic
654	C₄	#395418	1	2	yes	—
655	C₂	#72225622	1	1	yes	characteristic
656	C₂	#72240116	1	1	yes	characteristic
657	C₂	#72249705	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#154094
commutator	C₂⁴×C₄	C₂³×C₄
Frattini, cosocle	C₂⁵×C₄	C₂⁴
prime radical	#72236235	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72250009	C₂	semidirect
2. #72249967	C₂	semidirect
3. #72238180	C₂	semidirect
4. #72238135	C₂	semidirect
5. #72238128	C₂	semidirect
6. #72238122	C₂	semidirect
7. #72236752	C₂	semidirect
9. #72235750	C₂	semidirect
10. #72235087	C₂	semidirect
11. C₂×#185643	C₂	semidirect
14. C₂³×#418	C₂²	semidirect
15. #7645254	C₄	semidirect
17. #7631363	C₄	semidirect
18. #7631334	C₄	semidirect
21. #6374576	C₂²	semidirect
22. C₂×#20983	C₂²	semidirect
23. C₂×#20973	C₂²	semidirect
24. C₂×#20912	C₂²	semidirect
25. #6371361	C₂²	semidirect
28. C₂×#20598	C₂²	semidirect
29. C₂×#20598	C₂²	semidirect
30. C₂×#17577	C₂²	semidirect
31. #6342640	C₂²	semidirect
34. C₂×#10717	C₂²	semidirect
35. C₂×#10716	C₂²	semidirect
36. #181125	C₂²	semidirect
37. #181125	C₂²	semidirect
80. C₂×#1869	D₄	semidirect
81. #20722	D₄	semidirect
82. C₂×#1459	D₄	semidirect
83. #13871	D₄	semidirect
98. C₂⁴×C₈	C₂×D₄	semidirect
104. C₂²×#127	C₂²⋊C₄	semidirect
105. C₂²×#124	C₂²⋊C₄	semidirect
142. C₂²×#117	C₂×D₄	semidirect
148. C₂²×C₂²⋊C₈	C₂×D₄	semidirect
149. C₂²×C₂²⋊C₈	C₂×D₄	semidirect
150. #2591	C₂²⋊C₄	semidirect
151. #2539	C₂²⋊C₄	semidirect
154. C₂×#390	C₂²⋊C₄	semidirect
155. C₂×#390	C₂²⋊C₄	semidirect
160. #2280	C₂×D₄	semidirect
161. C₂×#380	C₂×D₄	semidirect
599. C₂³	#10987	semidirect
614. C₂×C₄	#10987	semidirect

Maximal subgroups

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

subgroup	index
#72250009	2
#72249967	2
#72238180	2
#72238135	2
#72238128	2
#72238122	2
#72236752	2
#72236235	2
#72235750	2
#72235087	2
C₂×#185643	2