Subgroups — #72337507

Order 2048.

At least 10471 conjugacy classes of subgroups; only the 1288 normal subgroups are shown.

The subgroup lattice has 632 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337506	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337494	C₂	1	1	yes	maximal, characteristic, complemented
3	#72337490	C₂	1	1	yes	maximal, characteristic, complemented
4	#72337434	C₂	1	8	yes	maximal, complemented
5	#72337425	C₂	1	8	yes	maximal, complemented
6	C₂×#6437988	C₂	1	1	yes	maximal, characteristic, complemented
7	#72312834	C₂	1	1	yes	maximal, characteristic, complemented
8	#72312734	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#494752	C₂	1	1	yes	maximal, characteristic, complemented
10	#72312291	C₂	1	1	yes	maximal, characteristic, complemented
11	#72303729	C₂	1	1	yes	maximal, characteristic, complemented
12	#72303176	C₂	1	1	yes	maximal, characteristic, complemented
13	#72280895	C₂	1	1	yes	maximal, characteristic, complemented
14	#72280610	C₂	1	1	yes	maximal, characteristic, complemented
15	#72279469	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#33553	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²×#12755	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#63090	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#3537	C₂²	1	1	yes	characteristic
20	C₂²×#3027	C₂²	1	1	yes	characteristic
21	C₂²×#3025	C₂²	1	1	yes	characteristic
22	C₂×#60369	C₂²	1	1	yes	characteristic
23	C₂×#60369	C₂²	1	1	yes	characteristic
24	C₂²×#3021	C₂²	1	1	yes	characteristic
25	C₂×#60076	C₂²	1	1	yes	characteristic, complemented
26	#7629120	C₂²	1	1	yes	characteristic
27	C₂×#33553	C₂²	1	8	yes	complemented
28	C₂²×#2134	C₂²	1	1	yes	characteristic
29	C₂²×#2128	C₂²	1	1	yes	characteristic
30	C₂²×#2122	C₂²	1	1	yes	characteristic
31	C₂²×#2122	C₂²	1	1	yes	characteristic
32	C₂²×#2120	C₂²	1	1	yes	characteristic
33	C₂²×#2120	C₂²	1	1	yes	characteristic
34	#6499824	C₂²	1	4	yes	complemented
35	#6499823	C₂²	1	4	yes	complemented
36	C₂×#24422	C₂²	1	4	yes	complemented
37	C₂×#24420	C₂²	1	4	yes	complemented
38	#6374164	C₂²	1	1	yes	characteristic
39	#6374164	C₂²	1	1	yes	characteristic
40	#6374112	C₂²	1	1	yes	characteristic
41	C₂²×#1914	C₂²	1	1	yes	characteristic
42	C₂²×#1910	C₂²	1	1	yes	characteristic
43	C₂²×#1891	C₂²	1	1	yes	characteristic
44	C₂×#20860	C₂²	1	1	yes	characteristic, complemented
45	C₂×#20623	C₂²	1	1	yes	characteristic, complemented
46	C₂×#20620	C₂²	1	1	yes	characteristic, complemented
47	#6370746	C₂²	1	1	yes	characteristic, complemented
48	C₂×#20612	C₂²	1	1	yes	characteristic, complemented
49	#6370692	C₂²	1	1	yes	characteristic, complemented
50	C₂×#13099	C₂²	1	4	yes	complemented
51	C₂×#13098	C₂²	1	4	yes	complemented
52	#494684	C₂²	1	4	yes	complemented
53	#494683	C₂²	1	4	yes	complemented
54	C₂×#12755	C₂²	1	8	yes	complemented
55	C₂×#12589	C₂²	1	1	yes	characteristic, complemented
56	C₂×#12589	C₂²	1	1	yes	characteristic, complemented
57	C₂×#12579	C₂²	1	1	yes	characteristic, complemented
58	C₂×#12577	C₂²	1	1	yes	characteristic, complemented
59	#492886	C₂²	1	1	yes	characteristic, complemented
60	#492886	C₂²	1	1	yes	characteristic, complemented
61	#394091	C₂²	1	4	yes	complemented
62	#394090	C₂²	1	4	yes	complemented
63	#394081	C₂²	1	4	yes	complemented
64	#394080	C₂²	1	4	yes	complemented
65	#353434	C₂²	1	1	yes	characteristic, complemented
66	#353422	C₂²	1	1	yes	characteristic, complemented
67	#148474	C₂²	1	8	yes	complemented
68	#148265	C₂²	1	8	yes	complemented
69	#148260	C₂²	1	8	yes	complemented
70	#147762	C₂²	1	8	yes	complemented
71	#147759	C₂²	1	8	yes	complemented
72	#147751	C₂²	1	8	yes	complemented
73	#147586	C₂²	1	4	yes	complemented
74	#147585	C₂²	1	4	yes	complemented
75	C₂²×#585	C₂³	1	1	yes	characteristic
76	C₂²×#585	C₂³	1	1	yes	characteristic
77	C₂×#3586	C₂³	1	2	yes	complemented
78	C₂×#3585	C₂³	1	2	yes	complemented
79	C₂³×AGL(1,ℤ/8)	D₄	1	1	yes	characteristic, complemented
80	C₂³×AGL(1,ℤ/8)	D₄	1	1	yes	characteristic, complemented
81	C₂³×AGL(1,ℤ/8)	D₄	1	1	yes	characteristic, complemented
82	C₂³×AGL(1,ℤ/8)	D₄	1	1	yes	characteristic, complemented
83	C₂²×#568	C₂³	1	1	yes	characteristic
84	C₂×#3537	C₂³	1	4	yes	—
85	C₂²×#535	C₂³	1	1	yes	characteristic
86	C₂²×#535	C₂³	1	1	yes	characteristic
87	C₂²×#525	D₄	1	1	yes	characteristic, complemented
88	C₂²×#525	D₄	1	1	yes	characteristic, complemented
89	C₂²×#525	C₂³	1	1	yes	characteristic
90	C₂×#3027	C₂³	1	4	yes	—
91	C₂×#3025	C₂³	1	4	yes	—
92	C₂×#3021	D₄	1	8	yes	complemented
93	C₂×#3021	C₂³	1	4	yes	—
94	C₂×#3010	D₄	1	2	yes	complemented
95	C₂×#3010	D₄	1	2	yes	complemented
96	C₂×#3010	C₂³	1	2	yes	—
97	C₂×#3010	C₂³	1	2	yes	—
98	C₂×#3008	D₄	1	2	yes	complemented
99	C₂×#3008	D₄	1	2	yes	complemented
100	C₂×#3008	C₂³	1	2	yes	—
101	C₂×#3008	C₂³	1	2	yes	—
102	C₂×#2896	D₄	1	1	yes	characteristic, complemented
103	C₂×#2896	D₄	1	1	yes	characteristic, complemented
104	C₂×#2896	D₄	1	1	yes	characteristic, complemented
105	C₂×#2896	D₄	1	1	yes	characteristic, complemented
106	C₂×#2889	C₂³	1	2	yes	complemented
107	C₂×#2888	C₂³	1	2	yes	complemented
108	C₂²×#413	C₂³	1	1	yes	characteristic
109	C₂²×#413	C₂³	1	1	yes	characteristic
110	C₂²×#413	C₂³	1	1	yes	characteristic
111	C₂²×#410	C₂³	1	1	yes	characteristic
112	C₂²×#410	C₂³	1	1	yes	characteristic
113	C₂²×#410	C₂³	1	1	yes	characteristic
114	C₂×#2879	C₂³	1	1	yes	characteristic
115	C₂×#2879	C₂³	1	1	yes	characteristic
116	#35659	D₄	1	2	yes	complemented
117	#35659	C₂³	1	2	yes	—
118	#35512	D₄	1	2	yes	complemented
119	#35512	C₂³	1	2	yes	—
120	#34044	D₄	1	1	yes	characteristic, complemented
121	#34044	D₄	1	1	yes	characteristic, complemented
122	#34044	C₂³	1	1	yes	characteristic
123	C₂×#2134	C₂³	1	4	yes	—
124	C₂×#2128	C₂³	1	4	yes	—
125	C₂×#2122	D₄	1	8	yes	complemented
126	C₂×#2122	C₂³	1	4	yes	—
127	C₂×#2122	C₂³	1	4	yes	—
128	C₂×#2120	D₄	1	4	yes	—
129	C₂×#2120	C₂³	1	4	yes	—
130	C₂×#2120	C₂³	1	4	yes	—
131	C₂²×#352	D₄	1	2	yes	—
132	C₂²×AΣL(1,𝔽₁₆)	D₄	1	2	yes	—
133	#21770	C₂³	1	4	yes	—
134	#21770	C₂³	1	4	yes	—
135	#21744	C₂³	1	4	yes	—
136	C₂×#1914	D₄	1	8	yes	complemented
137	C₂×#1914	C₂³	1	4	yes	—
138	C₂×#1910	C₂³	1	4	yes	—
139	C₂×#1891	C₂³	1	4	yes	—
140	C₂×#1858	C₂³	1	4	yes	complemented
141	C₂×#1796	C₂³	1	2	yes	complemented
142	C₂×#1795	C₂³	1	2	yes	complemented
143	C₂×#1794	C₂³	1	4	yes	complemented
144	#20618	C₂³	1	4	yes	complemented
145	C₂×#1791	C₂³	1	4	yes	complemented
146	#20587	C₂³	1	2	yes	complemented
147	#20586	C₂³	1	2	yes	complemented
148	C₂×#1395	C₂³	1	4	yes	complemented
149	C₂×#1395	C₂³	1	4	yes	complemented
150	C₂×#1392	C₂³	1	4	yes	complemented
151	C₂×#1391	C₂³	1	4	yes	complemented
152	#12575	C₂³	1	4	yes	complemented
153	#12575	C₂³	1	4	yes	complemented
154	#10748	C₂³	1	4	yes	complemented
155	#10742	C₂³	1	4	yes	complemented
156	C₂×#585	C₂×D₄	1	4	yes	—
157	C₂×#585	C₂⁴	1	2	yes	—
158	C₂²×2₋¹⁺⁴	C₂×D₄	1	1	yes	characteristic
159	C₂²×2₋¹⁺⁴	C₂×D₄	1	1	yes	characteristic
160	C₂²×2₋¹⁺⁴	C₂×D₄	1	1	yes	characteristic
161	C₂²×2₋¹⁺⁴	C₂⁴	1	1	yes	characteristic
162	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
163	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
164	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
165	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
166	C₂³×#41	C₂×D₄	1	1	yes	characteristic
167	C₂³×#41	C₂×D₄	1	1	yes	characteristic
168	C₂³×#41	C₂×D₄	1	1	yes	characteristic
169	C₂³×#41	C₂×D₄	1	1	yes	characteristic
170	C₂³×#41	C₂×D₄	1	1	yes	characteristic
171	C₂³×#41	C₂×D₄	1	1	yes	characteristic
172	C₂³×#41	C₂⁴	1	1	yes	characteristic
173	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
175	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
176	C₂²×AGL(1,ℤ/8)	C₂×D₄	1	2	yes	complemented
177	C₂²×AGL(1,ℤ/8)	C₂×D₄	1	2	yes	complemented
178	C₂²×AGL(1,ℤ/8)	C₂×D₄	1	2	yes	complemented
179	C₂²×AGL(1,ℤ/8)	C₂×D₄	1	2	yes	complemented
180	C₂³×SD₁₆	C₂×D₄	1	2	yes	complemented
181	C₂³×SD₁₆	C₂×D₄	1	2	yes	complemented
182	C₂³×D₈	C₂×D₄	1	2	yes	complemented
183	C₂³×D₈	C₂×D₄	1	2	yes	complemented
184	C₂×#568	C₂×D₄	1	4	yes	—
185	C₂×#568	C₂⁴	1	2	yes	—
186	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
187	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
188	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
189	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
190	C₂×#535	C₂×D₄	1	4	yes	—
191	C₂×#535	C₂×D₄	1	4	yes	—
192	C₂×#535	C₂×D₄	1	4	yes	—
193	C₂×#535	C₂×D₄	1	2	yes	—
194	C₂×#535	C₂×D₄	1	2	yes	—
195	C₂×#535	#41	1	4	yes	—
196	C₂×#535	C₂⁴	1	2	yes	—
197	C₂×#535	C₂⁴	1	2	yes	—
198	C₂×#525	C₂×D₄	1	2	yes	—
199	C₂×#525	C₂×D₄	1	2	yes	—
200	C₂×#525	C₂×D₄	1	2	yes	complemented
201	C₂×#525	C₂×D₄	1	2	yes	complemented
202	C₂×#525	#41	1	2	yes	—
203	C₂×#525	C₂⁴	1	2	yes	—
204	C₂²×#123	C₂×D₄	1	1	yes	characteristic
205	C₂²×#123	#41	1	1	yes	characteristic
206	C₂²×C₂²⋊Q₈	C₂×D₄	1	2	yes	complemented
207	C₂²×#121	C₂×D₄	1	2	yes	complemented
208	C₂²×#120	C₂×D₄	1	1	yes	characteristic
209	C₂²×#120	#41	1	1	yes	characteristic
210	C₂²×#117	C₂×D₄	1	1	yes	characteristic
211	C₂²×#117	C₂×D₄	1	1	yes	characteristic
212	C₂²×#117	C₂×D₄	1	1	yes	characteristic
213	C₂²×#117	C₂×D₄	1	1	yes	characteristic
214	C₂²×#117	C₂×D₄	1	1	yes	characteristic
215	#3021	C₂×D₄	1	16	yes	complemented
216	C₂×#419	C₂×D₄	1	2	yes	complemented
217	C₂×#419	C₂×D₄	1	2	yes	complemented
218	C₂×#418	C₂×D₄	1	2	yes	complemented
219	C₂×#418	C₂×D₄	1	2	yes	complemented
220	C₂×#416	C₂×D₄	1	2	yes	complemented
221	C₂×#416	C₂×D₄	1	2	yes	complemented
222	C₂×#416	C₂×D₄	1	2	yes	complemented
223	C₂×#416	C₂×D₄	1	2	yes	complemented
224	C₂×#413	C₂×D₄	1	4	yes	—
225	C₂×#413	C₂×D₄	1	4	yes	—
226	C₂×#413	C₂×D₄	1	4	yes	—
227	C₂×#413	C₂×D₄	1	2	yes	—
228	C₂×#413	C₂⁴	1	2	yes	—
229	C₂×#413	C₂⁴	1	2	yes	—
230	C₂×#413	C₂⁴	1	2	yes	—
231	C₂²×#100	C₂×D₄	1	2	yes	—
232	C₂×#410	C₂×D₄	1	4	yes	—
233	C₂×#410	C₂×D₄	1	2	yes	—
234	C₂×#410	C₂⁴	1	2	yes	—
235	C₂×#410	C₂⁴	1	2	yes	—
236	C₂×#410	C₂⁴	1	2	yes	—
237	C₂²×#99	C₂×D₄	1	2	yes	—
238	C₂×#408	C₂×D₄	1	2	yes	—
239	C₂×#408	C₂×D₄	1	2	yes	—
240	C₂×#408	C₂⁴	1	2	yes	—
241	C₂×#408	C₂⁴	1	2	yes	—
242	#2311	C₂×D₄	1	2	yes	—
243	#2311	C₂×D₄	1	2	yes	complemented
244	#2311	C₂×D₄	1	2	yes	complemented
245	#2311	C₂⁴	1	2	yes	—
246	#2310	C₂×D₄	1	2	yes	complemented
247	#2309	C₂×D₄	1	2	yes	complemented
248	#2122	C₂×D₄	1	16	yes	complemented
249	C₂×#352	C₂×D₄	1	8	yes	—
250	C₂×AΣL(1,𝔽₁₆)	C₂×D₄	1	8	yes	—
251	#1914	C₂×D₄	1	16	yes	complemented
252	C₂⁶	#121	1	1	yes	characteristic
253	#585	C₂²×D₄	1	4	yes	—
254	C₂²×#41	#120	1	1	yes	characteristic
255	C₂²×#41	#120	1	1	yes	characteristic
256	C₂²×#41	#120	1	1	yes	characteristic
257	C₂²×#41	#120	1	1	yes	characteristic
258	C₂²×#41	#120	1	1	yes	characteristic
259	C₂²×#41	#120	1	1	yes	characteristic
260	C₂²×#41	#120	1	1	yes	characteristic
261	C₂²×#41	#120	1	1	yes	characteristic
262	C₂²×#41	#120	1	1	yes	characteristic
263	C₂²×#41	#120	1	1	yes	characteristic
264	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
265	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
266	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
267	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
268	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
269	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
270	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
271	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
272	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
273	C₂²×#41	C₂²×D₄	1	2	yes	—
274	C₂²×#41	C₂²×D₄	1	2	yes	—
275	C₂²×#41	C₂²×D₄	1	2	yes	—
276	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
277	C₂²×#41	C₂×#41	1	1	yes	characteristic
278	C₂²×#41	C₂×#41	1	2	yes	—
279	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
280	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
281	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
282	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
283	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
284	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂²×#41	C₂⁵	1	1	yes	characteristic
286	C₂³×Q₈	#120	1	1	yes	characteristic
287	C₂³×Q₈	#120	1	1	yes	characteristic
288	C₂³×Q₈	#120	1	1	yes	characteristic
289	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
290	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
291	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
292	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
293	C₂³×D₄	#120	1	1	yes	characteristic
294	C₂³×D₄	#120	1	1	yes	characteristic
295	C₂³×D₄	#120	1	1	yes	characteristic
296	C₂³×D₄	#120	1	4	yes	—
297	C₂³×D₄	#121	1	4	yes	—
298	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
299	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
300	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
301	C₂³×D₄	C₂²×D₄	1	2	yes	—
302	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
303	C₂³×D₄	C₂²×D₄	1	2	yes	—
304	C₂³×D₄	C₂×#41	1	1	yes	characteristic
305	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
306	C₂⁴×C₄	#121	1	2	yes	—
307	C₂⁴×C₄	#121	1	1	yes	characteristic
308	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
309	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
310	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
311	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
312	#568	C₂²×D₄	1	4	yes	—
313	C₂²×M₁₆	C₂²×D₄	1	2	yes	—
314	C₂²×M₁₆	C₂²×D₄	1	2	yes	—
315	C₂³×C₈	C₂²×D₄	1	2	yes	—
316	C₂³×C₈	C₂²×D₄	1	2	yes	—
317	#535	#120	1	8	yes	—
318	#535	#121	1	8	yes	—
319	#535	C₂²×D₄	1	4	yes	—
320	#535	C₂²×D₄	1	4	yes	—
321	#535	C₂²×D₄	1	4	yes	—
322	#535	C₂×#41	1	4	yes	—
323	C₂×#123	#121	1	2	yes	—
324	C₂×#123	#121	1	2	yes	—
325	C₂×#123	C₂²×D₄	1	2	yes	—
326	C₂×#123	C₂×#41	1	2	yes	—
327	C₂×#120	#121	1	2	yes	—
328	C₂×#120	#121	1	2	yes	—
329	C₂×#120	C₂²×D₄	1	2	yes	—
330	C₂×#120	C₂×#41	1	2	yes	—
331	C₂×#117	#120	1	4	yes	—
332	C₂×#117	#121	1	4	yes	—
333	C₂×#117	#121	1	2	yes	—
334	C₂×#117	C₂²×D₄	1	2	yes	—
335	C₂×#117	C₂²×D₄	1	2	yes	—
336	C₂×#117	C₂²×D₄	1	2	yes	—
337	C₂×#117	C₂²×D₄	1	2	yes	—
338	C₂²×C₄⋊C₄	C₂²×D₄	1	2	yes	—
339	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
340	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
341	#413	#120	1	8	yes	—
342	#413	C₂²×D₄	1	4	yes	—
343	#413	C₂²×D₄	1	4	yes	—
344	#413	C₂²×D₄	1	4	yes	—
345	C₂×#100	#120	1	8	yes	—
346	C₂×#100	C₂²×D₄	1	4	yes	—
347	#410	C₂²×D₄	1	4	yes	—
348	C₂×#99	C₂²×D₄	1	4	yes	—
349	C₂⁵	C₂×#121	1	1	yes	characteristic
350	C₂⁵	C₂×#121	1	2	yes	—
351	C₂⁵	D₄²	1	1	yes	characteristic
352	C₂⁵	#546	1	1	yes	characteristic
353	C₂×#41	B(4,𝔽₂)	1	2	yes	—
354	C₂×#41	B(4,𝔽₂)	1	2	yes	—
355	C₂×#41	B(4,𝔽₂)	1	4	yes	—
356	C₂×#41	C₂×#120	1	2	yes	—
357	C₂×#41	C₂×#120	1	2	yes	—
358	C₂×#41	C₂×#121	1	2	yes	—
359	C₂×#41	#525	1	2	yes	—
360	C₂×#41	#534	1	2	yes	—
361	C₂×#41	#534	1	2	yes	—
362	C₂×#41	#534	1	2	yes	—
363	C₂×#41	#535	1	2	yes	—
364	C₂×#41	#535	1	2	yes	—
365	C₂×#41	#535	1	2	yes	—
366	C₂×#41	#538	1	2	yes	—
367	C₂×#41	#538	1	2	yes	—
368	C₂×#41	D₄²	1	2	yes	—
369	C₂×#41	D₄²	1	2	yes	—
370	C₂×#41	D₄²	1	2	yes	—
371	C₂×#41	D₄²	1	2	yes	—
372	C₂×#41	D₄²	1	2	yes	—
373	C₂×#41	D₄²	1	2	yes	—
374	C₂×#41	D₄²	1	2	yes	—
375	C₂×#41	D₄²	1	2	yes	—
376	C₂×#41	#546	1	2	yes	—
377	C₂×#41	#546	1	2	yes	—
378	C₂×#41	#546	1	2	yes	—
379	C₂×#41	#546	1	2	yes	—
380	C₂×#41	#546	1	2	yes	—
381	C₂×#41	#546	1	2	yes	—
382	C₂×#41	#560	1	2	yes	—
383	C₂×#41	#560	1	2	yes	—
384	C₂×#41	C₂³×D₄	1	1	yes	characteristic
385	C₂×#41	C₂³×D₄	1	1	yes	characteristic
386	C₂×#41	C₂³×D₄	1	1	yes	characteristic
387	C₂×#41	C₂²×#41	1	1	yes	characteristic
388	C₂×#41	C₂×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×#41	C₂×H(5,𝔽₂)	1	1	yes	characteristic
390	C₂×#41	C₂×H(5,𝔽₂)	1	1	yes	characteristic
391	C₂×#41	C₂×H(5,𝔽₂)	1	1	yes	characteristic
392	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
393	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
394	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
395	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
396	C₂²×Q₈	#534	1	2	yes	—
397	C₂²×Q₈	#534	1	1	yes	characteristic
398	C₂²×Q₈	#538	1	2	yes	—
399	C₂²×Q₈	D₄²	1	2	yes	—
400	C₂²×Q₈	D₄²	1	1	yes	characteristic
401	C₂²×Q₈	#546	1	1	yes	characteristic
402	C₂²×Q₈	#560	1	2	yes	—
403	C₂²×Q₈	C₂×H(5,𝔽₂)	1	1	yes	characteristic
404	C₂²×D₄	C₂×#120	1	2	yes	—
405	C₂²×D₄	C₂×#120	1	1	yes	characteristic
406	C₂²×D₄	C₂×#120	1	1	yes	characteristic
407	C₂²×D₄	C₂×#120	1	1	yes	characteristic
408	C₂²×D₄	C₂×#120	1	1	yes	characteristic
409	C₂²×D₄	C₂×#120	1	4	yes	—
410	C₂²×D₄	C₂×#121	1	4	yes	—
411	C₂²×D₄	C₂×#121	1	1	yes	characteristic
412	C₂²×D₄	C₂×#121	1	1	yes	characteristic
413	C₂²×D₄	#525	1	2	yes	—
414	C₂²×D₄	#534	1	1	yes	characteristic
415	C₂²×D₄	#535	1	2	yes	—
416	C₂²×D₄	#535	1	2	yes	—
417	C₂²×D₄	#535	1	2	yes	—
418	C₂²×D₄	#538	1	2	yes	—
419	C₂²×D₄	D₄²	1	2	yes	—
420	C₂²×D₄	D₄²	1	1	yes	characteristic
421	C₂²×D₄	D₄²	1	1	yes	characteristic
422	C₂²×D₄	D₄²	1	1	yes	characteristic
423	C₂²×D₄	#546	1	1	yes	characteristic
424	C₂²×D₄	#546	1	1	yes	characteristic
425	C₂²×D₄	#546	1	2	yes	—
426	C₂²×D₄	#546	1	2	yes	—
427	C₂²×D₄	#546	1	2	yes	—
428	C₂²×D₄	#546	1	2	yes	—
429	C₂²×D₄	#546	1	1	yes	characteristic
430	C₂²×D₄	#560	1	2	yes	—
431	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
432	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
433	C₂²×D₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
434	C₂³×C₄	C₂×#120	1	2	yes	—
435	C₂³×C₄	C₂×#120	1	1	yes	characteristic
436	C₂³×C₄	C₂×#120	1	1	yes	characteristic
437	C₂³×C₄	C₂×#120	1	1	yes	characteristic
438	C₂³×C₄	C₂×#120	1	1	yes	characteristic
439	C₂³×C₄	C₂×#121	1	1	yes	characteristic
440	C₂³×C₄	C₂×#121	1	2	yes	—
441	C₂³×C₄	C₂×#121	1	2	yes	—
442	C₂³×C₄	#534	1	2	yes	—
443	C₂³×C₄	#534	1	2	yes	—
444	C₂³×C₄	#534	1	1	yes	characteristic
445	C₂³×C₄	#534	1	1	yes	characteristic
446	C₂³×C₄	D₄²	1	2	yes	—
447	C₂³×C₄	D₄²	1	2	yes	—
448	C₂³×C₄	D₄²	1	2	yes	—
449	C₂³×C₄	D₄²	1	1	yes	characteristic
450	C₂³×C₄	D₄²	1	1	yes	characteristic
451	C₂³×C₄	D₄²	1	2	yes	—
452	C₂³×C₄	D₄²	1	2	yes	—
453	C₂³×C₄	D₄²	1	1	yes	characteristic
454	C₂³×C₄	D₄²	1	2	yes	—
455	C₂³×C₄	#546	1	1	yes	characteristic
456	C₂³×C₄	#546	1	2	yes	—
457	C₂³×C₄	#546	1	2	yes	—
458	C₂³×C₄	#546	1	1	yes	characteristic
459	C₂³×C₄	#546	1	1	yes	characteristic
460	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
461	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
462	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
463	#117	C₂×#120	1	4	yes	—
464	#117	C₂×#121	1	4	yes	—
465	C₂×C₂²⋊C₄	C₂×#121	1	2	yes	—
466	C₂×C₂²⋊C₄	C₂×#121	1	2	yes	—
467	#100	C₂×#120	1	8	yes	—
468	C₂⁴	#2404	1	2	yes	—
469	C₂⁴	#2408	1	2	yes	—
470	C₂⁴	#2408	1	2	yes	—
471	C₂⁴	#2410	1	1	yes	characteristic
472	C₂⁴	#2410	1	1	yes	characteristic
473	C₂⁴	#2410	1	1	yes	characteristic
474	C₂⁴	#2410	1	1	yes	characteristic
475	C₂⁴	#2424	1	2	yes	—
476	C₂⁴	#2669	1	2	yes	—
477	C₂⁴	#2678	1	2	yes	—
478	C₂⁴	C₂²×#121	1	1	yes	characteristic
479	C₂⁴	C₂×D₄²	1	1	yes	characteristic
480	C₂⁴	C₂×#546	1	1	yes	characteristic
481	C₂×Q₈	#2403	1	2	yes	—
482	C₂×Q₈	#2424	1	2	yes	—
483	C₂×Q₈	#2668	1	2	yes	—
484	C₂×Q₈	#2668	1	2	yes	—
485	C₂×Q₈	#2669	1	2	yes	—
486	C₂×Q₈	#2678	1	2	yes	—
487	C₂×Q₈	C₂×B(4,𝔽₂)	1	2	yes	—
488	C₂×Q₈	C₂×B(4,𝔽₂)	1	2	yes	—
489	C₂×Q₈	C₂×#534	1	1	yes	characteristic
490	C₂×Q₈	C₂×#538	1	1	yes	characteristic
491	C₂×Q₈	C₂×D₄²	1	1	yes	characteristic
492	C₂×Q₈	C₂×#560	1	1	yes	characteristic
493	C₂×D₄	#2403	1	2	yes	—
494	C₂×D₄	#2404	1	2	yes	—
495	C₂×D₄	#2404	1	2	yes	—
496	C₂×D₄	#2408	1	2	yes	—
497	C₂×D₄	#2408	1	2	yes	—
498	C₂×D₄	#2408	1	2	yes	—
499	C₂×D₄	#2408	1	2	yes	—
500	C₂×D₄	#2424	1	2	yes	—
501	C₂×D₄	#2424	1	2	yes	—
502	C₂×D₄	#2424	1	2	yes	—
503	C₂×D₄	#2668	1	2	yes	—
504	C₂×D₄	#2668	1	2	yes	—
505	C₂×D₄	#2669	1	2	yes	—
506	C₂×D₄	#2669	1	2	yes	—
507	C₂×D₄	#2669	1	2	yes	—
508	C₂×D₄	#2678	1	2	yes	—
509	C₂×D₄	#2678	1	2	yes	—
510	C₂×D₄	#2678	1	2	yes	—
511	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
512	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
513	C₂×D₄	C₂²×#120	1	1	yes	characteristic
514	C₂×D₄	C₂×#525	1	1	yes	characteristic
515	C₂×D₄	C₂×#535	1	1	yes	characteristic
516	C₂×D₄	C₂×#535	1	1	yes	characteristic
517	C₂×D₄	C₂×#535	1	1	yes	characteristic
518	C₂×D₄	C₂×#538	1	1	yes	characteristic
519	C₂×D₄	C₂×D₄²	1	1	yes	characteristic
520	C₂×D₄	C₂×#546	1	1	yes	characteristic
521	C₂×D₄	C₂×#546	1	1	yes	characteristic
522	C₂×D₄	C₂×#546	1	1	yes	characteristic
523	C₂×D₄	C₂×#546	1	1	yes	characteristic
524	C₂×D₄	C₂×#560	1	1	yes	characteristic
525	C₂²×C₄	#2403	1	2	yes	—
526	C₂²×C₄	#2403	1	2	yes	—
527	C₂²×C₄	#2403	1	2	yes	—
528	C₂²×C₄	#2403	1	2	yes	—
529	C₂²×C₄	#2403	1	2	yes	—
530	C₂²×C₄	#2403	1	2	yes	—
531	C₂²×C₄	#2404	1	2	yes	—
532	C₂²×C₄	#2408	1	2	yes	—
533	C₂²×C₄	#2408	1	2	yes	—
534	C₂²×C₄	#2410	1	2	yes	—
535	C₂²×C₄	#2410	1	2	yes	—
536	C₂²×C₄	#2410	1	2	yes	—
537	C₂²×C₄	#2410	1	1	yes	characteristic
538	C₂²×C₄	#2410	1	1	yes	characteristic
539	C₂²×C₄	#2410	1	1	yes	characteristic
540	C₂²×C₄	#2410	1	2	yes	—
541	C₂²×C₄	#2410	1	1	yes	characteristic
542	C₂²×C₄	#2424	1	2	yes	—
543	C₂²×C₄	#2424	1	2	yes	—
544	C₂²×C₄	#2424	1	2	yes	—
545	C₂²×C₄	#2668	1	2	yes	—
546	C₂²×C₄	#2668	1	2	yes	—
547	C₂²×C₄	#2668	1	2	yes	—
548	C₂²×C₄	#2668	1	2	yes	—
549	C₂²×C₄	#2669	1	2	yes	—
550	C₂²×C₄	#2669	1	2	yes	—
551	C₂²×C₄	#2669	1	2	yes	—
552	C₂²×C₄	#2678	1	2	yes	—
553	C₂²×C₄	#2678	1	2	yes	—
554	C₂²×C₄	#2678	1	2	yes	—
555	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
556	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
557	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
558	C₂²×C₄	#3026	1	1	yes	characteristic
559	C₂²×C₄	#3026	1	2	yes	—
560	C₂²×C₄	#3026	1	1	yes	characteristic
561	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
562	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
563	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
564	C₂²×C₄	C₂×#534	1	1	yes	characteristic
565	C₂²×C₄	C₂×#534	1	1	yes	characteristic
566	C₂²×C₄	C₂×#534	1	1	yes	characteristic
567	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
568	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
569	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
570	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
571	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
572	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
573	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
574	C₂²×C₄	C₂×#546	1	1	yes	characteristic
575	C₂²×C₄	C₂×#546	1	1	yes	characteristic
576	C₂²×C₄	C₂×#546	1	1	yes	characteristic
577	C₂³	#15257	1	2	yes	—
578	C₂³	#15257	1	2	yes	—
579	C₂³	#15257	1	2	yes	—
580	C₂³	#15257	1	2	yes	—
581	C₂³	#24404	1	1	yes	characteristic
582	C₂³	#24404	1	1	yes	characteristic
583	C₂³	#24404	1	2	yes	—
584	C₂³	C₂×#2410	1	1	yes	characteristic
585	C₂³	C₂×#2424	1	1	yes	characteristic
586	C₂³	C₂×#2408	1	1	yes	characteristic
587	C₂³	C₂×#2408	1	1	yes	characteristic
588	C₂³	C₂×#2404	1	1	yes	characteristic
589	C₂³	C₂×#2669	1	1	yes	characteristic
590	C₂³	C₂×#2678	1	1	yes	characteristic
591	C₂×C₄	#15257	1	2	yes	—
592	C₂×C₄	#15257	1	2	yes	—
593	C₂×C₄	#15257	1	2	yes	—
594	C₂×C₄	#15257	1	2	yes	—
595	C₂×C₄	#23900	1	2	yes	—
596	C₂×C₄	#23900	1	2	yes	—
597	C₂×C₄	#23903	1	2	yes	—
598	C₂×C₄	#23903	1	2	yes	—
599	C₂×C₄	#24404	1	2	yes	—
600	C₂×C₄	#24404	1	2	yes	—
601	C₂×C₄	#24404	1	2	yes	—
602	C₂×C₄	#24404	1	2	yes	—
603	C₂×C₄	#24404	1	2	yes	—
604	C₂×C₄	#24404	1	1	yes	characteristic
605	C₂×C₄	#24404	1	1	yes	characteristic
606	C₂×C₄	C₂×#2403	1	1	yes	characteristic
607	C₂×C₄	C₂×#2403	1	1	yes	characteristic
608	C₂×C₄	C₂×#2410	1	1	yes	characteristic
609	C₂×C₄	C₂×#2424	1	1	yes	characteristic
610	C₂×C₄	C₂×#2668	1	1	yes	characteristic
611	C₂×C₄	C₂×#2668	1	1	yes	characteristic
612	C₂×C₄	C₂×#2669	1	1	yes	characteristic
613	C₂×C₄	C₂×#2678	1	1	yes	characteristic
614	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
615	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
616	C₂×C₄	C₂×#3026	1	1	yes	characteristic
617	C₂×C₄	C₂×#3026	1	1	yes	characteristic
618	C₂²	#417508	1	2	yes	—
619	C₂²	#417508	1	2	yes	—
620	C₂²	C₂×#15257	1	1	yes	characteristic
621	C₂²	C₂×#24404	1	1	yes	characteristic
622	C₂²	C₂×#24404	1	1	yes	characteristic
623	C₄	#417508	1	2	yes	—
624	C₄	#417508	1	2	yes	—
625	C₄	C₂×#23900	1	1	yes	characteristic
626	C₄	C₂×#23903	1	1	yes	characteristic
627	C₄	C₂×#24404	1	1	yes	characteristic
628	C₄	C₂×#24404	1	1	yes	characteristic
629	C₂	#72237549	1	2	yes	—
630	C₂	C₂×#417508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#24404
commutator, Frattini, cosocle	C₂²×#41	C₂⁵
LCS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#585 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337506	C₂	semidirect
2. #72337494	C₂	semidirect
3. #72337490	C₂	semidirect
4. #72337434	C₂	semidirect
5. #72337425	C₂	semidirect
6. C₂×#6437988	C₂	semidirect
7. #72312834	C₂	semidirect
8. #72312734	C₂	semidirect
9. C₂×#494752	C₂	semidirect
10. #72312291	C₂	semidirect
11. #72303729	C₂	semidirect
12. #72303176	C₂	semidirect
13. #72280895	C₂	semidirect
14. #72280610	C₂	semidirect
15. #72279469	C₂	semidirect
16. C₂²×#33553	C₂	semidirect
17. C₂²×#12755	C₂	semidirect
18. C₂×#63090	C₂²	semidirect
25. C₂×#60076	C₂²	semidirect
27. C₂×#33553	C₂²	semidirect
34. #6499824	C₂²	semidirect
35. #6499823	C₂²	semidirect
36. C₂×#24422	C₂²	semidirect
37. C₂×#24420	C₂²	semidirect
44. C₂×#20860	C₂²	semidirect
45. C₂×#20623	C₂²	semidirect
46. C₂×#20620	C₂²	semidirect
47. #6370746	C₂²	semidirect
48. C₂×#20612	C₂²	semidirect
49. #6370692	C₂²	semidirect
50. C₂×#13099	C₂²	semidirect
51. C₂×#13098	C₂²	semidirect
52. #494684	C₂²	semidirect
53. #494683	C₂²	semidirect
54. C₂×#12755	C₂²	semidirect
55. C₂×#12589	C₂²	semidirect
56. C₂×#12589	C₂²	semidirect
57. C₂×#12579	C₂²	semidirect
58. C₂×#12577	C₂²	semidirect
59. #492886	C₂²	semidirect
60. #492886	C₂²	semidirect
61. #394091	C₂²	semidirect
62. #394090	C₂²	semidirect
63. #394081	C₂²	semidirect
64. #394080	C₂²	semidirect
65. #353434	C₂²	semidirect
66. #353422	C₂²	semidirect
67. #148474	C₂²	semidirect
68. #148265	C₂²	semidirect
69. #148260	C₂²	semidirect
70. #147762	C₂²	semidirect
71. #147759	C₂²	semidirect
72. #147751	C₂²	semidirect
73. #147586	C₂²	semidirect
74. #147585	C₂²	semidirect
77. C₂×#3586	C₂³	semidirect
78. C₂×#3585	C₂³	semidirect
79. C₂³×AGL(1,ℤ/8)	D₄	semidirect
80. C₂³×AGL(1,ℤ/8)	D₄	semidirect
81. C₂³×AGL(1,ℤ/8)	D₄	semidirect
82. C₂³×AGL(1,ℤ/8)	D₄	semidirect
87. C₂²×#525	D₄	semidirect
88. C₂²×#525	D₄	semidirect
92. C₂×#3021	D₄	semidirect
94. C₂×#3010	D₄	semidirect
95. C₂×#3010	D₄	semidirect
98. C₂×#3008	D₄	semidirect
99. C₂×#3008	D₄	semidirect
102. C₂×#2896	D₄	semidirect
103. C₂×#2896	D₄	semidirect
104. C₂×#2896	D₄	semidirect
105. C₂×#2896	D₄	semidirect
106. C₂×#2889	C₂³	semidirect
107. C₂×#2888	C₂³	semidirect
116. #35659	D₄	semidirect
118. #35512	D₄	semidirect
120. #34044	D₄	semidirect
121. #34044	D₄	semidirect
125. C₂×#2122	D₄	semidirect
136. C₂×#1914	D₄	semidirect
140. C₂×#1858	C₂³	semidirect
141. C₂×#1796	C₂³	semidirect
142. C₂×#1795	C₂³	semidirect
143. C₂×#1794	C₂³	semidirect
144. #20618	C₂³	semidirect
145. C₂×#1791	C₂³	semidirect
146. #20587	C₂³	semidirect
147. #20586	C₂³	semidirect
148. C₂×#1395	C₂³	semidirect
149. C₂×#1395	C₂³	semidirect
150. C₂×#1392	C₂³	semidirect
151. C₂×#1391	C₂³	semidirect
152. #12575	C₂³	semidirect
153. #12575	C₂³	semidirect
154. #10748	C₂³	semidirect
155. #10742	C₂³	semidirect
176. C₂²×AGL(1,ℤ/8)	C₂×D₄	semidirect
177. C₂²×AGL(1,ℤ/8)	C₂×D₄	semidirect
178. C₂²×AGL(1,ℤ/8)	C₂×D₄	semidirect
179. C₂²×AGL(1,ℤ/8)	C₂×D₄	semidirect
180. C₂³×SD₁₆	C₂×D₄	semidirect
181. C₂³×SD₁₆	C₂×D₄	semidirect
182. C₂³×D₈	C₂×D₄	semidirect
183. C₂³×D₈	C₂×D₄	semidirect
200. C₂×#525	C₂×D₄	semidirect
201. C₂×#525	C₂×D₄	semidirect
206. C₂²×C₂²⋊Q₈	C₂×D₄	semidirect
207. C₂²×#121	C₂×D₄	semidirect
215. #3021	C₂×D₄	semidirect
216. C₂×#419	C₂×D₄	semidirect
217. C₂×#419	C₂×D₄	semidirect
218. C₂×#418	C₂×D₄	semidirect
219. C₂×#418	C₂×D₄	semidirect
220. C₂×#416	C₂×D₄	semidirect
221. C₂×#416	C₂×D₄	semidirect
222. C₂×#416	C₂×D₄	semidirect
223. C₂×#416	C₂×D₄	semidirect
243. #2311	C₂×D₄	semidirect
244. #2311	C₂×D₄	semidirect
246. #2310	C₂×D₄	semidirect
247. #2309	C₂×D₄	semidirect
248. #2122	C₂×D₄	semidirect
251. #1914	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72337506	2
#72337494	2
#72337490	2
#72337434	2
#72337425	2
C₂×#6437988	2
#72312834	2
#72312734	2
C₂×#494752	2
#72312291	2
#72303729	2
#72303176	2
#72280895	2
#72280610	2
#72279469	2
C₂²×#33553	2
C₂²×#12755	2