Subgroups — #72263736

Order 2048.

At least 9103 conjugacy classes of subgroups; only the 1450 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	#72263735	C₂	1	1	yes	maximal, characteristic, complemented
2	#72263734	C₂	1	2	yes	maximal, complemented
3	#72263731	C₂	1	1	yes	maximal, characteristic
4	#72263730	C₂	1	2	yes	maximal, complemented
5	#72263728	C₂	1	2	yes	maximal, complemented
6	#72263724	C₂	1	2	yes	maximal, complemented
7	#72263635	C₂	1	1	yes	maximal, characteristic, complemented
8	#72263552	C₂	1	2	yes	maximal, complemented
9	#72263551	C₂	1	2	yes	maximal, complemented
10	#72263422	C₂	1	2	yes	maximal, complemented
11	#72236741	C₂	1	2	yes	maximal, complemented
12	#72236739	C₂	1	2	yes	maximal, complemented
13	#72236736	C₂	1	1	yes	maximal, characteristic, complemented
14	#72236734	C₂	1	1	yes	maximal, characteristic, complemented
15	#72236732	C₂	1	2	yes	maximal, complemented
16	C₂×#139777	C₂	1	2	yes	maximal, complemented
17	#72234608	C₂	1	1	yes	maximal, characteristic, complemented
18	#72233659	C₂	1	1	yes	maximal, characteristic, complemented
19	#72210479	C₂	1	2	yes	maximal, complemented
20	#10586472	C₂²	1	1	yes	characteristic
21	C₂×#37677	C₂²	1	1	yes	characteristic
22	#6595632	C₄	1	8	yes	complemented
23	#6573679	C₄	1	8	yes	complemented
24	#6490937	C₂²	1	2	yes	—
25	#6470020	C₂²	1	2	yes	—
26	#6470019	C₂²	1	2	yes	—
27	#6453842	C₂²	1	2	yes	—
28	#6437445	C₂²	1	2	yes	—
29	C₂×#23907	C₂²	1	2	yes	—
30	#6385983	C₂²	1	1	yes	characteristic
31	#6385971	C₂²	1	1	yes	characteristic
32	#6385971	C₂²	1	1	yes	characteristic
33	#606706	C₂²	1	1	yes	characteristic, complemented
34	#606701	C₂²	1	1	yes	characteristic, complemented
35	#582589	C₂²	1	1	yes	characteristic, complemented
36	#581737	C₂²	1	1	yes	characteristic, complemented
37	#581603	C₂²	1	1	yes	characteristic, complemented
38	#559850	C₂²	1	1	yes	characteristic, complemented
39	#553146	C₂²	1	1	yes	characteristic
40	#552942	C₂²	1	1	yes	characteristic, complemented
41	#552678	C₂²	1	1	yes	characteristic, complemented
42	#538835	C₂²	1	1	yes	characteristic, complemented
43	#538267	C₂²	1	1	yes	characteristic, complemented
44	#537841	C₂²	1	1	yes	characteristic, complemented
45	#518019	C₂²	1	1	yes	characteristic
46	C₂×#14169	C₂²	1	1	yes	characteristic, complemented
47	#496307	C₂²	1	2	yes	complemented
48	#496191	C₂²	1	2	yes	complemented
49	#496189	C₂²	1	2	yes	complemented
50	#493963	C₂²	1	2	yes	complemented
51	#493370	C₂²	1	2	yes	complemented
52	C₂×#12694	C₂²	1	2	yes	complemented
53	#187946	C₂²	1	2	yes	—
54	#187944	C₂²	1	2	yes	—
55	#187933	C₂²	1	2	yes	—
56	#187930	C₂²	1	2	yes	—
57	#187928	C₂²	1	2	yes	—
58	#187928	C₂²	1	2	yes	—
59	#187928	C₂²	1	2	yes	—
60	#187928	C₂²	1	2	yes	—
61	#187925	C₂²	1	2	yes	—
62	C₂×#8534	C₂²	1	2	yes	—
63	C₂×#8534	C₂²	1	2	yes	—
64	C₂×#8534	C₂²	1	2	yes	—
65	#139819	C₂²	1	4	yes	complemented
66	#139818	C₂²	1	4	yes	complemented
67	#139779	C₂²	1	4	yes	complemented
68	#139779	C₂²	1	4	yes	complemented
69	#139778	C₂²	1	4	yes	complemented
70	#139777	C₂²	1	4	yes	complemented
71	#139777	C₂²	1	4	yes	complemented
72	#139777	C₂²	1	4	yes	complemented
73	#139777	C₂²	1	4	yes	complemented
74	#139776	C₂²	1	4	yes	complemented
75	#139768	C₂²	1	4	yes	complemented
76	#139767	C₂²	1	4	yes	complemented
77	#139707	C₂²	1	4	yes	complemented
78	#139707	C₂²	1	4	yes	complemented
79	#139706	C₂²	1	4	yes	complemented
80	#139706	C₂²	1	4	yes	complemented
81	#128082	C₂²	1	2	yes	complemented
82	#128081	C₂²	1	2	yes	complemented
83	#128073	C₂²	1	2	yes	complemented
84	#128073	C₂²	1	2	yes	complemented
85	#128073	C₂²	1	2	yes	complemented
86	#128073	C₂²	1	2	yes	complemented
87	#128044	C₂²	1	2	yes	complemented
88	#128043	C₂²	1	2	yes	complemented
89	#128033	C₂²	1	2	yes	complemented
90	C₂×#7304	C₂²	1	2	yes	complemented
91	C₂×#7304	C₂²	1	2	yes	complemented
92	C₂×#7304	C₂²	1	2	yes	complemented
93	#62983	C₂×C₄	1	4	yes	complemented
94	D₄×H(5,𝔽₂)	C₂×C₄	1	4	yes	complemented
95	#62773	C₂³	1	2	yes	—
96	C₂×#3484	C₂³	1	2	yes	—
97	C₂×#3484	C₂³	1	2	yes	—
98	C₂²×#548	D₄	1	1	yes	characteristic
99	C₂²×D₄²	C₂³	1	1	yes	characteristic
100	C₂²×#535	C₂³	1	1	yes	characteristic
101	C₂³×#124	C₂³	1	1	yes	characteristic
102	C₂²×#518	D₄	1	1	yes	characteristic
103	C₂×#3019	C₂×C₄	1	4	yes	complemented
104	C₂²×#455	C₂×C₄	1	4	yes	complemented
105	#37677	C₂×C₄	1	8	yes	—
106	#37677	D₄	1	8	yes	—
107	C₂×#2856	D₄	1	1	yes	characteristic
108	C₂×#2548	D₄	1	1	yes	characteristic
109	C₂×#2544	C₂³	1	1	yes	characteristic
110	C₂×#2431	D₄	1	1	yes	characteristic
111	C₂×#2436	D₄	1	1	yes	characteristic
112	C₂×#2435	C₂³	1	1	yes	characteristic
113	C₂×#2433	C₂³	1	1	yes	characteristic
114	C₂×#2657	D₄	1	1	yes	characteristic
115	C₂×#2620	D₄	1	1	yes	characteristic
116	C₂×D₄×C₂²⋊C₄	D₄	1	1	yes	characteristic
117	C₂×D₄×C₂²⋊C₄	D₄	1	1	yes	characteristic
118	C₂×#2326	C₂³	1	1	yes	characteristic
119	C₂×#2308	D₄	1	1	yes	characteristic
120	C₂×#2290	D₄	1	1	yes	characteristic
121	#32993	C₂×C₄	1	4	yes	complemented
122	#32990	C₂×C₄	1	4	yes	complemented
123	#32924	C₂×C₄	1	8	yes	complemented
124	#32891	C₂×C₄	1	4	yes	complemented
125	#32888	C₂×C₄	1	4	yes	complemented
126	#25922	C₂×C₄	1	4	yes	complemented
127	#25743	C₂×C₄	1	4	yes	complemented
128	#24413	C₂×C₄	1	8	yes	complemented
129	#24412	C₂×C₄	1	8	yes	complemented
130	#24318	C₂×C₄	1	4	yes	complemented
131	#24228	C₂×C₄	1	4	yes	complemented
132	#24018	C₂³	1	4	yes	—
133	#23912	C₂×C₄	1	8	yes	complemented
134	#23907	C₂×C₄	1	8	yes	complemented
135	#23907	C₂³	1	4	yes	—
136	#23907	C₂³	1	4	yes	—
137	#23906	C₂³	1	4	yes	—
138	D₄×#100	C₂×C₄	1	8	yes	complemented
139	#21474	C₂×C₄	1	8	yes	complemented
140	#20953	C₂×C₄	1	4	yes	complemented
141	C₂×#1885	C₂×C₄	1	4	yes	complemented
142	#20726	C₂³	1	2	yes	—
143	#20722	C₂³	1	2	yes	—
144	#20722	C₂³	1	2	yes	—
145	C₂×#1783	C₂³	1	2	yes	—
146	C₂×#1783	C₂³	1	2	yes	—
147	C₂×#1781	C₂³	1	2	yes	—
148	C₂×#1781	C₂³	1	2	yes	—
149	C₂×#1781	C₂³	1	2	yes	—
150	C₂×#1781	C₂³	1	2	yes	—
151	#14667	C₂³	1	1	yes	characteristic
152	#14662	C₂³	1	1	yes	characteristic
153	#14662	C₂³	1	1	yes	characteristic
154	#14661	C₂³	1	1	yes	characteristic
155	#14659	D₄	1	1	yes	characteristic
156	#14657	D₄	1	1	yes	characteristic
157	#14503	C₂³	1	1	yes	characteristic
158	#14492	C₂³	1	1	yes	characteristic
159	#14492	C₂³	1	1	yes	characteristic
160	#14445	C₂³	1	1	yes	characteristic
161	#14419	C₂³	1	1	yes	characteristic
162	#14419	C₂³	1	1	yes	characteristic
163	#14344	C₂³	1	1	yes	characteristic
164	#14335	C₂³	1	1	yes	characteristic
165	#14290	D₄	1	1	yes	characteristic
166	#14257	D₄	1	1	yes	characteristic
167	#14255	D₄	1	1	yes	characteristic
168	#14255	D₄	1	1	yes	characteristic
169	#14255	D₄	1	1	yes	characteristic
170	#14255	D₄	1	1	yes	characteristic
171	#14169	C₂³	1	1	yes	characteristic
172	#14169	C₂³	1	1	yes	characteristic
173	#14169	C₂³	1	1	yes	characteristic
174	#14169	C₂³	1	1	yes	characteristic
175	#14157	C₂³	1	1	yes	characteristic
176	#14154	C₂³	1	1	yes	characteristic
177	#14136	D₄	1	1	yes	characteristic
178	#14134	D₄	1	1	yes	characteristic
179	#14134	D₄	1	1	yes	characteristic
180	#14134	D₄	1	1	yes	characteristic
181	#13940	C₂³	1	1	yes	characteristic
182	#13871	C₂³	1	1	yes	characteristic
183	#13867	C₂³	1	1	yes	characteristic
184	#13867	C₂³	1	1	yes	characteristic
185	#13867	C₂³	1	1	yes	characteristic
186	#13867	C₂³	1	1	yes	characteristic
187	#13839	C₂³	1	1	yes	characteristic
188	C₂×#1438	C₂³	1	1	yes	characteristic
189	C₂×#1438	C₂³	1	1	yes	characteristic
190	C₂×#1438	C₂³	1	1	yes	characteristic
191	#12856	C₂³	1	4	yes	—
192	#12694	C₂³	1	4	yes	—
193	#12694	C₂³	1	4	yes	—
194	#12693	C₂³	1	4	yes	—
195	#8534	C₂³	1	4	yes	—
196	#8534	C₂³	1	4	yes	—
197	#8534	C₂³	1	4	yes	—
198	#8534	C₂³	1	4	yes	—
199	#8534	C₂³	1	4	yes	—
200	#8534	C₂³	1	4	yes	—
201	#8534	C₂³	1	4	yes	—
202	#8534	C₂³	1	4	yes	—
203	#7304	C₂³	1	4	yes	—
204	#7304	C₂³	1	4	yes	—
205	#7304	C₂³	1	4	yes	—
206	#7304	C₂³	1	4	yes	—
207	#7304	C₂³	1	4	yes	—
208	#7304	C₂³	1	4	yes	—
209	#7304	C₂³	1	4	yes	—
210	#7304	C₂³	1	4	yes	—
211	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
212	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
213	C₂³×#41	C₂×D₄	1	1	yes	characteristic
214	C₂⁴×Q₈	C₂×D₄	1	1	yes	characteristic
215	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
216	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
217	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
218	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
219	#3501	C₂²×C₄	1	2	yes	—
220	#3496	C₂²×C₄	1	2	yes	—
221	#3484	C₂²×C₄	1	4	yes	—
222	#3484	C₂²×C₄	1	4	yes	—
223	#3484	C₂⁴	1	4	yes	—
224	C₂×#548	C₂²⋊C₄	1	8	yes	—
225	C₂×D₄²	C₂²×C₄	1	4	yes	—
226	C₂×D₄²	C₂×D₄	1	4	yes	—
227	C₂×D₄²	#41	1	4	yes	—
228	C₂×D₄²	#41	1	4	yes	—
229	C₂×#535	C₂²×C₄	1	4	yes	—
230	C₂×#535	C₂×D₄	1	4	yes	—
231	C₂²×#127	C₂×D₄	1	1	yes	characteristic
232	C₂²×#127	C₂×D₄	1	1	yes	characteristic
233	C₂²×#127	C₂×D₄	1	1	yes	characteristic
234	C₂²×#127	C₂⁴	1	1	yes	characteristic
235	C₂²×#124	C₂²×C₄	1	4	yes	—
236	C₂²×#124	C₂×D₄	1	1	yes	characteristic
237	C₂²×#124	C₂×D₄	1	1	yes	characteristic
238	C₂²×#124	C₂×D₄	1	4	yes	—
239	C₂²×#124	C₂×D₄	1	1	yes	characteristic
240	C₂²×#124	C₂×D₄	1	1	yes	characteristic
241	C₂²×#124	C₂×D₄	1	1	yes	characteristic
242	C₂²×#124	C₂⁴	1	1	yes	characteristic
243	C₂²×#124	C₂⁴	1	1	yes	characteristic
244	C₂²×C₂²⋊Q₈	C₂×D₄	1	1	yes	characteristic
245	C₂²×#121	C₂×D₄	1	1	yes	characteristic
246	C₂²×#121	C₂×D₄	1	1	yes	characteristic
247	C₂²×#121	C₂×D₄	1	1	yes	characteristic
248	C₂²×#121	C₂×D₄	1	1	yes	characteristic
249	C₂²×#121	C₂×D₄	1	1	yes	characteristic
250	C₂²×#120	C₂×D₄	1	1	yes	characteristic
251	C₂²×#120	C₂×D₄	1	1	yes	characteristic
252	C₂²×#120	C₂×D₄	1	1	yes	characteristic
253	C₂²×#120	C₂×D₄	1	1	yes	characteristic
254	C₄×H(5,𝔽₂)	C₂²×C₄	1	4	yes	—
255	C₂×#518	C₂²×C₄	1	2	yes	—
256	C₂×#518	C₂²×C₄	1	2	yes	—
257	C₂²×C₄×Q₈	C₂×D₄	1	1	yes	characteristic
258	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
259	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
260	C₂²×#117	C₂×D₄	1	1	yes	characteristic
261	C₂²×#117	C₂×D₄	1	1	yes	characteristic
262	C₂²×#117	C₂×D₄	1	1	yes	characteristic
263	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
264	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
265	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
266	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
267	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
268	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
269	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
270	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
271	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
272	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
273	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
274	C₂×#455	C₂²×C₄	1	8	yes	—
275	C₂×#455	C₂²×C₄	1	8	yes	—
276	C₂×#453	C₂²×C₄	1	8	yes	—
277	C₂²×#100	C₂²×C₄	1	4	yes	—
278	C₂²×#100	C₂²×C₄	1	4	yes	—
279	#2856	C₂²⋊C₄	1	8	yes	—
280	#2657	C₂²⋊C₄	1	8	yes	—
281	#2620	C₂²⋊C₄	1	8	yes	—
282	#2548	C₂²⋊C₄	1	8	yes	—
283	#2544	C₂²×C₄	1	4	yes	—
284	#2544	C₂×D₄	1	4	yes	—
285	#2436	C₂²⋊C₄	1	8	yes	—
286	#2435	C₂²×C₄	1	4	yes	—
287	#2435	C₂×D₄	1	4	yes	—
288	#2433	C₂²×C₄	1	4	yes	—
289	#2433	C₂×D₄	1	4	yes	—
290	#2431	C₂²⋊C₄	1	8	yes	—
291	#2326	C₂²×C₄	1	4	yes	—
292	#2326	C₂×D₄	1	4	yes	—
293	C₂×#391	C₂×D₄	1	1	yes	characteristic
294	C₂×#391	C₂×D₄	1	1	yes	characteristic
295	C₂×#390	C₂×D₄	1	1	yes	characteristic
296	C₂×#390	C₂×D₄	1	1	yes	characteristic
297	C₂×#390	C₂×D₄	1	1	yes	characteristic
298	C₂×#390	C₂⁴	1	1	yes	characteristic
299	#2290	C₂²⋊C₄	1	8	yes	—
300	C₂×#384	C₂×D₄	1	1	yes	characteristic
301	C₂×#381	C₂×D₄	1	1	yes	characteristic
302	C₂×#381	C₂×D₄	1	1	yes	characteristic
303	C₂×#380	C₂×D₄	1	1	yes	characteristic
304	C₂×#380	C₂×D₄	1	1	yes	characteristic
305	C₂×#379	C₂×D₄	1	1	yes	characteristic
306	C₂×#379	C₂×D₄	1	1	yes	characteristic
307	C₂×#379	C₂×D₄	1	1	yes	characteristic
308	C₂×#379	C₂×D₄	1	1	yes	characteristic
309	C₂×#379	C₂×D₄	1	1	yes	characteristic
310	C₂×#379	C₂×D₄	1	1	yes	characteristic
311	C₂×C₄×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
312	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
313	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
314	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
315	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
316	C₂×#376	C₂×D₄	1	1	yes	characteristic
317	#1887	C₂²×C₄	1	8	yes	—
318	#1885	C₂²×C₄	1	8	yes	—
319	#1885	C₂²×C₄	1	8	yes	—
320	#1856	C₂²×C₄	1	8	yes	—
321	#1800	C₂²×C₄	1	4	yes	—
322	#1792	C₂²×C₄	1	8	yes	—
323	#1783	C₂²×C₄	1	4	yes	—
324	#1783	C₂²×C₄	1	4	yes	—
325	#1783	C₂⁴	1	4	yes	—
326	#1781	C₂⁴	1	4	yes	—
327	#1781	C₂⁴	1	4	yes	—
328	C₄×#100	C₂²×C₄	1	4	yes	—
329	#1440	C₂×D₄	1	1	yes	characteristic
330	#1440	C₂×D₄	1	1	yes	characteristic
331	#1440	C₂×D₄	1	1	yes	characteristic
332	#1440	C₂×D₄	1	1	yes	characteristic
333	#1440	C₂×D₄	1	1	yes	characteristic
334	#1440	C₂×D₄	1	1	yes	characteristic
335	#1440	C₂×D₄	1	1	yes	characteristic
336	#1440	C₂×D₄	1	1	yes	characteristic
337	#1438	C₂×D₄	1	1	yes	characteristic
338	#1438	C₂×D₄	1	1	yes	characteristic
339	#1438	C₂×D₄	1	1	yes	characteristic
340	#1438	C₂×D₄	1	1	yes	characteristic
341	#1438	C₂×D₄	1	1	yes	characteristic
342	#1438	C₂×D₄	1	1	yes	characteristic
343	#1438	C₂×D₄	1	1	yes	characteristic
344	#1438	C₂×D₄	1	1	yes	characteristic
345	#1438	C₂×D₄	1	1	yes	characteristic
346	#1438	C₂×D₄	1	1	yes	characteristic
347	#1438	C₂×D₄	1	1	yes	characteristic
348	#1438	C₂×D₄	1	1	yes	characteristic
349	#1438	C₂×D₄	1	1	yes	characteristic
350	#1438	C₂×D₄	1	1	yes	characteristic
351	#1438	C₂×D₄	1	1	yes	characteristic
352	#1438	C₂×D₄	1	1	yes	characteristic
353	#1438	C₂⁴	1	1	yes	characteristic
354	#1438	C₂⁴	1	1	yes	characteristic
355	#1438	C₂⁴	1	1	yes	characteristic
356	#1438	C₂⁴	1	1	yes	characteristic
357	#1438	C₂⁴	1	1	yes	characteristic
358	#1438	C₂⁴	1	1	yes	characteristic
359	#1438	C₂⁴	1	1	yes	characteristic
360	#1438	C₂⁴	1	1	yes	characteristic
361	C₂⁶	C₂²×D₄	1	1	yes	characteristic
362	C₂⁶	C₂²×D₄	1	1	yes	characteristic
363	C₂⁶	C₂²×D₄	1	1	yes	characteristic
364	C₂⁶	C₂²×D₄	1	1	yes	characteristic
365	C₂×H(5,𝔽₂)	C₂³×C₄	1	4	yes	—
366	C₂²×#41	C₂×C₂²⋊C₄	1	4	yes	—
367	C₂³×Q₈	C₂×C₂²⋊C₄	1	4	yes	—
368	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
369	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
370	C₂³×D₄	C₄×D₄	1	4	yes	—
371	C₂³×D₄	C₄×D₄	1	4	yes	—
372	C₂³×D₄	#120	1	1	yes	characteristic
373	C₂³×D₄	#120	1	1	yes	characteristic
374	C₂³×D₄	#120	1	1	yes	characteristic
375	C₂³×D₄	#120	1	1	yes	characteristic
376	C₂³×D₄	#120	1	1	yes	characteristic
377	C₂³×D₄	#120	1	1	yes	characteristic
378	C₂³×D₄	#120	1	1	yes	characteristic
379	C₂³×D₄	#120	1	1	yes	characteristic
380	C₂³×D₄	#120	1	1	yes	characteristic
381	C₂³×D₄	#121	1	4	yes	—
382	C₂³×D₄	#121	1	4	yes	—
383	C₂³×D₄	C₂³×C₄	1	2	yes	—
384	C₂³×D₄	C₂³×C₄	1	2	yes	—
385	C₂³×D₄	C₂²×D₄	1	2	yes	—
386	C₂³×D₄	C₂²×D₄	1	2	yes	—
387	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
388	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
389	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
390	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
391	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
392	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
393	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
394	C₂³×D₄	C₂×#41	1	2	yes	—
395	C₂³×D₄	C₂×#41	1	2	yes	—
396	C₂³×D₄	C₂×#41	1	2	yes	—
397	C₂³×D₄	C₂×#41	1	2	yes	—
398	C₂³×D₄	C₂⁵	1	1	yes	characteristic
399	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
400	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
401	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
402	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
403	C₂×#127	C₂×C₂²⋊C₄	1	4	yes	—
404	C₂×#127	C₂×C₂²⋊C₄	1	4	yes	—
405	C₂×#127	C₂×C₂²⋊C₄	1	4	yes	—
406	C₂×#127	#123	1	4	yes	—
407	C₂×#127	C₂³×C₄	1	2	yes	—
408	C₂×#127	C₂²×D₄	1	2	yes	—
409	C₂×#124	C₂×C₂²⋊C₄	1	4	yes	—
410	C₂×#124	C₂×C₂²⋊C₄	1	4	yes	—
411	C₂×#124	C₂×C₂²⋊C₄	1	4	yes	—
412	C₂×#124	C₂×C₂²⋊C₄	1	4	yes	—
413	C₂×#124	C₂×C₂²⋊C₄	1	4	yes	—
414	C₂×#124	C₂³×C₄	1	2	yes	—
415	C₂×#124	C₂³×C₄	1	2	yes	—
416	C₂×#124	C₂²×D₄	1	2	yes	—
417	C₂×#124	C₂²×D₄	1	2	yes	—
418	C₂×C₂²⋊Q₈	C₂×C₂²⋊C₄	1	4	yes	—
419	C₂×#121	C₂×C₂²⋊C₄	1	4	yes	—
420	C₂×#121	C₂×C₂²⋊C₄	1	4	yes	—
421	C₂×#121	C₂×C₂²⋊C₄	1	4	yes	—
422	C₂×#121	C₄×D₄	1	4	yes	—
423	C₂×#121	C₄×D₄	1	4	yes	—
424	C₂×#121	#121	1	4	yes	—
425	C₂×#121	#121	1	4	yes	—
426	C₂×#121	#123	1	4	yes	—
427	C₂×#121	#123	1	4	yes	—
428	C₂×#120	C₄×D₄	1	4	yes	—
429	C₂×#120	C₄×D₄	1	4	yes	—
430	C₂×#120	C₄×D₄	1	4	yes	—
431	C₂×#120	C₄×D₄	1	4	yes	—
432	C₂×#120	#121	1	4	yes	—
433	C₂×#120	#121	1	4	yes	—
434	C₂×#120	#121	1	4	yes	—
435	C₂×#120	#121	1	4	yes	—
436	#518	C₂³×C₄	1	4	yes	—
437	C₂×C₄×Q₈	C₂×C₂²⋊C₄	1	4	yes	—
438	C₂×C₄×D₄	C₂×C₂²⋊C₄	1	4	yes	—
439	C₂×C₄×D₄	#123	1	4	yes	—
440	C₂×C₄×D₄	C₂³×C₄	1	2	yes	—
441	C₂×C₄×D₄	C₂²×D₄	1	2	yes	—
442	C₂×C₄×D₄	C₂×#41	1	2	yes	—
443	C₂×C₄×D₄	C₂×#41	1	2	yes	—
444	C₂×#117	C₂×C₂²⋊C₄	1	4	yes	—
445	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
446	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
447	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
448	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
449	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
450	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
451	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
452	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
453	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
454	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
455	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
456	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
457	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
458	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
459	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
460	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
461	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
462	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
463	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
464	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
465	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
466	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
467	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
468	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
469	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
470	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
471	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
472	C₂²×C₄²	C₂×C₂²⋊C₄	1	4	yes	—
473	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
474	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
475	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
476	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
477	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
478	C₂×#100	C₂³×C₄	1	8	yes	—
479	#391	C₂×C₂²⋊C₄	1	4	yes	—
480	#391	C₂×C₂²⋊C₄	1	4	yes	—
481	#390	C₂×C₂²⋊C₄	1	4	yes	—
482	#390	C₂×C₂²⋊C₄	1	4	yes	—
483	#390	C₂×C₂²⋊C₄	1	4	yes	—
484	#390	C₂³×C₄	1	2	yes	—
485	#390	C₂²×D₄	1	2	yes	—
486	#384	C₂×C₂²⋊C₄	1	4	yes	—
487	C₄×C₄⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
488	C₄×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
489	C₄×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
490	#376	C₂×C₂²⋊C₄	1	4	yes	—
491	C₂⁵	C₂×C₄×D₄	1	2	yes	—
492	C₂⁵	C₂×C₄×D₄	1	2	yes	—
493	C₂⁵	C₂×C₄×D₄	1	2	yes	—
494	C₂⁵	C₂×C₄×D₄	1	2	yes	—
495	C₂⁵	C₂×#120	1	1	yes	characteristic
496	C₂⁵	C₂×#120	1	1	yes	characteristic
497	C₂⁵	C₂×#120	1	1	yes	characteristic
498	C₂⁵	C₂×#120	1	1	yes	characteristic
499	C₂⁵	C₂×#120	1	1	yes	characteristic
500	C₂⁵	C₂×#120	1	1	yes	characteristic
501	C₂⁵	C₂×#121	1	2	yes	—
502	C₂⁵	C₂×#121	1	2	yes	—
503	C₂⁵	C₂×#121	1	2	yes	—
504	C₂⁵	C₂×#121	1	2	yes	—
505	C₂⁵	C₂³×D₄	1	1	yes	characteristic
506	C₂⁵	C₂³×D₄	1	1	yes	characteristic
507	C₂²×Q₈	C₂²×C₂²⋊C₄	1	2	yes	—
508	C₂²×Q₈	C₂²×C₂²⋊C₄	1	2	yes	—
509	C₂²×D₄	#386	1	4	yes	—
510	C₂²×D₄	#386	1	4	yes	—
511	C₂²×D₄	#386	1	4	yes	—
512	C₂²×D₄	#386	1	4	yes	—
513	C₂²×D₄	#386	1	4	yes	—
514	C₂²×D₄	#386	1	4	yes	—
515	C₂²×D₄	#386	1	4	yes	—
516	C₂²×D₄	#386	1	4	yes	—
517	C₂²×D₄	#386	1	4	yes	—
518	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
519	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
520	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
521	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
522	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
523	C₂²×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
524	C₂²×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
525	C₂²×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
526	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
527	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
528	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
529	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
530	C₂²×D₄	C₂×#121	1	2	yes	—
531	C₂²×D₄	C₂×#121	1	2	yes	—
532	C₂²×D₄	C₂×#121	1	2	yes	—
533	C₂²×D₄	C₂×#121	1	2	yes	—
534	C₂²×D₄	C₂×#123	1	2	yes	—
535	C₂²×D₄	C₂×#123	1	2	yes	—
536	C₂²×D₄	C₂⁴×C₄	1	1	yes	characteristic
537	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
538	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
539	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
540	C₂³×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
541	C₂³×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
542	C₂³×C₄	C₂×C₄×D₄	1	2	yes	—
543	C₂³×C₄	C₂×C₄×D₄	1	2	yes	—
544	C₂³×C₄	C₂×#120	1	1	yes	characteristic
545	C₂³×C₄	C₂×#120	1	1	yes	characteristic
546	C₂³×C₄	C₂×#120	1	1	yes	characteristic
547	C₂³×C₄	C₂×#120	1	1	yes	characteristic
548	C₂³×C₄	C₂×#120	1	1	yes	characteristic
549	C₂³×C₄	C₂×#120	1	1	yes	characteristic
550	C₂³×C₄	C₂×#121	1	2	yes	—
551	C₂³×C₄	C₂×#121	1	2	yes	—
552	C₂³×C₄	C₂×#123	1	2	yes	—
553	C₂³×C₄	C₂×#123	1	2	yes	—
554	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
555	C₂×C₄⋊C₄	#386	1	4	yes	—
556	C₂×C₄⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
557	C₂×C₄⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
558	C₂×C₄⋊C₄	C₂×#123	1	2	yes	—
559	C₂×C₂²⋊C₄	#386	1	4	yes	—
560	C₂×C₂²⋊C₄	#386	1	4	yes	—
561	C₂×C₂²⋊C₄	#386	1	4	yes	—
562	C₂×C₂²⋊C₄	#386	1	4	yes	—
563	C₂×C₂²⋊C₄	#386	1	4	yes	—
564	C₂×C₂²⋊C₄	#386	1	4	yes	—
565	C₂×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
566	C₂×C₂²⋊C₄	C₂×C₄×D₄	1	2	yes	—
567	C₂×C₂²⋊C₄	C₂×#121	1	2	yes	—
568	C₂×C₂²⋊C₄	C₂×#121	1	2	yes	—
569	C₂×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
570	C₂×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
571	C₂×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
572	C₂×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
573	C₂×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
574	C₂×C₄²	C₂×#123	1	2	yes	—
575	C₂⁴	C₂×#386	1	2	yes	—
576	C₂⁴	C₂×#386	1	2	yes	—
577	C₂⁴	C₂×#386	1	2	yes	—
578	C₂⁴	C₂×#386	1	2	yes	—
579	C₂⁴	C₂×#386	1	2	yes	—
580	C₂⁴	C₂×#386	1	2	yes	—
581	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
582	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
583	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
584	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
585	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
586	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
587	C₂⁴	C₂²×C₄×D₄	1	1	yes	characteristic
588	C₂⁴	C₂²×C₄×D₄	1	1	yes	characteristic
589	C₂⁴	C₂²×#120	1	1	yes	characteristic
590	C₂⁴	C₂²×#121	1	1	yes	characteristic
591	C₂⁴	C₂²×#121	1	1	yes	characteristic
592	C₂×D₄	#1889	1	4	yes	—
593	C₂×D₄	#1889	1	4	yes	—
594	C₂×D₄	#1889	1	4	yes	—
595	C₂×D₄	#1889	1	4	yes	—
596	C₂²×C₄	C₂×#386	1	2	yes	—
597	C₂²×C₄	C₂×#386	1	2	yes	—
598	C₂²×C₄	C₂×#386	1	2	yes	—
599	C₂²×C₄	C₂×#386	1	2	yes	—
600	C₂²×C₄	C₂×#386	1	2	yes	—
601	C₂²×C₄	C₂×#386	1	2	yes	—
602	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
603	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
604	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
605	C₂²×C₄	C₂³×C₂²⋊C₄	1	1	yes	characteristic
606	C₂²×C₄	C₂²×#123	1	1	yes	characteristic
607	C₄²	#2883	1	4	yes	complemented
608	C₂³	C₂×#1889	1	2	yes	—
609	C₂³	C₂×#1889	1	2	yes	—
610	C₂³	C₂×#1889	1	2	yes	—
611	C₂³	C₂×#1889	1	2	yes	—
612	C₂³	#33551	1	2	yes	—
613	C₂³	#33552	1	2	yes	—
614	C₂³	C₂²×#386	1	1	yes	characteristic
615	C₂³	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
616	C₂³	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
617	C₂×C₄	C₂×#1889	1	2	yes	—
618	C₂×C₄	C₂×#1889	1	2	yes	—
619	C₂×C₄	C₂×#2883	1	2	yes	—
620	C₂²	#494600	1	4	yes	—
621	C₂²	C₂²×#1889	1	1	yes	characteristic
622	C₂²	#6371504	1	1	yes	characteristic
623	C₂²	#6371551	1	1	yes	characteristic
624	C₂²	C₂×#33551	1	1	yes	characteristic
625	C₂²	C₂×#33552	1	1	yes	characteristic
626	C₂²	#7622881	1	1	yes	characteristic
627	C₂²	#7622881	1	1	yes	characteristic
628	C₂	C₂×#494600	1	1	yes	characteristic
629	C₂	#72251232	1	1	yes	characteristic
630	C₂	#72256625	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#1889
commutator	C₂²×D₄	C₂⁴×C₄
Frattini, cosocle	C₂³×D₄	C₂⁵
prime radical	#72263731	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72263735	C₂	semidirect
2. #72263734	C₂	semidirect
4. #72263730	C₂	semidirect
5. #72263728	C₂	semidirect
6. #72263724	C₂	semidirect
7. #72263635	C₂	semidirect
8. #72263552	C₂	semidirect
9. #72263551	C₂	semidirect
10. #72263422	C₂	semidirect
11. #72236741	C₂	semidirect
12. #72236739	C₂	semidirect
13. #72236736	C₂	semidirect
14. #72236734	C₂	semidirect
15. #72236732	C₂	semidirect
16. C₂×#139777	C₂	semidirect
17. #72234608	C₂	semidirect
18. #72233659	C₂	semidirect
19. #72210479	C₂	semidirect
22. #6595632	C₄	semidirect
23. #6573679	C₄	semidirect
33. #606706	C₂²	semidirect
34. #606701	C₂²	semidirect
35. #582589	C₂²	semidirect
36. #581737	C₂²	semidirect
37. #581603	C₂²	semidirect
38. #559850	C₂²	semidirect
40. #552942	C₂²	semidirect
41. #552678	C₂²	semidirect
42. #538835	C₂²	semidirect
43. #538267	C₂²	semidirect
44. #537841	C₂²	semidirect
46. C₂×#14169	C₂²	semidirect
47. #496307	C₂²	semidirect
48. #496191	C₂²	semidirect
49. #496189	C₂²	semidirect
50. #493963	C₂²	semidirect
51. #493370	C₂²	semidirect
52. C₂×#12694	C₂²	semidirect
65. #139819	C₂²	semidirect
66. #139818	C₂²	semidirect
67. #139779	C₂²	semidirect
68. #139779	C₂²	semidirect
69. #139778	C₂²	semidirect
70. #139777	C₂²	semidirect
71. #139777	C₂²	semidirect
72. #139777	C₂²	semidirect
73. #139777	C₂²	semidirect
74. #139776	C₂²	semidirect
75. #139768	C₂²	semidirect
76. #139767	C₂²	semidirect
77. #139707	C₂²	semidirect
78. #139707	C₂²	semidirect
79. #139706	C₂²	semidirect
80. #139706	C₂²	semidirect
81. #128082	C₂²	semidirect
82. #128081	C₂²	semidirect
83. #128073	C₂²	semidirect
84. #128073	C₂²	semidirect
85. #128073	C₂²	semidirect
86. #128073	C₂²	semidirect
87. #128044	C₂²	semidirect
88. #128043	C₂²	semidirect
89. #128033	C₂²	semidirect
90. C₂×#7304	C₂²	semidirect
91. C₂×#7304	C₂²	semidirect
92. C₂×#7304	C₂²	semidirect
93. #62983	C₂×C₄	semidirect
94. D₄×H(5,𝔽₂)	C₂×C₄	semidirect
103. C₂×#3019	C₂×C₄	semidirect
104. C₂²×#455	C₂×C₄	semidirect
121. #32993	C₂×C₄	semidirect
122. #32990	C₂×C₄	semidirect
123. #32924	C₂×C₄	semidirect
124. #32891	C₂×C₄	semidirect
125. #32888	C₂×C₄	semidirect
126. #25922	C₂×C₄	semidirect
127. #25743	C₂×C₄	semidirect
128. #24413	C₂×C₄	semidirect
129. #24412	C₂×C₄	semidirect
130. #24318	C₂×C₄	semidirect
131. #24228	C₂×C₄	semidirect
133. #23912	C₂×C₄	semidirect
134. #23907	C₂×C₄	semidirect
138. D₄×#100	C₂×C₄	semidirect
139. #21474	C₂×C₄	semidirect
140. #20953	C₂×C₄	semidirect
141. C₂×#1885	C₂×C₄	semidirect
607. C₄²	#2883	semidirect

Maximal subgroups

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

subgroup	index
#72263735	2
#72263734	2
#72263731	2
#72263730	2
#72263728	2
#72263724	2
#72263635	2
#72263552	2
#72263551	2
#72263422	2
#72236741	2
#72236739	2
#72236736	2
#72236734	2
#72236732	2
C₂×#139777	2
#72234608	2
#72233659	2
#72210479	2