Subgroups — #72313956

Order 2048.

At least 9780 conjugacy classes of subgroups; only the 834 normal subgroups are shown.

The subgroup lattice has 524 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72313926	C₂	1	1	yes	maximal, characteristic, complemented
2	#72313907	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#486533	C₂	1	1	yes	maximal, characteristic, complemented
4	#72313847	C₂	1	1	yes	maximal, characteristic, complemented
5	#72313844	C₂	1	1	yes	maximal, characteristic, complemented
6	#72313840	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#500489	C₂	1	1	yes	maximal, characteristic, complemented
8	#72313793	C₂	1	1	yes	maximal, characteristic, complemented
9	#72313565	C₂	1	1	yes	maximal, characteristic, complemented
10	#72313557	C₂	1	1	yes	maximal, characteristic, complemented
11	#72313443	C₂	1	4	yes	maximal, complemented
12	#72313440	C₂	1	4	yes	maximal, complemented
13	#72313388	C₂	1	4	yes	maximal, complemented
14	#72313386	C₂	1	4	yes	maximal, complemented
15	#72312715	C₂	1	1	yes	maximal, characteristic, complemented
16	#72311797	C₂	1	1	yes	maximal, characteristic, complemented
17	#72306052	C₂	1	1	yes	maximal, characteristic, complemented
18	#72306044	C₂	1	1	yes	maximal, characteristic, complemented
19	#72282436	C₂	1	1	yes	maximal, characteristic, complemented
20	#10586067	C₂²	1	1	yes	characteristic
21	#7672442	C₂²	1	1	yes	characteristic
22	C₂×#59945	C₂²	1	1	yes	characteristic, complemented
23	#7586780	C₂²	1	2	yes	complemented
24	#7576412	C₂²	1	2	yes	complemented
25	#7481510	C₂²	1	2	yes	complemented
26	#7454388	C₂²	1	2	yes	complemented
27	#6640543	C₂²	1	1	yes	characteristic, complemented
28	#6632008	C₂²	1	1	yes	characteristic, complemented
29	C₂×#32976	C₂²	1	1	yes	characteristic
30	C₂×#32926	C₂²	1	1	yes	characteristic
31	#6628968	C₂²	1	1	yes	characteristic
32	#6573698	C₂²	1	2	yes	complemented
33	#6573693	C₂²	1	2	yes	complemented
34	#6573670	C₂²	1	2	yes	complemented
35	D₄×#455	C₂²	1	2	yes	complemented
36	#6508610	C₂²	1	1	yes	characteristic
37	#6499761	C₂²	1	1	yes	characteristic
38	#6443328	C₂²	1	1	yes	characteristic
39	C₂×#24224	C₂²	1	1	yes	characteristic
40	#6408606	C₂²	1	1	yes	characteristic, complemented
41	#6379031	C₂²	1	1	yes	characteristic
42	C₂×C₄×#455	C₂²	1	1	yes	characteristic
43	C₂×#18203	C₂²	1	1	yes	characteristic
44	C₂×#18194	C₂²	1	1	yes	characteristic
45	#500489	C₂²	1	4	yes	complemented
46	#500489	C₂²	1	4	yes	complemented
47	C₂×#13292	C₂²	1	1	yes	characteristic, complemented
48	C₂×#13218	C₂²	1	1	yes	characteristic, complemented
49	C₂×#12798	C₂²	1	1	yes	characteristic, complemented
50	#486533	C₂²	1	4	yes	complemented
51	#486533	C₂²	1	4	yes	complemented
52	C₂×#11903	C₂²	1	1	yes	characteristic, complemented
53	C₂×#11880	C₂²	1	1	yes	characteristic, complemented
54	#482050	C₂²	1	1	yes	characteristic, complemented
55	#482050	C₂²	1	1	yes	characteristic, complemented
56	C₂×#11477	C₂²	1	1	yes	characteristic, complemented
57	#346546	C₂²	1	4	yes	complemented
58	#346542	C₂²	1	4	yes	complemented
59	#346522	C₂²	1	4	yes	—
60	#346518	C₂²	1	4	yes	—
61	#185627	C₂²	1	1	yes	characteristic
62	#185627	C₂²	1	1	yes	characteristic
63	#185331	C₂²	1	1	yes	characteristic
64	#185331	C₂²	1	1	yes	characteristic
65	#184905	C₂²	1	1	yes	characteristic
66	#184905	C₂²	1	1	yes	characteristic
67	#144271	C₂²	1	4	yes	complemented
68	#144270	C₂²	1	4	yes	complemented
69	#144230	C₂²	1	4	yes	complemented
70	#144228	C₂²	1	4	yes	complemented
71	#142735	C₂²	1	4	yes	complemented
72	#142734	C₂²	1	4	yes	complemented
73	#142605	C₂²	1	4	yes	complemented
74	#142604	C₂²	1	4	yes	complemented
75	#139617	C₂²	1	4	yes	complemented
76	#139615	C₂²	1	4	yes	complemented
77	#138967	C₂²	1	4	yes	complemented
78	#138966	C₂²	1	4	yes	complemented
79	#138881	C₂²	1	4	yes	complemented
80	#138880	C₂²	1	4	yes	complemented
81	#136852	C₂²	1	4	yes	complemented
82	#136850	C₂²	1	4	yes	complemented
83	#130775	C₂²	1	2	yes	complemented
84	#130773	C₂²	1	2	yes	complemented
85	#130771	C₂²	1	2	yes	complemented
86	#130769	C₂²	1	2	yes	complemented
87	#127353	C₂²	1	1	yes	characteristic, complemented
88	#127353	C₂²	1	1	yes	characteristic, complemented
89	#127067	C₂²	1	1	yes	characteristic, complemented
90	#127067	C₂²	1	1	yes	characteristic, complemented
91	#62988	C₂³	1	2	yes	—
92	D₄×H(5,𝔽₂)	C₂³	1	2	yes	—
93	C₂×#3497	D₄	1	1	yes	characteristic, complemented
94	C₂×#3496	C₂³	1	1	yes	characteristic
95	C₂×#3496	C₂³	1	1	yes	characteristic
96	C₂×C₄×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
97	C₂²×#481	D₄	1	1	yes	characteristic, complemented
98	C₂²×#481	D₄	1	1	yes	characteristic, complemented
99	C₂²×#455	D₄	1	1	yes	characteristic
100	C₂²×#447	D₄	1	1	yes	characteristic
101	C₂²×#447	D₄	1	1	yes	characteristic
102	#59945	C₂³	1	2	yes	complemented
103	#59945	C₂³	1	2	yes	complemented
104	#39361	D₄	1	1	yes	characteristic, complemented
105	#39361	C₂³	1	1	yes	characteristic
106	#39360	D₄	1	1	yes	characteristic, complemented
107	#39360	C₂³	1	1	yes	characteristic
108	#36644	D₄	1	1	yes	characteristic, complemented
109	#36125	D₄	1	1	yes	characteristic, complemented
110	#36125	C₂³	1	1	yes	characteristic
111	D₄×#124	D₄	1	1	yes	characteristic, complemented
112	D₄×#124	C₂³	1	1	yes	characteristic
113	#35178	D₄	1	1	yes	characteristic, complemented
114	#35165	C₂³	1	1	yes	characteristic
115	C₂×#2487	C₂³	1	1	yes	characteristic
116	C₂×#2408	C₂³	1	1	yes	characteristic
117	C₄×#532	D₄	1	1	yes	characteristic, complemented
118	C₂×C₄×#124	C₂³	1	1	yes	characteristic
119	#32976	C₂³	1	4	yes	—
120	#32926	C₂³	1	4	yes	—
121	#32922	C₂³	1	2	yes	—
122	#32920	C₂³	1	2	yes	—
123	#32260	C₂³	1	2	yes	complemented
124	#32028	C₂³	1	2	yes	complemented
125	#31260	C₂³	1	2	yes	complemented
126	#30499	C₂³	1	2	yes	complemented
127	#25739	C₂³	1	2	yes	—
128	#25738	C₂³	1	2	yes	—
129	#24224	D₄	1	2	yes	complemented
130	#24224	D₄	1	2	yes	complemented
131	#24224	C₂³	1	2	yes	—
132	#24224	C₂³	1	2	yes	—
133	#24089	D₄	1	2	yes	complemented
134	#24089	C₂³	1	2	yes	—
135	#24033	D₄	1	2	yes	complemented
136	#24033	C₂³	1	2	yes	—
137	#23960	D₄	1	2	yes	—
138	#23960	C₂³	1	2	yes	—
139	#23848	D₄	1	2	yes	—
140	#23848	C₂³	1	2	yes	—
141	C₂×#2040	D₄	1	2	yes	complemented
142	C₂×#2007	D₄	1	2	yes	complemented
143	#23352	C₂³	1	2	yes	complemented
144	#23273	C₂³	1	2	yes	complemented
145	#22411	C₂³	1	2	yes	—
146	D₄×#100	C₂³	1	2	yes	—
147	#21758	D₄	1	1	yes	characteristic
148	#21758	D₄	1	1	yes	characteristic
149	#21758	D₄	1	1	yes	characteristic, complemented
150	#21758	D₄	1	1	yes	characteristic, complemented
151	#21738	C₂³	1	1	yes	characteristic
152	#21738	C₂³	1	1	yes	characteristic
153	C₄×#455	D₄	1	2	yes	—
154	C₄×#455	D₄	1	2	yes	—
155	C₄×#455	C₂³	1	2	yes	—
156	C₄×#455	C₂³	1	2	yes	—
157	C₂×#1885	D₄	1	1	yes	characteristic
158	C₂×#1856	D₄	1	1	yes	characteristic
159	C₂×#1800	C₂³	1	1	yes	characteristic
160	C₂×#1799	C₂³	1	1	yes	characteristic
161	C₂×#1799	C₂³	1	1	yes	characteristic
162	C₂×#1792	D₄	1	1	yes	characteristic
163	C₂×C₄×#100	C₂³	1	1	yes	characteristic
164	#18203	C₂³	1	4	yes	—
165	#18194	C₂³	1	4	yes	—
166	C₂×#1459	C₂³	1	1	yes	characteristic
167	C₂×#1459	C₂³	1	1	yes	characteristic
168	#13292	C₂³	1	4	yes	complemented
169	#13218	C₂³	1	4	yes	complemented
170	#12798	C₂³	1	2	yes	complemented
171	#12798	C₂³	1	2	yes	complemented
172	#11903	C₂³	1	4	yes	complemented
173	#11880	C₂³	1	4	yes	complemented
174	#11878	C₂³	1	2	yes	complemented
175	#11878	C₂³	1	2	yes	complemented
176	#11877	C₂³	1	2	yes	complemented
177	#11877	C₂³	1	2	yes	complemented
178	#11477	C₂³	1	2	yes	complemented
179	#11477	C₂³	1	2	yes	complemented
180	#8376	C₂³	1	2	yes	—
181	#8376	C₂³	1	2	yes	—
182	#8375	C₂³	1	2	yes	—
183	#8375	C₂³	1	2	yes	—
184	#8350	C₂³	1	4	yes	—
185	#8350	C₂³	1	4	yes	—
186	#8316	C₂³	1	4	yes	—
187	#8316	C₂³	1	4	yes	—
188	#7236	C₂³	1	4	yes	complemented
189	#7236	C₂³	1	4	yes	complemented
190	#7203	C₂³	1	4	yes	complemented
191	#7203	C₂³	1	4	yes	complemented
192	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
193	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
194	C₂³×D₈	C₂×D₄	1	2	yes	—
195	#3496	C₂×D₄	1	2	yes	—
196	#3496	C₂⁴	1	2	yes	—
197	C₂×#550	C₂×D₄	1	1	yes	characteristic, complemented
198	C₂×#550	C₂×D₄	1	1	yes	characteristic, complemented
199	C₂×D₄×Q₈	C₂×D₄	1	1	yes	characteristic, complemented
200	C₂×D₄×Q₈	C₂×D₄	1	1	yes	characteristic, complemented
201	C₂×#547	C₂×D₄	1	1	yes	characteristic
202	C₂×#547	C₂×D₄	1	1	yes	characteristic
203	C₂×#547	C₂×D₄	1	1	yes	characteristic
204	C₂×#547	C₂⁴	1	1	yes	characteristic
205	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
206	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
208	C₂×D₄²	C₂⁴	1	1	yes	characteristic
209	C₂×#537	C₂×D₄	1	2	yes	—
210	C₂×#532	C₂×D₄	1	1	yes	characteristic
211	C₂×#532	C₂×D₄	1	1	yes	characteristic
212	C₂×#532	C₂×D₄	1	1	yes	characteristic
213	C₂²×#124	C₂×D₄	1	1	yes	characteristic
214	C₂²×#121	C₂×D₄	1	1	yes	characteristic
215	C₂²×#121	C₂×D₄	1	1	yes	characteristic
216	C₂²×#121	C₂×D₄	1	1	yes	characteristic
217	C₂²×#121	C₂×D₄	1	1	yes	characteristic
218	C₄×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
219	C₄×H(5,𝔽₂)	C₂⁴	1	2	yes	—
220	C₂×#518	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×Q₈	C₂×D₄	1	1	yes	characteristic
222	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
223	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
224	C₂×#481	C₂×D₄	1	2	yes	complemented
225	C₂×#481	C₂×D₄	1	2	yes	complemented
226	C₂×#455	C₂×D₄	1	2	yes	—
227	C₂×#455	C₂×D₄	1	2	yes	—
228	C₂×#447	C₂×D₄	1	2	yes	—
229	C₂×#447	C₂×D₄	1	2	yes	—
230	C₂²×#106	C₂×D₄	1	2	yes	complemented
231	C₂²×#102	C₂×D₄	1	2	yes	—
232	C₂²×#102	C₂×D₄	1	2	yes	complemented
233	C₂²×#100	C₂×D₄	1	1	yes	characteristic
234	C₂²×#100	C₂×D₄	1	1	yes	characteristic
235	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic
236	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic
237	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic
238	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic
239	#2868	C₂×D₄	1	1	yes	characteristic, complemented
240	#2867	C₂×D₄	1	1	yes	characteristic, complemented
241	#2858	C₂×D₄	1	1	yes	characteristic, complemented
242	#2858	C₂×D₄	1	1	yes	characteristic, complemented
243	#2548	C₂×D₄	1	1	yes	characteristic, complemented
244	#2548	C₂×D₄	1	1	yes	characteristic, complemented
245	#2547	C₂×D₄	1	1	yes	characteristic
246	#2547	C₂⁴	1	1	yes	characteristic
247	#2546	C₂×D₄	1	1	yes	characteristic, complemented
248	#2546	C₂×D₄	1	1	yes	characteristic, complemented
249	#2541	C₂×D₄	1	1	yes	characteristic
250	#2541	C₂⁴	1	1	yes	characteristic
251	#2487	C₂×D₄	1	2	yes	—
252	#2487	C₂⁴	1	2	yes	—
253	#2408	C₂×D₄	1	2	yes	—
254	#2408	C₂⁴	1	2	yes	—
255	C₄×#124	C₂×D₄	1	1	yes	characteristic
256	C₄×#124	C₂×D₄	1	1	yes	characteristic
257	C₄×#124	C₂×D₄	1	1	yes	characteristic
258	C₄×#124	C₂×D₄	1	1	yes	characteristic
259	C₄×#124	C₂×D₄	1	1	yes	characteristic
260	C₄×#124	C₂×D₄	1	1	yes	characteristic
261	C₄×#124	C₂⁴	1	1	yes	characteristic
262	C₄×#124	C₂⁴	1	1	yes	characteristic
263	C₂×#391	C₂×D₄	1	1	yes	characteristic
264	#2293	C₂×D₄	1	1	yes	characteristic, complemented
265	#2293	C₂×D₄	1	1	yes	characteristic, complemented
266	C₂×#390	C₂×D₄	1	1	yes	characteristic
267	#2285	C₂×D₄	1	1	yes	characteristic
268	C₂×#382	C₂×D₄	1	1	yes	characteristic
269	C₂×#380	C₂×D₄	1	1	yes	characteristic
270	C₂×#380	C₂×D₄	1	1	yes	characteristic
271	C₄²×D₄	C₂×D₄	1	1	yes	characteristic, complemented
272	C₄²×D₄	C₂×D₄	1	1	yes	characteristic, complemented
273	C₂×C₄³	C₂×D₄	1	1	yes	characteristic
274	#2040	C₂×D₄	1	4	yes	complemented
275	#2040	C₂×D₄	1	4	yes	complemented
276	#2007	C₂×D₄	1	4	yes	complemented
277	#2007	C₂×D₄	1	4	yes	complemented
278	#1893	C₂×D₄	1	2	yes	—
279	#1893	C₂×D₄	1	2	yes	—
280	#1893	C₂×D₄	1	2	yes	complemented
281	#1893	C₂×D₄	1	2	yes	complemented
282	#1885	C₂×D₄	1	2	yes	—
283	#1885	C₂×D₄	1	2	yes	—
284	#1869	C₂×D₄	1	2	yes	—
285	#1869	C₂×D₄	1	2	yes	—
286	#1869	C₂⁴	1	2	yes	—
287	#1869	C₂⁴	1	2	yes	—
288	#1856	C₂×D₄	1	2	yes	—
289	#1856	C₂×D₄	1	2	yes	—
290	#1823	C₂×D₄	1	2	yes	—
291	#1823	C₂×D₄	1	2	yes	complemented
292	#1800	C₂×D₄	1	2	yes	—
293	#1800	C₂⁴	1	2	yes	—
294	#1799	C₂⁴	1	2	yes	—
295	#1799	C₂⁴	1	2	yes	—
296	#1792	C₂×D₄	1	2	yes	—
297	#1792	C₂×D₄	1	2	yes	—
298	#1786	C₂×D₄	1	2	yes	—
299	#1786	C₂×D₄	1	2	yes	complemented
300	C₄×#100	C₂×D₄	1	2	yes	—
301	C₄×#100	C₂⁴	1	2	yes	—
302	C₂×#343	C₂×D₄	1	2	yes	—
303	#1459	C₂⁴	1	2	yes	—
304	#1459	C₂⁴	1	2	yes	—
305	C₂×H(5,𝔽₂)	AGL(1,ℤ/8)	1	4	yes	—
306	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
307	C₂³×Q₈	#120	1	1	yes	characteristic
308	C₂³×D₄	#120	1	1	yes	characteristic
309	C₂³×D₄	#120	1	1	yes	characteristic
310	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
311	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
312	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
313	C₂³×D₄	C₂²×D₄	1	2	yes	—
314	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
315	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
316	C₂³×C₈	C₂²×D₄	1	2	yes	—
317	C₂³×C₈	C₂²×D₄	1	2	yes	—
318	#537	C₂²×D₄	1	4	yes	—
319	C₂×C₄⋊Q₈	#120	1	1	yes	characteristic
320	C₂×C₄⋊Q₈	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄⋊Q₈	C₂²×D₄	1	1	yes	characteristic
322	C₂×#127	#120	1	1	yes	characteristic
323	C₂×#127	C₂²×D₄	1	1	yes	characteristic
324	C₂×#127	C₂²×D₄	1	1	yes	characteristic
325	C₂×#124	#120	1	1	yes	characteristic
326	C₂×#124	#120	1	1	yes	characteristic
327	C₂×#124	C₂²×D₄	1	1	yes	characteristic
328	C₂×#124	C₂²×D₄	1	1	yes	characteristic
329	C₂×#121	#120	1	1	yes	characteristic
330	C₂×#121	#120	1	1	yes	characteristic
331	C₂×#121	#120	1	1	yes	characteristic
332	C₂×#121	#120	1	1	yes	characteristic
333	C₂×#121	#120	1	1	yes	characteristic
334	C₂×#121	#120	1	1	yes	characteristic
335	C₂×#121	#120	1	1	yes	characteristic
336	C₂×#121	#120	1	1	yes	characteristic
337	C₂×#121	C₂²×D₄	1	1	yes	characteristic
338	C₂×#121	C₂²×D₄	1	1	yes	characteristic
339	C₂×#121	C₂²×D₄	1	1	yes	characteristic
340	C₂×#121	C₂²×D₄	1	1	yes	characteristic
341	#518	C₂²×D₄	1	2	yes	—
342	C₂×C₄×Q₈	#120	1	1	yes	characteristic
343	C₂×C₄×Q₈	#120	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×D₄	1	1	yes	characteristic
345	C₂×C₄×Q₈	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×D₄	#120	1	1	yes	characteristic
347	C₂×C₄×D₄	#120	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
349	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
350	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
351	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
352	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
353	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
354	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
355	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
356	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
357	C₂×C₄×D₄	C₂⁵	1	1	yes	characteristic
358	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
359	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
361	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
362	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
363	C₂²×C₄⋊C₄	C₂²×D₄	1	2	yes	—
364	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
365	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
366	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
367	C₂²×C₄²	#120	1	1	yes	characteristic
368	C₂²×C₄²	#120	1	1	yes	characteristic
369	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
370	C₂×#100	AGL(1,ℤ/8)	1	4	yes	—
371	C₂×#100	C₂²×D₄	1	2	yes	—
372	C₂×#100	C₂²×D₄	1	2	yes	—
373	C₂×C₂²⋊C₈	C₂²×D₄	1	2	yes	—
374	C₂×C₂²⋊C₈	C₂²×D₄	1	2	yes	—
375	#391	#120	1	1	yes	characteristic
376	#391	#120	1	1	yes	characteristic
377	#391	C₂²×D₄	1	1	yes	characteristic
378	#391	C₂²×D₄	1	1	yes	characteristic
379	#390	#120	1	1	yes	characteristic
380	#390	#120	1	1	yes	characteristic
381	#390	C₂²×D₄	1	1	yes	characteristic
382	#390	C₂²×D₄	1	1	yes	characteristic
383	#382	#120	1	1	yes	characteristic
384	#382	#120	1	1	yes	characteristic
385	#382	C₂²×D₄	1	1	yes	characteristic
386	#382	C₂²×D₄	1	1	yes	characteristic
387	#380	C₂²×D₄	1	2	yes	—
388	#380	C₂²×D₄	1	2	yes	—
389	C₄×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
390	C₄×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
391	C₄³	#120	1	1	yes	characteristic
392	C₄³	#120	1	1	yes	characteristic
393	C₄³	C₂²×D₄	1	1	yes	characteristic
394	C₄³	C₂²×D₄	1	1	yes	characteristic
395	#343	C₂²×D₄	1	4	yes	—
396	C₂⁵	D₄²	1	1	yes	characteristic
397	C₂⁵	D₄²	1	1	yes	characteristic
398	H(5,𝔽₂)	C₂×AGL(1,ℤ/8)	1	4	yes	—
399	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
400	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
401	C₂²×D₄	C₂×#120	1	1	yes	characteristic
402	C₂²×D₄	C₂×#120	1	1	yes	characteristic
403	C₂²×D₄	C₂×#120	1	1	yes	characteristic
404	C₂²×D₄	C₂×#120	1	1	yes	characteristic
405	C₂²×D₄	C₂×#120	1	1	yes	characteristic
406	C₂²×D₄	D₄²	1	1	yes	characteristic
407	C₂²×D₄	D₄²	1	1	yes	characteristic
408	C₂²×D₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
409	C₂²×D₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
410	C₂²×D₄	#576	1	2	yes	—
411	C₂²×D₄	#577	1	2	yes	—
412	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
413	C₂³×C₄	C₂×#120	1	1	yes	characteristic
414	C₂³×C₄	C₂×#120	1	1	yes	characteristic
415	C₂³×C₄	C₂×#120	1	1	yes	characteristic
416	C₂³×C₄	C₂×#120	1	1	yes	characteristic
417	C₂³×C₄	#534	1	1	yes	characteristic
418	C₂³×C₄	D₄²	1	1	yes	characteristic
419	C₂³×C₄	D₄²	1	1	yes	characteristic
420	C₂³×C₄	D₄²	1	1	yes	characteristic
421	C₂³×C₄	D₄²	1	1	yes	characteristic
422	C₂³×C₄	D₄²	1	1	yes	characteristic
423	C₂³×C₄	D₄²	1	1	yes	characteristic
424	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
425	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
426	#121	B(4,𝔽₂)	1	2	yes	—
427	#121	B(4,𝔽₂)	1	2	yes	—
428	#121	B(4,𝔽₂)	1	2	yes	—
429	#121	B(4,𝔽₂)	1	2	yes	—
430	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
437	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
438	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×D₄	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
444	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
445	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
446	C₂×C₂²⋊C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
447	C₂×C₂²⋊C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
448	C₂×C₄²	C₂×#120	1	1	yes	characteristic
449	C₂×C₄²	C₂×#120	1	1	yes	characteristic
450	C₂×C₄²	C₂×#120	1	1	yes	characteristic
451	C₂×C₄²	C₂×#120	1	1	yes	characteristic
452	C₂×C₄²	C₂×#120	1	1	yes	characteristic
453	C₂×C₄²	C₂×#120	1	1	yes	characteristic
454	C₂×C₄²	#534	1	1	yes	characteristic
455	C₂×C₄²	C₂³×D₄	1	1	yes	characteristic
456	#100	C₂×AGL(1,ℤ/8)	1	4	yes	—
457	C₂⁴	#2403	1	1	yes	characteristic
458	C₂⁴	#2403	1	1	yes	characteristic
459	C₂⁴	#2403	1	1	yes	characteristic
460	C₂⁴	#3264	1	2	yes	—
461	C₂⁴	#3265	1	2	yes	—
462	C₂⁴	#3272	1	2	yes	—
463	C₂⁴	#3274	1	2	yes	—
464	C₂⁴	C₂×D₄²	1	1	yes	characteristic
465	C₂⁴	C₂×D₄²	1	1	yes	characteristic
466	C₂×D₄	#3003	1	2	yes	—
467	C₂×D₄	#3012	1	2	yes	—
468	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
469	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
470	C₂×D₄	#3264	1	2	yes	—
471	C₂×D₄	#3265	1	2	yes	—
472	C₂×D₄	#3272	1	2	yes	—
473	C₂×D₄	#3274	1	2	yes	—
474	C₂×D₄	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
475	C₂×D₄	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
476	C₂×D₄	C₂×#576	1	1	yes	characteristic
477	C₂×D₄	C₂×#577	1	1	yes	characteristic
478	C₂²×C₄	#2403	1	1	yes	characteristic
479	C₂²×C₄	#2403	1	1	yes	characteristic
480	C₂²×C₄	#2403	1	1	yes	characteristic
481	C₂²×C₄	#2403	1	1	yes	characteristic
482	C₂²×C₄	#2403	1	1	yes	characteristic
483	C₂²×C₄	#3003	1	2	yes	—
484	C₂²×C₄	#3012	1	2	yes	—
485	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
486	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
487	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
488	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
489	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
490	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
491	C₂²×C₄	C₂×#534	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
494	C₄⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
495	C₄⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
496	C₂³	#23860	1	2	yes	—
497	C₂³	#23860	1	2	yes	—
498	C₂³	#23860	1	2	yes	—
499	C₂³	#23882	1	2	yes	—
500	C₂³	#23900	1	1	yes	characteristic
501	C₂³	#23900	1	1	yes	characteristic
502	C₂³	C₂×#2403	1	1	yes	characteristic
503	C₂³	C₂×#3264	1	1	yes	characteristic
504	C₂³	C₂×#3265	1	1	yes	characteristic
505	C₂³	C₂×#3272	1	1	yes	characteristic
506	C₂³	C₂×#3274	1	1	yes	characteristic
507	C₂×C₄	#23860	1	2	yes	—
508	C₂×C₄	#23900	1	1	yes	characteristic
509	C₂×C₄	#23900	1	1	yes	characteristic
510	C₂×C₄	C₂×#3003	1	1	yes	characteristic
511	C₂×C₄	C₂×#3012	1	1	yes	characteristic
512	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
513	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
514	C₂²	#248551	1	2	yes	—
515	C₂²	#248551	1	2	yes	—
516	C₂²	C₂×#23860	1	1	yes	characteristic
517	C₂²	C₂×#23882	1	1	yes	characteristic
518	C₂²	C₂×#23900	1	1	yes	characteristic
519	C₄	#7622881	1	2	yes	—
520	C₂	C₂×#248551	1	1	yes	characteristic
521	C₂	#72236949	1	1	yes	characteristic
522	C₂	#72306415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#23900
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁵
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#3496 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72313926	C₂	semidirect
2. #72313907	C₂	semidirect
3. C₂×#486533	C₂	semidirect
4. #72313847	C₂	semidirect
5. #72313844	C₂	semidirect
6. #72313840	C₂	semidirect
7. C₂×#500489	C₂	semidirect
8. #72313793	C₂	semidirect
9. #72313565	C₂	semidirect
10. #72313557	C₂	semidirect
11. #72313443	C₂	semidirect
12. #72313440	C₂	semidirect
13. #72313388	C₂	semidirect
14. #72313386	C₂	semidirect
15. #72312715	C₂	semidirect
16. #72311797	C₂	semidirect
17. #72306052	C₂	semidirect
18. #72306044	C₂	semidirect
19. #72282436	C₂	semidirect
22. C₂×#59945	C₂²	semidirect
23. #7586780	C₂²	semidirect
24. #7576412	C₂²	semidirect
25. #7481510	C₂²	semidirect
26. #7454388	C₂²	semidirect
27. #6640543	C₂²	semidirect
28. #6632008	C₂²	semidirect
32. #6573698	C₂²	semidirect
33. #6573693	C₂²	semidirect
34. #6573670	C₂²	semidirect
35. D₄×#455	C₂²	semidirect
40. #6408606	C₂²	semidirect
45. #500489	C₂²	semidirect
46. #500489	C₂²	semidirect
47. C₂×#13292	C₂²	semidirect
48. C₂×#13218	C₂²	semidirect
49. C₂×#12798	C₂²	semidirect
50. #486533	C₂²	semidirect
51. #486533	C₂²	semidirect
52. C₂×#11903	C₂²	semidirect
53. C₂×#11880	C₂²	semidirect
54. #482050	C₂²	semidirect
55. #482050	C₂²	semidirect
56. C₂×#11477	C₂²	semidirect
57. #346546	C₂²	semidirect
58. #346542	C₂²	semidirect
67. #144271	C₂²	semidirect
68. #144270	C₂²	semidirect
69. #144230	C₂²	semidirect
70. #144228	C₂²	semidirect
71. #142735	C₂²	semidirect
72. #142734	C₂²	semidirect
73. #142605	C₂²	semidirect
74. #142604	C₂²	semidirect
75. #139617	C₂²	semidirect
76. #139615	C₂²	semidirect
77. #138967	C₂²	semidirect
78. #138966	C₂²	semidirect
79. #138881	C₂²	semidirect
80. #138880	C₂²	semidirect
81. #136852	C₂²	semidirect
82. #136850	C₂²	semidirect
83. #130775	C₂²	semidirect
84. #130773	C₂²	semidirect
85. #130771	C₂²	semidirect
86. #130769	C₂²	semidirect
87. #127353	C₂²	semidirect
88. #127353	C₂²	semidirect
89. #127067	C₂²	semidirect
90. #127067	C₂²	semidirect
93. C₂×#3497	D₄	semidirect
97. C₂²×#481	D₄	semidirect
98. C₂²×#481	D₄	semidirect
102. #59945	C₂³	semidirect
103. #59945	C₂³	semidirect
104. #39361	D₄	semidirect
106. #39360	D₄	semidirect
108. #36644	D₄	semidirect
109. #36125	D₄	semidirect
111. D₄×#124	D₄	semidirect
113. #35178	D₄	semidirect
117. C₄×#532	D₄	semidirect
123. #32260	C₂³	semidirect
124. #32028	C₂³	semidirect
125. #31260	C₂³	semidirect
126. #30499	C₂³	semidirect
129. #24224	D₄	semidirect
130. #24224	D₄	semidirect
133. #24089	D₄	semidirect
135. #24033	D₄	semidirect
141. C₂×#2040	D₄	semidirect
142. C₂×#2007	D₄	semidirect
143. #23352	C₂³	semidirect
144. #23273	C₂³	semidirect
149. #21758	D₄	semidirect
150. #21758	D₄	semidirect
168. #13292	C₂³	semidirect
169. #13218	C₂³	semidirect
170. #12798	C₂³	semidirect
171. #12798	C₂³	semidirect
172. #11903	C₂³	semidirect
173. #11880	C₂³	semidirect
174. #11878	C₂³	semidirect
175. #11878	C₂³	semidirect
176. #11877	C₂³	semidirect
177. #11877	C₂³	semidirect
178. #11477	C₂³	semidirect
179. #11477	C₂³	semidirect
188. #7236	C₂³	semidirect
189. #7236	C₂³	semidirect
190. #7203	C₂³	semidirect
191. #7203	C₂³	semidirect
197. C₂×#550	C₂×D₄	semidirect
198. C₂×#550	C₂×D₄	semidirect
199. C₂×D₄×Q₈	C₂×D₄	semidirect
200. C₂×D₄×Q₈	C₂×D₄	semidirect
224. C₂×#481	C₂×D₄	semidirect
225. C₂×#481	C₂×D₄	semidirect
230. C₂²×#106	C₂×D₄	semidirect
232. C₂²×#102	C₂×D₄	semidirect
239. #2868	C₂×D₄	semidirect
240. #2867	C₂×D₄	semidirect
241. #2858	C₂×D₄	semidirect
242. #2858	C₂×D₄	semidirect
243. #2548	C₂×D₄	semidirect
244. #2548	C₂×D₄	semidirect
247. #2546	C₂×D₄	semidirect
248. #2546	C₂×D₄	semidirect
264. #2293	C₂×D₄	semidirect
265. #2293	C₂×D₄	semidirect
271. C₄²×D₄	C₂×D₄	semidirect
272. C₄²×D₄	C₂×D₄	semidirect
274. #2040	C₂×D₄	semidirect
275. #2040	C₂×D₄	semidirect
276. #2007	C₂×D₄	semidirect
277. #2007	C₂×D₄	semidirect
280. #1893	C₂×D₄	semidirect
281. #1893	C₂×D₄	semidirect
291. #1823	C₂×D₄	semidirect
299. #1786	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72313926	2
#72313907	2
C₂×#486533	2
#72313847	2
#72313844	2
#72313840	2
C₂×#500489	2
#72313793	2
#72313565	2
#72313557	2
#72313443	2
#72313440	2
#72313388	2
#72313386	2
#72312715	2
#72311797	2
#72306052	2
#72306044	2
#72282436	2