Subgroups — #72341513

Order 4096.

Only the 1037 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 513 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341503	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341497	C₂	1	1	yes	maximal, characteristic, complemented
3	#72341472	C₂	1	2	yes	maximal, complemented
4	#72341469	C₂	1	2	yes	maximal, complemented
5	#72341467	C₂	1	2	yes	maximal, complemented
6	#72341466	C₂²	1	2	yes	complemented
7	#72341465	C₂²	1	2	yes	complemented
8	#72341456	C₂	1	2	yes	maximal, complemented
9	#72341456	C₂	1	2	yes	maximal, complemented
10	#72341453	C₂²	1	2	yes	complemented
11	#72341399	C₂	1	1	yes	maximal, characteristic, complemented
12	#72341387	C₂	1	1	yes	maximal, characteristic, complemented
13	#72341386	C₂²	1	1	yes	characteristic, complemented
14	#72341371	C₂	1	1	yes	maximal, characteristic, complemented
15	#72341366	C₂²	1	1	yes	characteristic, complemented
16	#72341366	C₂²	1	1	yes	characteristic, complemented
17	#72341363	C₂²	1	1	yes	characteristic, complemented
18	#72341362	C₂²	1	1	yes	characteristic, complemented
19	#72341307	C₂	1	1	yes	maximal, characteristic, complemented
20	#72341301	C₂²	1	2	yes	complemented
21	#72341298	C₂²	1	1	yes	characteristic
22	#72341294	C₂²	1	1	yes	characteristic, complemented
23	#72340210	C₂	1	2	yes	maximal, complemented
24	#72337574	C₂	1	2	yes	maximal, complemented
25	#72337559	C₂²	1	2	yes	complemented
26	#72337559	C₂²	1	2	yes	complemented
27	#72312619	C₂²	1	1	yes	characteristic
28	#72312142	C₂	1	1	yes	maximal, characteristic, complemented
29	#72312133	C₂²	1	1	yes	characteristic, complemented
30	#72311618	C₂²	1	2	yes	complemented
31	#72311618	C₂²	1	2	yes	complemented
32	#72311606	C₂²	1	2	yes	complemented
33	#72311606	C₂²	1	2	yes	complemented
34	#72311370	C₂²	1	1	yes	characteristic, complemented
35	#72311366	C₂²	1	1	yes	characteristic
36	#72311270	C₂	1	2	yes	maximal, complemented
37	#72311266	C₂²	1	2	yes	complemented
38	#72311264	C₂²	1	2	yes	complemented
39	#72311147	C₂²	1	1	yes	characteristic, complemented
40	#72311145	C₂²	1	2	yes	complemented
41	#72311145	C₂²	1	2	yes	complemented
42	#72311140	C₂²	1	2	yes	complemented
43	#72311055	C₂²	1	2	yes	complemented
44	#72311055	C₂²	1	2	yes	complemented
45	#72310930	C₂²	1	2	yes	complemented
46	#72310916	C₂²	1	1	yes	characteristic, complemented
47	#72306417	C₂²	1	2	yes	—
48	#72306417	C₂²	1	2	yes	complemented
49	#72303086	C₂²	1	2	yes	complemented
50	#72303081	C₂²	1	1	yes	characteristic, complemented
51	#72302950	C₂²	1	2	yes	complemented
52	#72284724	C₂²	1	2	yes	complemented
53	#72260657	C₂²	1	2	yes	complemented
54	#72241340	C₂	1	2	yes	maximal, complemented
55	#72241338	C₂	1	2	yes	maximal, complemented
56	#72241337	C₂²	1	4	yes	complemented
57	#72241337	C₂²	1	4	yes	complemented
58	#72241334	C₂	1	2	yes	maximal, complemented
59	#72241333	C₂²	1	2	yes	complemented
60	#72241332	C₂²	1	4	yes	complemented
61	#72241332	C₂²	1	4	yes	complemented
62	#72241315	C₂²	1	2	yes	—
63	#72241315	C₂²	1	2	yes	complemented
64	#72241304	C₂²	1	1	yes	characteristic
65	#72241303	C₂²	1	1	yes	characteristic
66	#72241301	C₂²	1	1	yes	characteristic
67	#72238017	C₂	1	2	yes	maximal, complemented
68	#72238016	C₂²	1	2	yes	complemented
69	#72238015	C₂²	1	4	yes	complemented
70	#72238015	C₂²	1	4	yes	complemented
71	#72238014	C₂²	1	2	yes	complemented
72	#72238013	C₂²	1	4	yes	complemented
73	#72238013	C₂²	1	4	yes	complemented
74	#72238012	C₂²	1	1	yes	characteristic
75	#72238010	C₂²	1	2	yes	complemented
76	#72236925	C₂²	1	4	yes	complemented
77	#72236711	C₂²	1	2	yes	complemented
78	#72236702	C₂²	1	2	yes	complemented
79	#72236657	C₂²	1	2	yes	complemented
80	#72234146	C₂²	1	2	yes	—
81	#72234146	C₂²	1	2	yes	complemented
82	#72233917	C₂²	1	2	yes	complemented
83	#72227731	C₂²	1	1	yes	characteristic, complemented
84	#72213523	C₂²	1	4	yes	complemented
85	#72213513	C₂²	1	4	yes	complemented
86	#72213504	C₂²	1	4	yes	complemented
87	#72213504	C₂²	1	4	yes	complemented
88	#72213504	C₂²	1	4	yes	complemented
89	#72213504	C₂²	1	4	yes	complemented
90	#72210025	C₂²	1	4	yes	complemented
91	H(9,𝔽₂)	C₂³	1	1	yes	characteristic
92	#10586106	C₂³	1	1	yes	characteristic
93	#10586097	C₂³	1	1	yes	characteristic
94	#10574841	C₂³	1	1	yes	characteristic
95	#10574841	C₂³	1	1	yes	characteristic
96	#10574841	C₂³	1	1	yes	characteristic
97	#10574841	C₂³	1	1	yes	characteristic
98	C₂×#60391	D₄	1	1	yes	characteristic, complemented
99	C₂×#60389	D₄	1	1	yes	characteristic, complemented
100	C₂×#60389	C₂³	1	1	yes	characteristic
101	C₂×#60389	C₂³	1	1	yes	characteristic
102	C₂×#60388	C₂³	1	1	yes	characteristic
103	C₂×#60388	C₂³	1	1	yes	characteristic
104	#10574064	C₂³	1	1	yes	characteristic
105	#10574064	C₂³	1	1	yes	characteristic
106	#8928621	C₂³	1	2	yes	—
107	#7742598	C₂³	1	2	yes	—
108	#7737411	C₂³	1	2	yes	—
109	#7672684	C₂³	1	1	yes	characteristic
110	C₂²×#2199	D₄	1	4	yes	complemented
111	C₂²×#2196	D₄	1	4	yes	complemented
112	#6628972	D₄	1	1	yes	characteristic, complemented
113	#6628971	D₄	1	1	yes	characteristic, complemented
114	#6628902	C₂³	1	1	yes	characteristic
115	#6628896	C₂³	1	4	yes	—
116	#6628896	C₂³	1	4	yes	—
117	#6628896	C₂³	1	4	yes	—
118	#6628874	D₄	1	1	yes	characteristic, complemented
119	#6628873	D₄	1	1	yes	characteristic, complemented
120	#6628873	C₂³	1	1	yes	characteristic
121	#6628873	C₂³	1	1	yes	characteristic
122	#6470234	C₂³	1	4	yes	—
123	#6470234	C₂³	1	4	yes	—
124	#6443332	D₄	1	1	yes	characteristic, complemented
125	#6443330	D₄	1	1	yes	characteristic, complemented
126	#6442555	D₄	1	1	yes	characteristic, complemented
127	#6442555	D₄	1	1	yes	characteristic, complemented
128	#6437971	D₄	1	1	yes	characteristic, complemented
129	#6437970	D₄	1	1	yes	characteristic, complemented
130	#6437970	C₂³	1	1	yes	characteristic
131	#6437970	C₂³	1	1	yes	characteristic
132	#6437630	C₂³	1	1	yes	characteristic
133	#6437455	D₄	1	1	yes	characteristic, complemented
134	#6437453	D₄	1	1	yes	characteristic, complemented
135	#6437453	C₂³	1	1	yes	characteristic
136	#6437452	C₂³	1	1	yes	characteristic
137	#6385981	C₂³	1	2	yes	—
138	#6385980	C₂³	1	2	yes	—
139	#6379034	C₂³	1	1	yes	characteristic
140	#6379033	C₂³	1	1	yes	characteristic
141	C₄×#3018	D₄	1	1	yes	characteristic, complemented
142	C₄×#3018	D₄	1	1	yes	characteristic, complemented
143	#498927	D₄	1	8	yes	complemented
144	#498927	D₄	1	8	yes	complemented
145	#496240	C₂³	1	4	yes	—
146	#496240	C₂³	1	4	yes	—
147	#496205	C₂³	1	2	yes	—
148	#496205	C₂³	1	2	yes	—
149	#494682	C₂³	1	4	yes	—
150	#494682	C₂³	1	4	yes	—
151	#494682	C₂³	1	4	yes	—
152	#494682	C₂³	1	4	yes	—
153	#494597	D₄	1	4	yes	complemented
154	#494596	D₄	1	4	yes	complemented
155	#493433	C₂³	1	2	yes	—
156	#493433	C₂³	1	2	yes	—
157	#493430	C₂³	1	2	yes	—
158	#493430	C₂³	1	2	yes	—
159	#493397	C₂³	1	4	yes	—
160	#493397	C₂³	1	4	yes	—
161	#449493	D₄	1	2	yes	complemented
162	#417810	D₄	1	2	yes	complemented
163	#417568	D₄	1	2	yes	complemented
164	#417566	D₄	1	2	yes	complemented
165	#392138	C₂³	1	1	yes	characteristic
166	#392138	C₂³	1	1	yes	characteristic
167	#392136	C₂³	1	1	yes	characteristic
168	#392135	C₂³	1	4	yes	—
169	#392135	C₂³	1	4	yes	—
170	#392135	C₂³	1	4	yes	—
171	#374010	C₂³	1	1	yes	characteristic
172	#360531	C₂³	1	1	yes	characteristic
173	#360531	C₂³	1	1	yes	characteristic
174	#359606	C₂³	1	1	yes	characteristic
175	#359605	C₂³	1	1	yes	characteristic
176	#359529	C₂³	1	1	yes	characteristic
177	#359529	C₂³	1	1	yes	characteristic
178	#356056	C₂³	1	1	yes	characteristic
179	#213862	C₂³	1	2	yes	—
180	#213854	C₂³	1	4	yes	—
181	#146577	C₂³	1	4	yes	—
182	#146577	C₂³	1	4	yes	—
183	#146577	C₂³	1	4	yes	—
184	#146577	C₂³	1	4	yes	—
185	#128079	C₂³	1	4	yes	—
186	#128079	C₂³	1	4	yes	—
187	#128062	C₂³	1	4	yes	—
188	C₂×2₋¹⁺⁶	C₂×D₄	1	1	yes	characteristic, complemented
189	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
190	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
191	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
192	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
193	#62859	C₂×D₄	1	1	yes	characteristic, complemented
194	#62852	C₂×D₄	1	1	yes	characteristic
195	#62852	C₂⁴	1	1	yes	characteristic
196	#62852	C₂⁴	1	1	yes	characteristic
197	#62812	C₂×D₄	1	1	yes	characteristic
198	#62812	C₂×D₄	1	1	yes	characteristic, complemented
199	C₂×#3500	C₂×D₄	1	1	yes	characteristic
200	C₂×#3500	C₂⁴	1	1	yes	characteristic
201	C₂×#3499	C₂×D₄	1	1	yes	characteristic
202	C₂×#3498	C₂⁴	1	1	yes	characteristic
203	C₂²×D₄²	C₂×D₄	1	2	yes	—
204	C₂²×H(3,𝔽₄)	C₂×D₄	1	2	yes	—
205	C₄×#585	C₂×D₄	1	1	yes	characteristic
206	C₄×#585	C₂×D₄	1	1	yes	characteristic, complemented
207	C₂²×B(4,𝔽₂)	C₂×D₄	1	8	yes	—
208	C₂²×B(4,𝔽₂)	C₂×D₄	1	8	yes	complemented
209	C₂×#3022	C₂×D₄	1	1	yes	characteristic
210	C₂×#3022	C₂×D₄	1	1	yes	characteristic
211	C₂×#3022	C₂×D₄	1	1	yes	characteristic
212	C₂×#3022	C₂×D₄	1	1	yes	characteristic
213	C₂×#3022	C₂×D₄	1	1	yes	characteristic
214	C₂×#3022	C₂×D₄	1	1	yes	characteristic
215	C₂×#3022	C₂×D₄	1	1	yes	characteristic, complemented
216	C₂×#3022	C₂×D₄	1	1	yes	characteristic, complemented
217	C₂×#3022	C₂×D₄	1	1	yes	characteristic, complemented
218	C₂×#3021	C₂×D₄	1	1	yes	characteristic
219	C₂×#3021	C₂×D₄	1	1	yes	characteristic
220	C₂×#3021	C₂×D₄	1	1	yes	characteristic
221	C₂×#3019	C₂×D₄	1	2	yes	—
222	C₂×#3019	C₂×D₄	1	4	yes	—
223	C₂×#3019	C₂×D₄	1	4	yes	—
224	C₂×#3019	C₂×D₄	1	2	yes	complemented
225	C₂×#3018	C₂×D₄	1	1	yes	characteristic
226	C₂×#3018	C₂×D₄	1	1	yes	characteristic
227	C₂×#3018	C₂×D₄	1	1	yes	characteristic
228	C₂×#3018	C₂×D₄	1	1	yes	characteristic
229	C₂×#3018	C₂×D₄	1	1	yes	characteristic
230	C₂×#3018	C₂×D₄	1	1	yes	characteristic
231	C₂²×#455	C₂×D₄	1	4	yes	—
232	C₂²×#455	C₂×D₄	1	2	yes	complemented
233	C₂²×#453	C₂×D₄	1	2	yes	—
234	C₂²×#453	C₂×D₄	1	4	yes	—
235	C₂×#2889	C₂×D₄	1	1	yes	characteristic, complemented
236	C₂×#2888	C₂×D₄	1	1	yes	characteristic
237	C₂×#2888	C₂⁴	1	1	yes	characteristic
238	C₂×#2888	C₂⁴	1	1	yes	characteristic
239	C₂×#2888	C₂⁴	1	1	yes	characteristic
240	C₂×#2888	C₂⁴	1	1	yes	characteristic
241	#36702	C₂×D₄	1	4	yes	—
242	#36702	C₂×D₄	1	4	yes	—
243	C₂×#2858	C₂×D₄	1	1	yes	characteristic
244	#35676	C₂×D₄	1	1	yes	characteristic
245	#35676	C₂×D₄	1	1	yes	characteristic
246	#35616	C₂×D₄	1	1	yes	characteristic
247	#35616	C₂×D₄	1	2	yes	complemented
248	#35616	C₂⁴	1	1	yes	characteristic
249	#35346	C₂⁴	1	4	yes	—
250	#35327	C₂×D₄	1	1	yes	characteristic
251	#35327	C₂×D₄	1	1	yes	characteristic
252	#35316	C₂×D₄	1	1	yes	characteristic
253	#35272	C₂×D₄	1	1	yes	characteristic
254	#34366	C₂×D₄	1	2	yes	—
255	#34364	C₂×D₄	1	2	yes	—
256	C₄×#532	C₂×D₄	1	1	yes	characteristic
257	C₄×#532	C₂×D₄	1	1	yes	characteristic
258	C₂×#2127	C₂×D₄	1	4	yes	—
259	C₂×#2127	C₂×D₄	1	4	yes	—
260	C₂²×#354	C₂×D₄	1	4	yes	—
261	C₂²×#353	C₂×D₄	1	4	yes	—
262	C₂²×AΣL(1,𝔽₁₆)	C₂×D₄	1	8	yes	—
263	C₂²×AΣL(1,𝔽₁₆)	C₂×D₄	1	8	yes	complemented
264	#20956	C₂×D₄	1	1	yes	characteristic
265	#20955	C₂×D₄	1	1	yes	characteristic
266	#20955	C₂×D₄	1	1	yes	characteristic
267	#20955	C₂×D₄	1	1	yes	characteristic, complemented
268	#20952	C₂×D₄	1	1	yes	characteristic
269	#20952	C₂×D₄	1	1	yes	characteristic
270	#20855	C₂×D₄	1	1	yes	characteristic
271	#20855	C₂×D₄	1	1	yes	characteristic, complemented
272	#20637	C₂×D₄	1	1	yes	characteristic
273	#20637	C₂×D₄	1	1	yes	characteristic, complemented
274	#20615	C₂×D₄	1	1	yes	characteristic
275	#20615	C₂×D₄	1	1	yes	characteristic
276	#20615	C₂×D₄	1	1	yes	characteristic
277	#20615	C₂×D₄	1	1	yes	characteristic, complemented
278	#20615	C₂×D₄	1	1	yes	characteristic, complemented
279	#20615	C₂×D₄	1	1	yes	characteristic, complemented
280	#20609	C₂×D₄	1	8	yes	—
281	#20609	C₂×D₄	1	8	yes	complemented
282	#20593	C₂×D₄	1	1	yes	characteristic, complemented
283	#20592	C₂×D₄	1	1	yes	characteristic
284	#20592	C₂⁴	1	1	yes	characteristic
285	#20592	C₂⁴	1	1	yes	characteristic
286	#20586	C₂×D₄	1	2	yes	complemented
287	#20586	C₂⁴	1	1	yes	characteristic
288	#20585	C₂⁴	1	4	yes	—
289	#20585	C₂⁴	1	4	yes	—
290	#20585	C₂⁴	1	4	yes	—
291	C₄×#413	C₂×D₄	1	1	yes	characteristic
292	C₄×#413	C₂×D₄	1	1	yes	characteristic, complemented
293	#12574	C₂×D₄	1	8	yes	—
294	#12574	C₂×D₄	1	8	yes	complemented
295	#11167	C₂×D₄	1	2	yes	complemented
296	#10989	C₂×D₄	1	2	yes	—
297	#10989	C₂×D₄	1	2	yes	complemented
298	#10982	C₂×D₄	1	2	yes	complemented
299	#10977	C₂×D₄	1	2	yes	complemented
300	#10975	C₂×D₄	1	2	yes	—
301	C₂×#585	C₂²×D₄	1	1	yes	characteristic
302	C₂×#585	C₂²×D₄	1	1	yes	characteristic
303	C₂×#585	C₂²×D₄	1	1	yes	characteristic
304	C₂×#585	C₂²×D₄	1	1	yes	characteristic
305	C₂×#585	C₂²×D₄	1	1	yes	characteristic
306	C₂×#585	C₂²×D₄	1	1	yes	characteristic
307	C₂²×2₋¹⁺⁴	#120	1	1	yes	characteristic, complemented
308	C₂²×2₋¹⁺⁴	C₂²×D₄	1	1	yes	characteristic
309	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
310	C₂²×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
311	C₂²×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
312	C₂²×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
313	C₂²×H(5,𝔽₂)	C₂⁵	1	1	yes	characteristic
314	C₂×#536	#120	1	1	yes	characteristic, complemented
315	C₂×#536	C₂²×D₄	1	1	yes	characteristic
316	C₂×#536	C₂²×D₄	1	1	yes	characteristic
317	C₂×#536	C₂²×D₄	1	1	yes	characteristic
318	C₂×#536	H(5,𝔽₂)	1	2	yes	—
319	C₂×#535	#120	1	1	yes	characteristic, complemented
320	C₂×#535	C₂²×D₄	1	1	yes	characteristic
321	C₂×#535	C₂²×D₄	1	1	yes	characteristic
322	C₂×#535	C₂²×D₄	1	1	yes	characteristic
323	C₂×#535	C₂²×D₄	1	2	yes	—
324	C₂×#535	C₂²×D₄	1	2	yes	—
325	C₂×#535	H(5,𝔽₂)	1	2	yes	—
326	C₂×#532	C₂²×D₄	1	1	yes	characteristic
327	C₂×#532	C₂²×D₄	1	1	yes	characteristic
328	C₂×#532	H(5,𝔽₂)	1	2	yes	—
329	C₂×#532	H(5,𝔽₂)	1	2	yes	—
330	C₂²×C₄⋊Q₈	#120	1	1	yes	characteristic, complemented
331	C₂²×#127	C₂²×D₄	1	2	yes	—
332	C₂²×#127	C₂²×D₄	1	1	yes	characteristic
333	C₂²×#124	#120	1	1	yes	characteristic, complemented
334	C₂²×#124	#120	1	1	yes	characteristic, complemented
335	C₂²×#124	C₂²×D₄	1	2	yes	—
336	C₂²×#124	C₂²×D₄	1	1	yes	characteristic
337	C₂²×#124	C₂²×D₄	1	1	yes	characteristic
338	C₂²×#120	C₂²×D₄	1	4	yes	—
339	C₂²×#120	C₂²×D₄	1	4	yes	—
340	C₂²×#120	H(5,𝔽₂)	1	4	yes	—
341	#3428	C₂²×D₄	1	1	yes	characteristic
342	#3428	C₂²×D₄	1	1	yes	characteristic
343	C₂×C₄×#41	#120	1	1	yes	characteristic, complemented
344	C₂×C₄×#41	#120	1	1	yes	characteristic, complemented
345	C₂×C₄×#41	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×#41	C₂²×D₄	1	1	yes	characteristic
347	C₂×#455	#120	1	2	yes	—
348	C₂×#455	#120	1	4	yes	—
349	C₂×#455	#120	1	4	yes	—
350	C₂×#455	#120	1	2	yes	—
351	C₂×#455	#120	1	2	yes	—
352	C₂×#455	#120	1	2	yes	—
353	C₂×#453	#120	1	2	yes	—
354	C₂×#453	#120	1	4	yes	—
355	C₂×#453	#120	1	4	yes	—
356	C₂×#453	#120	1	2	yes	—
357	C₂×#453	#120	1	2	yes	—
358	C₂×#453	#120	1	2	yes	—
359	C₂×#413	C₂²×D₄	1	1	yes	characteristic
360	C₂×#413	C₂²×D₄	1	1	yes	characteristic
361	C₂×#413	C₂²×D₄	1	1	yes	characteristic
362	C₂×#413	C₂²×D₄	1	1	yes	characteristic
363	C₂×#413	C₂²×D₄	1	1	yes	characteristic
364	C₂×#413	C₂²×D₄	1	1	yes	characteristic
365	C₂×#413	C₂²×D₄	1	1	yes	characteristic
366	C₂×#413	C₂²×D₄	1	1	yes	characteristic
367	C₂²×#100	C₂²×D₄	1	2	yes	—
368	C₂²×#100	C₂²×D₄	1	2	yes	—
369	C₂²×#100	C₂²×D₄	1	4	yes	—
370	C₂²×#100	C₂²×D₄	1	4	yes	—
371	C₂²×#99	C₂²×D₄	1	8	yes	—
372	C₂²×#99	C₂²×D₄	1	8	yes	—
373	#2310	#120	1	1	yes	characteristic, complemented
374	#2310	#120	1	1	yes	characteristic, complemented
375	#2309	#120	1	1	yes	characteristic, complemented
376	#2309	C₂²×D₄	1	1	yes	characteristic
377	#2309	C₂²×D₄	1	2	yes	—
378	#2308	C₂²×D₄	1	4	yes	—
379	#2308	C₂²×D₄	1	4	yes	—
380	#2308	H(5,𝔽₂)	1	4	yes	—
381	#2297	#120	1	1	yes	characteristic, complemented
382	#2297	#120	1	1	yes	characteristic, complemented
383	#2297	C₂²×D₄	1	1	yes	characteristic
384	#2297	C₂²×D₄	1	1	yes	characteristic
385	#2285	#120	1	1	yes	characteristic, complemented
386	#2285	#120	1	1	yes	characteristic, complemented
387	C₂×C₄³	#120	1	1	yes	characteristic, complemented
388	C₂⁶	D₄²	1	4	yes	—
389	C₂×H(5,𝔽₂)	C₂×#120	1	2	yes	—
390	C₂×H(5,𝔽₂)	C₂×#120	1	2	yes	—
391	C₂×H(5,𝔽₂)	C₂×#120	1	2	yes	—
392	C₂×H(5,𝔽₂)	C₂×#120	1	2	yes	—
393	C₂²×#41	C₂×#120	1	1	yes	characteristic
394	C₂²×#41	C₂×#120	1	1	yes	characteristic
395	C₂²×#41	D₄²	1	2	yes	—
396	C₂²×#41	D₄²	1	2	yes	—
397	C₂²×#41	C₂³×D₄	1	1	yes	characteristic
398	C₂²×#41	C₂³×D₄	1	1	yes	characteristic
399	C₂²×#41	C₂×H(5,𝔽₂)	1	2	yes	—
400	C₂²×#41	C₂×H(5,𝔽₂)	1	2	yes	—
401	C₂³×Q₈	C₂×#120	1	1	yes	characteristic
402	C₂³×Q₈	C₂×#120	1	1	yes	characteristic
403	C₂³×Q₈	D₄²	1	2	yes	—
404	C₂³×D₄	C₂×#120	1	1	yes	characteristic
405	C₂³×D₄	D₄²	1	2	yes	—
406	C₂³×D₄	C₂³×D₄	1	1	yes	characteristic
407	C₂³×D₄	C₂³×D₄	1	2	yes	—
408	C₂³×D₄	C₂³×D₄	1	2	yes	—
409	C₂³×D₄	C₂×H(5,𝔽₂)	1	4	yes	—
410	C₂×C₄⋊Q₈	#534	1	1	yes	characteristic
411	C₂×C₄⋊Q₈	#534	1	1	yes	characteristic
412	C₂×C₄⋊Q₈	#534	1	2	yes	—
413	C₂×#127	C₂×#120	1	1	yes	characteristic
414	C₂×#127	C₂×#120	1	2	yes	—
415	C₂×#127	C₂×#120	1	1	yes	characteristic
416	C₂×#127	D₄²	1	2	yes	—
417	C₂×#127	D₄²	1	2	yes	—
418	C₂×#127	D₄²	1	2	yes	—
419	C₂×#127	D₄²	1	2	yes	—
420	C₂×#124	C₂×#120	1	1	yes	characteristic
421	C₂×#124	C₂×#120	1	2	yes	—
422	C₂×#124	C₂×#120	1	1	yes	characteristic
423	C₂×#124	#534	1	1	yes	characteristic
424	C₂×#124	#534	1	1	yes	characteristic
425	C₂×#124	#534	1	2	yes	—
426	C₂×#124	D₄²	1	2	yes	—
427	C₂×#124	D₄²	1	2	yes	—
428	C₂×#124	D₄²	1	2	yes	—
429	C₂×#124	D₄²	1	2	yes	—
430	C₂×#117	C₂×#120	1	1	yes	characteristic
431	C₂×#117	C₂×#120	1	1	yes	characteristic
432	C₂×#117	C₂×#120	1	1	yes	characteristic
433	C₂×#117	C₂×#120	1	1	yes	characteristic
434	C₂×#117	D₄²	1	2	yes	—
435	C₂×#117	D₄²	1	2	yes	—
436	C₂²×C₂²⋊C₄	D₄²	1	4	yes	—
437	C₂²×C₂²⋊C₄	D₄²	1	4	yes	—
438	C₂²×C₂²⋊C₄	D₄²	1	4	yes	—
439	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
440	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
441	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
442	C₂²×C₄²	D₄²	1	2	yes	—
443	C₂²×C₄²	D₄²	1	2	yes	—
444	#455	B(4,𝔽₂)	1	8	yes	—
445	#453	B(4,𝔽₂)	1	8	yes	—
446	C₂×#100	C₂×#120	1	2	yes	—
447	C₂×#100	C₂×#120	1	2	yes	—
448	C₂×#100	C₂×#120	1	2	yes	—
449	C₂×#100	C₂×#120	1	2	yes	—
450	C₂⁵	C₂×D₄²	1	4	yes	—
451	C₂⁵	#3484	1	2	yes	—
452	H(5,𝔽₂)	C₂×B(4,𝔽₂)	1	4	yes	—
453	H(5,𝔽₂)	C₂×B(4,𝔽₂)	1	4	yes	—
454	C₂²×Q₈	#2403	1	2	yes	—
455	C₂²×Q₈	#2403	1	1	yes	characteristic
456	C₂²×Q₈	#2403	1	1	yes	characteristic
457	C₂²×Q₈	C₂×#534	1	1	yes	characteristic
458	C₂²×Q₈	C₂×#534	1	1	yes	characteristic
459	C₂²×D₄	#2403	1	2	yes	—
460	C₂²×D₄	#2403	1	1	yes	characteristic
461	C₂²×D₄	#2403	1	1	yes	characteristic
462	C₂²×D₄	C₂²×#120	1	1	yes	characteristic
463	C₂²×D₄	C₂²×#120	1	1	yes	characteristic
464	C₂²×D₄	C₂×D₄²	1	2	yes	—
465	C₂²×D₄	C₂×D₄²	1	2	yes	—
466	C₂²×D₄	#3484	1	2	yes	—
467	C₂²×D₄	#3484	1	2	yes	—
468	C₂²×D₄	#3484	1	4	yes	—
469	C₂²×D₄	#3484	1	4	yes	—
470	C₂³×C₄	C₂²×#120	1	1	yes	characteristic
471	C₂³×C₄	C₂×D₄²	1	2	yes	—
472	C₂³×C₄	C₂×D₄²	1	2	yes	—
473	C₂³×C₄	#3532	1	4	yes	—
474	#127	C₂×B(4,𝔽₂)	1	4	yes	—
475	#124	C₂×B(4,𝔽₂)	1	4	yes	—
476	C₂×C₄²	#2403	1	1	yes	characteristic
477	C₂×C₄²	#2403	1	1	yes	characteristic
478	C₂×C₄²	#2403	1	2	yes	—
479	C₂×C₄²	#2403	1	2	yes	—
480	C₂×C₄²	#2403	1	1	yes	characteristic
481	C₂×C₄²	#2403	1	1	yes	characteristic
482	#100	C₂×B(4,𝔽₂)	1	4	yes	—
483	#100	C₂×B(4,𝔽₂)	1	4	yes	—
484	C₂⁴	#35346	1	1	yes	characteristic
485	C₂⁴	D₄×#120	1	2	yes	—
486	C₂⁴	D₄×#120	1	2	yes	—
487	C₂⁴	#58990	1	4	yes	—
488	C₂×Q₈	#23900	1	2	yes	—
489	C₂×D₄	#23900	1	2	yes	—
490	C₂×D₄	C₂²×B(4,𝔽₂)	1	2	yes	—
491	C₂×D₄	C₂²×B(4,𝔽₂)	1	2	yes	—
492	C₂×D₄	#60399	1	2	yes	—
493	C₂×D₄	#60399	1	2	yes	—
494	C₂²×C₄	C₂×#2403	1	1	yes	characteristic
495	C₂²×C₄	C₂×#2403	1	1	yes	characteristic
496	C₂²×C₄	#37644	1	2	yes	—
497	C₂²×C₄	#37644	1	2	yes	—
498	C₄²	#23900	1	2	yes	—
499	C₄²	#23900	1	2	yes	—
500	C₂³	#647656	1	1	yes	characteristic
501	C₂³	#6499874	1	2	yes	—
502	C₂³	D₄×B(4,𝔽₂)	1	4	yes	—
503	C₂³	D₄×B(4,𝔽₂)	1	4	yes	—
504	C₂×C₄	C₂×#23900	1	1	yes	characteristic
505	C₂×C₄	C₂×#23900	1	1	yes	characteristic
506	C₂×C₄	#6437632	1	1	yes	characteristic
507	C₂×C₄	#6437632	1	1	yes	characteristic
508	C₂²	#72216646	1	1	yes	characteristic
509	C₂²	#72216660	1	1	yes	characteristic
510	C₂²	#72216660	1	1	yes	characteristic
511	C₂	#72216662	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216662
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁵
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72341503	C₂	semidirect
2. #72341497	C₂	semidirect
3. #72341472	C₂	semidirect
4. #72341469	C₂	semidirect
5. #72341467	C₂	semidirect
6. #72341466	C₂²	semidirect
7. #72341465	C₂²	semidirect
8. #72341456	C₂	semidirect
9. #72341456	C₂	semidirect
10. #72341453	C₂²	semidirect
11. #72341399	C₂	semidirect
12. #72341387	C₂	semidirect
13. #72341386	C₂²	semidirect
14. #72341371	C₂	semidirect
15. #72341366	C₂²	semidirect
16. #72341366	C₂²	semidirect
17. #72341363	C₂²	semidirect
18. #72341362	C₂²	semidirect
19. #72341307	C₂	semidirect
20. #72341301	C₂²	semidirect
22. #72341294	C₂²	semidirect
23. #72340210	C₂	semidirect
24. #72337574	C₂	semidirect
25. #72337559	C₂²	semidirect
26. #72337559	C₂²	semidirect
28. #72312142	C₂	semidirect
29. #72312133	C₂²	semidirect
30. #72311618	C₂²	semidirect
31. #72311618	C₂²	semidirect
32. #72311606	C₂²	semidirect
33. #72311606	C₂²	semidirect
34. #72311370	C₂²	semidirect
36. #72311270	C₂	semidirect
37. #72311266	C₂²	semidirect
38. #72311264	C₂²	semidirect
39. #72311147	C₂²	semidirect
40. #72311145	C₂²	semidirect
41. #72311145	C₂²	semidirect
42. #72311140	C₂²	semidirect
43. #72311055	C₂²	semidirect
44. #72311055	C₂²	semidirect
45. #72310930	C₂²	semidirect
46. #72310916	C₂²	semidirect
48. #72306417	C₂²	semidirect
49. #72303086	C₂²	semidirect
50. #72303081	C₂²	semidirect
51. #72302950	C₂²	semidirect
52. #72284724	C₂²	semidirect
53. #72260657	C₂²	semidirect
54. #72241340	C₂	semidirect
55. #72241338	C₂	semidirect
56. #72241337	C₂²	semidirect
57. #72241337	C₂²	semidirect
58. #72241334	C₂	semidirect
59. #72241333	C₂²	semidirect
60. #72241332	C₂²	semidirect
61. #72241332	C₂²	semidirect
63. #72241315	C₂²	semidirect
67. #72238017	C₂	semidirect
68. #72238016	C₂²	semidirect
69. #72238015	C₂²	semidirect
70. #72238015	C₂²	semidirect
71. #72238014	C₂²	semidirect
72. #72238013	C₂²	semidirect
73. #72238013	C₂²	semidirect
75. #72238010	C₂²	semidirect
76. #72236925	C₂²	semidirect
77. #72236711	C₂²	semidirect
78. #72236702	C₂²	semidirect
79. #72236657	C₂²	semidirect
81. #72234146	C₂²	semidirect
82. #72233917	C₂²	semidirect
83. #72227731	C₂²	semidirect
84. #72213523	C₂²	semidirect
85. #72213513	C₂²	semidirect
86. #72213504	C₂²	semidirect
87. #72213504	C₂²	semidirect
88. #72213504	C₂²	semidirect
89. #72213504	C₂²	semidirect
90. #72210025	C₂²	semidirect
98. C₂×#60391	D₄	semidirect
99. C₂×#60389	D₄	semidirect
110. C₂²×#2199	D₄	semidirect
111. C₂²×#2196	D₄	semidirect
112. #6628972	D₄	semidirect
113. #6628971	D₄	semidirect
118. #6628874	D₄	semidirect
119. #6628873	D₄	semidirect
124. #6443332	D₄	semidirect
125. #6443330	D₄	semidirect
126. #6442555	D₄	semidirect
127. #6442555	D₄	semidirect
128. #6437971	D₄	semidirect
129. #6437970	D₄	semidirect
133. #6437455	D₄	semidirect
134. #6437453	D₄	semidirect
141. C₄×#3018	D₄	semidirect
142. C₄×#3018	D₄	semidirect
143. #498927	D₄	semidirect
144. #498927	D₄	semidirect
153. #494597	D₄	semidirect
154. #494596	D₄	semidirect
161. #449493	D₄	semidirect
162. #417810	D₄	semidirect
163. #417568	D₄	semidirect
164. #417566	D₄	semidirect
188. C₂×2₋¹⁺⁶	C₂×D₄	semidirect
193. #62859	C₂×D₄	semidirect
198. #62812	C₂×D₄	semidirect
206. C₄×#585	C₂×D₄	semidirect
208. C₂²×B(4,𝔽₂)	C₂×D₄	semidirect
215. C₂×#3022	C₂×D₄	semidirect
216. C₂×#3022	C₂×D₄	semidirect
217. C₂×#3022	C₂×D₄	semidirect
224. C₂×#3019	C₂×D₄	semidirect
232. C₂²×#455	C₂×D₄	semidirect
235. C₂×#2889	C₂×D₄	semidirect
247. #35616	C₂×D₄	semidirect
263. C₂²×AΣL(1,𝔽₁₆)	C₂×D₄	semidirect
267. #20955	C₂×D₄	semidirect
271. #20855	C₂×D₄	semidirect
273. #20637	C₂×D₄	semidirect
277. #20615	C₂×D₄	semidirect
278. #20615	C₂×D₄	semidirect
279. #20615	C₂×D₄	semidirect
281. #20609	C₂×D₄	semidirect
282. #20593	C₂×D₄	semidirect
286. #20586	C₂×D₄	semidirect
292. C₄×#413	C₂×D₄	semidirect
294. #12574	C₂×D₄	semidirect
295. #11167	C₂×D₄	semidirect
297. #10989	C₂×D₄	semidirect
298. #10982	C₂×D₄	semidirect
299. #10977	C₂×D₄	semidirect
307. C₂²×2₋¹⁺⁴	#120	semidirect
314. C₂×#536	#120	semidirect
319. C₂×#535	#120	semidirect
330. C₂²×C₄⋊Q₈	#120	semidirect
333. C₂²×#124	#120	semidirect
334. C₂²×#124	#120	semidirect
343. C₂×C₄×#41	#120	semidirect
344. C₂×C₄×#41	#120	semidirect
373. #2310	#120	semidirect
374. #2310	#120	semidirect
375. #2309	#120	semidirect
381. #2297	#120	semidirect
382. #2297	#120	semidirect
385. #2285	#120	semidirect
386. #2285	#120	semidirect
387. C₂×C₄³	#120	semidirect

Maximal subgroups

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

subgroup	index
#72341503	2
#72341497	2
#72341472	2
#72341469	2
#72341467	2
#72341456	2
#72341399	2
#72341387	2
#72341371	2
#72341307	2
#72340210	2
#72337574	2
#72312142	2
#72311270	2
#72241340	2
#72241338	2
#72241334	2
#72238017	2