Subgroups — #72311184

Order 2048.

At least 9535 conjugacy classes of subgroups; only the 758 normal subgroups are shown.

The subgroup lattice has 587 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72311180	C₂	1	1	yes	maximal, characteristic
2	#72311172	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#128079	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#359529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#146577	C₂	1	1	yes	maximal, characteristic, complemented
6	#72306279	C₂	1	1	yes	maximal, characteristic, complemented
7	#72303066	C₂	1	4	yes	maximal, complemented
8	C₂×#493397	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238015	C₂	1	4	yes	maximal, complemented
10	C₂×#62852	C₂²	1	1	yes	characteristic
11	C₂²×#2888	C₂²	1	1	yes	characteristic
12	#7630442	C₂²	1	1	yes	characteristic
13	#6470235	C₄	1	2	yes	complemented
14	#6470235	C₂²	1	2	yes	—
15	#6470234	C₄	1	2	yes	complemented
16	#6470234	C₂²	1	2	yes	—
17	#6440460	C₄	1	2	yes	complemented
18	#6437621	C₄	1	2	yes	complemented
19	C₂×#20592	C₂²	1	1	yes	characteristic
20	C₂×#20585	C₂²	1	1	yes	characteristic
21	C₂×#20585	C₂²	1	1	yes	characteristic
22	C₂×#20585	C₂²	1	1	yes	characteristic
23	#496241	C₂²	1	2	yes	complemented
24	#496240	C₂²	1	2	yes	complemented
25	#494682	C₂²	1	2	yes	complemented
26	#494681	C₂²	1	2	yes	complemented
27	#493397	C₂²	1	4	yes	—
28	#359529	C₂²	1	4	yes	—
29	#146577	C₂²	1	4	yes	complemented
30	#128079	C₂²	1	4	yes	complemented
31	C₂²×#585	D₄	1	1	yes	characteristic
32	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
33	#62852	C₂×C₄	1	2	yes	—
34	#62852	D₄	1	2	yes	—
35	#62852	D₄	1	2	yes	—
36	#62852	C₂³	1	2	yes	—
37	C₂×#3484	C₂×C₄	1	2	yes	—
38	C₂×#3484	D₄	1	2	yes	—
39	C₂²×#534	C₂×C₄	1	1	yes	characteristic
40	C₂²×#534	D₄	1	1	yes	characteristic
41	C₂²×#534	D₄	1	1	yes	characteristic
42	C₂×#3428	D₄	1	1	yes	characteristic
43	C₂×#2888	D₄	1	2	yes	complemented
44	C₂×#2888	C₂³	1	2	yes	—
45	C₂²×#413	D₄	1	1	yes	characteristic, complemented
46	C₂²×#413	D₄	1	1	yes	characteristic, complemented
47	C₂³×#100	D₄	1	2	yes	complemented
48	C₂³×#99	D₄	1	2	yes	—
49	C₂³×#99	D₄	1	2	yes	complemented
50	#35446	C₂×C₄	1	1	yes	characteristic
51	#35446	D₄	1	1	yes	characteristic
52	#35446	D₄	1	1	yes	characteristic
53	#35446	C₂³	1	1	yes	characteristic
54	#35346	C₂×C₄	1	1	yes	characteristic
55	#35346	D₄	1	1	yes	characteristic
56	#35346	D₄	1	1	yes	characteristic
57	#35346	C₂³	1	1	yes	characteristic
58	C₂×#2308	C₂×C₄	1	1	yes	characteristic
59	C₂×#2308	D₄	1	1	yes	characteristic
60	C₂×#2308	D₄	1	1	yes	characteristic
61	#24058	C₂×C₄	1	4	yes	complemented
62	#24058	C₂×C₄	1	4	yes	complemented
63	#23901	C₂×C₄	1	4	yes	complemented
64	#23900	C₂×C₄	1	4	yes	complemented
65	#20592	D₄	1	2	yes	complemented
66	#20592	C₂³	1	2	yes	—
67	#20585	C₂×C₄	1	2	yes	—
68	#20585	D₄	1	2	yes	—
69	#20585	D₄	1	2	yes	—
70	#20585	D₄	1	2	yes	complemented
71	#20585	D₄	1	2	yes	complemented
72	#20585	C₂³	1	2	yes	—
73	#20585	C₂³	1	2	yes	—
74	#20585	C₂³	1	2	yes	—
75	C₂×#1782	D₄	1	2	yes	—
76	C₂×#1782	D₄	1	2	yes	complemented
77	C₂×#1781	C₂×C₄	1	2	yes	—
78	C₂×#585	C₂²⋊C₄	1	2	yes	—
79	C₂×#585	C₂×D₄	1	2	yes	—
80	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
81	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
93	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
97	C₂³×#41	C₂×D₄	1	1	yes	characteristic
98	C₂³×#41	C₂×D₄	1	1	yes	characteristic
99	C₂³×#41	C₂×D₄	1	1	yes	characteristic
100	C₂³×#41	C₂×D₄	1	1	yes	characteristic
101	C₂³×#41	#41	1	1	yes	characteristic
102	C₂³×#41	#41	1	1	yes	characteristic
103	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
110	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
111	C₂⁴×D₄	#41	1	1	yes	characteristic
112	C₂⁴×D₄	#41	1	1	yes	characteristic
113	#3484	C₂²⋊C₄	1	4	yes	—
114	#3484	C₂²⋊C₄	1	4	yes	—
115	#3484	C₂²×C₄	1	4	yes	—
116	#3484	C₂×D₄	1	4	yes	—
117	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
121	C₂²×#123	C₂×D₄	1	1	yes	characteristic
122	C₂²×#123	C₂×D₄	1	1	yes	characteristic
123	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
127	C₂²×#120	C₂×D₄	1	1	yes	characteristic
128	C₂²×#120	C₂×D₄	1	1	yes	characteristic
129	#3428	C₂²⋊C₄	1	2	yes	—
130	#3428	C₂×D₄	1	2	yes	—
131	C₂×#518	C₂²⋊C₄	1	2	yes	—
132	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
133	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×#413	C₂×D₄	1	2	yes	—
136	C₂×#413	C₂×D₄	1	2	yes	—
137	C₂²×#100	C₂×D₄	1	4	yes	—
138	C₂²×#99	C₂²⋊C₄	1	4	yes	—
139	C₂²×#99	C₂²⋊C₄	1	4	yes	—
140	C₂²×#99	C₂×D₄	1	4	yes	—
141	C₂²×#99	C₂×D₄	1	4	yes	—
142	#2308	C₂²⋊C₄	1	1	yes	characteristic
143	#2308	C₂²⋊C₄	1	1	yes	characteristic
144	#2308	C₂²⋊C₄	1	1	yes	characteristic
145	#2308	C₂²⋊C₄	1	1	yes	characteristic
146	#2308	C₂²⋊C₄	1	1	yes	characteristic
147	#2308	C₂²⋊C₄	1	1	yes	characteristic
148	#2308	C₂²×C₄	1	1	yes	characteristic
149	#2308	C₂²×C₄	1	1	yes	characteristic
150	#2308	C₂×D₄	1	1	yes	characteristic
151	#2308	C₂×D₄	1	1	yes	characteristic
152	#2308	C₂×D₄	1	1	yes	characteristic
153	#2308	C₂×D₄	1	1	yes	characteristic
154	#1782	C₂²⋊C₄	1	4	yes	—
155	#1782	C₂×D₄	1	4	yes	—
156	#1782	C₂×D₄	1	4	yes	—
157	#1781	C₂²⋊C₄	1	4	yes	—
158	#1781	C₂²×C₄	1	4	yes	—
159	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁶	C₄×D₄	1	1	yes	characteristic
162	C₂⁶	#120	1	1	yes	characteristic
163	C₂⁶	#121	1	1	yes	characteristic
164	C₂⁶	#121	1	1	yes	characteristic
165	C₂⁶	#121	1	1	yes	characteristic
166	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	4	yes	—
167	C₂²×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×#41	C₄×D₄	1	1	yes	characteristic
170	C₂²×#41	C₄×D₄	1	1	yes	characteristic
171	C₂²×#41	C₄×D₄	1	1	yes	characteristic
172	C₂²×#41	C₄×D₄	1	1	yes	characteristic
173	C₂²×#41	#120	1	1	yes	characteristic
174	C₂²×#41	#120	1	1	yes	characteristic
175	C₂²×#41	#120	1	1	yes	characteristic
176	C₂²×#41	#120	1	1	yes	characteristic
177	C₂²×#41	#120	1	1	yes	characteristic
178	C₂²×#41	#120	1	1	yes	characteristic
179	C₂²×#41	#120	1	1	yes	characteristic
180	C₂²×#41	#120	1	1	yes	characteristic
181	C₂²×#41	#120	1	1	yes	characteristic
182	C₂²×#41	#121	1	1	yes	characteristic
183	C₂²×#41	#121	1	1	yes	characteristic
184	C₂²×#41	#121	1	1	yes	characteristic
185	C₂²×#41	#121	1	1	yes	characteristic
186	C₂²×#41	#121	1	1	yes	characteristic
187	C₂²×#41	#121	1	1	yes	characteristic
188	C₂²×#41	#121	1	1	yes	characteristic
189	C₂²×#41	#121	1	1	yes	characteristic
190	C₂²×#41	#121	1	1	yes	characteristic
191	C₂²×#41	#121	1	1	yes	characteristic
192	C₂²×#41	#121	1	1	yes	characteristic
193	C₂²×#41	#121	1	1	yes	characteristic
194	C₂²×#41	#121	1	1	yes	characteristic
195	C₂²×#41	#121	1	1	yes	characteristic
196	C₂²×#41	#121	1	1	yes	characteristic
197	C₂²×#41	#121	1	1	yes	characteristic
198	C₂²×#41	#123	1	1	yes	characteristic
199	C₂²×#41	#123	1	1	yes	characteristic
200	C₂²×#41	#123	1	1	yes	characteristic
201	C₂²×#41	#123	1	1	yes	characteristic
202	C₂²×#41	#123	1	1	yes	characteristic
203	C₂²×#41	#124	1	1	yes	characteristic
204	C₂²×#41	#124	1	1	yes	characteristic
205	C₂²×#41	#124	1	1	yes	characteristic
206	C₂²×#41	#127	1	1	yes	characteristic
207	C₂²×#41	#127	1	1	yes	characteristic
208	C₂²×#41	#127	1	1	yes	characteristic
209	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
210	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
211	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
212	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
213	C₂²×#41	C₂×#41	1	1	yes	characteristic
214	C₂²×#41	C₂×#41	1	1	yes	characteristic
215	C₂³×Q₈	#121	1	1	yes	characteristic
216	C₂³×Q₈	#121	1	1	yes	characteristic
217	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
227	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
228	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
229	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
230	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
231	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
232	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
233	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
234	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
235	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
236	C₂³×D₄	#120	1	1	yes	characteristic
237	C₂³×D₄	#120	1	1	yes	characteristic
238	C₂³×D₄	#120	1	1	yes	characteristic
239	C₂³×D₄	#120	1	1	yes	characteristic
240	C₂³×D₄	#120	1	1	yes	characteristic
241	C₂³×D₄	#120	1	1	yes	characteristic
242	C₂³×D₄	#120	1	1	yes	characteristic
243	C₂³×D₄	#120	1	1	yes	characteristic
244	C₂³×D₄	#120	1	1	yes	characteristic
245	C₂³×D₄	#120	1	1	yes	characteristic
246	C₂³×D₄	#120	1	1	yes	characteristic
247	C₂³×D₄	#120	1	1	yes	characteristic
248	C₂³×D₄	#120	1	1	yes	characteristic
249	C₂³×D₄	#120	1	1	yes	characteristic
250	C₂³×D₄	#120	1	1	yes	characteristic
251	C₂³×D₄	#120	1	1	yes	characteristic
252	C₂³×D₄	#120	1	1	yes	characteristic
253	C₂³×D₄	#120	1	1	yes	characteristic
254	C₂³×D₄	#120	1	1	yes	characteristic
255	C₂³×D₄	#120	1	1	yes	characteristic
256	C₂³×D₄	#121	1	1	yes	characteristic
257	C₂³×D₄	#121	1	1	yes	characteristic
258	C₂³×D₄	#121	1	1	yes	characteristic
259	C₂³×D₄	#121	1	1	yes	characteristic
260	C₂³×D₄	#121	1	1	yes	characteristic
261	C₂³×D₄	#121	1	1	yes	characteristic
262	C₂³×D₄	#121	1	1	yes	characteristic
263	C₂³×D₄	#121	1	1	yes	characteristic
264	C₂³×D₄	#121	1	1	yes	characteristic
265	C₂³×D₄	#121	1	1	yes	characteristic
266	C₂³×D₄	#121	1	1	yes	characteristic
267	C₂³×D₄	#121	1	1	yes	characteristic
268	C₂³×D₄	#121	1	1	yes	characteristic
269	C₂³×D₄	#121	1	1	yes	characteristic
270	C₂³×D₄	#121	1	1	yes	characteristic
271	C₂³×D₄	#121	1	1	yes	characteristic
272	C₂³×D₄	#121	1	1	yes	characteristic
273	C₂³×D₄	#121	1	1	yes	characteristic
274	C₂³×D₄	#121	1	1	yes	characteristic
275	C₂³×D₄	#121	1	1	yes	characteristic
276	C₂³×D₄	#121	1	1	yes	characteristic
277	C₂³×D₄	#121	1	1	yes	characteristic
278	C₂³×D₄	#121	1	1	yes	characteristic
279	C₂³×D₄	#121	1	1	yes	characteristic
280	C₂³×D₄	#123	1	1	yes	characteristic
281	C₂³×D₄	#123	1	1	yes	characteristic
282	C₂³×D₄	#123	1	1	yes	characteristic
283	C₂³×D₄	#123	1	1	yes	characteristic
284	C₂³×D₄	#123	1	1	yes	characteristic
285	C₂³×D₄	#123	1	1	yes	characteristic
286	C₂³×D₄	#123	1	1	yes	characteristic
287	C₂³×D₄	#123	1	1	yes	characteristic
288	C₂³×D₄	#123	1	1	yes	characteristic
289	C₂³×D₄	#123	1	1	yes	characteristic
290	C₂³×D₄	#124	1	1	yes	characteristic
291	C₂³×D₄	#124	1	1	yes	characteristic
292	C₂³×D₄	#124	1	1	yes	characteristic
293	C₂³×D₄	#124	1	1	yes	characteristic
294	C₂³×D₄	#127	1	1	yes	characteristic
295	C₂³×D₄	#127	1	1	yes	characteristic
296	C₂³×D₄	#127	1	1	yes	characteristic
297	C₂³×D₄	#127	1	1	yes	characteristic
298	C₂³×D₄	C₂³×C₄	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	1	yes	characteristic
302	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
303	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
304	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
305	C₂³×D₄	C₂×#41	1	1	yes	characteristic
306	C₂³×D₄	C₂×#41	1	1	yes	characteristic
307	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
309	C₂⁴×C₄	#120	1	1	yes	characteristic
310	C₂⁴×C₄	#120	1	1	yes	characteristic
311	C₂⁴×C₄	#121	1	1	yes	characteristic
312	C₂⁴×C₄	#121	1	1	yes	characteristic
313	C₂⁴×C₄	#121	1	1	yes	characteristic
314	C₂⁴×C₄	#123	1	1	yes	characteristic
315	C₂⁴×C₄	#124	1	1	yes	characteristic
316	C₂⁴×C₄	#127	1	1	yes	characteristic
317	#518	C₂×C₂²⋊C₄	1	4	yes	—
318	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
319	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
320	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
321	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
322	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
323	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
324	C₂×#99	C₂×C₂²⋊C₄	1	8	yes	—
325	C₂⁵	#379	1	1	yes	characteristic
326	C₂⁵	#379	1	1	yes	characteristic
327	C₂⁵	#379	1	1	yes	characteristic
328	C₂⁵	#386	1	1	yes	characteristic
329	C₂⁵	#386	1	1	yes	characteristic
330	C₂⁵	#386	1	1	yes	characteristic
331	C₂⁵	#386	1	1	yes	characteristic
332	C₂⁵	#386	1	1	yes	characteristic
333	C₂⁵	#386	1	1	yes	characteristic
334	C₂⁵	#386	1	1	yes	characteristic
335	C₂⁵	#386	1	1	yes	characteristic
336	C₂⁵	#386	1	1	yes	characteristic
337	C₂⁵	#390	1	1	yes	characteristic
338	C₂⁵	#390	1	1	yes	characteristic
339	C₂⁵	#390	1	1	yes	characteristic
340	C₂⁵	#392	1	1	yes	characteristic
341	C₂⁵	#392	1	1	yes	characteristic
342	C₂⁵	#392	1	1	yes	characteristic
343	C₂⁵	#392	1	1	yes	characteristic
344	C₂⁵	#392	1	1	yes	characteristic
345	C₂⁵	#392	1	1	yes	characteristic
346	C₂⁵	#394	1	1	yes	characteristic
347	C₂⁵	#394	1	1	yes	characteristic
348	C₂⁵	#394	1	1	yes	characteristic
349	C₂⁵	#394	1	1	yes	characteristic
350	C₂⁵	#394	1	1	yes	characteristic
351	C₂⁵	#394	1	1	yes	characteristic
352	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
353	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
354	C₂⁵	C₂×#120	1	1	yes	characteristic
355	C₂⁵	C₂×#121	1	1	yes	characteristic
356	C₂⁵	C₂×#121	1	1	yes	characteristic
357	C₂⁵	C₂×#121	1	1	yes	characteristic
358	C₂²×Q₈	#386	1	1	yes	characteristic
359	C₂²×Q₈	#390	1	1	yes	characteristic
360	C₂²×Q₈	#392	1	1	yes	characteristic
361	C₂²×Q₈	#392	1	1	yes	characteristic
362	C₂²×Q₈	#394	1	1	yes	characteristic
363	C₂²×Q₈	#394	1	1	yes	characteristic
364	C₂²×Q₈	C₂×#121	1	1	yes	characteristic
365	C₂²×Q₈	C₂×#121	1	1	yes	characteristic
366	C₂²×D₄	#379	1	1	yes	characteristic
367	C₂²×D₄	#379	1	1	yes	characteristic
368	C₂²×D₄	#379	1	1	yes	characteristic
369	C₂²×D₄	#379	1	1	yes	characteristic
370	C₂²×D₄	#386	1	1	yes	characteristic
371	C₂²×D₄	#386	1	1	yes	characteristic
372	C₂²×D₄	#386	1	1	yes	characteristic
373	C₂²×D₄	#386	1	1	yes	characteristic
374	C₂²×D₄	#386	1	1	yes	characteristic
375	C₂²×D₄	#386	1	1	yes	characteristic
376	C₂²×D₄	#386	1	1	yes	characteristic
377	C₂²×D₄	#386	1	1	yes	characteristic
378	C₂²×D₄	#386	1	1	yes	characteristic
379	C₂²×D₄	#386	1	1	yes	characteristic
380	C₂²×D₄	#386	1	1	yes	characteristic
381	C₂²×D₄	#386	1	1	yes	characteristic
382	C₂²×D₄	#386	1	1	yes	characteristic
383	C₂²×D₄	#386	1	1	yes	characteristic
384	C₂²×D₄	#386	1	1	yes	characteristic
385	C₂²×D₄	#390	1	1	yes	characteristic
386	C₂²×D₄	#390	1	1	yes	characteristic
387	C₂²×D₄	#390	1	1	yes	characteristic
388	C₂²×D₄	#390	1	1	yes	characteristic
389	C₂²×D₄	#390	1	1	yes	characteristic
390	C₂²×D₄	#390	1	1	yes	characteristic
391	C₂²×D₄	#390	1	1	yes	characteristic
392	C₂²×D₄	#392	1	1	yes	characteristic
393	C₂²×D₄	#392	1	1	yes	characteristic
394	C₂²×D₄	#392	1	1	yes	characteristic
395	C₂²×D₄	#392	1	1	yes	characteristic
396	C₂²×D₄	#392	1	1	yes	characteristic
397	C₂²×D₄	#392	1	1	yes	characteristic
398	C₂²×D₄	#392	1	1	yes	characteristic
399	C₂²×D₄	#392	1	1	yes	characteristic
400	C₂²×D₄	#392	1	1	yes	characteristic
401	C₂²×D₄	#392	1	1	yes	characteristic
402	C₂²×D₄	#392	1	1	yes	characteristic
403	C₂²×D₄	#392	1	1	yes	characteristic
404	C₂²×D₄	#392	1	1	yes	characteristic
405	C₂²×D₄	#392	1	1	yes	characteristic
406	C₂²×D₄	#394	1	1	yes	characteristic
407	C₂²×D₄	#394	1	1	yes	characteristic
408	C₂²×D₄	#394	1	1	yes	characteristic
409	C₂²×D₄	#394	1	1	yes	characteristic
410	C₂²×D₄	#394	1	1	yes	characteristic
411	C₂²×D₄	#394	1	1	yes	characteristic
412	C₂²×D₄	#394	1	1	yes	characteristic
413	C₂²×D₄	#394	1	1	yes	characteristic
414	C₂²×D₄	#394	1	1	yes	characteristic
415	C₂²×D₄	#394	1	1	yes	characteristic
416	C₂²×D₄	#394	1	1	yes	characteristic
417	C₂²×D₄	#394	1	1	yes	characteristic
418	C₂²×D₄	#394	1	1	yes	characteristic
419	C₂²×D₄	#394	1	1	yes	characteristic
420	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
421	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
422	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
423	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
424	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
425	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
426	C₂²×D₄	#458	1	2	yes	—
427	C₂²×D₄	#458	1	2	yes	—
428	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
429	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
430	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
431	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
432	C₂²×D₄	C₂×#120	1	1	yes	characteristic
433	C₂²×D₄	C₂×#120	1	1	yes	characteristic
434	C₂²×D₄	C₂×#120	1	1	yes	characteristic
435	C₂²×D₄	C₂×#120	1	1	yes	characteristic
436	C₂²×D₄	C₂×#120	1	1	yes	characteristic
437	C₂²×D₄	C₂×#121	1	1	yes	characteristic
438	C₂²×D₄	C₂×#121	1	1	yes	characteristic
439	C₂²×D₄	C₂×#121	1	1	yes	characteristic
440	C₂²×D₄	C₂×#121	1	1	yes	characteristic
441	C₂²×D₄	C₂×#123	1	1	yes	characteristic
442	C₂²×D₄	C₂×#123	1	1	yes	characteristic
443	C₂²×D₄	C₂×#123	1	1	yes	characteristic
444	C₂²×D₄	C₂×#124	1	1	yes	characteristic
445	C₂²×D₄	C₂×#127	1	1	yes	characteristic
446	C₂³×C₄	#379	1	1	yes	characteristic
447	C₂³×C₄	#386	1	1	yes	characteristic
448	C₂³×C₄	#386	1	1	yes	characteristic
449	C₂³×C₄	#386	1	1	yes	characteristic
450	C₂³×C₄	#386	1	1	yes	characteristic
451	C₂³×C₄	#386	1	1	yes	characteristic
452	C₂³×C₄	#386	1	1	yes	characteristic
453	C₂³×C₄	#386	1	1	yes	characteristic
454	C₂³×C₄	#390	1	1	yes	characteristic
455	C₂³×C₄	#390	1	1	yes	characteristic
456	C₂³×C₄	#390	1	1	yes	characteristic
457	C₂³×C₄	#390	1	1	yes	characteristic
458	C₂³×C₄	#390	1	1	yes	characteristic
459	C₂³×C₄	#392	1	1	yes	characteristic
460	C₂³×C₄	#392	1	1	yes	characteristic
461	C₂³×C₄	#392	1	1	yes	characteristic
462	C₂³×C₄	#392	1	1	yes	characteristic
463	C₂³×C₄	#392	1	1	yes	characteristic
464	C₂³×C₄	#392	1	1	yes	characteristic
465	C₂³×C₄	#392	1	1	yes	characteristic
466	C₂³×C₄	#392	1	1	yes	characteristic
467	C₂³×C₄	#392	1	1	yes	characteristic
468	C₂³×C₄	#392	1	1	yes	characteristic
469	C₂³×C₄	#394	1	1	yes	characteristic
470	C₂³×C₄	#394	1	1	yes	characteristic
471	C₂³×C₄	#394	1	1	yes	characteristic
472	C₂³×C₄	#394	1	1	yes	characteristic
473	C₂³×C₄	#394	1	1	yes	characteristic
474	C₂³×C₄	#394	1	1	yes	characteristic
475	C₂³×C₄	#394	1	1	yes	characteristic
476	C₂³×C₄	#394	1	1	yes	characteristic
477	C₂³×C₄	#394	1	1	yes	characteristic
478	C₂³×C₄	#394	1	1	yes	characteristic
479	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
480	C₂³×C₄	C₂×#120	1	1	yes	characteristic
481	C₂³×C₄	C₂×#120	1	1	yes	characteristic
482	C₂³×C₄	C₂×#121	1	1	yes	characteristic
483	C₂³×C₄	C₂×#121	1	1	yes	characteristic
484	C₂³×C₄	C₂×#121	1	1	yes	characteristic
485	C₂³×C₄	C₂×#123	1	1	yes	characteristic
486	C₂³×C₄	C₂×#124	1	1	yes	characteristic
487	C₂³×C₄	C₂×#127	1	1	yes	characteristic
488	C₂⁴	#1438	1	1	yes	characteristic
489	C₂⁴	#1438	1	1	yes	characteristic
490	C₂⁴	#1438	1	1	yes	characteristic
491	C₂⁴	#1438	1	1	yes	characteristic
492	C₂⁴	#1438	1	1	yes	characteristic
493	C₂⁴	#1438	1	1	yes	characteristic
494	C₂⁴	#1438	1	1	yes	characteristic
495	C₂⁴	#1438	1	1	yes	characteristic
496	C₂⁴	#1438	1	1	yes	characteristic
497	C₂⁴	#1438	1	1	yes	characteristic
498	C₂⁴	#1790	1	2	yes	—
499	C₂⁴	#1790	1	2	yes	—
500	C₂⁴	#1889	1	2	yes	—
501	C₂⁴	#1889	1	2	yes	—
502	C₂⁴	#1913	1	2	yes	—
503	C₂⁴	#1913	1	2	yes	—
504	C₂⁴	#2021	1	2	yes	—
505	C₂⁴	#2021	1	2	yes	—
506	C₂⁴	#2044	1	2	yes	—
507	C₂⁴	#2044	1	2	yes	—
508	C₂⁴	C₂×#379	1	1	yes	characteristic
509	C₂⁴	C₂×#386	1	1	yes	characteristic
510	C₂⁴	C₂×#386	1	1	yes	characteristic
511	C₂⁴	C₂×#386	1	1	yes	characteristic
512	C₂⁴	C₂×#390	1	1	yes	characteristic
513	C₂⁴	C₂×#392	1	1	yes	characteristic
514	C₂⁴	C₂×#392	1	1	yes	characteristic
515	C₂⁴	C₂×#394	1	1	yes	characteristic
516	C₂⁴	C₂×#394	1	1	yes	characteristic
517	C₂×D₄	#1790	1	2	yes	—
518	C₂×D₄	#1790	1	2	yes	—
519	C₂×D₄	#1889	1	2	yes	—
520	C₂×D₄	#1889	1	2	yes	—
521	C₂×D₄	#2021	1	2	yes	—
522	C₂×D₄	#2021	1	2	yes	—
523	C₂×D₄	#2021	1	2	yes	—
524	C₂×D₄	#2021	1	2	yes	—
525	C₂×D₄	#2021	1	2	yes	—
526	C₂×D₄	#2022	1	2	yes	—
527	C₂×D₄	#2044	1	2	yes	—
528	C₂×D₄	#2044	1	2	yes	—
529	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
530	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
531	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
532	C₂×D₄	C₂×#458	1	2	yes	—
533	C₂²×C₄	#1438	1	1	yes	characteristic
534	C₂²×C₄	#1438	1	1	yes	characteristic
535	C₂²×C₄	#1438	1	1	yes	characteristic
536	C₂²×C₄	#1438	1	1	yes	characteristic
537	C₂²×C₄	#1438	1	1	yes	characteristic
538	C₂²×C₄	#1438	1	1	yes	characteristic
539	C₂²×C₄	#2021	1	1	yes	characteristic
540	C₂²×C₄	#2021	1	1	yes	characteristic
541	C₂²×C₄	#2021	1	1	yes	characteristic
542	C₂²×C₄	#2021	1	1	yes	characteristic
543	C₂²×C₄	#2021	1	1	yes	characteristic
544	C₂²×C₄	#2021	1	1	yes	characteristic
545	C₂²×C₄	#2022	1	1	yes	characteristic
546	C₂²×C₄	#2022	1	1	yes	characteristic
547	C₂²×C₄	C₂×#386	1	1	yes	characteristic
548	C₂²×C₄	C₂×#390	1	1	yes	characteristic
549	C₂²×C₄	C₂×#392	1	1	yes	characteristic
550	C₂²×C₄	C₂×#392	1	1	yes	characteristic
551	C₂²×C₄	C₂×#394	1	1	yes	characteristic
552	C₂²×C₄	C₂×#394	1	1	yes	characteristic
553	C₂³	#10972	1	2	yes	—
554	C₂³	#10972	1	1	yes	characteristic
555	C₂³	#10972	1	1	yes	characteristic
556	C₂³	#10991	1	2	yes	—
557	C₂³	#10991	1	2	yes	—
558	C₂³	#10991	1	1	yes	characteristic
559	C₂³	#10991	1	1	yes	characteristic
560	C₂³	#10995	1	2	yes	—
561	C₂³	#10995	1	2	yes	—
562	C₂³	#10995	1	1	yes	characteristic
563	C₂³	#10995	1	1	yes	characteristic
564	C₂³	C₂×#1438	1	1	yes	characteristic
565	C₂³	C₂×#1790	1	2	yes	—
566	C₂³	C₂×#1889	1	2	yes	—
567	C₂³	C₂×#1913	1	2	yes	—
568	C₂³	C₂×#2021	1	2	yes	—
569	C₂³	C₂×#2044	1	2	yes	—
570	C₂×C₄	#10991	1	1	yes	characteristic
571	C₂×C₄	#10991	1	1	yes	characteristic
572	C₂×C₄	#10995	1	1	yes	characteristic
573	C₂×C₄	#10995	1	1	yes	characteristic
574	C₂×C₄	C₂×#2021	1	1	yes	characteristic
575	C₂×C₄	C₂×#2021	1	1	yes	characteristic
576	C₂×C₄	C₂×#2021	1	1	yes	characteristic
577	C₂×C₄	C₂×#2022	1	1	yes	characteristic
578	C₂²	#115204	1	1	yes	characteristic
579	C₂²	#115204	1	1	yes	characteristic
580	C₂²	#115204	1	1	yes	characteristic
581	C₂²	#115204	1	1	yes	characteristic
582	C₂²	C₂×#10972	1	1	yes	characteristic
583	C₂²	C₂×#10995	1	1	yes	characteristic
584	C₂²	C₂×#10991	1	1	yes	characteristic
585	C₂	C₂×#115204	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#115204
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
prime radical	#72311180	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×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
2. #72311172	C₂	semidirect
3. C₂×#128079	C₂	semidirect
4. C₂×#359529	C₂	semidirect
5. C₂×#146577	C₂	semidirect
6. #72306279	C₂	semidirect
7. #72303066	C₂	semidirect
8. C₂×#493397	C₂	semidirect
9. #72238015	C₂	semidirect
13. #6470235	C₄	semidirect
15. #6470234	C₄	semidirect
17. #6440460	C₄	semidirect
18. #6437621	C₄	semidirect
23. #496241	C₂²	semidirect
24. #496240	C₂²	semidirect
25. #494682	C₂²	semidirect
26. #494681	C₂²	semidirect
29. #146577	C₂²	semidirect
30. #128079	C₂²	semidirect
43. C₂×#2888	D₄	semidirect
45. C₂²×#413	D₄	semidirect
46. C₂²×#413	D₄	semidirect
47. C₂³×#100	D₄	semidirect
49. C₂³×#99	D₄	semidirect
61. #24058	C₂×C₄	semidirect
62. #24058	C₂×C₄	semidirect
63. #23901	C₂×C₄	semidirect
64. #23900	C₂×C₄	semidirect
65. #20592	D₄	semidirect
70. #20585	D₄	semidirect
71. #20585	D₄	semidirect
76. C₂×#1782	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72311180	2
#72311172	2
C₂×#128079	2
C₂×#359529	2
C₂×#146577	2
#72306279	2
#72303066	2
C₂×#493397	2
#72238015	2