Subgroups — #72240605

Order 2048.

At least 7290 conjugacy classes of subgroups; only the 888 normal subgroups are shown.

The subgroup lattice has 480 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493995	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#480376	C₂	1	1	yes	maximal, characteristic, complemented
3	#72240579	C₂	1	1	yes	maximal, characteristic, complemented
4	#72240339	C₂	1	1	yes	maximal, characteristic, complemented
5	#72240324	C₂	1	8	yes	maximal, complemented
6	C₂×#647663	C₂	1	1	yes	maximal, characteristic
7	C₂×#185755	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#185643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄²	C₂²	1	1	yes	characteristic
10	C₂⁴×#121	C₄	1	1	yes	characteristic, complemented
11	C₂²×#2847	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₄×#120	C₂²	1	1	yes	characteristic
13	C₂×#20726	C₂²	1	1	yes	characteristic
14	C₂×#20722	C₂²	1	1	yes	characteristic
15	C₂×#17364	C₂²	1	1	yes	characteristic
16	C₂×#17362	C₂²	1	1	yes	characteristic
17	#647663	C₄	1	4	yes	complemented
18	#647663	C₂²	1	4	yes	—
19	C₂×#14137	C₄	1	1	yes	characteristic, complemented
20	C₂×#13902	C₄	1	1	yes	characteristic, complemented
21	C₂×#13872	C₂²	1	1	yes	characteristic
22	C₂×#12793	C₂²	1	4	yes	complemented
23	C₂×#11471	C₂²	1	4	yes	complemented
24	C₂×#8383	C₂²	1	4	yes	complemented
25	C₂×#8375	C₂²	1	4	yes	complemented
26	#127619	C₂²	1	4	yes	complemented
27	#127615	C₂²	1	4	yes	complemented
28	C₂⁵×D₄	C₂×C₄	1	1	yes	characteristic
29	C₂⁵×D₄	D₄	1	1	yes	characteristic
30	C₂⁶×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄²	C₂×C₄	1	2	yes	—
32	C₂²×D₄²	D₄	1	2	yes	—
33	C₂²×D₄²	D₄	1	2	yes	—
34	C₂²×D₄²	C₂³	1	2	yes	—
35	C₂³×#127	D₄	1	1	yes	characteristic
36	C₂³×#121	C₂×C₄	1	1	yes	characteristic
37	C₂³×#121	C₂×C₄	1	4	yes	—
38	C₂³×#121	D₄	1	1	yes	characteristic
39	C₂³×#120	D₄	1	1	yes	characteristic
40	C₂³×C₄×D₄	D₄	1	1	yes	characteristic
41	C₂³×C₄×D₄	C₂³	1	1	yes	characteristic
42	C₂⁴×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×#2847	C₂×C₄	1	4	yes	—
44	C₂×C₄×#120	C₂×C₄	1	2	yes	—
45	C₂×C₄×#120	D₄	1	2	yes	—
46	C₂×C₄×#120	D₄	1	2	yes	—
47	C₂×C₄×#120	C₂³	1	2	yes	—
48	C₂²×#386	D₄	1	1	yes	characteristic
49	C₂²×#385	D₄	1	1	yes	characteristic
50	C₂²×#380	C₂×C₄	1	1	yes	characteristic
51	C₂²×#379	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
53	#20726	D₄	1	4	yes	complemented
54	#20722	D₄	1	4	yes	complemented
55	C₂×#1800	C₂³	1	2	yes	—
56	C₂×#1799	C₂³	1	2	yes	—
57	#17364	D₄	1	4	yes	complemented
58	#17362	D₄	1	4	yes	complemented
59	C₂×#1460	C₂³	1	2	yes	—
60	C₂×#1459	C₂³	1	2	yes	—
61	#14137	C₂×C₄	1	4	yes	—
62	#13902	C₂×C₄	1	4	yes	—
63	#13872	C₂×C₄	1	2	yes	—
64	#13872	D₄	1	2	yes	—
65	#13872	D₄	1	2	yes	—
66	#13872	C₂³	1	2	yes	—
67	C₂×#1441	D₄	1	2	yes	—
68	C₂×#1439	D₄	1	2	yes	—
69	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
71	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
72	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂²×C₄	1	2	yes	—
76	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
77	C₂⁴×D₄	C₂×D₄	1	2	yes	—
78	C₂⁵×C₄	C₂²⋊C₄	1	2	yes	—
79	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂⁵×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂⁵×C₄	C₂²×C₄	1	2	yes	—
82	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×#127	C₂²⋊C₄	1	2	yes	—
84	C₂²×#127	C₂×D₄	1	2	yes	—
85	C₂²×#121	C₂²⋊C₄	1	2	yes	—
86	C₂²×#121	C₂²⋊C₄	1	2	yes	—
87	C₂²×#121	C₂²×C₄	1	2	yes	—
88	C₂²×#121	C₂×D₄	1	2	yes	—
89	C₂²×#120	C₂²⋊C₄	1	2	yes	—
90	C₂²×#120	C₂×D₄	1	2	yes	—
91	C₂²×C₄×D₄	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×D₄	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
94	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
99	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
100	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
103	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
104	C₂²×C₄×D₄	#41	1	2	yes	—
105	C₂²×C₄×D₄	#41	1	2	yes	—
106	C₂²×C₄×D₄	#41	1	2	yes	—
107	C₂²×C₄×D₄	#41	1	2	yes	—
108	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
109	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
111	C₂³×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
112	C₂³×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
119	C₂×#386	C₂²⋊C₄	1	2	yes	—
120	C₂×#386	C₂×D₄	1	2	yes	—
121	C₂×#385	C₂²⋊C₄	1	2	yes	—
122	C₂×#385	C₂×D₄	1	2	yes	—
123	C₂×#380	C₂²⋊C₄	1	2	yes	—
124	C₂×#380	C₂²×C₄	1	2	yes	—
125	C₂×#379	C₂²⋊C₄	1	2	yes	—
126	C₂×#379	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
129	C₂²×#95	C₂²⋊C₄	1	2	yes	—
130	#1800	C₂×D₄	1	4	yes	complemented
131	#1799	C₂×D₄	1	4	yes	complemented
132	#1460	C₂×D₄	1	4	yes	complemented
133	#1459	C₂×D₄	1	4	yes	complemented
134	#1441	C₂²⋊C₄	1	4	yes	—
135	#1441	C₂×D₄	1	4	yes	—
136	#1439	C₂²⋊C₄	1	4	yes	—
137	#1439	C₂×D₄	1	4	yes	—
138	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂⁶	#120	1	1	yes	characteristic
141	C₂³×D₄	#99	1	2	yes	—
142	C₂³×D₄	#99	1	2	yes	—
143	C₂³×D₄	#99	1	4	yes	—
144	C₂³×D₄	#99	1	2	yes	—
145	C₂³×D₄	#99	1	2	yes	—
146	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
147	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂³×D₄	#120	1	1	yes	characteristic
151	C₂³×D₄	#120	1	1	yes	characteristic
152	C₂³×D₄	#120	1	2	yes	—
153	C₂³×D₄	#121	1	2	yes	—
154	C₂³×D₄	#121	1	2	yes	—
155	C₂³×D₄	#123	1	2	yes	—
156	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
157	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁴×C₄	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂⁴×C₄	C₄×D₄	1	2	yes	—
165	C₂⁴×C₄	C₄×D₄	1	2	yes	—
166	C₂⁴×C₄	C₄×D₄	1	2	yes	—
167	C₂⁴×C₄	C₄×D₄	1	2	yes	—
168	C₂⁴×C₄	#120	1	1	yes	characteristic
169	C₂⁴×C₄	#120	1	1	yes	characteristic
170	C₂⁴×C₄	#120	1	1	yes	characteristic
171	C₂⁴×C₄	#120	1	1	yes	characteristic
172	C₂⁴×C₄	#120	1	2	yes	—
173	C₂⁴×C₄	#120	1	1	yes	characteristic
174	C₂⁴×C₄	#121	1	2	yes	—
175	C₂⁴×C₄	#121	1	2	yes	—
176	C₂⁴×C₄	#123	1	2	yes	—
177	C₂⁴×C₄	C₂³×C₄	1	1	yes	characteristic
178	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
179	C₂×#120	AGL(1,ℤ/8)	1	8	yes	—
180	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
181	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
182	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
183	C₂×C₄×D₄	C₄×D₄	1	2	yes	—
184	C₂×C₄×D₄	#120	1	2	yes	—
185	C₂×C₄×D₄	#120	1	2	yes	—
186	C₂×C₄×D₄	#120	1	2	yes	—
187	C₂×C₄×D₄	#120	1	2	yes	—
188	C₂×C₄×D₄	#121	1	2	yes	—
189	C₂×C₄×D₄	#121	1	2	yes	—
190	C₂×C₄×D₄	#121	1	2	yes	—
191	C₂×C₄×D₄	#121	1	2	yes	—
192	C₂×C₄×D₄	#121	1	2	yes	—
193	C₂×C₄×D₄	#121	1	2	yes	—
194	C₂×C₄×D₄	#121	1	2	yes	—
195	C₂×C₄×D₄	#121	1	2	yes	—
196	C₂×C₄×D₄	#121	1	2	yes	—
197	C₂×C₄×D₄	#121	1	2	yes	—
198	C₂×C₄×D₄	#121	1	2	yes	—
199	C₂×C₄×D₄	#121	1	2	yes	—
200	C₂×C₄×D₄	#123	1	2	yes	—
201	C₂×C₄×D₄	#123	1	2	yes	—
202	C₂×C₄×D₄	#123	1	2	yes	—
203	C₂×C₄×D₄	#123	1	2	yes	—
204	C₂×C₄×D₄	#124	1	2	yes	—
205	C₂×C₄×D₄	#124	1	2	yes	—
206	C₂×C₄×D₄	#127	1	2	yes	—
207	C₂×C₄×D₄	#127	1	2	yes	—
208	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
216	C₂²×C₄⋊C₄	#99	1	4	yes	—
217	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
219	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
221	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
222	C₂²×C₄⋊C₄	#121	1	2	yes	—
223	C₂²×C₄⋊C₄	#121	1	2	yes	—
224	C₂²×C₄⋊C₄	#124	1	2	yes	—
225	C₂²×C₄⋊C₄	#127	1	2	yes	—
226	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
227	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₂²⋊C₄	#120	1	2	yes	—
232	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
233	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
234	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
235	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
236	C₂²×C₂²⋊C₄	#121	1	2	yes	—
237	C₂²×C₂²⋊C₄	#121	1	2	yes	—
238	C₂²×C₂²⋊C₄	#121	1	2	yes	—
239	C₂²×C₂²⋊C₄	#121	1	2	yes	—
240	C₂²×C₂²⋊C₄	#123	1	2	yes	—
241	C₂²×C₂²⋊C₄	#124	1	2	yes	—
242	C₂²×C₂²⋊C₄	#127	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
244	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
245	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₄²	#120	1	2	yes	—
247	C₂²×C₄²	#120	1	1	yes	characteristic
248	C₂²×C₄²	#120	1	1	yes	characteristic
249	C₂²×C₄²	#121	1	2	yes	—
250	C₂²×C₄²	#121	1	2	yes	—
251	C₂²×C₄²	#123	1	2	yes	—
252	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
253	C₂×#95	C₂×C₂²⋊C₄	1	4	yes	—
254	C₂⁵	#379	1	1	yes	characteristic
255	C₂⁵	#379	1	1	yes	characteristic
256	C₂⁵	#379	1	1	yes	characteristic
257	C₂⁵	#386	1	2	yes	—
258	C₂⁵	#386	1	2	yes	—
259	C₂⁵	#392	1	2	yes	—
260	C₂⁵	#394	1	2	yes	—
261	C₂⁵	#418	1	4	yes	complemented
262	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
263	C₂⁵	C₂×#120	1	1	yes	characteristic
264	C₂²×D₄	AΣL(1,𝔽₁₆)	1	8	yes	complemented
265	C₂²×D₄	#386	1	2	yes	—
266	C₂²×D₄	#386	1	2	yes	—
267	C₂²×D₄	#392	1	2	yes	—
268	C₂²×D₄	#392	1	2	yes	—
269	C₂²×D₄	#394	1	2	yes	—
270	C₂²×D₄	#394	1	2	yes	—
271	C₂²×D₄	C₂×#99	1	4	yes	—
272	C₂²×D₄	C₂×#99	1	4	yes	—
273	C₂²×D₄	C₂×#99	1	4	yes	—
274	C₂²×D₄	C₂×#120	1	1	yes	characteristic
275	C₂²×D₄	C₂×#121	1	1	yes	characteristic
276	C₂²×D₄	C₂×#121	1	1	yes	characteristic
277	C₂²×D₄	C₂×#123	1	1	yes	characteristic
278	C₂²×D₄	C₂×AGL(1,ℤ/8)	1	4	yes	—
279	C₂²×D₄	#575	1	4	yes	—
280	C₂³×C₄	#379	1	1	yes	characteristic
281	C₂³×C₄	#379	1	1	yes	characteristic
282	C₂³×C₄	#379	1	1	yes	characteristic
283	C₂³×C₄	#379	1	1	yes	characteristic
284	C₂³×C₄	#379	1	1	yes	characteristic
285	C₂³×C₄	#386	1	2	yes	—
286	C₂³×C₄	#386	1	2	yes	—
287	C₂³×C₄	#386	1	2	yes	—
288	C₂³×C₄	#386	1	2	yes	—
289	C₂³×C₄	#386	1	2	yes	—
290	C₂³×C₄	#386	1	2	yes	—
291	C₂³×C₄	#386	1	2	yes	—
292	C₂³×C₄	#386	1	2	yes	—
293	C₂³×C₄	#390	1	2	yes	—
294	C₂³×C₄	#390	1	2	yes	—
295	C₂³×C₄	#390	1	2	yes	—
296	C₂³×C₄	#390	1	2	yes	—
297	C₂³×C₄	#392	1	2	yes	—
298	C₂³×C₄	#392	1	2	yes	—
299	C₂³×C₄	#392	1	2	yes	—
300	C₂³×C₄	#392	1	2	yes	—
301	C₂³×C₄	#392	1	2	yes	—
302	C₂³×C₄	#394	1	2	yes	—
303	C₂³×C₄	#394	1	2	yes	—
304	C₂³×C₄	#394	1	2	yes	—
305	C₂³×C₄	#394	1	2	yes	—
306	C₂³×C₄	#394	1	2	yes	—
307	C₂³×C₄	C₂×#99	1	1	yes	characteristic
308	C₂³×C₄	C₂×#99	1	2	yes	—
309	C₂³×C₄	C₂×#99	1	1	yes	characteristic
310	C₂³×C₄	C₂×#99	1	1	yes	characteristic
311	C₂³×C₄	C₂×#99	1	2	yes	—
312	C₂³×C₄	C₂×#99	1	1	yes	characteristic
313	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
314	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
315	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
316	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
317	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
318	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
319	C₂³×C₄	C₂×#120	1	1	yes	characteristic
320	C₂³×C₄	C₂×#120	1	1	yes	characteristic
321	C₂³×C₄	C₂×#120	1	1	yes	characteristic
322	C₂³×C₄	C₂×#120	1	1	yes	characteristic
323	C₂³×C₄	C₂×#121	1	1	yes	characteristic
324	C₂³×C₄	C₂×#121	1	1	yes	characteristic
325	C₂³×C₄	C₂×#123	1	1	yes	characteristic
326	C₂×C₄⋊C₄	#390	1	2	yes	—
327	C₂×C₄⋊C₄	#390	1	2	yes	—
328	C₂×C₄⋊C₄	#392	1	2	yes	—
329	C₂×C₄⋊C₄	#392	1	2	yes	—
330	C₂×C₄⋊C₄	#394	1	2	yes	—
331	C₂×C₄⋊C₄	#394	1	2	yes	—
332	C₂×C₄⋊C₄	C₂×#99	1	4	yes	—
333	C₂×C₄⋊C₄	C₂×#121	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂×#121	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂×#124	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂×#127	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	#386	1	2	yes	—
338	C₂×C₂²⋊C₄	#386	1	2	yes	—
339	C₂×C₂²⋊C₄	#390	1	2	yes	—
340	C₂×C₂²⋊C₄	#390	1	2	yes	—
341	C₂×C₂²⋊C₄	#392	1	2	yes	—
342	C₂×C₂²⋊C₄	#392	1	2	yes	—
343	C₂×C₂²⋊C₄	#392	1	2	yes	—
344	C₂×C₂²⋊C₄	#392	1	2	yes	—
345	C₂×C₂²⋊C₄	#394	1	2	yes	—
346	C₂×C₂²⋊C₄	#394	1	2	yes	—
347	C₂×C₂²⋊C₄	#394	1	2	yes	—
348	C₂×C₂²⋊C₄	#394	1	2	yes	—
349	C₂×C₂²⋊C₄	#418	1	4	yes	complemented
350	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
355	C₂×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
356	C₂×C₂²⋊C₄	C₂×#124	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
358	C₂×C₄²	#386	1	2	yes	—
359	C₂×C₄²	#386	1	2	yes	—
360	C₂×C₄²	#392	1	2	yes	—
361	C₂×C₄²	#392	1	2	yes	—
362	C₂×C₄²	#394	1	2	yes	—
363	C₂×C₄²	#394	1	2	yes	—
364	C₂×C₄²	C₂×#120	1	1	yes	characteristic
365	C₂×C₄²	C₂×#121	1	1	yes	characteristic
366	C₂×C₄²	C₂×#121	1	1	yes	characteristic
367	C₂×C₄²	C₂×#123	1	1	yes	characteristic
368	C₂⁴	#1438	1	2	yes	—
369	C₂⁴	#1438	1	2	yes	—
370	C₂⁴	#1438	1	2	yes	—
371	C₂⁴	#1781	1	2	yes	—
372	C₂⁴	#1781	1	1	yes	characteristic
373	C₂⁴	#1781	1	1	yes	characteristic
374	C₂⁴	#1795	1	2	yes	—
375	C₂⁴	#1795	1	2	yes	—
376	C₂⁴	C₂×#379	1	1	yes	characteristic
377	C₂⁴	C₂×#386	1	1	yes	characteristic
378	C₂⁴	C₂×#386	1	1	yes	characteristic
379	C₂⁴	C₂×#392	1	1	yes	characteristic
380	C₂⁴	C₂×#394	1	1	yes	characteristic
381	C₂⁴	C₂×#418	1	2	yes	—
382	C₂⁴	#2896	1	2	yes	—
383	C₂⁴	#3003	1	2	yes	—
384	C₂⁴	#3005	1	2	yes	—
385	C₂×D₄	C₂×AΣL(1,𝔽₁₆)	1	8	yes	—
386	C₂×D₄	#3006	1	4	yes	—
387	C₂×D₄	#3055	1	4	yes	—
388	C₂×D₄	#3055	1	4	yes	—
389	C₂×D₄	#3094	1	4	yes	—
390	C₂²×C₄	#1438	1	2	yes	—
391	C₂²×C₄	#1438	1	2	yes	—
392	C₂²×C₄	#1438	1	2	yes	—
393	C₂²×C₄	#1438	1	2	yes	—
394	C₂²×C₄	#1438	1	2	yes	—
395	C₂²×C₄	#1781	1	2	yes	—
396	C₂²×C₄	#1781	1	1	yes	characteristic
397	C₂²×C₄	#1781	1	1	yes	characteristic
398	C₂²×C₄	#1898	1	2	yes	—
399	C₂²×C₄	#1898	1	2	yes	—
400	C₂²×C₄	#1898	1	2	yes	—
401	C₂²×C₄	#1898	1	2	yes	—
402	C₂²×C₄	#1898	1	2	yes	—
403	C₂²×C₄	#1898	1	2	yes	—
404	C₂²×C₄	#1898	1	2	yes	—
405	C₂²×C₄	#1898	1	2	yes	—
406	C₂²×C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
407	C₂²×C₄	#2121	1	4	yes	—
408	C₂²×C₄	C₂×#386	1	1	yes	characteristic
409	C₂²×C₄	C₂×#386	1	1	yes	characteristic
410	C₂²×C₄	C₂×#390	1	1	yes	characteristic
411	C₂²×C₄	C₂×#390	1	1	yes	characteristic
412	C₂²×C₄	C₂×#392	1	1	yes	characteristic
413	C₂²×C₄	C₂×#392	1	1	yes	characteristic
414	C₂²×C₄	C₂×#392	1	1	yes	characteristic
415	C₂²×C₄	C₂×#394	1	1	yes	characteristic
416	C₂²×C₄	C₂×#394	1	1	yes	characteristic
417	C₂²×C₄	C₂×#394	1	1	yes	characteristic
418	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
419	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
420	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
421	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
422	C₂²×C₄	#3003	1	2	yes	—
423	C₂²×C₄	#3005	1	2	yes	—
424	C₂³	#10965	1	2	yes	—
425	C₂³	#10965	1	2	yes	—
426	C₂³	#10984	1	4	yes	—
427	C₂³	#12687	1	2	yes	—
428	C₂³	#12690	1	2	yes	—
429	C₂³	#12711	1	2	yes	—
430	C₂³	#12713	1	2	yes	—
431	C₂³	C₂×#1438	1	1	yes	characteristic
432	C₂³	C₂×#1781	1	1	yes	characteristic
433	C₂³	C₂×#1781	1	1	yes	characteristic
434	C₂³	#20598	1	1	yes	characteristic
435	C₂³	#20600	1	1	yes	characteristic
436	C₂³	C₂×#1795	1	1	yes	characteristic
437	C₂³	C₂×#1795	1	1	yes	characteristic
438	C₂³	#20912	1	2	yes	—
439	C₂³	#20973	1	2	yes	—
440	C₂³	#23391	1	2	yes	—
441	C₂³	#23480	1	2	yes	—
442	C₂×C₄	#10965	1	2	yes	—
443	C₂×C₄	#10965	1	2	yes	—
444	C₂×C₄	#12687	1	2	yes	—
445	C₂×C₄	#12690	1	2	yes	—
446	C₂×C₄	#13082	1	8	yes	—
447	C₂×C₄	#13086	1	8	yes	—
448	C₂×C₄	C₂×#1898	1	1	yes	characteristic
449	C₂×C₄	C₂×#1898	1	1	yes	characteristic
450	C₂×C₄	C₂×#1898	1	1	yes	characteristic
451	C₂×C₄	C₂×#1898	1	1	yes	characteristic
452	C₂×C₄	#23391	1	2	yes	—
453	C₂×C₄	#23480	1	2	yes	—
454	C₂×C₄	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
455	C₂×C₄	C₂×#2121	1	2	yes	—
456	C₂²	#147555	1	4	yes	—
457	C₂²	#147573	1	4	yes	—
458	C₂²	#185643	1	1	yes	characteristic
459	C₂²	#185643	1	1	yes	characteristic
460	C₂²	#185652	1	1	yes	characteristic
461	C₂²	#185652	1	1	yes	characteristic
462	C₂²	C₂×#10965	1	1	yes	characteristic
463	C₂²	#357433	1	1	yes	characteristic
464	C₂²	C₂×#10984	1	1	yes	characteristic
465	C₂²	C₂×#12687	1	1	yes	characteristic
466	C₂²	C₂×#12690	1	1	yes	characteristic
467	C₂²	C₂×#12711	1	1	yes	characteristic
468	C₂²	C₂×#12713	1	1	yes	characteristic
469	C₄	#147555	1	4	yes	—
470	C₄	C₂×#13082	1	2	yes	—
471	C₄	C₂×#13086	1	2	yes	—
472	C₂	#72231038	1	1	yes	characteristic
473	C₂	C₂×#147555	1	1	yes	characteristic
474	C₂	C₂×#147573	1	1	yes	characteristic
475	C₂	#72240212	1	1	yes	characteristic
476	C₂	#72240228	1	1	yes	characteristic
477	C₂	#72240250	1	1	yes	characteristic
478	C₂	#72240305	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×#1781
commutator	C₂⁴×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
prime radical	C₂×#647663	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#493995	C₂	semidirect
2. C₂×#480376	C₂	semidirect
3. #72240579	C₂	semidirect
4. #72240339	C₂	semidirect
5. #72240324	C₂	semidirect
7. C₂×#185755	C₂	semidirect
8. C₂×#185643	C₂	semidirect
10. C₂⁴×#121	C₄	semidirect
11. C₂²×#2847	C₄	semidirect
17. #647663	C₄	semidirect
19. C₂×#14137	C₄	semidirect
20. C₂×#13902	C₄	semidirect
22. C₂×#12793	C₂²	semidirect
23. C₂×#11471	C₂²	semidirect
24. C₂×#8383	C₂²	semidirect
25. C₂×#8375	C₂²	semidirect
26. #127619	C₂²	semidirect
27. #127615	C₂²	semidirect
53. #20726	D₄	semidirect
54. #20722	D₄	semidirect
57. #17364	D₄	semidirect
58. #17362	D₄	semidirect
130. #1800	C₂×D₄	semidirect
131. #1799	C₂×D₄	semidirect
132. #1460	C₂×D₄	semidirect
133. #1459	C₂×D₄	semidirect
261. C₂⁵	#418	semidirect
264. C₂²×D₄	AΣL(1,𝔽₁₆)	semidirect
349. C₂×C₂²⋊C₄	#418	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493995	2
C₂×#480376	2
#72240579	2
#72240339	2
#72240324	2
C₂×#647663	2
C₂×#185755	2
C₂×#185643	2