Subgroups — C₂×#72260421 #72261084

Order 2048.

At least 9281 conjugacy classes of subgroups; only the 1592 normal subgroups are shown.

The subgroup lattice has 672 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×#111959	C₂	1	2	yes	maximal, complemented
2	C₂×#6343491	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#116147	C₂	1	2	yes	maximal, complemented
4	#72260421	C₂	1	16	yes	maximal, direct product factor
5	C₂×#686216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6470228	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#114565	C₂	1	2	yes	maximal, complemented
8	C₂×#201819	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#6437449	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#359513	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#234591	C₂	1	2	yes	maximal, complemented
12	C₂×#106626	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#62811	C₂²	1	1	yes	characteristic
14	C₂³×#535	C₂²	1	1	yes	characteristic
15	C₂²×#3003	C₂²	1	2	yes	—
16	C₂×#60064	C₂²	1	1	yes	characteristic
17	C₂×#35050	C₂²	1	2	yes	—
18	C₂²×#2404	C₂²	1	1	yes	characteristic
19	C₂³×#392	C₂²	1	1	yes	characteristic, complemented
20	C₂²×#2339	C₂²	1	1	yes	characteristic
21	#6470228	C₂²	1	8	yes	complemented
22	#6437449	C₂²	1	8	yes	complemented
23	C₂×#21748	C₂²	1	2	yes	—
24	C₂×#20979	C₂²	1	2	yes	—
25	C₂×#20598	C₂²	1	2	yes	complemented
26	C₂×#20591	C₂²	1	1	yes	characteristic
27	C₂×#20584	C₂²	1	1	yes	characteristic
28	#6343491	C₂²	1	8	yes	complemented
29	C₂×#17587	C₂²	1	2	yes	—
30	C₂²×#1504	C₂²	1	2	yes	—
31	C₂²×#1462	C₂²	1	2	yes	complemented
32	#686216	C₂²	1	8	yes	complemented
33	C₂×#14335	C₂²	1	1	yes	characteristic, complemented
34	C₂×#13878	C₂²	1	1	yes	characteristic, complemented
35	C₂×#13868	C₂²	1	1	yes	characteristic, complemented
36	#359513	C₂²	1	8	yes	complemented
37	C₂×#10716	C₂²	1	2	yes	complemented
38	#234591	C₂²	1	16	yes	complemented
39	#201819	C₂²	1	8	yes	complemented
40	C₂×#8020	C₂²	1	2	yes	complemented
41	C₂×#8114	C₂²	1	2	yes	complemented
42	C₂×#7943	C₂²	1	2	yes	complemented
43	#116147	C₂²	1	16	yes	complemented
44	#114565	C₂²	1	16	yes	complemented
45	#111959	C₂²	1	16	yes	complemented
46	#106626	C₂²	1	8	yes	complemented
47	C₂²×#585	D₄	1	1	yes	characteristic, complemented
48	C₂⁴×#41	C₂³	1	1	yes	characteristic
49	C₂⁵×D₄	C₂³	1	1	yes	characteristic
50	#62811	D₄	1	4	yes	complemented
51	#62811	C₂³	1	4	yes	—
52	C₂×#3449	D₄	1	2	yes	complemented
53	C₂²×#535	D₄	1	4	yes	complemented
54	C₂²×#535	C₂³	1	4	yes	—
55	C₂³×#127	D₄	1	2	yes	complemented
56	C₂³×#124	D₄	1	2	yes	complemented
57	C₂²×#525	C₂³	1	2	yes	—
58	C₂³×#120	D₄	1	1	yes	characteristic
59	C₂³×#120	D₄	1	1	yes	characteristic
60	C₂×#3428	D₄	1	1	yes	characteristic, complemented
61	C₂³×#117	C₂³	1	1	yes	characteristic
62	C₂⁴×C₂²⋊C₄	C₂³	1	1	yes	characteristic
63	C₂×#3003	C₂³	1	8	yes	—
64	C₂²×#410	D₄	1	2	yes	complemented
65	C₂²×#410	D₄	1	2	yes	complemented
66	#60064	C₂³	1	4	yes	—
67	C₂²×#407	C₂³	1	2	yes	—
68	C₂²×#407	C₂³	1	2	yes	—
69	#35050	C₂³	1	8	yes	—
70	C₂×#2404	D₄	1	4	yes	—
71	C₂×#2404	C₂³	1	4	yes	—
72	C₂²×#392	C₂³	1	4	yes	complemented
73	C₂×#2339	D₄	1	4	yes	complemented
74	C₂×#2339	C₂³	1	4	yes	—
75	C₂²×#386	D₄	1	1	yes	characteristic
76	C₂²×#386	D₄	1	1	yes	characteristic
77	C₂×#2280	C₂³	1	2	yes	—
78	C₂²×#381	D₄	1	1	yes	characteristic, complemented
79	C₂²×#381	C₂³	1	1	yes	characteristic
80	C₂²×#381	C₂³	1	1	yes	characteristic
81	C₂²×#380	D₄	1	1	yes	characteristic, complemented
82	C₂²×C₄×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
83	C₂³×#95	D₄	1	1	yes	characteristic, complemented
84	C₂³×#95	C₂³	1	1	yes	characteristic
85	#21748	C₂³	1	8	yes	—
86	#20979	C₂³	1	8	yes	—
87	#20598	C₂³	1	8	yes	complemented
88	#20591	D₄	1	4	yes	complemented
89	#20591	C₂³	1	4	yes	—
90	C₂×#1785	D₄	1	2	yes	complemented
91	C₂×#1785	D₄	1	2	yes	complemented
92	#20584	D₄	1	4	yes	complemented
93	#20584	C₂³	1	4	yes	—
94	#17587	C₂³	1	8	yes	—
95	C₂×#1504	D₄	1	8	yes	complemented
96	C₂×#1504	C₂³	1	8	yes	—
97	C₂²×#328	D₄	1	4	yes	complemented
98	C₂²×#327	D₄	1	4	yes	complemented
99	C₂×#1462	C₂³	1	8	yes	complemented
100	#14335	C₂³	1	4	yes	complemented
101	#13878	C₂³	1	4	yes	complemented
102	#13868	C₂³	1	4	yes	complemented
103	#10716	C₂³	1	8	yes	complemented
104	#8114	C₂³	1	8	yes	complemented
105	#8020	C₂³	1	8	yes	complemented
106	#7943	C₂³	1	8	yes	complemented
107	C₂⁷	C₂×D₄	1	1	yes	characteristic
108	C₂×#585	C₂×D₄	1	4	yes	complemented
109	C₂³×#41	C₂×D₄	1	2	yes	—
110	C₂³×#41	C₂×D₄	1	2	yes	—
111	C₂³×#41	C₂×D₄	1	2	yes	—
112	C₂³×#41	C₂×D₄	1	1	yes	characteristic
113	C₂³×#41	C₂×D₄	1	1	yes	characteristic
114	C₂³×#41	C₂×D₄	1	2	yes	—
115	C₂³×#41	C₂×D₄	1	1	yes	characteristic
116	C₂³×#41	C₂×D₄	1	1	yes	characteristic
117	C₂³×#41	C₂×D₄	1	2	yes	—
118	C₂³×#41	C₂×D₄	1	1	yes	characteristic
119	C₂³×#41	C₂×D₄	1	1	yes	characteristic
120	C₂³×#41	C₂×D₄	1	2	yes	—
121	C₂³×#41	C₂×D₄	1	2	yes	—
122	C₂³×#41	#41	1	2	yes	—
123	C₂³×#41	#41	1	1	yes	characteristic
124	C₂³×#41	#41	1	1	yes	characteristic
125	C₂³×#41	C₂⁴	1	2	yes	—
126	C₂⁴×Q₈	C₂×D₄	1	1	yes	characteristic
127	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
128	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
129	C₂⁴×D₄	C₂×D₄	1	2	yes	—
130	C₂⁴×D₄	C₂×D₄	1	2	yes	—
131	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
132	C₂⁴×D₄	C₂⁴	1	2	yes	—
133	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂⁵×C₄	C₂⁴	1	1	yes	characteristic
137	#3449	C₂×D₄	1	8	yes	complemented
138	C₂²×#127	C₂×D₄	1	8	yes	complemented
139	C₂²×#124	C₂×D₄	1	8	yes	complemented
140	C₂×#525	C₂×D₄	1	4	yes	—
141	C₂×#525	C₂×D₄	1	4	yes	—
142	C₂×#525	#41	1	4	yes	—
143	C₂×#525	C₂⁴	1	4	yes	—
144	C₂²×C₂²⋊Q₈	C₂×D₄	1	2	yes	—
145	C₂²×C₂²⋊Q₈	#41	1	2	yes	—
146	C₂²×#121	C₂×D₄	1	2	yes	—
147	C₂²×#121	#41	1	2	yes	—
148	C₂²×#120	C₂×D₄	1	4	yes	—
149	C₂²×#120	C₂×D₄	1	4	yes	—
150	#3428	C₂×D₄	1	4	yes	complemented
151	C₂²×#117	C₂×D₄	1	2	yes	—
152	C₂²×#117	C₂×D₄	1	4	yes	—
153	C₂²×#117	C₂×D₄	1	2	yes	—
154	C₂²×#117	C₂⁴	1	2	yes	—
155	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
156	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
157	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
158	C₂³×C₂²⋊C₄	C₂⁴	1	2	yes	—
159	C₂³×C₄²	C₂×D₄	1	2	yes	—
160	C₂×#410	C₂×D₄	1	8	yes	complemented
161	C₂×#410	C₂×D₄	1	8	yes	complemented
162	C₂×#407	C₂×D₄	1	4	yes	—
163	C₂×#407	C₂×D₄	1	4	yes	—
164	C₂×#407	C₂×D₄	1	4	yes	—
165	C₂×#407	C₂⁴	1	4	yes	—
166	C₂×#407	C₂⁴	1	4	yes	—
167	C₂²×C₂²⋊C₈	C₂×D₄	1	4	yes	—
168	C₂×#386	C₂×D₄	1	4	yes	—
169	C₂×#386	C₂×D₄	1	4	yes	—
170	#2280	C₂×D₄	1	4	yes	—
171	#2280	C₂⁴	1	4	yes	—
172	C₂×#381	C₂×D₄	1	2	yes	—
173	C₂×#381	C₂×D₄	1	2	yes	—
174	C₂×#381	C₂×D₄	1	4	yes	complemented
175	C₂×#381	C₂⁴	1	2	yes	—
176	C₂×#381	C₂⁴	1	2	yes	—
177	C₂×#380	C₂×D₄	1	4	yes	complemented
178	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	complemented
179	C₂²×#95	C₂×D₄	1	1	yes	characteristic
180	C₂²×#95	C₂×D₄	1	1	yes	characteristic
181	C₂²×#95	C₂×D₄	1	1	yes	characteristic
182	C₂²×#95	C₂×D₄	1	1	yes	characteristic
183	C₂²×#95	C₂×D₄	1	4	yes	—
184	C₂²×#95	C₂×D₄	1	4	yes	—
185	C₂²×#95	C₂×D₄	1	2	yes	—
186	C₂²×#95	C₂×D₄	1	4	yes	complemented
187	C₂²×#95	C₂⁴	1	2	yes	—
188	#1785	C₂×D₄	1	8	yes	complemented
189	#1785	C₂×D₄	1	8	yes	complemented
190	C₂×#328	C₂×D₄	1	16	yes	complemented
191	C₂×#327	C₂×D₄	1	16	yes	complemented
192	C₂⁶	#120	1	1	yes	characteristic
193	C₂⁶	#120	1	1	yes	characteristic
194	C₂⁶	C₂²×D₄	1	1	yes	characteristic
195	C₂⁶	C₂²×D₄	1	2	yes	—
196	C₂²×#41	#120	1	2	yes	—
197	C₂²×#41	#120	1	2	yes	—
198	C₂²×#41	#120	1	2	yes	—
199	C₂²×#41	#120	1	2	yes	—
200	C₂²×#41	#120	1	2	yes	—
201	C₂²×#41	#120	1	2	yes	—
202	C₂²×#41	#120	1	2	yes	—
203	C₂²×#41	#120	1	2	yes	—
204	C₂²×#41	#121	1	2	yes	—
205	C₂²×#41	#121	1	2	yes	—
206	C₂²×#41	#121	1	2	yes	—
207	C₂²×#41	#121	1	2	yes	—
208	C₂²×#41	#121	1	2	yes	—
209	C₂²×#41	#121	1	2	yes	—
210	C₂²×#41	#121	1	2	yes	—
211	C₂²×#41	#121	1	2	yes	—
212	C₂²×#41	#121	1	2	yes	—
213	C₂²×#41	#121	1	2	yes	—
214	C₂²×#41	#121	1	2	yes	—
215	C₂²×#41	#121	1	2	yes	—
216	C₂²×#41	#121	1	4	yes	complemented
217	C₂²×#41	#127	1	2	yes	—
218	C₂²×#41	#127	1	2	yes	—
219	C₂²×#41	#127	1	2	yes	—
220	C₂²×#41	#127	1	2	yes	—
221	C₂²×#41	C₂²×D₄	1	4	yes	—
222	C₂²×#41	C₂²×D₄	1	2	yes	—
223	C₂²×#41	C₂²×D₄	1	2	yes	—
224	C₂²×#41	C₂²×D₄	1	2	yes	—
225	C₂²×#41	C₂²×D₄	1	2	yes	—
226	C₂²×#41	C₂²×D₄	1	2	yes	—
227	C₂²×#41	C₂²×D₄	1	2	yes	—
228	C₂²×#41	C₂×#41	1	2	yes	—
229	C₂²×#41	C₂×#41	1	2	yes	—
230	C₂³×Q₈	#120	1	2	yes	—
231	C₂³×Q₈	#120	1	1	yes	characteristic
232	C₂³×Q₈	#120	1	1	yes	characteristic
233	C₂³×Q₈	#120	1	2	yes	—
234	C₂³×Q₈	#121	1	1	yes	characteristic
235	C₂³×Q₈	#121	1	1	yes	characteristic
236	C₂³×Q₈	#121	1	2	yes	—
237	C₂³×Q₈	#121	1	2	yes	complemented
238	C₂³×Q₈	C₂²×D₄	1	2	yes	—
239	C₂³×D₄	#120	1	1	yes	characteristic
240	C₂³×D₄	#120	1	2	yes	—
241	C₂³×D₄	#120	1	1	yes	characteristic
242	C₂³×D₄	#120	1	2	yes	—
243	C₂³×D₄	#120	1	2	yes	—
244	C₂³×D₄	#120	1	2	yes	—
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	2	yes	—
249	C₂³×D₄	#120	1	1	yes	characteristic
250	C₂³×D₄	#120	1	2	yes	—
251	C₂³×D₄	#121	1	1	yes	characteristic
252	C₂³×D₄	#121	1	1	yes	characteristic
253	C₂³×D₄	#121	1	2	yes	—
254	C₂³×D₄	#121	1	1	yes	characteristic
255	C₂³×D₄	#121	1	1	yes	characteristic
256	C₂³×D₄	#121	1	2	yes	—
257	C₂³×D₄	#121	1	1	yes	characteristic
258	C₂³×D₄	#121	1	1	yes	characteristic
259	C₂³×D₄	#121	1	2	yes	—
260	C₂³×D₄	#121	1	2	yes	complemented
261	C₂³×D₄	C₂²×D₄	1	2	yes	—
262	C₂³×D₄	C₂²×D₄	1	2	yes	—
263	C₂³×D₄	C₂²×D₄	1	4	yes	—
264	C₂³×D₄	C₂²×D₄	1	2	yes	—
265	C₂⁴×C₄	#120	1	1	yes	characteristic
266	C₂⁴×C₄	#120	1	1	yes	characteristic
267	C₂⁴×C₄	#120	1	2	yes	—
268	C₂⁴×C₄	#120	1	2	yes	—
269	C₂⁴×C₄	#121	1	1	yes	characteristic
270	C₂⁴×C₄	#121	1	1	yes	characteristic
271	C₂⁴×C₄	#121	1	1	yes	characteristic
272	C₂⁴×C₄	#121	1	1	yes	characteristic
273	C₂⁴×C₄	#121	1	2	yes	—
274	C₂⁴×C₄	#121	1	2	yes	—
275	C₂⁴×C₄	#127	1	1	yes	characteristic
276	C₂⁴×C₄	#127	1	1	yes	characteristic
277	C₂⁴×C₄	#127	1	1	yes	characteristic
278	C₂⁴×C₄	#127	1	1	yes	characteristic
279	C₂⁴×C₄	#127	1	2	yes	—
280	C₂⁴×C₄	#127	1	2	yes	—
281	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
282	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
283	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
284	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
285	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
286	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
287	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
288	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
289	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
290	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
291	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
293	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
294	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
295	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
296	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
297	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
298	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
299	C₂⁴×C₄	C₂×#41	1	2	yes	—
300	C₂⁴×C₄	C₂×#41	1	2	yes	—
301	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
302	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
303	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
304	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
305	C₂⁴×C₄	C₂⁵	1	1	yes	characteristic
306	C₂×C₂²⋊Q₈	#121	1	4	yes	complemented
307	C₂×C₂²⋊Q₈	#121	1	4	yes	complemented
308	C₂×C₂²⋊Q₈	C₂²×D₄	1	4	yes	—
309	C₂×C₂²⋊Q₈	C₂×#41	1	4	yes	—
310	C₂×#121	#121	1	4	yes	complemented
311	C₂×#121	#121	1	4	yes	complemented
312	C₂×#121	C₂²×D₄	1	4	yes	—
313	C₂×#121	C₂×#41	1	4	yes	—
314	C₂×#117	#121	1	4	yes	complemented
315	C₂×#117	C₂²×D₄	1	8	yes	—
316	C₂×#117	C₂²×D₄	1	4	yes	—
317	C₂²×C₄⋊C₄	#121	1	2	yes	complemented
318	C₂²×C₄⋊C₄	#121	1	2	yes	complemented
319	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
320	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
321	C₂²×C₄²	C₂²×D₄	1	4	yes	—
322	#407	C₂²×D₄	1	8	yes	—
323	C₂×C₂²⋊C₈	C₂²×D₄	1	8	yes	—
324	C₂×#95	#120	1	8	yes	—
325	C₂×#95	#120	1	2	yes	—
326	C₂×#95	#120	1	2	yes	—
327	C₂×#95	#120	1	2	yes	—
328	C₂×#95	#120	1	2	yes	—
329	C₂×#95	C₂²×D₄	1	2	yes	—
330	C₂×#95	C₂²×D₄	1	2	yes	—
331	C₂×#95	C₂²×D₄	1	8	yes	—
332	C₂×#95	C₂²×D₄	1	2	yes	—
333	C₂×#95	C₂²×D₄	1	2	yes	—
334	C₂⁵	#392	1	1	yes	characteristic
335	C₂⁵	#392	1	1	yes	characteristic
336	C₂⁵	#392	1	1	yes	characteristic
337	C₂⁵	#392	1	2	yes	—
338	C₂⁵	#392	1	1	yes	characteristic
339	C₂⁵	#392	1	2	yes	—
340	C₂⁵	C₂×#120	1	1	yes	characteristic
341	C₂⁵	C₂×#120	1	2	yes	—
342	C₂⁵	C₂×#120	1	1	yes	characteristic
343	C₂⁵	C₂×#120	1	1	yes	characteristic
344	C₂⁵	C₂×#120	1	2	yes	—
345	C₂⁵	C₂×#120	1	1	yes	characteristic
346	C₂⁵	C₂×#121	1	1	yes	characteristic
347	C₂⁵	C₂×#121	1	1	yes	characteristic
348	C₂⁵	D₄²	1	2	yes	—
349	C₂⁵	D₄²	1	2	yes	—
350	C₂⁵	#546	1	2	yes	—
351	C₂⁵	#546	1	2	yes	—
352	C₂⁵	C₂³×D₄	1	1	yes	characteristic
353	C₂²×Q₈	#392	1	2	yes	—
354	C₂²×Q₈	#392	1	2	yes	—
355	C₂²×Q₈	#392	1	2	yes	—
356	C₂²×Q₈	#392	1	2	yes	—
357	C₂²×Q₈	C₂×#120	1	2	yes	—
358	C₂²×Q₈	C₂×#120	1	2	yes	—
359	C₂²×Q₈	C₂×#121	1	2	yes	—
360	C₂²×Q₈	C₂×#121	1	2	yes	—
361	C₂²×Q₈	C₂×#121	1	4	yes	complemented
362	C₂²×D₄	#392	1	2	yes	—
363	C₂²×D₄	#392	1	2	yes	—
364	C₂²×D₄	#392	1	2	yes	—
365	C₂²×D₄	#392	1	2	yes	—
366	C₂²×D₄	#392	1	2	yes	—
367	C₂²×D₄	#392	1	2	yes	—
368	C₂²×D₄	#392	1	2	yes	—
369	C₂²×D₄	#392	1	2	yes	—
370	C₂²×D₄	#392	1	2	yes	—
371	C₂²×D₄	#392	1	2	yes	—
372	C₂²×D₄	#392	1	2	yes	—
373	C₂²×D₄	#392	1	2	yes	—
374	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
375	C₂²×D₄	C₂×#120	1	2	yes	—
376	C₂²×D₄	C₂×#120	1	2	yes	—
377	C₂²×D₄	C₂×#120	1	2	yes	—
378	C₂²×D₄	C₂×#120	1	2	yes	—
379	C₂²×D₄	C₂×#120	1	2	yes	—
380	C₂²×D₄	C₂×#120	1	2	yes	—
381	C₂²×D₄	C₂×#121	1	2	yes	—
382	C₂²×D₄	C₂×#121	1	2	yes	—
383	C₂²×D₄	C₂×#121	1	2	yes	—
384	C₂²×D₄	C₂×#121	1	2	yes	—
385	C₂²×D₄	C₂×#121	1	2	yes	—
386	C₂²×D₄	C₂×#121	1	2	yes	—
387	C₂²×D₄	C₂×#121	1	4	yes	complemented
388	C₂³×C₄	#392	1	1	yes	characteristic
389	C₂³×C₄	#392	1	1	yes	characteristic
390	C₂³×C₄	#392	1	1	yes	characteristic
391	C₂³×C₄	#392	1	2	yes	—
392	C₂³×C₄	#392	1	2	yes	—
393	C₂³×C₄	#392	1	2	yes	—
394	C₂³×C₄	#392	1	2	yes	—
395	C₂³×C₄	#392	1	2	yes	—
396	C₂³×C₄	#392	1	1	yes	characteristic
397	C₂³×C₄	#392	1	2	yes	—
398	C₂³×C₄	#392	1	2	yes	—
399	C₂³×C₄	#392	1	2	yes	—
400	C₂³×C₄	#392	1	2	yes	—
401	C₂³×C₄	#392	1	2	yes	—
402	C₂³×C₄	C₂×#120	1	2	yes	—
403	C₂³×C₄	C₂×#120	1	1	yes	characteristic
404	C₂³×C₄	C₂×#120	1	1	yes	characteristic
405	C₂³×C₄	C₂×#120	1	1	yes	characteristic
406	C₂³×C₄	C₂×#120	1	1	yes	characteristic
407	C₂³×C₄	C₂×#120	1	2	yes	—
408	C₂³×C₄	C₂×#120	1	1	yes	characteristic
409	C₂³×C₄	C₂×#120	1	2	yes	—
410	C₂³×C₄	C₂×#120	1	1	yes	characteristic
411	C₂³×C₄	C₂×#120	1	1	yes	characteristic
412	C₂³×C₄	C₂×#120	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×#121	1	2	yes	—
416	C₂³×C₄	C₂×#121	1	2	yes	—
417	C₂³×C₄	C₂×#121	1	1	yes	characteristic
418	C₂³×C₄	C₂×#121	1	1	yes	characteristic
419	C₂³×C₄	C₂×#121	1	2	yes	—
420	C₂³×C₄	C₂×#121	1	2	yes	—
421	C₂³×C₄	C₂×#121	1	1	yes	characteristic
422	C₂³×C₄	C₂×#121	1	1	yes	characteristic
423	C₂³×C₄	C₂×#121	1	2	yes	—
424	C₂³×C₄	C₂×#121	1	2	yes	—
425	C₂³×C₄	C₂×#121	1	1	yes	characteristic
426	C₂³×C₄	C₂×#121	1	1	yes	characteristic
427	C₂³×C₄	C₂×#121	1	2	yes	—
428	C₂³×C₄	C₂×#121	1	2	yes	—
429	C₂³×C₄	C₂×#121	1	1	yes	characteristic
430	C₂³×C₄	C₂×#121	1	1	yes	characteristic
431	C₂³×C₄	C₂×#121	1	1	yes	characteristic
432	C₂³×C₄	C₂×#121	1	1	yes	characteristic
433	C₂³×C₄	#525	1	1	yes	characteristic
434	C₂³×C₄	#525	1	2	yes	—
435	C₂³×C₄	#525	1	1	yes	characteristic
436	C₂³×C₄	#525	1	2	yes	—
437	C₂³×C₄	#525	1	2	yes	—
438	C₂³×C₄	#525	1	2	yes	—
439	C₂³×C₄	#525	1	2	yes	—
440	C₂³×C₄	C₂×#127	1	2	yes	—
441	C₂³×C₄	C₂×#127	1	2	yes	—
442	C₂³×C₄	C₂×#127	1	1	yes	characteristic
443	C₂³×C₄	C₂×#127	1	1	yes	characteristic
444	C₂³×C₄	C₂×#127	1	2	yes	—
445	C₂³×C₄	C₂×#127	1	2	yes	—
446	C₂³×C₄	C₂×#127	1	1	yes	characteristic
447	C₂³×C₄	C₂×#127	1	1	yes	characteristic
448	C₂³×C₄	#534	1	2	yes	—
449	C₂³×C₄	#534	1	2	yes	—
450	C₂³×C₄	#534	1	1	yes	characteristic
451	C₂³×C₄	#534	1	2	yes	—
452	C₂³×C₄	#534	1	1	yes	characteristic
453	C₂³×C₄	#534	1	2	yes	—
454	C₂³×C₄	#534	1	2	yes	—
455	C₂³×C₄	#535	1	1	yes	characteristic
456	C₂³×C₄	#535	1	2	yes	—
457	C₂³×C₄	#535	1	1	yes	characteristic
458	C₂³×C₄	#538	1	2	yes	—
459	C₂³×C₄	#538	1	2	yes	—
460	C₂³×C₄	#538	1	2	yes	—
461	C₂³×C₄	#538	1	2	yes	—
462	C₂³×C₄	D₄²	1	2	yes	—
463	C₂³×C₄	D₄²	1	2	yes	—
464	C₂³×C₄	D₄²	1	2	yes	—
465	C₂³×C₄	D₄²	1	2	yes	—
466	C₂³×C₄	D₄²	1	2	yes	—
467	C₂³×C₄	D₄²	1	2	yes	—
468	C₂³×C₄	D₄²	1	2	yes	—
469	C₂³×C₄	D₄²	1	2	yes	—
470	C₂³×C₄	D₄²	1	2	yes	—
471	C₂³×C₄	D₄²	1	2	yes	—
472	C₂³×C₄	D₄²	1	2	yes	—
473	C₂³×C₄	D₄²	1	2	yes	—
474	C₂³×C₄	D₄²	1	2	yes	—
475	C₂³×C₄	D₄²	1	2	yes	—
476	C₂³×C₄	#546	1	2	yes	—
477	C₂³×C₄	#546	1	2	yes	—
478	C₂³×C₄	#546	1	2	yes	—
479	C₂³×C₄	#546	1	2	yes	—
480	C₂³×C₄	#546	1	2	yes	—
481	C₂³×C₄	#546	1	2	yes	—
482	C₂³×C₄	#546	1	2	yes	—
483	C₂³×C₄	#546	1	2	yes	—
484	C₂³×C₄	#546	1	2	yes	—
485	C₂³×C₄	#546	1	2	yes	—
486	C₂³×C₄	#546	1	2	yes	—
487	C₂³×C₄	#546	1	2	yes	—
488	C₂³×C₄	#546	1	2	yes	—
489	C₂³×C₄	#546	1	2	yes	—
490	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
491	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
492	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
493	C₂³×C₄	C₂³×D₄	1	2	yes	—
494	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
495	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
496	C₂³×C₄	C₂²×#41	1	1	yes	characteristic
497	C₂³×C₄	C₂²×#41	1	2	yes	—
498	C₂³×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
499	C₂³×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
500	C₂×C₄⋊C₄	C₂×#121	1	4	yes	complemented
501	C₂×C₄⋊C₄	C₂×#121	1	4	yes	complemented
502	#95	C₂×#120	1	4	yes	—
503	#95	C₂×#120	1	4	yes	—
504	C₂⁴	#2021	1	4	yes	—
505	C₂⁴	C₂×#392	1	1	yes	characteristic
506	C₂⁴	C₂×#392	1	1	yes	characteristic
507	C₂⁴	C₂×#392	1	2	yes	—
508	C₂⁴	C₂×#392	1	2	yes	—
509	C₂⁴	C₂×#392	1	2	yes	—
510	C₂⁴	C₂×#392	1	1	yes	characteristic
511	C₂⁴	C₂×#392	1	2	yes	—
512	C₂⁴	C₂×#392	1	1	yes	characteristic
513	C₂⁴	#2403	1	1	yes	characteristic
514	C₂⁴	#2403	1	2	yes	—
515	C₂⁴	#2403	1	1	yes	characteristic
516	C₂⁴	#2404	1	1	yes	characteristic
517	C₂⁴	#2404	1	2	yes	—
518	C₂⁴	#2404	1	1	yes	characteristic
519	C₂⁴	#2410	1	2	yes	—
520	C₂⁴	#2410	1	2	yes	—
521	C₂⁴	#2410	1	2	yes	—
522	C₂⁴	#2410	1	2	yes	—
523	C₂⁴	#2424	1	2	yes	—
524	C₂⁴	#2424	1	2	yes	—
525	C₂⁴	#2424	1	2	yes	—
526	C₂⁴	#2424	1	2	yes	—
527	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
528	C₂⁴	#3026	1	4	yes	—
529	C₂⁴	C₂²×#120	1	1	yes	characteristic
530	C₂⁴	C₂²×#120	1	1	yes	characteristic
531	C₂⁴	C₂²×#121	1	1	yes	characteristic
532	C₂⁴	C₂×D₄²	1	2	yes	—
533	C₂⁴	C₂×#546	1	2	yes	—
534	C₂×D₄	#2021	1	8	yes	—
535	C₂×D₄	C₂×B(4,𝔽₂)	1	8	yes	—
536	C₂²×C₄	#2021	1	4	yes	—
537	C₂²×C₄	C₂×#392	1	2	yes	—
538	C₂²×C₄	C₂×#392	1	2	yes	—
539	C₂²×C₄	C₂×#392	1	2	yes	—
540	C₂²×C₄	C₂×#392	1	1	yes	characteristic
541	C₂²×C₄	C₂×#392	1	1	yes	characteristic
542	C₂²×C₄	C₂×#392	1	1	yes	characteristic
543	C₂²×C₄	C₂×#392	1	1	yes	characteristic
544	C₂²×C₄	C₂×#392	1	2	yes	—
545	C₂²×C₄	#2403	1	1	yes	characteristic
546	C₂²×C₄	#2403	1	2	yes	—
547	C₂²×C₄	#2403	1	1	yes	characteristic
548	C₂²×C₄	#2404	1	1	yes	characteristic
549	C₂²×C₄	#2404	1	2	yes	—
550	C₂²×C₄	#2404	1	1	yes	characteristic
551	C₂²×C₄	#2410	1	2	yes	—
552	C₂²×C₄	#2410	1	2	yes	—
553	C₂²×C₄	#2410	1	2	yes	—
554	C₂²×C₄	#2410	1	2	yes	—
555	C₂²×C₄	#2410	1	2	yes	—
556	C₂²×C₄	#2410	1	2	yes	—
557	C₂²×C₄	#2410	1	2	yes	—
558	C₂²×C₄	#2410	1	2	yes	—
559	C₂²×C₄	#2410	1	2	yes	—
560	C₂²×C₄	#2410	1	2	yes	—
561	C₂²×C₄	#2410	1	2	yes	—
562	C₂²×C₄	#2410	1	2	yes	—
563	C₂²×C₄	#2424	1	2	yes	—
564	C₂²×C₄	#2424	1	2	yes	—
565	C₂²×C₄	#2424	1	2	yes	—
566	C₂²×C₄	#2424	1	2	yes	—
567	C₂²×C₄	#2424	1	2	yes	—
568	C₂²×C₄	#2424	1	2	yes	—
569	C₂²×C₄	#2424	1	2	yes	—
570	C₂²×C₄	#2424	1	2	yes	—
571	C₂²×C₄	#2424	1	2	yes	—
572	C₂²×C₄	#2424	1	2	yes	—
573	C₂²×C₄	#2424	1	2	yes	—
574	C₂²×C₄	#2424	1	2	yes	—
575	C₂²×C₄	#2433	1	2	yes	—
576	C₂²×C₄	#2433	1	1	yes	characteristic
577	C₂²×C₄	#2433	1	1	yes	characteristic
578	C₂²×C₄	#2433	1	1	yes	characteristic
579	C₂²×C₄	#2433	1	2	yes	—
580	C₂²×C₄	#2433	1	1	yes	characteristic
581	C₂²×C₄	#2539	1	2	yes	—
582	C₂²×C₄	#2539	1	1	yes	characteristic
583	C₂²×C₄	#2539	1	1	yes	characteristic
584	C₂²×C₄	#2539	1	1	yes	characteristic
585	C₂²×C₄	#2539	1	2	yes	—
586	C₂²×C₄	#2539	1	1	yes	characteristic
587	C₂²×C₄	#2617	1	2	yes	—
588	C₂²×C₄	#2617	1	1	yes	characteristic
589	C₂²×C₄	#2617	1	1	yes	characteristic
590	C₂²×C₄	#2617	1	1	yes	characteristic
591	C₂²×C₄	#2617	1	1	yes	characteristic
592	C₂²×C₄	#2617	1	2	yes	—
593	C₂²×C₄	#2847	1	2	yes	—
594	C₂²×C₄	#2847	1	1	yes	characteristic
595	C₂²×C₄	#2847	1	1	yes	characteristic
596	C₂²×C₄	#2847	1	1	yes	characteristic
597	C₂²×C₄	#2847	1	1	yes	characteristic
598	C₂²×C₄	#2847	1	2	yes	—
599	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
600	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
601	C₂²×C₄	C₂²×#121	1	2	yes	—
602	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
603	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
604	C₂²×C₄	C₂×#525	1	1	yes	characteristic
605	C₂²×C₄	C₂×#525	1	2	yes	—
606	C₂²×C₄	C₂²×#127	1	1	yes	characteristic
607	C₂²×C₄	C₂×#534	1	2	yes	—
608	C₂²×C₄	C₂×#534	1	1	yes	characteristic
609	C₂²×C₄	C₂×#535	1	1	yes	characteristic
610	C₂²×C₄	C₂×#538	1	2	yes	—
611	C₂²×C₄	C₂×D₄²	1	2	yes	—
612	C₂²×C₄	C₂×D₄²	1	2	yes	—
613	C₂²×C₄	C₂×D₄²	1	2	yes	—
614	C₂²×C₄	C₂×#546	1	2	yes	—
615	C₂²×C₄	C₂×#546	1	2	yes	—
616	C₂²×C₄	C₂×#546	1	2	yes	—
617	C₂³	#14716	1	2	yes	—
618	C₂³	#14716	1	1	yes	characteristic
619	C₂³	#14716	1	1	yes	characteristic
620	C₂³	#14716	1	2	yes	—
621	C₂³	#14716	1	1	yes	characteristic
622	C₂³	#14716	1	1	yes	characteristic
623	C₂³	C₂×#2021	1	2	yes	—
624	C₂³	C₂×#2021	1	4	yes	—
625	C₂³	C₂×#2021	1	2	yes	—
626	C₂³	#23444	1	2	yes	—
627	C₂³	#23444	1	2	yes	—
628	C₂³	#23882	1	2	yes	—
629	C₂³	#23884	1	2	yes	—
630	C₂³	#23990	1	4	yes	—
631	C₂³	#24404	1	8	yes	—
632	C₂³	#24404	1	8	yes	—
633	C₂³	C₂²×#392	1	1	yes	characteristic
634	C₂³	C₂×#2403	1	1	yes	characteristic
635	C₂³	C₂×#2410	1	2	yes	—
636	C₂³	C₂×#2424	1	2	yes	—
637	C₂³	C₂×#2404	1	1	yes	characteristic
638	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
639	C₂³	C₂×#3026	1	2	yes	—
640	C₂×C₄	#14716	1	1	yes	characteristic
641	C₂×C₄	#14716	1	2	yes	—
642	C₂×C₄	#14716	1	1	yes	characteristic
643	C₂×C₄	#14716	1	1	yes	characteristic
644	C₂×C₄	#14716	1	2	yes	—
645	C₂×C₄	#14716	1	1	yes	characteristic
646	C₂×C₄	C₂×#2021	1	4	yes	—
647	C₂×C₄	C₂×#2617	1	1	yes	characteristic
648	C₂×C₄	C₂×#2410	1	2	yes	—
649	C₂×C₄	C₂×#2424	1	2	yes	—
650	C₂×C₄	C₂×#2433	1	1	yes	characteristic
651	C₂×C₄	C₂×#2539	1	1	yes	characteristic
652	C₂×C₄	C₂×#2847	1	1	yes	characteristic
653	C₂²	#394227	1	2	yes	—
654	C₂²	#394227	1	2	yes	—
655	C₂²	#394262	1	2	yes	—
656	C₂²	#394262	1	2	yes	—
657	C₂²	#394267	1	2	yes	—
658	C₂²	#394267	1	4	yes	—
659	C₂²	#394267	1	2	yes	—
660	C₂²	C₂×#14716	1	1	yes	characteristic
661	C₂²	C₂²×#2021	1	1	yes	characteristic
662	C₂²	C₂×#23444	1	1	yes	characteristic
663	C₂²	C₂×#23882	1	1	yes	characteristic
664	C₂²	C₂×#23884	1	1	yes	characteristic
665	C₂²	C₂×#24404	1	4	yes	—
666	C₂²	C₂×#23990	1	2	yes	—
667	C₂	#72260421	1	4	yes	direct product factor
668	C₂	C₂×#394262	1	1	yes	characteristic
669	C₂	C₂×#394267	1	1	yes	characteristic
670	C₂	C₂×#394227	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14716
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁵
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#111959	C₂	semidirect
2. C₂×#6343491	C₂	semidirect
3. C₂×#116147	C₂	semidirect
4. #72260421	C₂	direct (N × H)
5. C₂×#686216	C₂	semidirect
6. C₂×#6470228	C₂	semidirect
7. C₂×#114565	C₂	semidirect
8. C₂×#201819	C₂	semidirect
9. C₂×#6437449	C₂	semidirect
10. C₂×#359513	C₂	semidirect
11. C₂×#234591	C₂	semidirect
12. C₂×#106626	C₂	semidirect
19. C₂³×#392	C₂²	semidirect
21. #6470228	C₂²	semidirect
22. #6437449	C₂²	semidirect
25. C₂×#20598	C₂²	semidirect
28. #6343491	C₂²	semidirect
31. C₂²×#1462	C₂²	semidirect
32. #686216	C₂²	semidirect
33. C₂×#14335	C₂²	semidirect
34. C₂×#13878	C₂²	semidirect
35. C₂×#13868	C₂²	semidirect
36. #359513	C₂²	semidirect
37. C₂×#10716	C₂²	semidirect
38. #234591	C₂²	semidirect
39. #201819	C₂²	semidirect
40. C₂×#8020	C₂²	semidirect
41. C₂×#8114	C₂²	semidirect
42. C₂×#7943	C₂²	semidirect
43. #116147	C₂²	semidirect
44. #114565	C₂²	semidirect
45. #111959	C₂²	semidirect
46. #106626	C₂²	semidirect
47. C₂²×#585	D₄	semidirect
50. #62811	D₄	semidirect
52. C₂×#3449	D₄	semidirect
53. C₂²×#535	D₄	semidirect
55. C₂³×#127	D₄	semidirect
56. C₂³×#124	D₄	semidirect
60. C₂×#3428	D₄	semidirect
64. C₂²×#410	D₄	semidirect
65. C₂²×#410	D₄	semidirect
72. C₂²×#392	C₂³	semidirect
73. C₂×#2339	D₄	semidirect
78. C₂²×#381	D₄	semidirect
81. C₂²×#380	D₄	semidirect
82. C₂²×C₄×C₂²⋊C₄	D₄	semidirect
83. C₂³×#95	D₄	semidirect
87. #20598	C₂³	semidirect
88. #20591	D₄	semidirect
90. C₂×#1785	D₄	semidirect
91. C₂×#1785	D₄	semidirect
92. #20584	D₄	semidirect
95. C₂×#1504	D₄	semidirect
97. C₂²×#328	D₄	semidirect
98. C₂²×#327	D₄	semidirect
99. C₂×#1462	C₂³	semidirect
100. #14335	C₂³	semidirect
101. #13878	C₂³	semidirect
102. #13868	C₂³	semidirect
103. #10716	C₂³	semidirect
104. #8114	C₂³	semidirect
105. #8020	C₂³	semidirect
106. #7943	C₂³	semidirect
108. C₂×#585	C₂×D₄	semidirect
137. #3449	C₂×D₄	semidirect
138. C₂²×#127	C₂×D₄	semidirect
139. C₂²×#124	C₂×D₄	semidirect
150. #3428	C₂×D₄	semidirect
160. C₂×#410	C₂×D₄	semidirect
161. C₂×#410	C₂×D₄	semidirect
174. C₂×#381	C₂×D₄	semidirect
177. C₂×#380	C₂×D₄	semidirect
178. C₂×C₄×C₂²⋊C₄	C₂×D₄	semidirect
186. C₂²×#95	C₂×D₄	semidirect
188. #1785	C₂×D₄	semidirect
189. #1785	C₂×D₄	semidirect
190. C₂×#328	C₂×D₄	semidirect
191. C₂×#327	C₂×D₄	semidirect
216. C₂²×#41	#121	semidirect
237. C₂³×Q₈	#121	semidirect
260. C₂³×D₄	#121	semidirect
306. C₂×C₂²⋊Q₈	#121	semidirect
307. C₂×C₂²⋊Q₈	#121	semidirect
310. C₂×#121	#121	semidirect
311. C₂×#121	#121	semidirect
314. C₂×#117	#121	semidirect
317. C₂²×C₄⋊C₄	#121	semidirect
318. C₂²×C₄⋊C₄	#121	semidirect
361. C₂²×Q₈	C₂×#121	semidirect
387. C₂²×D₄	C₂×#121	semidirect
500. C₂×C₄⋊C₄	C₂×#121	semidirect
501. C₂×C₄⋊C₄	C₂×#121	semidirect
667. C₂	#72260421	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#111959	2
C₂×#6343491	2
C₂×#116147	2
#72260421	2
C₂×#686216	2
C₂×#6470228	2
C₂×#114565	2
C₂×#201819	2
C₂×#6437449	2
C₂×#359513	2
C₂×#234591	2
C₂×#106626	2

Subgroups — C2×#72260421 #72261084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72260421 #72261084