Subgroups — C₂×#72234076 #72314567

Order 2048.

At least 7943 conjugacy classes of subgroups; only the 768 normal subgroups are shown.

The subgroup lattice has 589 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×#185652	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#128179	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#128050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#482360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#482360	C₂	1	1	yes	maximal, characteristic, complemented
6	#72234076	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4421184	C₂	1	1	yes	maximal, characteristic
8	C₂×#185475	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×#546	C₂²	1	1	yes	characteristic
10	C₂²×#2535	C₄	1	1	yes	characteristic, complemented
11	C₂²×#2535	C₂²	1	1	yes	characteristic
12	C₂²×#2631	C₄	1	1	yes	characteristic, complemented
13	C₂×#20722	C₂²	1	1	yes	characteristic
14	C₂×#20722	C₂²	1	1	yes	characteristic
15	C₂×#17362	C₂²	1	1	yes	characteristic
16	C₂×#17362	C₂²	1	1	yes	characteristic
17	#4421184	C₄	1	4	yes	complemented
18	#4421184	C₂²	1	4	yes	—
19	C₂×#14662	C₄	1	1	yes	characteristic, complemented
20	C₂×#14345	C₄	1	1	yes	characteristic, complemented
21	C₂×#14345	C₂²	1	1	yes	characteristic
22	#482360	C₂²	1	4	yes	complemented
23	#482360	C₂²	1	4	yes	complemented
24	#185652	C₂²	1	4	yes	complemented
25	#185475	C₂²	1	4	yes	complemented
26	#128179	C₂²	1	4	yes	complemented
27	#128050	C₂²	1	4	yes	complemented
28	C₂⁴×#41	D₄	1	1	yes	characteristic
29	C₂⁵×D₄	D₄	1	1	yes	characteristic
30	C₂²×#546	C₂×C₄	1	2	yes	—
31	C₂²×#546	D₄	1	2	yes	—
32	C₂²×#546	D₄	1	2	yes	—
33	C₂²×#546	C₂³	1	2	yes	—
34	C₂³×#124	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₂²⋊Q₈	D₄	1	1	yes	characteristic
36	C₂³×#121	D₄	1	1	yes	characteristic
37	C₂³×C₄×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×D₄	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×D₄	C₂³	1	1	yes	characteristic
40	C₂×#2535	C₂×C₄	1	2	yes	—
41	C₂×#2535	C₂×C₄	1	4	yes	complemented
42	C₂×#2535	D₄	1	2	yes	—
43	C₂×#2535	D₄	1	2	yes	—
44	C₂×#2535	C₂³	1	2	yes	—
45	C₂×#2631	C₂×C₄	1	4	yes	complemented
46	C₂²×#397	D₄	1	1	yes	characteristic
47	C₂²×#397	D₄	1	1	yes	characteristic
48	C₂²×#392	D₄	1	1	yes	characteristic
49	C₂²×#392	D₄	1	1	yes	characteristic
50	C₂²×#383	C₂×C₄	1	1	yes	characteristic
51	C₂²×#382	C₂×C₄	1	1	yes	characteristic
52	C₂²×#382	C₂×C₄	1	1	yes	characteristic
53	#20722	D₄	1	2	yes	complemented
54	#20722	D₄	1	2	yes	complemented
55	#20722	C₂³	1	2	yes	—
56	#20722	C₂³	1	2	yes	—
57	C₂×#1799	D₄	1	2	yes	complemented
58	C₂×C₄×#99	D₄	1	2	yes	complemented
59	#17362	D₄	1	2	yes	complemented
60	#17362	D₄	1	2	yes	complemented
61	#17362	C₂³	1	2	yes	—
62	#17362	C₂³	1	2	yes	—
63	C₂×#1465	D₄	1	2	yes	complemented
64	C₂×#1459	D₄	1	2	yes	complemented
65	#14662	C₂×C₄	1	4	yes	complemented
66	#14345	C₂×C₄	1	2	yes	—
67	#14345	C₂×C₄	1	4	yes	complemented
68	#14345	D₄	1	2	yes	—
69	#14345	D₄	1	2	yes	—
70	#14345	C₂³	1	2	yes	—
71	C₂×#1445	D₄	1	2	yes	—
72	C₂×#1438	D₄	1	2	yes	—
73	C₂³×#41	C₂²⋊C₄	1	2	yes	—
74	C₂³×#41	C₂×D₄	1	2	yes	—
75	C₂⁴×Q₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
77	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	2	yes	—
82	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×#124	C₂²⋊C₄	1	2	yes	—
85	C₂²×#124	C₂²×C₄	1	2	yes	—
86	C₂²×C₂²⋊Q₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₂²⋊Q₈	C₂×D₄	1	2	yes	—
88	C₂²×#121	C₂²⋊C₄	1	2	yes	—
89	C₂²×#121	C₂×D₄	1	2	yes	—
90	C₂²×C₄×D₄	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×D₄	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×D₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×D₄	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×D₄	C₂²×C₄	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	1	yes	characteristic
99	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
103	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×D₄	#41	1	1	yes	characteristic
110	C₂²×C₄×D₄	#41	1	1	yes	characteristic
111	C₂²×C₄×D₄	#41	1	1	yes	characteristic
112	C₂²×C₄×D₄	#41	1	1	yes	characteristic
113	C₂²×C₄×D₄	#41	1	1	yes	characteristic
114	C₂²×C₄×D₄	#41	1	1	yes	characteristic
115	C₂²×C₄×D₄	#41	1	1	yes	characteristic
116	C₂²×C₄×D₄	#41	1	1	yes	characteristic
117	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
118	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
123	C₂³×C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₂×#397	C₂²⋊C₄	1	2	yes	—
125	C₂×#397	C₂²⋊C₄	1	2	yes	—
126	C₂×#397	C₂×D₄	1	2	yes	—
127	C₂×#397	C₂×D₄	1	2	yes	—
128	C₂×#392	C₂²⋊C₄	1	2	yes	—
129	C₂×#392	C₂²⋊C₄	1	2	yes	—
130	C₂×#392	C₂×D₄	1	2	yes	—
131	C₂×#392	C₂×D₄	1	2	yes	—
132	C₂×#383	C₂²⋊C₄	1	2	yes	—
133	C₂×#383	C₂²×C₄	1	2	yes	—
134	C₂×#382	C₂²⋊C₄	1	2	yes	—
135	C₂×#382	C₂²⋊C₄	1	2	yes	—
136	C₂×#382	C₂²×C₄	1	2	yes	—
137	C₂×#382	C₂²×C₄	1	2	yes	—
138	C₂²×#95	C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×#95	C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×#95	C₂²⋊C₄	1	2	yes	—
141	C₂²×#95	C₂²⋊C₄	1	2	yes	—
142	#1799	C₂×D₄	1	4	yes	complemented
143	C₄×#99	C₂×D₄	1	4	yes	complemented
144	#1465	C₂×D₄	1	4	yes	complemented
145	#1459	C₂×D₄	1	4	yes	complemented
146	#1445	C₂²⋊C₄	1	4	yes	—
147	#1445	C₂×D₄	1	4	yes	—
148	#1438	C₂²⋊C₄	1	4	yes	—
149	#1438	C₂×D₄	1	4	yes	—
150	C₂⁶	#120	1	1	yes	characteristic
151	C₂⁶	#120	1	1	yes	characteristic
152	C₂³×Q₈	#99	1	2	yes	—
153	C₂³×Q₈	#99	1	2	yes	—
154	C₂³×Q₈	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂³×D₄	#99	1	2	yes	—
156	C₂³×D₄	#99	1	2	yes	—
157	C₂³×D₄	#99	1	2	yes	—
158	C₂³×D₄	#99	1	2	yes	—
159	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂³×D₄	#120	1	1	yes	characteristic
164	C₂³×D₄	#120	1	1	yes	characteristic
165	C₂³×D₄	#121	1	1	yes	characteristic
166	C₂³×D₄	#121	1	1	yes	characteristic
167	C₂³×D₄	#121	1	1	yes	characteristic
168	C₂³×D₄	#121	1	1	yes	characteristic
169	C₂³×D₄	#124	1	1	yes	characteristic
170	C₂³×D₄	#124	1	1	yes	characteristic
171	C₂³×D₄	#127	1	1	yes	characteristic
172	C₂³×D₄	#127	1	1	yes	characteristic
173	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
174	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂⁴×C₄	#120	1	1	yes	characteristic
180	C₂⁴×C₄	#120	1	1	yes	characteristic
181	C₂⁴×C₄	#120	1	1	yes	characteristic
182	C₂⁴×C₄	#120	1	1	yes	characteristic
183	C₂⁴×C₄	#120	1	1	yes	characteristic
184	C₂⁴×C₄	#120	1	1	yes	characteristic
185	C₂⁴×C₄	#120	1	1	yes	characteristic
186	C₂⁴×C₄	#120	1	1	yes	characteristic
187	C₂⁴×C₄	#120	1	1	yes	characteristic
188	C₂⁴×C₄	#120	1	1	yes	characteristic
189	C₂⁴×C₄	#121	1	1	yes	characteristic
190	C₂⁴×C₄	#121	1	1	yes	characteristic
191	C₂⁴×C₄	#121	1	1	yes	characteristic
192	C₂⁴×C₄	#121	1	1	yes	characteristic
193	C₂⁴×C₄	#121	1	1	yes	characteristic
194	C₂⁴×C₄	#121	1	1	yes	characteristic
195	C₂⁴×C₄	#121	1	1	yes	characteristic
196	C₂⁴×C₄	#121	1	1	yes	characteristic
197	C₂⁴×C₄	#123	1	1	yes	characteristic
198	C₂⁴×C₄	#123	1	1	yes	characteristic
199	C₂⁴×C₄	#123	1	1	yes	characteristic
200	C₂⁴×C₄	#123	1	1	yes	characteristic
201	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
202	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×D₄	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×D₄	#120	1	1	yes	characteristic
212	C₂×C₄×D₄	#120	1	1	yes	characteristic
213	C₂×C₄×D₄	#120	1	1	yes	characteristic
214	C₂×C₄×D₄	#120	1	1	yes	characteristic
215	C₂×C₄×D₄	#120	1	1	yes	characteristic
216	C₂×C₄×D₄	#120	1	1	yes	characteristic
217	C₂×C₄×D₄	#120	1	1	yes	characteristic
218	C₂×C₄×D₄	#120	1	1	yes	characteristic
219	C₂×C₄×D₄	#121	1	1	yes	characteristic
220	C₂×C₄×D₄	#121	1	1	yes	characteristic
221	C₂×C₄×D₄	#121	1	1	yes	characteristic
222	C₂×C₄×D₄	#121	1	1	yes	characteristic
223	C₂×C₄×D₄	#121	1	1	yes	characteristic
224	C₂×C₄×D₄	#121	1	1	yes	characteristic
225	C₂×C₄×D₄	#121	1	1	yes	characteristic
226	C₂×C₄×D₄	#121	1	1	yes	characteristic
227	C₂×C₄×D₄	#121	1	1	yes	characteristic
228	C₂×C₄×D₄	#121	1	1	yes	characteristic
229	C₂×C₄×D₄	#121	1	1	yes	characteristic
230	C₂×C₄×D₄	#121	1	1	yes	characteristic
231	C₂×C₄×D₄	#121	1	1	yes	characteristic
232	C₂×C₄×D₄	#121	1	1	yes	characteristic
233	C₂×C₄×D₄	#121	1	1	yes	characteristic
234	C₂×C₄×D₄	#121	1	1	yes	characteristic
235	C₂×C₄×D₄	#121	1	1	yes	characteristic
236	C₂×C₄×D₄	#121	1	1	yes	characteristic
237	C₂×C₄×D₄	#121	1	1	yes	characteristic
238	C₂×C₄×D₄	#121	1	1	yes	characteristic
239	C₂×C₄×D₄	#121	1	1	yes	characteristic
240	C₂×C₄×D₄	#121	1	1	yes	characteristic
241	C₂×C₄×D₄	#121	1	1	yes	characteristic
242	C₂×C₄×D₄	#121	1	1	yes	characteristic
243	C₂×C₄×D₄	#123	1	1	yes	characteristic
244	C₂×C₄×D₄	#123	1	1	yes	characteristic
245	C₂×C₄×D₄	#123	1	1	yes	characteristic
246	C₂×C₄×D₄	#123	1	1	yes	characteristic
247	C₂×C₄×D₄	#123	1	1	yes	characteristic
248	C₂×C₄×D₄	#123	1	1	yes	characteristic
249	C₂×C₄×D₄	#123	1	1	yes	characteristic
250	C₂×C₄×D₄	#123	1	1	yes	characteristic
251	C₂×C₄×D₄	#124	1	1	yes	characteristic
252	C₂×C₄×D₄	#124	1	1	yes	characteristic
253	C₂×C₄×D₄	#124	1	1	yes	characteristic
254	C₂×C₄×D₄	#124	1	1	yes	characteristic
255	C₂×C₄×D₄	#127	1	1	yes	characteristic
256	C₂×C₄×D₄	#127	1	1	yes	characteristic
257	C₂×C₄×D₄	#127	1	1	yes	characteristic
258	C₂×C₄×D₄	#127	1	1	yes	characteristic
259	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	#99	1	2	yes	—
268	C₂²×C₄⋊C₄	#99	1	2	yes	—
269	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
273	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	#121	1	1	yes	characteristic
275	C₂²×C₄⋊C₄	#121	1	1	yes	characteristic
276	C₂²×C₄⋊C₄	#121	1	1	yes	characteristic
277	C₂²×C₄⋊C₄	#121	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	#124	1	1	yes	characteristic
279	C₂²×C₄⋊C₄	#124	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	#127	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	#127	1	1	yes	characteristic
282	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
283	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
284	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
285	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
286	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
287	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
288	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
289	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
290	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
291	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
292	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
293	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
294	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
295	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
297	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
299	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
300	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
301	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
302	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
303	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
306	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
307	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
309	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
311	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
312	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
313	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
314	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
315	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
316	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
317	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
318	C₂²×C₄²	C₂³×C₄	1	1	yes	characteristic
319	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	—
320	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	—
321	C₂×#95	C₂×C₂²⋊C₄	1	4	yes	—
322	C₂×#95	C₂×C₂²⋊C₄	1	4	yes	—
323	C₂⁵	#379	1	1	yes	characteristic
324	C₂⁵	#379	1	1	yes	characteristic
325	C₂⁵	#379	1	1	yes	characteristic
326	C₂⁵	#392	1	1	yes	characteristic
327	C₂⁵	#392	1	1	yes	characteristic
328	C₂⁵	#392	1	1	yes	characteristic
329	C₂⁵	#392	1	1	yes	characteristic
330	C₂⁵	#394	1	1	yes	characteristic
331	C₂⁵	#394	1	1	yes	characteristic
332	C₂⁵	#394	1	1	yes	characteristic
333	C₂⁵	#394	1	1	yes	characteristic
334	C₂⁵	C₂×#120	1	1	yes	characteristic
335	C₂⁵	C₂×#120	1	1	yes	characteristic
336	C₂²×Q₈	C₂×#99	1	4	yes	—
337	C₂²×D₄	#390	1	1	yes	characteristic
338	C₂²×D₄	#390	1	1	yes	characteristic
339	C₂²×D₄	#390	1	1	yes	characteristic
340	C₂²×D₄	#390	1	1	yes	characteristic
341	C₂²×D₄	#392	1	1	yes	characteristic
342	C₂²×D₄	#392	1	1	yes	characteristic
343	C₂²×D₄	#392	1	1	yes	characteristic
344	C₂²×D₄	#392	1	1	yes	characteristic
345	C₂²×D₄	#394	1	1	yes	characteristic
346	C₂²×D₄	#394	1	1	yes	characteristic
347	C₂²×D₄	#394	1	1	yes	characteristic
348	C₂²×D₄	#394	1	1	yes	characteristic
349	C₂²×D₄	C₂×#99	1	4	yes	—
350	C₂²×D₄	C₂×#99	1	4	yes	—
351	C₂²×D₄	C₂×#121	1	1	yes	characteristic
352	C₂²×D₄	C₂×#121	1	1	yes	characteristic
353	C₂²×D₄	C₂×#124	1	1	yes	characteristic
354	C₂²×D₄	C₂×#127	1	1	yes	characteristic
355	C₂³×C₄	#379	1	1	yes	characteristic
356	C₂³×C₄	#379	1	1	yes	characteristic
357	C₂³×C₄	#379	1	1	yes	characteristic
358	C₂³×C₄	#379	1	1	yes	characteristic
359	C₂³×C₄	#379	1	1	yes	characteristic
360	C₂³×C₄	#386	1	1	yes	characteristic
361	C₂³×C₄	#386	1	1	yes	characteristic
362	C₂³×C₄	#386	1	1	yes	characteristic
363	C₂³×C₄	#386	1	1	yes	characteristic
364	C₂³×C₄	#386	1	1	yes	characteristic
365	C₂³×C₄	#386	1	1	yes	characteristic
366	C₂³×C₄	#386	1	1	yes	characteristic
367	C₂³×C₄	#386	1	1	yes	characteristic
368	C₂³×C₄	#386	1	1	yes	characteristic
369	C₂³×C₄	#386	1	1	yes	characteristic
370	C₂³×C₄	#386	1	1	yes	characteristic
371	C₂³×C₄	#386	1	1	yes	characteristic
372	C₂³×C₄	#386	1	1	yes	characteristic
373	C₂³×C₄	#386	1	1	yes	characteristic
374	C₂³×C₄	#386	1	1	yes	characteristic
375	C₂³×C₄	#386	1	1	yes	characteristic
376	C₂³×C₄	#390	1	1	yes	characteristic
377	C₂³×C₄	#390	1	1	yes	characteristic
378	C₂³×C₄	#390	1	1	yes	characteristic
379	C₂³×C₄	#390	1	1	yes	characteristic
380	C₂³×C₄	#392	1	1	yes	characteristic
381	C₂³×C₄	#392	1	1	yes	characteristic
382	C₂³×C₄	#392	1	1	yes	characteristic
383	C₂³×C₄	#392	1	1	yes	characteristic
384	C₂³×C₄	#392	1	1	yes	characteristic
385	C₂³×C₄	#392	1	1	yes	characteristic
386	C₂³×C₄	#392	1	1	yes	characteristic
387	C₂³×C₄	#392	1	1	yes	characteristic
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	1	yes	characteristic
392	C₂³×C₄	#394	1	1	yes	characteristic
393	C₂³×C₄	#394	1	1	yes	characteristic
394	C₂³×C₄	#394	1	1	yes	characteristic
395	C₂³×C₄	#394	1	1	yes	characteristic
396	C₂³×C₄	#394	1	1	yes	characteristic
397	C₂³×C₄	#394	1	1	yes	characteristic
398	C₂³×C₄	#394	1	1	yes	characteristic
399	C₂³×C₄	#394	1	1	yes	characteristic
400	C₂³×C₄	#394	1	1	yes	characteristic
401	C₂³×C₄	#394	1	1	yes	characteristic
402	C₂³×C₄	#394	1	1	yes	characteristic
403	C₂³×C₄	#394	1	1	yes	characteristic
404	C₂³×C₄	C₂×#99	1	1	yes	characteristic
405	C₂³×C₄	C₂×#99	1	1	yes	characteristic
406	C₂³×C₄	C₂×#99	1	1	yes	characteristic
407	C₂³×C₄	C₂×#99	1	1	yes	characteristic
408	C₂³×C₄	C₂×#99	1	1	yes	characteristic
409	C₂³×C₄	C₂×#99	1	1	yes	characteristic
410	C₂³×C₄	C₂×#99	1	1	yes	characteristic
411	C₂³×C₄	C₂×#99	1	1	yes	characteristic
412	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
413	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
414	C₂³×C₄	C₂²×C₂²⋊C₄	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	C₂×#120	1	1	yes	characteristic
418	C₂³×C₄	C₂×#120	1	1	yes	characteristic
419	C₂³×C₄	C₂×#121	1	1	yes	characteristic
420	C₂³×C₄	C₂×#121	1	1	yes	characteristic
421	C₂³×C₄	C₂×#121	1	1	yes	characteristic
422	C₂³×C₄	C₂×#121	1	1	yes	characteristic
423	C₂³×C₄	C₂×#123	1	1	yes	characteristic
424	C₂³×C₄	C₂×#123	1	1	yes	characteristic
425	C₂×C₄⋊C₄	#390	1	1	yes	characteristic
426	C₂×C₄⋊C₄	#390	1	1	yes	characteristic
427	C₂×C₄⋊C₄	#390	1	1	yes	characteristic
428	C₂×C₄⋊C₄	#390	1	1	yes	characteristic
429	C₂×C₄⋊C₄	#392	1	1	yes	characteristic
430	C₂×C₄⋊C₄	#392	1	1	yes	characteristic
431	C₂×C₄⋊C₄	#392	1	1	yes	characteristic
432	C₂×C₄⋊C₄	#392	1	1	yes	characteristic
433	C₂×C₄⋊C₄	#394	1	1	yes	characteristic
434	C₂×C₄⋊C₄	#394	1	1	yes	characteristic
435	C₂×C₄⋊C₄	#394	1	1	yes	characteristic
436	C₂×C₄⋊C₄	#394	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂×#99	1	4	yes	—
438	C₂×C₄⋊C₄	C₂×#121	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂×#121	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂×#124	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂×#127	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
444	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
445	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
446	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
447	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
448	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
449	C₂×C₂²⋊C₄	#386	1	1	yes	characteristic
450	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
453	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
454	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
455	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
456	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	#392	1	1	yes	characteristic
458	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
459	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
460	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
461	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
462	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
464	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
465	C₂×C₂²⋊C₄	#394	1	1	yes	characteristic
466	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
467	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
468	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
469	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
470	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
471	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
472	C₂×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
473	C₂×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
474	C₂×C₄²	#386	1	1	yes	characteristic
475	C₂×C₄²	#386	1	1	yes	characteristic
476	C₂×C₄²	#386	1	1	yes	characteristic
477	C₂×C₄²	#386	1	1	yes	characteristic
478	C₂×C₄²	#386	1	1	yes	characteristic
479	C₂×C₄²	#386	1	1	yes	characteristic
480	C₂×C₄²	#386	1	1	yes	characteristic
481	C₂×C₄²	#386	1	1	yes	characteristic
482	C₂×C₄²	#390	1	1	yes	characteristic
483	C₂×C₄²	#390	1	1	yes	characteristic
484	C₂×C₄²	#390	1	1	yes	characteristic
485	C₂×C₄²	#390	1	1	yes	characteristic
486	C₂×C₄²	C₂×C₄×D₄	1	1	yes	characteristic
487	C₂×C₄²	C₂×C₄×D₄	1	1	yes	characteristic
488	C₂×C₄²	C₂×C₄×D₄	1	1	yes	characteristic
489	C₂×C₄²	C₂×C₄×D₄	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	#1781	1	1	yes	characteristic
497	C₂⁴	#1781	1	1	yes	characteristic
498	C₂⁴	#1781	1	1	yes	characteristic
499	C₂⁴	#1781	1	1	yes	characteristic
500	C₂⁴	#2008	1	4	yes	—
501	C₂⁴	C₂×#379	1	1	yes	characteristic
502	C₂⁴	C₂×#392	1	1	yes	characteristic
503	C₂⁴	C₂×#392	1	1	yes	characteristic
504	C₂⁴	C₂×#394	1	1	yes	characteristic
505	C₂⁴	C₂×#394	1	1	yes	characteristic
506	C₂²×C₄	#1438	1	1	yes	characteristic
507	C₂²×C₄	#1438	1	1	yes	characteristic
508	C₂²×C₄	#1438	1	1	yes	characteristic
509	C₂²×C₄	#1438	1	1	yes	characteristic
510	C₂²×C₄	#1438	1	1	yes	characteristic
511	C₂²×C₄	#1438	1	1	yes	characteristic
512	C₂²×C₄	#1438	1	1	yes	characteristic
513	C₂²×C₄	#1438	1	1	yes	characteristic
514	C₂²×C₄	#1438	1	1	yes	characteristic
515	C₂²×C₄	#1438	1	1	yes	characteristic
516	C₂²×C₄	#1781	1	1	yes	characteristic
517	C₂²×C₄	#1781	1	1	yes	characteristic
518	C₂²×C₄	#1781	1	1	yes	characteristic
519	C₂²×C₄	#1781	1	1	yes	characteristic
520	C₂²×C₄	#1898	1	1	yes	characteristic
521	C₂²×C₄	#1898	1	1	yes	characteristic
522	C₂²×C₄	#1898	1	1	yes	characteristic
523	C₂²×C₄	#1898	1	1	yes	characteristic
524	C₂²×C₄	#1898	1	1	yes	characteristic
525	C₂²×C₄	#1898	1	1	yes	characteristic
526	C₂²×C₄	#1898	1	1	yes	characteristic
527	C₂²×C₄	#1898	1	1	yes	characteristic
528	C₂²×C₄	#1898	1	1	yes	characteristic
529	C₂²×C₄	#1898	1	1	yes	characteristic
530	C₂²×C₄	#1898	1	1	yes	characteristic
531	C₂²×C₄	#1898	1	1	yes	characteristic
532	C₂²×C₄	#1898	1	1	yes	characteristic
533	C₂²×C₄	#1898	1	1	yes	characteristic
534	C₂²×C₄	#1898	1	1	yes	characteristic
535	C₂²×C₄	#1898	1	1	yes	characteristic
536	C₂²×C₄	C₂×#386	1	1	yes	characteristic
537	C₂²×C₄	C₂×#386	1	1	yes	characteristic
538	C₂²×C₄	C₂×#386	1	1	yes	characteristic
539	C₂²×C₄	C₂×#386	1	1	yes	characteristic
540	C₂²×C₄	C₂×#390	1	1	yes	characteristic
541	C₂²×C₄	C₂×#390	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×#394	1	1	yes	characteristic
545	C₂²×C₄	C₂×#394	1	1	yes	characteristic
546	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
547	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
548	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
549	C₂²×C₄	C₂²×#99	1	1	yes	characteristic
550	C₂³	#10965	1	1	yes	characteristic
551	C₂³	#10965	1	1	yes	characteristic
552	C₂³	#10965	1	1	yes	characteristic
553	C₂³	#10965	1	1	yes	characteristic
554	C₂³	#11003	1	2	yes	—
555	C₂³	#11004	1	2	yes	—
556	C₂³	#11004	1	2	yes	—
557	C₂³	#12707	1	4	yes	—
558	C₂³	#12711	1	2	yes	—
559	C₂³	#12713	1	2	yes	—
560	C₂³	C₂×#1438	1	1	yes	characteristic
561	C₂³	C₂×#1781	1	1	yes	characteristic
562	C₂³	C₂×#1781	1	1	yes	characteristic
563	C₂³	C₂×#2008	1	2	yes	—
564	C₂×C₄	#10965	1	1	yes	characteristic
565	C₂×C₄	#10965	1	1	yes	characteristic
566	C₂×C₄	#10965	1	1	yes	characteristic
567	C₂×C₄	#10965	1	1	yes	characteristic
568	C₂×C₄	C₂×#1898	1	1	yes	characteristic
569	C₂×C₄	C₂×#1898	1	1	yes	characteristic
570	C₂×C₄	C₂×#1898	1	1	yes	characteristic
571	C₂×C₄	C₂×#1898	1	1	yes	characteristic
572	C₂²	#114044	1	2	yes	—
573	C₂²	#114044	1	2	yes	—
574	C₂²	#147569	1	2	yes	—
575	C₂²	#147569	1	2	yes	—
576	C₂²	#147573	1	2	yes	—
577	C₂²	#147573	1	2	yes	—
578	C₂²	C₂×#10965	1	1	yes	characteristic
579	C₂²	C₂×#11003	1	1	yes	characteristic
580	C₂²	C₂×#11004	1	1	yes	characteristic
581	C₂²	C₂×#12707	1	2	yes	—
582	C₂²	C₂×#12711	1	1	yes	characteristic
583	C₂²	C₂×#12713	1	1	yes	characteristic
584	C₂	#72234076	1	4	yes	direct product factor
585	C₂	C₂×#147573	1	1	yes	characteristic
586	C₂	C₂×#147569	1	1	yes	characteristic
587	C₂	C₂×#114044	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#10965
commutator	C₂²×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
prime radical	C₂×#4421184	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#546 ◁ 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×#185652	C₂	semidirect
2. C₂×#128179	C₂	semidirect
3. C₂×#128050	C₂	semidirect
4. C₂×#482360	C₂	semidirect
5. C₂×#482360	C₂	semidirect
6. #72234076	C₂	direct (N × H)
8. C₂×#185475	C₂	semidirect
10. C₂²×#2535	C₄	semidirect
12. C₂²×#2631	C₄	semidirect
17. #4421184	C₄	semidirect
19. C₂×#14662	C₄	semidirect
20. C₂×#14345	C₄	semidirect
22. #482360	C₂²	semidirect
23. #482360	C₂²	semidirect
24. #185652	C₂²	semidirect
25. #185475	C₂²	semidirect
26. #128179	C₂²	semidirect
27. #128050	C₂²	semidirect
41. C₂×#2535	C₂×C₄	semidirect
45. C₂×#2631	C₂×C₄	semidirect
53. #20722	D₄	semidirect
54. #20722	D₄	semidirect
57. C₂×#1799	D₄	semidirect
58. C₂×C₄×#99	D₄	semidirect
59. #17362	D₄	semidirect
60. #17362	D₄	semidirect
63. C₂×#1465	D₄	semidirect
64. C₂×#1459	D₄	semidirect
65. #14662	C₂×C₄	semidirect
67. #14345	C₂×C₄	semidirect
142. #1799	C₂×D₄	semidirect
143. C₄×#99	C₂×D₄	semidirect
144. #1465	C₂×D₄	semidirect
145. #1459	C₂×D₄	semidirect
584. C₂	#72234076	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×#185652	2
C₂×#128179	2
C₂×#128050	2
C₂×#482360	2
#72234076	2
C₂×#4421184	2
C₂×#185475	2

Subgroups — C2×#72234076 #72314567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72234076 #72314567