Subgroups — C₂×#72231778 #72260679

Order 2048.

At least 8074 conjugacy classes of subgroups; only the 1832 normal subgroups are shown.

The subgroup lattice has 615 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×#234958	C₂	1	2	yes	maximal, complemented
2	C₂×#112557	C₂	1	2	yes	maximal, complemented
3	C₂×#187961	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#201446	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#108194	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#115640	C₂	1	2	yes	maximal, complemented
7	C₂×#394070	C₂	1	2	yes	maximal, complemented
8	C₂×#6487644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#7631334	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#6437510	C₂	1	1	yes	maximal, characteristic, complemented
11	#72231778	C₂	1	16	yes	maximal, direct product factor
12	C₂×#185643	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁴×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic, complemented
14	C₂²×#3496	C₂²	1	1	yes	characteristic
15	C₂³×D₄²	C₂²	1	1	yes	characteristic
16	C₂²×#3006	C₂²	1	1	yes	characteristic, complemented
17	#7631334	C₂²	1	8	yes	complemented
18	C₂×#35250	C₂²	1	2	yes	complemented
19	C₂×#35001	C₂²	1	2	yes	—
20	C₂×#34736	C₂²	1	1	yes	characteristic
21	C₂²×#2433	C₂²	1	2	yes	—
22	C₂×C₄×#518	C₂²	1	1	yes	characteristic
23	#6487644	C₂²	1	8	yes	complemented
24	#6437510	C₂²	1	8	yes	complemented
25	C₂×#21793	C₂²	1	2	yes	complemented
26	C₂²×#1900	C₂²	1	2	yes	complemented
27	C₂×#20973	C₂²	1	1	yes	characteristic, complemented
28	C₂×#20898	C₂²	1	1	yes	characteristic
29	C₂×#20722	C₂²	1	1	yes	characteristic
30	C₂²×#1789	C₂²	1	2	yes	complemented
31	C₂×#20598	C₂²	1	1	yes	characteristic, complemented
32	C₂×#20598	C₂²	1	2	yes	complemented
33	C₂×#17577	C₂²	1	2	yes	complemented
34	C₂×#17396	C₂²	1	2	yes	complemented
35	C₂×#17362	C₂²	1	1	yes	characteristic
36	#394070	C₂²	1	16	yes	complemented
37	C₂×#10732	C₂²	1	2	yes	complemented
38	#234958	C₂²	1	16	yes	complemented
39	#201446	C₂²	1	8	yes	complemented
40	#187961	C₂²	1	8	yes	complemented
41	#185643	C₂²	1	8	yes	complemented
42	C₂×#7942	C₂²	1	2	yes	complemented
43	C₂×#7942	C₂²	1	2	yes	complemented
44	#115640	C₂²	1	16	yes	complemented
45	#112557	C₂²	1	16	yes	complemented
46	#108194	C₂²	1	8	yes	complemented
47	C₂³×H(5,𝔽₂)	D₄	1	2	yes	—
48	C₂⁴×#41	C₂³	1	1	yes	characteristic
49	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
50	C₂⁵×D₄	C₂³	1	1	yes	characteristic
51	C₂³×AGL(1,ℤ/8)	C₂³	1	4	yes	complemented
52	C₂⁴×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×#3496	D₄	1	4	yes	—
54	C₂×#3496	C₂³	1	4	yes	—
55	C₂²×D₄²	D₄	1	4	yes	complemented
56	C₂²×D₄²	C₂³	1	4	yes	—
57	C₂²×#534	D₄	1	2	yes	—
58	C₂³×#127	D₄	1	1	yes	characteristic, complemented
59	C₂³×#127	D₄	1	4	yes	complemented
60	C₂³×#121	D₄	1	1	yes	characteristic, complemented
61	C₂³×#121	C₂³	1	1	yes	characteristic
62	C₂³×#121	C₂³	1	2	yes	—
63	C₂³×C₄×D₄	D₄	1	1	yes	characteristic, complemented
64	C₂³×C₄×D₄	C₂³	1	1	yes	characteristic
65	C₂×#3006	C₂³	1	4	yes	complemented
66	C₂²×#407	C₂³	1	2	yes	—
67	C₂²×#407	C₂³	1	1	yes	characteristic
68	#35250	C₂³	1	8	yes	complemented
69	#35001	D₄	1	8	yes	complemented
70	#35001	C₂³	1	8	yes	—
71	C₂×#2539	D₄	1	8	yes	complemented
72	#34736	D₄	1	4	yes	complemented
73	#34736	C₂³	1	4	yes	—
74	C₂×#2409	D₄	1	2	yes	complemented
75	C₂×#2433	D₄	1	8	yes	complemented
76	C₂×#2433	C₂³	1	8	yes	—
77	C₂×#2408	D₄	1	2	yes	complemented
78	C₂²×#390	D₄	1	4	yes	complemented
79	C₂×#2280	C₂³	1	1	yes	characteristic
80	C₂×#2280	C₂³	1	2	yes	—
81	C₄×#518	D₄	1	4	yes	complemented
82	C₄×#518	C₂³	1	4	yes	—
83	C₂×#2274	D₄	1	2	yes	complemented
84	C₂×#2273	D₄	1	2	yes	complemented
85	C₂²×C₄×C₂²⋊C₄	C₂³	1	2	yes	—
86	#21793	C₂³	1	8	yes	complemented
87	C₂×#1900	C₂³	1	8	yes	complemented
88	#20973	C₂³	1	4	yes	complemented
89	#20898	D₄	1	4	yes	complemented
90	#20898	C₂³	1	4	yes	—
91	C₂×#1870	D₄	1	4	yes	complemented
92	#20722	D₄	1	4	yes	complemented
93	#20722	C₂³	1	4	yes	—
94	C₂×#1799	D₄	1	4	yes	complemented
95	C₂×#1789	C₂³	1	8	yes	complemented
96	#20598	C₂³	1	4	yes	complemented
97	#20598	C₂³	1	8	yes	complemented
98	#17577	C₂³	1	8	yes	complemented
99	#17396	C₂³	1	8	yes	complemented
100	#17362	C₂³	1	4	yes	—
101	#10732	C₂³	1	8	yes	complemented
102	#7942	C₂³	1	8	yes	complemented
103	#7942	C₂³	1	8	yes	complemented
104	C₂²×H(5,𝔽₂)	C₂×D₄	1	8	yes	—
105	C₂³×#41	C₂×D₄	1	2	yes	complemented
106	C₂³×#41	C₂⁴	1	2	yes	—
107	C₂⁴×Q₈	C₂×D₄	1	1	yes	characteristic, complemented
108	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂⁴×D₄	C₂×D₄	1	4	yes	—
110	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
111	C₂⁴×D₄	C₂×D₄	1	4	yes	—
112	C₂⁴×D₄	C₂×D₄	1	2	yes	—
113	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
114	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
115	C₂⁴×D₄	C₂×D₄	1	2	yes	—
116	C₂⁴×D₄	C₂×D₄	1	4	yes	—
117	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic, complemented
118	C₂⁴×D₄	C₂×D₄	1	4	yes	complemented
119	C₂⁴×D₄	C₂⁴	1	2	yes	—
120	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂⁵×C₄	C₂⁴	1	1	yes	characteristic
122	C₂³×M₁₆	C₂×D₄	1	2	yes	complemented
123	C₂³×M₁₆	C₂×D₄	1	2	yes	complemented
124	C₂³×M₁₆	C₂⁴	1	2	yes	—
125	C₂×#534	C₂×D₄	1	8	yes	—
126	C₂²×#127	C₂×D₄	1	4	yes	complemented
127	C₂²×#127	C₂×D₄	1	16	yes	complemented
128	C₂²×#121	C₂×D₄	1	4	yes	—
129	C₂²×#121	C₂×D₄	1	4	yes	—
130	C₂²×#121	C₂×D₄	1	2	yes	—
131	C₂²×#121	C₂×D₄	1	4	yes	—
132	C₂²×#121	C₂×D₄	1	4	yes	—
133	C₂²×#121	C₂×D₄	1	4	yes	—
134	C₂²×#121	C₂×D₄	1	4	yes	complemented
135	C₂²×#121	C₂⁴	1	2	yes	—
136	C₂²×#121	C₂⁴	1	4	yes	—
137	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
138	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
139	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
140	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
141	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
142	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
143	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
144	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
145	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
146	C₂²×C₄×D₄	C₂×D₄	1	4	yes	complemented
147	C₂²×C₄×D₄	#41	1	2	yes	—
148	C₂²×C₄×D₄	#41	1	2	yes	—
149	C₂²×C₄×D₄	C₂⁴	1	2	yes	—
150	C₂³×C₄⋊C₄	C₂×D₄	1	2	yes	—
151	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
152	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
153	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
154	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
155	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
156	C₂³×C₄²	C₂×D₄	1	4	yes	—
157	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
158	C₂×#407	C₂×D₄	1	2	yes	complemented
159	C₂×#407	C₂⁴	1	4	yes	—
160	C₂×#407	C₂⁴	1	2	yes	—
161	C₂²×C₂²⋊C₈	C₂×D₄	1	2	yes	complemented
162	#2539	C₂×D₄	1	32	yes	complemented
163	#2409	C₂×D₄	1	8	yes	complemented
164	#2408	C₂×D₄	1	8	yes	complemented
165	C₂×#390	C₂×D₄	1	16	yes	complemented
166	#2280	C₂×D₄	1	2	yes	complemented
167	#2280	C₂×D₄	1	4	yes	complemented
168	#2280	C₂⁴	1	2	yes	—
169	#2280	C₂⁴	1	4	yes	—
170	C₂×#380	C₂×D₄	1	4	yes	complemented
171	C₂×#380	C₂×D₄	1	2	yes	complemented
172	#2274	C₂×D₄	1	8	yes	complemented
173	#2273	C₂×D₄	1	8	yes	complemented
174	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
175	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
176	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
177	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
178	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	4	yes	—
179	C₂×C₄×C₂²⋊C₄	C₂⁴	1	4	yes	—
180	#1870	C₂×D₄	1	16	yes	complemented
181	#1799	C₂×D₄	1	16	yes	complemented
182	C₂⁶	#120	1	1	yes	characteristic
183	C₂⁶	#120	1	4	yes	—
184	C₂⁶	C₂²×D₄	1	1	yes	characteristic
185	C₂³×Q₈	C₂²×D₄	1	2	yes	complemented
186	C₂³×D₄	#120	1	2	yes	—
187	C₂³×D₄	#120	1	2	yes	—
188	C₂³×D₄	#120	1	2	yes	—
189	C₂³×D₄	#120	1	2	yes	—
190	C₂³×D₄	#120	1	8	yes	—
191	C₂³×D₄	#120	1	8	yes	—
192	C₂³×D₄	#120	1	4	yes	—
193	C₂³×D₄	#120	1	2	yes	—
194	C₂³×D₄	#120	1	2	yes	—
195	C₂³×D₄	#120	1	2	yes	—
196	C₂³×D₄	#120	1	2	yes	—
197	C₂³×D₄	#120	1	2	yes	—
198	C₂³×D₄	#121	1	2	yes	—
199	C₂³×D₄	#121	1	2	yes	—
200	C₂³×D₄	C₂²×D₄	1	2	yes	—
201	C₂³×D₄	C₂²×D₄	1	8	yes	—
202	C₂³×D₄	C₂²×D₄	1	2	yes	—
203	C₂³×D₄	C₂²×D₄	1	8	yes	—
204	C₂³×D₄	C₂²×D₄	1	4	yes	—
205	C₂³×D₄	C₂²×D₄	1	2	yes	—
206	C₂³×D₄	C₂²×D₄	1	2	yes	—
207	C₂³×D₄	C₂²×D₄	1	2	yes	complemented
208	C₂⁴×C₄	#120	1	1	yes	characteristic
209	C₂⁴×C₄	#120	1	4	yes	—
210	C₂⁴×C₄	#120	1	2	yes	—
211	C₂⁴×C₄	#120	1	2	yes	—
212	C₂⁴×C₄	#120	1	1	yes	characteristic
213	C₂⁴×C₄	#120	1	2	yes	—
214	C₂⁴×C₄	#120	1	1	yes	characteristic
215	C₂⁴×C₄	#121	1	2	yes	—
216	C₂⁴×C₄	#121	1	2	yes	—
217	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
218	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
219	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
220	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
221	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
224	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
225	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
226	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
227	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
228	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
229	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
230	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
231	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
232	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
233	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
234	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
235	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
236	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
237	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
238	C₂⁴×C₄	H(5,𝔽₂)	1	2	yes	—
239	C₂⁴×C₄	C₂⁵	1	1	yes	characteristic
240	C₂²×M₁₆	C₂²×D₄	1	4	yes	complemented
241	C₂×#121	#120	1	8	yes	—
242	C₂×#121	C₂²×D₄	1	8	yes	—
243	C₂×C₄×D₄	#120	1	4	yes	—
244	C₂×C₄×D₄	#120	1	4	yes	—
245	C₂×C₄×D₄	#120	1	4	yes	—
246	C₂×C₄×D₄	#120	1	4	yes	—
247	C₂×C₄×D₄	#121	1	4	yes	—
248	C₂×C₄×D₄	#121	1	4	yes	—
249	C₂×C₄×D₄	#121	1	4	yes	—
250	C₂×C₄×D₄	#121	1	4	yes	—
251	C₂×C₄×D₄	#121	1	4	yes	—
252	C₂×C₄×D₄	#121	1	4	yes	—
253	C₂×C₄×D₄	#127	1	4	yes	—
254	C₂×C₄×D₄	#127	1	4	yes	—
255	C₂×C₄×D₄	C₂²×D₄	1	4	yes	—
256	C₂×C₄×D₄	C₂²×D₄	1	4	yes	—
257	C₂×C₄×D₄	C₂²×D₄	1	4	yes	—
258	C₂×C₄×D₄	C₂×#41	1	4	yes	—
259	C₂²×C₄⋊C₄	#120	1	4	yes	—
260	C₂²×C₄⋊C₄	#120	1	2	yes	—
261	C₂²×C₄⋊C₄	#120	1	2	yes	—
262	C₂²×C₄⋊C₄	#120	1	2	yes	—
263	C₂²×C₄⋊C₄	#121	1	2	yes	—
264	C₂²×C₄⋊C₄	#121	1	2	yes	—
265	C₂²×C₄⋊C₄	#127	1	2	yes	—
266	C₂²×C₄⋊C₄	#127	1	2	yes	—
267	C₂²×C₄⋊C₄	C₂²×D₄	1	4	yes	—
268	C₂²×C₄⋊C₄	C₂²×D₄	1	2	yes	—
269	C₂²×C₄⋊C₄	C₂²×D₄	1	2	yes	—
270	C₂²×C₂²⋊C₄	#120	1	8	yes	—
271	C₂²×C₂²⋊C₄	#120	1	2	yes	—
272	C₂²×C₂²⋊C₄	#120	1	2	yes	—
273	C₂²×C₂²⋊C₄	#120	1	2	yes	—
274	C₂²×C₂²⋊C₄	#120	1	2	yes	—
275	C₂²×C₂²⋊C₄	#121	1	2	yes	—
276	C₂²×C₂²⋊C₄	#121	1	2	yes	—
277	C₂²×C₂²⋊C₄	#121	1	2	yes	—
278	C₂²×C₂²⋊C₄	#121	1	2	yes	—
279	C₂²×C₂²⋊C₄	#127	1	2	yes	—
280	C₂²×C₂²⋊C₄	#127	1	2	yes	—
281	C₂²×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
282	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
283	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
284	C₂²×C₄²	#120	1	2	yes	—
285	C₂²×C₄²	#120	1	2	yes	—
286	C₂²×C₄²	#120	1	2	yes	—
287	C₂²×C₄²	#120	1	8	yes	—
288	C₂²×C₄²	#120	1	8	yes	—
289	C₂²×C₄²	#121	1	2	yes	—
290	C₂²×C₄²	#121	1	2	yes	—
291	C₂²×C₄²	C₂²×D₄	1	2	yes	—
292	C₂²×C₄²	C₂²×D₄	1	8	yes	—
293	C₂×C₂²⋊C₈	C₂²×D₄	1	4	yes	complemented
294	#380	C₂²×D₄	1	8	yes	complemented
295	#380	C₂²×D₄	1	4	yes	complemented
296	C₄×C₂²⋊C₄	#120	1	8	yes	—
297	C₄×C₂²⋊C₄	#120	1	8	yes	—
298	C₄×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
299	C₄×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
300	C₂⁵	#392	1	2	yes	—
301	C₂⁵	#392	1	2	yes	—
302	C₂⁵	C₂×#120	1	1	yes	characteristic
303	C₂⁵	C₂×#120	1	2	yes	—
304	C₂⁵	C₂×#120	1	2	yes	—
305	C₂⁵	C₂×#120	1	1	yes	characteristic
306	C₂⁵	C₂×#120	1	2	yes	—
307	C₂⁵	C₂×#120	1	1	yes	characteristic
308	C₂⁵	C₂×#120	1	8	yes	—
309	C₂⁵	C₂×#120	1	1	yes	characteristic
310	C₂⁵	C₂×#121	1	1	yes	characteristic
311	C₂⁵	C₂×#121	1	1	yes	characteristic
312	C₂⁵	#534	1	2	yes	—
313	C₂⁵	#534	1	2	yes	—
314	C₂⁵	C₂³×D₄	1	1	yes	characteristic
315	C₂²×D₄	#392	1	4	yes	—
316	C₂²×D₄	#392	1	4	yes	—
317	C₂²×D₄	#453	1	16	yes	—
318	C₂²×D₄	B(4,𝔽₂)	1	8	yes	complemented
319	C₂²×D₄	C₂×#120	1	4	yes	—
320	C₂²×D₄	C₂×#120	1	4	yes	—
321	C₂²×D₄	C₂×#120	1	4	yes	—
322	C₂²×D₄	C₂×#120	1	4	yes	—
323	C₂²×D₄	C₂×#121	1	4	yes	—
324	C₂³×C₄	#392	1	2	yes	—
325	C₂³×C₄	#392	1	2	yes	—
326	C₂³×C₄	#392	1	2	yes	—
327	C₂³×C₄	#392	1	2	yes	—
328	C₂³×C₄	#392	1	2	yes	—
329	C₂³×C₄	#392	1	4	yes	—
330	C₂³×C₄	#392	1	4	yes	—
331	C₂³×C₄	#392	1	2	yes	—
332	C₂³×C₄	C₂×#120	1	2	yes	—
333	C₂³×C₄	C₂×#120	1	1	yes	characteristic
334	C₂³×C₄	C₂×#120	1	2	yes	—
335	C₂³×C₄	C₂×#120	1	1	yes	characteristic
336	C₂³×C₄	C₂×#120	1	2	yes	—
337	C₂³×C₄	C₂×#120	1	1	yes	characteristic
338	C₂³×C₄	C₂×#120	1	1	yes	characteristic
339	C₂³×C₄	C₂×#120	1	2	yes	—
340	C₂³×C₄	C₂×#120	1	2	yes	—
341	C₂³×C₄	C₂×#120	1	1	yes	characteristic
342	C₂³×C₄	C₂×#120	1	1	yes	characteristic
343	C₂³×C₄	C₂×#120	1	8	yes	—
344	C₂³×C₄	C₂×#120	1	2	yes	—
345	C₂³×C₄	C₂×#120	1	4	yes	—
346	C₂³×C₄	C₂×#120	1	2	yes	—
347	C₂³×C₄	C₂×#120	1	1	yes	characteristic
348	C₂³×C₄	C₂×#120	1	2	yes	—
349	C₂³×C₄	C₂×#120	1	2	yes	—
350	C₂³×C₄	C₂×#120	1	2	yes	—
351	C₂³×C₄	C₂×#120	1	2	yes	—
352	C₂³×C₄	C₂×#120	1	1	yes	characteristic
353	C₂³×C₄	C₂×#120	1	1	yes	characteristic
354	C₂³×C₄	C₂×#120	1	1	yes	characteristic
355	C₂³×C₄	C₂×#120	1	1	yes	characteristic
356	C₂³×C₄	C₂×#120	1	1	yes	characteristic
357	C₂³×C₄	C₂×#121	1	1	yes	characteristic
358	C₂³×C₄	C₂×#121	1	1	yes	characteristic
359	C₂³×C₄	C₂×#121	1	4	yes	—
360	C₂³×C₄	C₂×#121	1	1	yes	characteristic
361	C₂³×C₄	C₂×#121	1	1	yes	characteristic
362	C₂³×C₄	C₂×#121	1	1	yes	characteristic
363	C₂³×C₄	C₂×#121	1	1	yes	characteristic
364	C₂³×C₄	C₂×#121	1	1	yes	characteristic
365	C₂³×C₄	C₂×#121	1	1	yes	characteristic
366	C₂³×C₄	C₂×#121	1	1	yes	characteristic
367	C₂³×C₄	C₂×#121	1	1	yes	characteristic
368	C₂³×C₄	C₂×#127	1	1	yes	characteristic
369	C₂³×C₄	C₂×#127	1	1	yes	characteristic
370	C₂³×C₄	C₂×#127	1	1	yes	characteristic
371	C₂³×C₄	C₂×#127	1	1	yes	characteristic
372	C₂³×C₄	#534	1	1	yes	characteristic
373	C₂³×C₄	#534	1	1	yes	characteristic
374	C₂³×C₄	#534	1	1	yes	characteristic
375	C₂³×C₄	#534	1	1	yes	characteristic
376	C₂³×C₄	#534	1	2	yes	—
377	C₂³×C₄	#534	1	2	yes	—
378	C₂³×C₄	#534	1	2	yes	—
379	C₂³×C₄	#534	1	2	yes	—
380	C₂³×C₄	#535	1	2	yes	—
381	C₂³×C₄	#535	1	2	yes	—
382	C₂³×C₄	#535	1	2	yes	—
383	C₂³×C₄	#535	1	2	yes	—
384	C₂³×C₄	D₄²	1	2	yes	—
385	C₂³×C₄	D₄²	1	2	yes	—
386	C₂³×C₄	D₄²	1	2	yes	—
387	C₂³×C₄	D₄²	1	2	yes	—
388	C₂³×C₄	D₄²	1	2	yes	—
389	C₂³×C₄	D₄²	1	2	yes	—
390	C₂³×C₄	D₄²	1	2	yes	—
391	C₂³×C₄	D₄²	1	2	yes	—
392	C₂³×C₄	D₄²	1	2	yes	—
393	C₂³×C₄	D₄²	1	2	yes	—
394	C₂³×C₄	D₄²	1	2	yes	—
395	C₂³×C₄	D₄²	1	2	yes	—
396	C₂³×C₄	D₄²	1	2	yes	—
397	C₂³×C₄	D₄²	1	2	yes	—
398	C₂³×C₄	D₄²	1	2	yes	—
399	C₂³×C₄	D₄²	1	2	yes	—
400	C₂³×C₄	D₄²	1	2	yes	—
401	C₂³×C₄	D₄²	1	2	yes	—
402	C₂³×C₄	D₄²	1	2	yes	—
403	C₂³×C₄	D₄²	1	2	yes	—
404	C₂³×C₄	D₄²	1	2	yes	—
405	C₂³×C₄	D₄²	1	2	yes	—
406	C₂³×C₄	D₄²	1	2	yes	—
407	C₂³×C₄	D₄²	1	2	yes	—
408	C₂³×C₄	#546	1	2	yes	—
409	C₂³×C₄	#546	1	2	yes	—
410	C₂³×C₄	#546	1	2	yes	—
411	C₂³×C₄	#546	1	2	yes	—
412	C₂³×C₄	#546	1	2	yes	—
413	C₂³×C₄	#546	1	2	yes	—
414	C₂³×C₄	#546	1	2	yes	—
415	C₂³×C₄	#546	1	2	yes	—
416	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
417	C₂³×C₄	C₂³×D₄	1	2	yes	—
418	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
419	C₂³×C₄	C₂³×D₄	1	2	yes	—
420	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
421	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
422	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
423	C₂³×C₄	C₂²×#41	1	1	yes	characteristic
424	C₂³×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
425	C₂³×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
426	C₂×C₄⋊C₄	#392	1	4	yes	—
427	C₂×C₄⋊C₄	#392	1	4	yes	—
428	C₂×C₄⋊C₄	C₂×#120	1	4	yes	—
429	C₂×C₄⋊C₄	C₂×#121	1	4	yes	—
430	C₂×C₄⋊C₄	C₂×#127	1	4	yes	—
431	C₂×C₂²⋊C₄	#392	1	4	yes	—
432	C₂×C₂²⋊C₄	#392	1	4	yes	—
433	C₂×C₂²⋊C₄	#392	1	4	yes	—
434	C₂×C₂²⋊C₄	#392	1	4	yes	—
435	C₂×C₂²⋊C₄	C₂×#120	1	4	yes	—
436	C₂×C₂²⋊C₄	C₂×#121	1	4	yes	—
437	C₂×C₂²⋊C₄	C₂×#121	1	4	yes	—
438	C₂×C₂²⋊C₄	C₂×#127	1	4	yes	—
439	C₂×C₄²	#392	1	4	yes	—
440	C₂×C₄²	#392	1	4	yes	—
441	C₂×C₄²	C₂×#120	1	4	yes	—
442	C₂×C₄²	C₂×#120	1	16	yes	—
443	C₂×C₄²	C₂×#121	1	4	yes	—
444	C₂⁴	#2002	1	8	yes	complemented
445	C₂⁴	C₂×#392	1	1	yes	characteristic
446	C₂⁴	C₂×#392	1	1	yes	characteristic
447	C₂⁴	C₂×#392	1	4	yes	—
448	C₂⁴	C₂×#392	1	1	yes	characteristic
449	C₂⁴	C₂×#392	1	1	yes	characteristic
450	C₂⁴	#2403	1	2	yes	—
451	C₂⁴	#2403	1	2	yes	—
452	C₂⁴	#2403	1	2	yes	—
453	C₂⁴	#2403	1	2	yes	—
454	C₂⁴	#2410	1	2	yes	—
455	C₂⁴	#2410	1	2	yes	—
456	C₂⁴	#2410	1	2	yes	—
457	C₂⁴	#2410	1	2	yes	—
458	C₂⁴	#2613	1	1	yes	characteristic
459	C₂⁴	#2613	1	1	yes	characteristic
460	C₂⁴	#2613	1	2	yes	—
461	C₂⁴	#2846	1	1	yes	characteristic
462	C₂⁴	#2846	1	2	yes	—
463	C₂⁴	#2846	1	1	yes	characteristic
464	C₂⁴	C₂×#453	1	8	yes	—
465	C₂⁴	C₂²×#120	1	2	yes	—
466	C₂⁴	C₂²×#120	1	1	yes	characteristic
467	C₂⁴	C₂²×#120	1	1	yes	characteristic
468	C₂⁴	C₂²×#121	1	1	yes	characteristic
469	C₂⁴	C₂×#534	1	2	yes	—
470	C₂×D₄	#2002	1	16	yes	complemented
471	C₂×D₄	C₂×#453	1	16	yes	—
472	C₂×D₄	C₂×B(4,𝔽₂)	1	8	yes	complemented
473	C₂²×C₄	#2002	1	8	yes	complemented
474	C₂²×C₄	C₂×#392	1	4	yes	—
475	C₂²×C₄	C₂×#392	1	1	yes	characteristic
476	C₂²×C₄	C₂×#392	1	1	yes	characteristic
477	C₂²×C₄	C₂×#392	1	4	yes	—
478	C₂²×C₄	C₂×#392	1	4	yes	—
479	C₂²×C₄	C₂×#392	1	1	yes	characteristic
480	C₂²×C₄	C₂×#392	1	1	yes	characteristic
481	C₂²×C₄	#2403	1	1	yes	characteristic
482	C₂²×C₄	#2403	1	1	yes	characteristic
483	C₂²×C₄	#2403	1	1	yes	characteristic
484	C₂²×C₄	#2403	1	1	yes	characteristic
485	C₂²×C₄	#2403	1	1	yes	characteristic
486	C₂²×C₄	#2403	1	2	yes	—
487	C₂²×C₄	#2403	1	2	yes	—
488	C₂²×C₄	#2403	1	1	yes	characteristic
489	C₂²×C₄	#2403	1	2	yes	—
490	C₂²×C₄	#2403	1	2	yes	—
491	C₂²×C₄	#2403	1	2	yes	—
492	C₂²×C₄	#2403	1	2	yes	—
493	C₂²×C₄	#2403	1	2	yes	—
494	C₂²×C₄	#2403	1	2	yes	—
495	C₂²×C₄	#2403	1	1	yes	characteristic
496	C₂²×C₄	#2403	1	1	yes	characteristic
497	C₂²×C₄	#2408	1	2	yes	—
498	C₂²×C₄	#2408	1	2	yes	—
499	C₂²×C₄	#2408	1	2	yes	—
500	C₂²×C₄	#2408	1	2	yes	—
501	C₂²×C₄	#2408	1	2	yes	—
502	C₂²×C₄	#2408	1	2	yes	—
503	C₂²×C₄	#2408	1	2	yes	—
504	C₂²×C₄	#2408	1	2	yes	—
505	C₂²×C₄	#2410	1	1	yes	characteristic
506	C₂²×C₄	#2410	1	1	yes	characteristic
507	C₂²×C₄	#2410	1	1	yes	characteristic
508	C₂²×C₄	#2410	1	1	yes	characteristic
509	C₂²×C₄	#2410	1	1	yes	characteristic
510	C₂²×C₄	#2410	1	2	yes	—
511	C₂²×C₄	#2410	1	2	yes	—
512	C₂²×C₄	#2410	1	1	yes	characteristic
513	C₂²×C₄	#2410	1	2	yes	—
514	C₂²×C₄	#2410	1	2	yes	—
515	C₂²×C₄	#2410	1	2	yes	—
516	C₂²×C₄	#2410	1	2	yes	—
517	C₂²×C₄	#2410	1	2	yes	—
518	C₂²×C₄	#2410	1	2	yes	—
519	C₂²×C₄	#2410	1	1	yes	characteristic
520	C₂²×C₄	#2410	1	1	yes	characteristic
521	C₂²×C₄	#2433	1	2	yes	—
522	C₂²×C₄	#2433	1	2	yes	—
523	C₂²×C₄	#2433	1	2	yes	—
524	C₂²×C₄	#2433	1	2	yes	—
525	C₂²×C₄	#2433	1	2	yes	—
526	C₂²×C₄	#2433	1	2	yes	—
527	C₂²×C₄	#2433	1	2	yes	—
528	C₂²×C₄	#2433	1	2	yes	—
529	C₂²×C₄	#2613	1	1	yes	characteristic
530	C₂²×C₄	#2613	1	1	yes	characteristic
531	C₂²×C₄	#2613	1	2	yes	—
532	C₂²×C₄	#2846	1	1	yes	characteristic
533	C₂²×C₄	#2846	1	2	yes	—
534	C₂²×C₄	#2846	1	1	yes	characteristic
535	C₂²×C₄	C₂×B(4,𝔽₂)	1	4	yes	—
536	C₂²×C₄	#3026	1	4	yes	—
537	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
538	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
539	C₂²×C₄	C₂²×#120	1	2	yes	—
540	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
541	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
542	C₂²×C₄	C₂²×#121	1	1	yes	characteristic
543	C₂²×C₄	C₂²×#127	1	1	yes	characteristic
544	C₂²×C₄	C₂×#534	1	1	yes	characteristic
545	C₂²×C₄	C₂×#534	1	1	yes	characteristic
546	C₂²×C₄	C₂×#535	1	2	yes	—
547	C₂²×C₄	C₂×D₄²	1	2	yes	—
548	C₂²×C₄	C₂×D₄²	1	2	yes	—
549	C₂²×C₄	C₂×D₄²	1	2	yes	—
550	C₂²×C₄	C₂×D₄²	1	2	yes	—
551	C₂²×C₄	C₂×D₄²	1	2	yes	—
552	C₂²×C₄	C₂×D₄²	1	2	yes	—
553	C₂²×C₄	C₂×#546	1	2	yes	—
554	C₂²×C₄	C₂×#546	1	2	yes	—
555	C₂³	#14715	1	2	yes	—
556	C₂³	#14715	1	1	yes	characteristic
557	C₂³	#14715	1	1	yes	characteristic
558	C₂³	#14715	1	2	yes	—
559	C₂³	#14715	1	1	yes	characteristic
560	C₂³	#14715	1	1	yes	characteristic
561	C₂³	C₂×#2002	1	4	yes	—
562	C₂³	C₂×#2002	1	4	yes	—
563	C₂³	C₂×#2002	1	8	yes	complemented
564	C₂³	#23906	1	8	yes	—
565	C₂³	#23906	1	8	yes	—
566	C₂³	#32898	1	4	yes	—
567	C₂³	C₂²×#392	1	1	yes	characteristic
568	C₂³	C₂×#2613	1	1	yes	characteristic
569	C₂³	C₂×#2403	1	2	yes	—
570	C₂³	C₂×#2410	1	2	yes	—
571	C₂³	C₂×#2846	1	1	yes	characteristic
572	C₂³	C₂²×#453	1	4	yes	—
573	C₂×C₄	#14715	1	1	yes	characteristic
574	C₂×C₄	#14715	1	1	yes	characteristic
575	C₂×C₄	#14715	1	2	yes	—
576	C₂×C₄	#14715	1	2	yes	—
577	C₂×C₄	#14715	1	1	yes	characteristic
578	C₂×C₄	#14715	1	1	yes	characteristic
579	C₂×C₄	C₂×#2002	1	8	yes	complemented
580	C₂×C₄	#23900	1	2	yes	—
581	C₂×C₄	#23900	1	2	yes	—
582	C₂×C₄	#23903	1	2	yes	—
583	C₂×C₄	#23903	1	2	yes	—
584	C₂×C₄	#24404	1	4	yes	—
585	C₂×C₄	#24404	1	4	yes	—
586	C₂×C₄	#32898	1	4	yes	—
587	C₂×C₄	C₂×#2403	1	1	yes	characteristic
588	C₂×C₄	C₂×#2403	1	1	yes	characteristic
589	C₂×C₄	C₂×#2410	1	1	yes	characteristic
590	C₂×C₄	C₂×#2410	1	1	yes	characteristic
591	C₂×C₄	C₂×#2408	1	2	yes	—
592	C₂×C₄	C₂×#2433	1	2	yes	—
593	C₂×C₄	C₂²×B(4,𝔽₂)	1	2	yes	—
594	C₂×C₄	C₂×#3026	1	2	yes	—
595	C₂²	#393958	1	2	yes	—
596	C₂²	#393958	1	4	yes	—
597	C₂²	#393958	1	2	yes	—
598	C₂²	#394066	1	2	yes	—
599	C₂²	#394066	1	2	yes	—
600	C₂²	#394086	1	2	yes	—
601	C₂²	C₂×#14715	1	1	yes	characteristic
602	C₂²	C₂²×#2002	1	2	yes	—
603	C₂²	C₂×#23906	1	4	yes	—
604	C₂²	C₂×#32898	1	2	yes	—
605	C₄	#394066	1	2	yes	—
606	C₄	#394066	1	2	yes	—
607	C₄	C₂×#23900	1	1	yes	characteristic
608	C₄	C₂×#23903	1	1	yes	characteristic
609	C₄	C₂×#24404	1	2	yes	—
610	C₂	#72231778	1	4	yes	direct product factor
611	C₂	C₂×#393958	1	1	yes	characteristic
612	C₂	C₂×#394066	1	1	yes	characteristic
613	C₂	C₂×#394086	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14715
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×#234958	C₂	semidirect
2. C₂×#112557	C₂	semidirect
3. C₂×#187961	C₂	semidirect
4. C₂×#201446	C₂	semidirect
5. C₂×#108194	C₂	semidirect
6. C₂×#115640	C₂	semidirect
7. C₂×#394070	C₂	semidirect
8. C₂×#6487644	C₂	semidirect
9. C₂×#7631334	C₂	semidirect
10. C₂×#6437510	C₂	semidirect
11. #72231778	C₂	direct (N × H)
12. C₂×#185643	C₂	semidirect
13. C₂⁴×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂²×#3006	C₂²	semidirect
17. #7631334	C₂²	semidirect
18. C₂×#35250	C₂²	semidirect
23. #6487644	C₂²	semidirect
24. #6437510	C₂²	semidirect
25. C₂×#21793	C₂²	semidirect
26. C₂²×#1900	C₂²	semidirect
27. C₂×#20973	C₂²	semidirect
30. C₂²×#1789	C₂²	semidirect
31. C₂×#20598	C₂²	semidirect
32. C₂×#20598	C₂²	semidirect
33. C₂×#17577	C₂²	semidirect
34. C₂×#17396	C₂²	semidirect
36. #394070	C₂²	semidirect
37. C₂×#10732	C₂²	semidirect
38. #234958	C₂²	semidirect
39. #201446	C₂²	semidirect
40. #187961	C₂²	semidirect
41. #185643	C₂²	semidirect
42. C₂×#7942	C₂²	semidirect
43. C₂×#7942	C₂²	semidirect
44. #115640	C₂²	semidirect
45. #112557	C₂²	semidirect
46. #108194	C₂²	semidirect
49. C₂⁵×D₄	D₄	semidirect
51. C₂³×AGL(1,ℤ/8)	C₂³	semidirect
55. C₂²×D₄²	D₄	semidirect
58. C₂³×#127	D₄	semidirect
59. C₂³×#127	D₄	semidirect
60. C₂³×#121	D₄	semidirect
63. C₂³×C₄×D₄	D₄	semidirect
65. C₂×#3006	C₂³	semidirect
68. #35250	C₂³	semidirect
69. #35001	D₄	semidirect
71. C₂×#2539	D₄	semidirect
72. #34736	D₄	semidirect
74. C₂×#2409	D₄	semidirect
75. C₂×#2433	D₄	semidirect
77. C₂×#2408	D₄	semidirect
78. C₂²×#390	D₄	semidirect
81. C₄×#518	D₄	semidirect
83. C₂×#2274	D₄	semidirect
84. C₂×#2273	D₄	semidirect
86. #21793	C₂³	semidirect
87. C₂×#1900	C₂³	semidirect
88. #20973	C₂³	semidirect
89. #20898	D₄	semidirect
91. C₂×#1870	D₄	semidirect
92. #20722	D₄	semidirect
94. C₂×#1799	D₄	semidirect
95. C₂×#1789	C₂³	semidirect
96. #20598	C₂³	semidirect
97. #20598	C₂³	semidirect
98. #17577	C₂³	semidirect
99. #17396	C₂³	semidirect
101. #10732	C₂³	semidirect
102. #7942	C₂³	semidirect
103. #7942	C₂³	semidirect
105. C₂³×#41	C₂×D₄	semidirect
107. C₂⁴×Q₈	C₂×D₄	semidirect
117. C₂⁴×D₄	C₂×D₄	semidirect
118. C₂⁴×D₄	C₂×D₄	semidirect
122. C₂³×M₁₆	C₂×D₄	semidirect
123. C₂³×M₁₆	C₂×D₄	semidirect
126. C₂²×#127	C₂×D₄	semidirect
127. C₂²×#127	C₂×D₄	semidirect
134. C₂²×#121	C₂×D₄	semidirect
146. C₂²×C₄×D₄	C₂×D₄	semidirect
158. C₂×#407	C₂×D₄	semidirect
161. C₂²×C₂²⋊C₈	C₂×D₄	semidirect
162. #2539	C₂×D₄	semidirect
163. #2409	C₂×D₄	semidirect
164. #2408	C₂×D₄	semidirect
165. C₂×#390	C₂×D₄	semidirect
166. #2280	C₂×D₄	semidirect
167. #2280	C₂×D₄	semidirect
170. C₂×#380	C₂×D₄	semidirect
171. C₂×#380	C₂×D₄	semidirect
172. #2274	C₂×D₄	semidirect
173. #2273	C₂×D₄	semidirect
180. #1870	C₂×D₄	semidirect
181. #1799	C₂×D₄	semidirect
185. C₂³×Q₈	C₂²×D₄	semidirect
207. C₂³×D₄	C₂²×D₄	semidirect
240. C₂²×M₁₆	C₂²×D₄	semidirect
293. C₂×C₂²⋊C₈	C₂²×D₄	semidirect
294. #380	C₂²×D₄	semidirect
295. #380	C₂²×D₄	semidirect
318. C₂²×D₄	B(4,𝔽₂)	semidirect
444. C₂⁴	#2002	semidirect
470. C₂×D₄	#2002	semidirect
472. C₂×D₄	C₂×B(4,𝔽₂)	semidirect
473. C₂²×C₄	#2002	semidirect
563. C₂³	C₂×#2002	semidirect
579. C₂×C₄	C₂×#2002	semidirect
610. C₂	#72231778	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×#234958	2
C₂×#112557	2
C₂×#187961	2
C₂×#201446	2
C₂×#108194	2
C₂×#115640	2
C₂×#394070	2
C₂×#6487644	2
C₂×#7631334	2
C₂×#6437510	2
#72231778	2
C₂×#185643	2

Subgroups — C2×#72231778 #72260679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72231778 #72260679