Subgroups — C₂×#72216528 #72251458

Order 4096.

At least 15687 conjugacy classes of subgroups; only the 1603 normal subgroups are shown.

The subgroup lattice has 597 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×#72216516	C₂	1	2	yes	maximal, complemented
2	C₂×#72216511	C₂	1	2	yes	maximal, complemented
3	C₂×#72216523	C₂	1	2	yes	maximal, complemented
4	C₂×#72216507	C₂	1	2	yes	maximal, complemented
5	C₂×#72213409	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#72213452	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#72213429	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#72215536	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#72213443	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#152071	C₂²	1	2	yes	complemented
11	C₂×#94424	C₂²	1	2	yes	complemented
12	C₂×#94424	C₂²	1	2	yes	complemented
13	C₂×#152047	C₂²	1	2	yes	complemented
14	C₂×#133350	C₂²	1	2	yes	—
15	C₂×#133350	C₂²	1	2	yes	—
16	C₂×#94425	C₂²	1	2	yes	complemented
17	C₂×#94425	C₂²	1	2	yes	complemented
18	C₂×#72213685	C₂	1	1	yes	maximal, characteristic
19	C₂×#246930	C₂²	1	2	yes	—
20	C₂×#246957	C₄	1	4	yes	complemented
21	C₂×#246927	C₂²	1	2	yes	—
22	C₂×#72210566	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂×#386041	C₂²	1	1	yes	characteristic, complemented
24	C₂×#391879	C₂²	1	1	yes	characteristic, complemented
25	C₂×#386039	C₂²	1	1	yes	characteristic, complemented
26	C₂×#133356	C₂²	1	2	yes	—
27	C₂×#133356	C₂²	1	2	yes	—
28	C₂×#493370	C₂²	1	1	yes	characteristic
29	C₂×#493385	C₂²	1	1	yes	characteristic
30	C₂×#200876	C₄	1	2	yes	complemented
31	C₂×#359513	C₂²	1	1	yes	characteristic
32	C₂×#6348967	C₄	1	2	yes	complemented
33	C₂×#359509	C₂²	1	1	yes	characteristic
34	C₂×#187811	C₂²	1	1	yes	characteristic
35	C₂×#10575547	C₂²	1	1	yes	characteristic
36	C₂²×#52004	C₂²	1	1	yes	characteristic
37	C₂×#375718	C₂²	1	1	yes	characteristic, complemented
38	#72216528	C₂	1	16	yes	maximal, direct product factor
39	#72216523	C₂²	1	16	yes	complemented
40	#72216516	C₂²	1	16	yes	complemented
41	#72216511	C₂²	1	16	yes	complemented
42	#72216507	C₂²	1	16	yes	complemented
43	#72215536	C₂²	1	8	yes	complemented
44	#72213685	C₄	1	8	yes	complemented
45	#72213685	C₂²	1	8	yes	—
46	#72213452	C₂²	1	8	yes	complemented
47	#72213443	C₂²	1	8	yes	complemented
48	#72213429	C₂²	1	8	yes	complemented
49	#72213409	C₂²	1	8	yes	complemented
50	#72210566	C₂²	1	8	yes	complemented
51	C₂×#62814	D₄	1	1	yes	characteristic
52	C₂³×D₄²	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂³×#535	C₂³	1	1	yes	characteristic
54	#10575547	C₂×C₄	1	4	yes	—
55	#10575547	D₄	1	4	yes	—
56	#10575547	D₄	1	4	yes	—
57	#10575547	C₂³	1	4	yes	—
58	C₂×#60806	C₂×C₄	1	2	yes	—
59	C₂×#60806	D₄	1	2	yes	—
60	C₂×#60799	D₄	1	2	yes	—
61	C₂×#60798	C₂×C₄	1	2	yes	—
62	C₂×#60644	D₄	1	1	yes	characteristic
63	C₂×#60644	C₂³	1	1	yes	characteristic
64	C₂×#60619	D₄	1	1	yes	characteristic
65	C₂×#60619	C₂³	1	1	yes	characteristic
66	C₂×#60124	D₄	1	1	yes	characteristic
67	C₂×#60064	D₄	1	1	yes	characteristic
68	C₂×#52004	C₂×C₄	1	4	yes	complemented
69	C₂×#52004	C₂×C₄	1	2	yes	complemented
70	C₂×#52004	C₂³	1	2	yes	—
71	C₂×#52004	C₂³	1	4	yes	—
72	C₂²×#2380	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×#34171	D₄	1	1	yes	characteristic
74	C₂²×#1783	D₄	1	1	yes	characteristic
75	C₂×#20584	D₄	1	1	yes	characteristic
76	#6348967	C₂×C₄	1	16	yes	complemented
77	C₂×#18529	C₂×C₄	1	4	yes	complemented
78	C₂×#18524	C₂×C₄	1	4	yes	complemented
79	C₂×#17741	C₂×C₄	1	2	yes	complemented
80	#493385	C₂³	1	4	yes	—
81	C₂×#12703	C₂³	1	2	yes	—
82	#493370	C₂³	1	4	yes	—
83	C₂×#12696	C₂³	1	2	yes	—
84	#391879	C₂³	1	4	yes	complemented
85	#386041	C₂³	1	4	yes	complemented
86	#386039	C₂³	1	4	yes	complemented
87	#375718	C₂³	1	4	yes	complemented
88	#359513	D₄	1	4	yes	—
89	#359513	C₂³	1	4	yes	—
90	#359509	D₄	1	4	yes	—
91	#359509	C₂³	1	4	yes	—
92	C₂×#10989	D₄	1	1	yes	characteristic
93	C₂×#10989	D₄	1	1	yes	characteristic
94	C₂×#10989	C₂³	1	1	yes	characteristic
95	C₂×#10989	C₂³	1	1	yes	characteristic
96	C₂×#10987	D₄	1	1	yes	characteristic
97	C₂×#10987	D₄	1	1	yes	characteristic
98	C₂×#10987	C₂³	1	1	yes	characteristic
99	C₂×#10987	C₂³	1	1	yes	characteristic
100	#246957	C₂×C₄	1	32	yes	complemented
101	#246930	C₂×C₄	1	8	yes	complemented
102	#246930	C₂³	1	8	yes	—
103	#246927	C₂×C₄	1	8	yes	complemented
104	#246927	C₂³	1	8	yes	—
105	#200876	C₂×C₄	1	16	yes	complemented
106	C₂×#9428	C₂×C₄	1	4	yes	complemented
107	C₂×#9425	C₂×C₄	1	4	yes	complemented
108	#187811	C₂×C₄	1	4	yes	complemented
109	#187811	C₂³	1	4	yes	—
110	C₂×#8522	C₂×C₄	1	2	yes	complemented
111	C₂×#8522	C₂³	1	2	yes	—
112	#152071	C₂³	1	8	yes	complemented
113	#152047	C₂³	1	8	yes	complemented
114	#133356	C₂³	1	8	yes	—
115	#133356	C₂³	1	8	yes	—
116	#133350	C₂³	1	8	yes	—
117	#133350	C₂³	1	8	yes	—
118	#94425	C₂³	1	8	yes	complemented
119	#94425	C₂³	1	8	yes	complemented
120	#94424	C₂³	1	8	yes	complemented
121	#94424	C₂³	1	8	yes	complemented
122	C₂⁴×#41	C₂×D₄	1	1	yes	characteristic
123	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
124	#62814	C₂²⋊C₄	1	4	yes	—
125	#62814	C₂×D₄	1	4	yes	—
126	C₂²×D₄²	C₂²×C₄	1	2	yes	—
127	C₂²×D₄²	C₂²×C₄	1	4	yes	complemented
128	C₂×#3455	C₂²⋊C₄	1	2	yes	—
129	C₂×#3458	C₂×D₄	1	1	yes	characteristic
130	C₂×#3449	C₂×D₄	1	1	yes	characteristic
131	C₂²×#535	C₂²×C₄	1	1	yes	characteristic
132	C₂²×#535	C₂²×C₄	1	2	yes	—
133	C₂²×#535	C₂×D₄	1	1	yes	characteristic
134	C₂²×#535	C₂×D₄	1	2	yes	—
135	C₂²×#535	C₂×D₄	1	1	yes	characteristic
136	C₂²×#535	C₂×D₄	1	2	yes	—
137	C₂²×#535	C₂×D₄	1	1	yes	characteristic
138	C₂²×#535	C₂×D₄	1	2	yes	—
139	C₂²×#535	C₂×D₄	1	1	yes	characteristic
140	C₂²×#535	C₂×D₄	1	2	yes	—
141	C₂²×#535	#41	1	1	yes	characteristic
142	C₂²×#535	#41	1	2	yes	—
143	C₂²×#535	#41	1	1	yes	characteristic
144	C₂²×#535	#41	1	2	yes	—
145	C₂²×#535	C₂⁴	1	1	yes	characteristic
146	C₂²×#535	C₂⁴	1	2	yes	—
147	C₂×#3444	C₂²⋊C₄	1	2	yes	—
148	C₂³×#127	C₂²×C₄	1	1	yes	characteristic
149	C₂³×#127	#41	1	1	yes	characteristic
150	C₂³×#124	C₂×D₄	1	1	yes	characteristic
151	C₂³×#124	#41	1	1	yes	characteristic
152	C₂³×#117	C₂×D₄	1	1	yes	characteristic
153	#60806	C₂²⋊C₄	1	8	yes	—
154	#60806	C₂²⋊C₄	1	8	yes	—
155	#60806	C₂²×C₄	1	8	yes	—
156	#60806	C₂×D₄	1	8	yes	—
157	#60799	C₂²⋊C₄	1	8	yes	—
158	#60799	C₂×D₄	1	8	yes	—
159	#60798	C₂²⋊C₄	1	8	yes	—
160	#60798	C₂²×C₄	1	8	yes	—
161	#60644	C₂²⋊C₄	1	4	yes	—
162	#60644	C₂²×C₄	1	2	yes	—
163	#60644	C₂×D₄	1	2	yes	—
164	#60644	C₂×D₄	1	4	yes	—
165	#60644	C₂×D₄	1	2	yes	—
166	#60644	C₂⁴	1	2	yes	—
167	C₂×#3153	C₂²⋊C₄	1	2	yes	—
168	C₂×#3153	C₂×D₄	1	2	yes	—
169	C₂×#3151	C₂²⋊C₄	1	2	yes	—
170	C₂×#3151	C₂²×C₄	1	2	yes	—
171	#60619	C₂²⋊C₄	1	4	yes	—
172	#60619	C₂²×C₄	1	2	yes	—
173	#60619	C₂×D₄	1	2	yes	—
174	#60619	C₂×D₄	1	4	yes	—
175	#60619	C₂×D₄	1	2	yes	—
176	#60619	C₂⁴	1	2	yes	—
177	C₂×#3140	C₂²⋊C₄	1	2	yes	—
178	C₂×#3140	C₂×D₄	1	2	yes	—
179	C₂×#3138	C₂²⋊C₄	1	2	yes	—
180	C₂×#3138	C₂²×C₄	1	2	yes	—
181	#60124	C₂²⋊C₄	1	4	yes	—
182	#60124	C₂×D₄	1	4	yes	—
183	C₂×#2921	C₂²⋊C₄	1	2	yes	—
184	C₂×#2920	C₂²⋊C₄	1	2	yes	—
185	C₂³×#100	C₂×D₄	1	1	yes	characteristic
186	C₂³×#100	C₂×D₄	1	1	yes	characteristic
187	C₂²×#410	C₂×D₄	1	1	yes	characteristic
188	C₂²×#410	C₂×D₄	1	1	yes	characteristic
189	#60064	C₂×D₄	1	4	yes	—
190	C₂×#2879	C₂×D₄	1	1	yes	characteristic
191	C₂×#2879	C₂×D₄	1	1	yes	characteristic
192	C₂×#2873	C₂×D₄	1	1	yes	characteristic
193	C₂×#2873	C₂×D₄	1	1	yes	characteristic
194	#52004	C₂²×C₄	1	4	yes	—
195	#52004	C₂²×C₄	1	8	yes	complemented
196	#52004	C₂⁴	1	4	yes	—
197	C₂×#2380	C₂²×C₄	1	2	yes	—
198	C₂×#2380	C₂²×C₄	1	4	yes	complemented
199	#34171	C₂×D₄	1	4	yes	—
200	C₂×#2309	C₂×D₄	1	1	yes	characteristic
201	C₂×#2309	C₂×D₄	1	1	yes	characteristic
202	C₂×#1783	C₂×D₄	1	4	yes	—
203	#20584	C₂×D₄	1	4	yes	—
204	#18529	C₂²×C₄	1	16	yes	complemented
205	#18524	C₂²×C₄	1	16	yes	complemented
206	#17741	C₂²×C₄	1	8	yes	complemented
207	C₂×#1555	C₂²×C₄	1	4	yes	—
208	#12703	C₂⁴	1	4	yes	—
209	#12696	C₂⁴	1	4	yes	—
210	#10989	C₂×D₄	1	4	yes	—
211	#10989	C₂×D₄	1	2	yes	—
212	#10989	C₂×D₄	1	2	yes	—
213	#10989	C₂×D₄	1	4	yes	—
214	#10989	C₂⁴	1	2	yes	—
215	#10989	C₂⁴	1	2	yes	—
216	#10987	C₂×D₄	1	4	yes	—
217	#10987	C₂×D₄	1	2	yes	—
218	#10987	C₂×D₄	1	2	yes	—
219	#10987	C₂×D₄	1	4	yes	—
220	#10987	C₂⁴	1	2	yes	—
221	#10987	C₂⁴	1	2	yes	—
222	#9428	C₂²×C₄	1	16	yes	complemented
223	#9425	C₂²×C₄	1	16	yes	complemented
224	#8522	C₂²×C₄	1	4	yes	—
225	#8522	C₂²×C₄	1	8	yes	complemented
226	#8522	C₂⁴	1	4	yes	—
227	C₂³×#41	C₄×D₄	1	1	yes	characteristic
228	C₂³×#41	C₄×D₄	1	2	yes	—
229	C₂³×#41	#120	1	1	yes	characteristic
230	C₂³×#41	#120	1	2	yes	—
231	C₂³×#41	#121	1	1	yes	characteristic
232	C₂³×#41	#121	1	2	yes	—
233	C₂³×#41	C₂²×D₄	1	1	yes	characteristic
234	C₂³×#41	C₂²×D₄	1	2	yes	—
235	C₂⁴×Q₈	#121	1	1	yes	characteristic
236	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
237	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
238	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
239	C₂⁴×D₄	C₄×D₄	1	2	yes	—
240	C₂⁴×D₄	#120	1	1	yes	characteristic
241	C₂⁴×D₄	#120	1	2	yes	—
242	C₂⁴×D₄	#121	1	1	yes	characteristic
243	C₂⁴×D₄	#121	1	1	yes	characteristic
244	C₂⁴×D₄	#121	1	2	yes	—
245	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
246	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
247	C₂⁵×C₄	#120	1	1	yes	characteristic
248	C₂×D₄²	C₂³×C₄	1	4	yes	—
249	#3458	C₂×C₂²⋊C₄	1	2	yes	—
250	#3458	C₂²×D₄	1	2	yes	—
251	#3455	C₂×C₂²⋊C₄	1	8	yes	—
252	#3449	C₂×C₂²⋊C₄	1	2	yes	—
253	#3449	C₂²×D₄	1	2	yes	—
254	C₂×#535	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂×#535	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×#535	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×#535	C₄×D₄	1	2	yes	—
258	C₂×#535	C₄×D₄	1	2	yes	—
259	C₂×#535	C₄×D₄	1	2	yes	—
260	C₂×#535	C₄×D₄	1	2	yes	—
261	C₂×#535	C₄×D₄	1	2	yes	—
262	C₂×#535	C₄×D₄	1	2	yes	—
263	C₂×#535	#120	1	2	yes	—
264	C₂×#535	#120	1	2	yes	—
265	C₂×#535	#120	1	2	yes	—
266	C₂×#535	#121	1	2	yes	—
267	C₂×#535	#121	1	2	yes	—
268	C₂×#535	#121	1	2	yes	—
269	C₂×#535	#121	1	2	yes	—
270	C₂×#535	#121	1	2	yes	—
271	C₂×#535	#121	1	2	yes	—
272	C₂×#535	#123	1	2	yes	—
273	C₂×#535	#123	1	2	yes	—
274	C₂×#535	#123	1	2	yes	—
275	C₂×#535	C₂³×C₄	1	1	yes	characteristic
276	C₂×#535	C₂³×C₄	1	2	yes	—
277	C₂×#535	C₂²×D₄	1	2	yes	—
278	C₂×#535	C₂²×D₄	1	1	yes	characteristic
279	C₂×#535	C₂²×D₄	1	2	yes	—
280	C₂×#535	C₂²×D₄	1	1	yes	characteristic
281	C₂×#535	C₂²×D₄	1	2	yes	—
282	C₂×#535	C₂²×D₄	1	1	yes	characteristic
283	C₂×#535	C₂²×D₄	1	2	yes	—
284	C₂×#535	C₂²×D₄	1	1	yes	characteristic
285	C₂×#535	C₂×#41	1	2	yes	—
286	C₂×#535	C₂×#41	1	1	yes	characteristic
287	C₂×#535	C₂×#41	1	2	yes	—
288	C₂×#535	C₂×#41	1	1	yes	characteristic
289	C₂×#535	C₂⁵	1	1	yes	characteristic
290	#3444	C₂×C₂²⋊C₄	1	8	yes	—
291	C₂×#533	C₂×C₂²⋊C₄	1	2	yes	—
292	C₂×#532	C₂×C₂²⋊C₄	1	2	yes	—
293	C₂²×#127	C₂×C₂²⋊C₄	1	2	yes	—
294	C₂²×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
295	C₂²×#127	C₄×D₄	1	1	yes	characteristic
296	C₂²×#127	C₄×D₄	1	1	yes	characteristic
297	C₂²×#127	C₄×D₄	1	2	yes	—
298	C₂²×#127	C₄×D₄	1	2	yes	—
299	C₂²×#127	C₄×D₄	1	1	yes	characteristic
300	C₂²×#127	C₄×D₄	1	2	yes	—
301	C₂²×#127	#121	1	1	yes	characteristic
302	C₂²×#127	#121	1	2	yes	—
303	C₂²×#127	#123	1	1	yes	characteristic
304	C₂²×#127	#123	1	2	yes	—
305	C₂²×#127	C₂³×C₄	1	1	yes	characteristic
306	C₂²×#127	C₂³×C₄	1	2	yes	—
307	C₂²×#127	C₂×#41	1	1	yes	characteristic
308	C₂²×#127	C₂×#41	1	2	yes	—
309	C₂²×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²×#124	C₂×C₂²⋊C₄	1	2	yes	—
311	C₂²×#124	C₄×D₄	1	1	yes	characteristic
312	C₂²×#124	C₄×D₄	1	2	yes	—
313	C₂²×#124	#121	1	1	yes	characteristic
314	C₂²×#124	#121	1	1	yes	characteristic
315	C₂²×#124	#121	1	1	yes	characteristic
316	C₂²×#124	#121	1	2	yes	—
317	C₂²×#124	#121	1	2	yes	—
318	C₂²×#124	#121	1	2	yes	—
319	C₂²×#124	#123	1	1	yes	characteristic
320	C₂²×#124	#123	1	2	yes	—
321	C₂²×#124	C₂²×D₄	1	1	yes	characteristic
322	C₂²×#124	C₂²×D₄	1	2	yes	—
323	C₂²×#124	C₂×#41	1	1	yes	characteristic
324	C₂²×#124	C₂×#41	1	2	yes	—
325	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
326	C₂²×#117	C₂×C₂²⋊C₄	1	2	yes	—
327	C₂²×#117	#120	1	1	yes	characteristic
328	C₂²×#117	#120	1	2	yes	—
329	C₂²×#117	#123	1	1	yes	characteristic
330	C₂²×#117	#123	1	2	yes	—
331	C₂²×#117	C₂²×D₄	1	1	yes	characteristic
332	C₂²×#117	C₂²×D₄	1	2	yes	—
333	C₂³×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
334	C₂³×C₄²	#123	1	1	yes	characteristic
335	#3153	C₂×C₂²⋊C₄	1	8	yes	—
336	#3153	C₂×C₂²⋊C₄	1	4	yes	—
337	#3153	C₂²×D₄	1	4	yes	—
338	#3151	#99	1	8	yes	complemented
339	#3151	C₂×C₂²⋊C₄	1	4	yes	—
340	#3151	C₂×C₂²⋊C₄	1	8	yes	—
341	#3151	C₂³×C₄	1	4	yes	—
342	C₂×#493	#99	1	4	yes	complemented
343	C₂×#492	#99	1	4	yes	complemented
344	#3140	C₂×C₂²⋊C₄	1	4	yes	—
345	#3140	C₂×C₂²⋊C₄	1	8	yes	—
346	#3140	C₂²×D₄	1	4	yes	—
347	#3138	C₂×C₂²⋊C₄	1	8	yes	—
348	#3138	C₂×C₂²⋊C₄	1	4	yes	—
349	#3138	C₂³×C₄	1	4	yes	—
350	#2921	C₂×C₂²⋊C₄	1	8	yes	—
351	#2920	C₂×C₂²⋊C₄	1	8	yes	—
352	C₂²×#100	C₂²×D₄	1	2	yes	—
353	C₂²×#100	C₂²×D₄	1	2	yes	—
354	C₂×#410	C₂²×D₄	1	2	yes	—
355	C₂×#410	C₂²×D₄	1	2	yes	—
356	#2879	C₂²×D₄	1	2	yes	—
357	#2879	C₂²×D₄	1	2	yes	—
358	#2873	C₂×C₂²⋊C₄	1	2	yes	—
359	#2873	C₂×C₂²⋊C₄	1	2	yes	—
360	#2873	C₂²×D₄	1	2	yes	—
361	#2873	C₂²×D₄	1	2	yes	—
362	C₂×C₄×M₁₆	C₂×C₂²⋊C₄	1	2	yes	—
363	C₂×C₄×M₁₆	C₂×C₂²⋊C₄	1	2	yes	—
364	#2380	C₂³×C₄	1	4	yes	—
365	#2309	C₂²×D₄	1	2	yes	—
366	#2309	C₂²×D₄	1	2	yes	—
367	#1555	C₂³×C₄	1	8	yes	—
368	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
369	C₂²×#41	#386	1	2	yes	—
370	C₂²×#41	#386	1	2	yes	—
371	C₂²×#41	#386	1	2	yes	—
372	C₂²×#41	C₂×C₄×D₄	1	1	yes	characteristic
373	C₂²×#41	C₂×C₄×D₄	1	2	yes	—
374	C₂²×#41	C₂×#120	1	2	yes	—
375	C₂²×#41	C₂×#120	1	1	yes	characteristic
376	C₂²×#41	C₂×#121	1	2	yes	—
377	C₂²×#41	C₂×#121	1	1	yes	characteristic
378	C₂²×#41	C₂³×D₄	1	1	yes	characteristic
379	C₂³×Q₈	#386	1	1	yes	characteristic
380	C₂³×Q₈	#386	1	2	yes	—
381	C₂³×Q₈	C₂×#121	1	1	yes	characteristic
382	C₂³×Q₈	C₂×#121	1	2	yes	—
383	C₂³×D₄	#386	1	1	yes	characteristic
384	C₂³×D₄	#386	1	2	yes	—
385	C₂³×D₄	#386	1	1	yes	characteristic
386	C₂³×D₄	#386	1	1	yes	characteristic
387	C₂³×D₄	#386	1	2	yes	—
388	C₂³×D₄	#386	1	2	yes	—
389	C₂³×D₄	#386	1	2	yes	—
390	C₂³×D₄	#386	1	2	yes	—
391	C₂³×D₄	#386	1	2	yes	—
392	C₂³×D₄	C₂×C₄×D₄	1	1	yes	characteristic
393	C₂³×D₄	C₂×C₄×D₄	1	1	yes	characteristic
394	C₂³×D₄	C₂×C₄×D₄	1	2	yes	—
395	C₂³×D₄	C₂×C₄×D₄	1	2	yes	—
396	C₂³×D₄	C₂×C₄×D₄	1	1	yes	characteristic
397	C₂³×D₄	C₂×C₄×D₄	1	2	yes	—
398	C₂³×D₄	C₂×#120	1	2	yes	—
399	C₂³×D₄	C₂×#120	1	1	yes	characteristic
400	C₂³×D₄	C₂×#121	1	1	yes	characteristic
401	C₂³×D₄	C₂×#121	1	2	yes	—
402	C₂³×D₄	C₂×#121	1	1	yes	characteristic
403	C₂³×D₄	C₂×#121	1	2	yes	—
404	C₂³×D₄	C₂³×D₄	1	1	yes	characteristic
405	C₂⁴×C₄	#386	1	1	yes	characteristic
406	C₂⁴×C₄	#386	1	1	yes	characteristic
407	C₂⁴×C₄	#386	1	2	yes	—
408	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
409	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
410	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
411	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
412	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
413	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
414	C₂⁴×C₄	C₂×#120	1	2	yes	—
415	#533	C₂²×C₂²⋊C₄	1	4	yes	—
416	#532	C₂×#99	1	4	yes	complemented
417	#532	C₂²×C₂²⋊C₄	1	4	yes	—
418	C₂×C₄⋊Q₈	C₂×#99	1	2	yes	complemented
419	C₂×#127	#386	1	2	yes	—
420	C₂×#127	#386	1	2	yes	—
421	C₂×#127	#386	1	2	yes	—
422	C₂×#127	#386	1	2	yes	—
423	C₂×#127	#386	1	2	yes	—
424	C₂×#127	#386	1	2	yes	—
425	C₂×#127	C₂×#99	1	4	yes	—
426	C₂×#127	C₂×#99	1	4	yes	—
427	C₂×#127	C₂×#99	1	2	yes	complemented
428	C₂×#127	C₂²×C₂²⋊C₄	1	1	yes	characteristic
429	C₂×#127	C₂²×C₂²⋊C₄	1	2	yes	—
430	C₂×#127	C₂×C₄×D₄	1	2	yes	—
431	C₂×#127	C₂×C₄×D₄	1	2	yes	—
432	C₂×#127	C₂×C₄×D₄	1	1	yes	characteristic
433	C₂×#127	C₂×C₄×D₄	1	1	yes	characteristic
434	C₂×#127	C₂×C₄×D₄	1	1	yes	characteristic
435	C₂×#127	C₂×C₄×D₄	1	2	yes	—
436	C₂×#127	C₂×#121	1	2	yes	—
437	C₂×#127	C₂×#121	1	1	yes	characteristic
438	C₂×#127	C₂×#123	1	2	yes	—
439	C₂×#127	C₂×#123	1	1	yes	characteristic
440	C₂×#127	C₂⁴×C₄	1	1	yes	characteristic
441	C₂×#127	C₂²×#41	1	1	yes	characteristic
442	C₂×#124	#386	1	2	yes	—
443	C₂×#124	#386	1	2	yes	—
444	C₂×#124	#386	1	2	yes	—
445	C₂×#124	#386	1	2	yes	—
446	C₂×#124	#386	1	2	yes	—
447	C₂×#124	#386	1	2	yes	—
448	C₂×#124	C₂²×C₂²⋊C₄	1	1	yes	characteristic
449	C₂×#124	C₂²×C₂²⋊C₄	1	2	yes	—
450	C₂×#124	C₂×C₄×D₄	1	1	yes	characteristic
451	C₂×#124	C₂×C₄×D₄	1	2	yes	—
452	C₂×#124	C₂×#121	1	1	yes	characteristic
453	C₂×#124	C₂×#121	1	2	yes	—
454	C₂×#124	C₂×#121	1	1	yes	characteristic
455	C₂×#124	C₂×#121	1	2	yes	—
456	C₂×#124	C₂×#121	1	2	yes	—
457	C₂×#124	C₂×#121	1	1	yes	characteristic
458	C₂×#124	C₂×#123	1	2	yes	—
459	C₂×#124	C₂×#123	1	1	yes	characteristic
460	C₂×#124	C₂³×D₄	1	1	yes	characteristic
461	C₂×#124	C₂²×#41	1	1	yes	characteristic
462	C₂×#117	#386	1	2	yes	—
463	C₂×#117	#386	1	2	yes	—
464	C₂×#117	#386	1	2	yes	—
465	C₂×#117	C₂²×C₂²⋊C₄	1	1	yes	characteristic
466	C₂×#117	C₂²×C₂²⋊C₄	1	2	yes	—
467	C₂×#117	C₂×#120	1	1	yes	characteristic
468	C₂×#117	C₂×#120	1	2	yes	—
469	C₂×#117	C₂×#123	1	1	yes	characteristic
470	C₂×#117	C₂×#123	1	2	yes	—
471	C₂×#117	C₂³×D₄	1	1	yes	characteristic
472	C₂²×C₄²	#386	1	1	yes	characteristic
473	C₂²×C₄²	#386	1	2	yes	—
474	C₂²×C₄²	#386	1	1	yes	characteristic
475	C₂²×C₄²	#386	1	2	yes	—
476	C₂²×C₄²	C₂²×C₂²⋊C₄	1	2	yes	—
477	C₂²×C₄²	C₂²×C₂²⋊C₄	1	1	yes	characteristic
478	C₂²×C₄²	C₂×#123	1	1	yes	characteristic
479	C₂²×C₄²	C₂×#123	1	2	yes	—
480	#493	C₂×#99	1	16	yes	complemented
481	#492	C₂×#99	1	16	yes	complemented
482	C₄×M₁₆	C₂×#99	1	4	yes	complemented
483	C₄×M₁₆	C₂²×C₂²⋊C₄	1	4	yes	—
484	C₄×M₁₆	C₂²×C₂²⋊C₄	1	4	yes	—
485	C₂×C₄×C₈	C₂×#99	1	4	yes	complemented
486	C₂⁵	#1889	1	1	yes	characteristic
487	C₂⁵	#1889	1	1	yes	characteristic
488	C₂⁵	#1889	1	2	yes	—
489	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
490	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
491	C₂²×Q₈	C₂×#386	1	2	yes	—
492	C₂²×Q₈	C₂×#386	1	1	yes	characteristic
493	C₂²×Q₈	C₂²×#121	1	1	yes	characteristic
494	C₂²×D₄	#1898	1	2	yes	—
495	C₂²×D₄	#1898	1	2	yes	—
496	C₂²×D₄	#1898	1	2	yes	—
497	C₂²×D₄	#1898	1	2	yes	—
498	C₂²×D₄	C₂×#386	1	1	yes	characteristic
499	C₂²×D₄	C₂×#386	1	1	yes	characteristic
500	C₂²×D₄	C₂×#386	1	2	yes	—
501	C₂²×D₄	C₂×#386	1	1	yes	characteristic
502	C₂²×D₄	C₂×#386	1	2	yes	—
503	C₂²×D₄	C₂×#386	1	2	yes	—
504	C₂²×D₄	C₂²×C₄×D₄	1	1	yes	characteristic
505	C₂²×D₄	C₂²×C₄×D₄	1	1	yes	characteristic
506	C₂²×D₄	C₂²×#121	1	1	yes	characteristic
507	C₂³×C₄	#1889	1	1	yes	characteristic
508	C₂³×C₄	#1889	1	1	yes	characteristic
509	C₂³×C₄	#1889	1	2	yes	—
510	C₂³×C₄	#1889	1	2	yes	—
511	C₂³×C₄	#1889	1	2	yes	—
512	C₂³×C₄	#1889	1	2	yes	—
513	C₂³×C₄	#1889	1	2	yes	—
514	C₂³×C₄	C₂×#386	1	1	yes	characteristic
515	C₂³×C₄	C₂×#386	1	1	yes	characteristic
516	C₂³×C₄	C₂×#386	1	2	yes	—
517	C₂³×C₄	C₂×#386	1	2	yes	—
518	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
519	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
520	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
521	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
522	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
523	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
524	C₂³×C₄	C₂²×#120	1	1	yes	characteristic
525	C₄⋊Q₈	C₂²×#99	1	4	yes	complemented
526	#127	#1898	1	8	yes	—
527	#127	C₂²×#99	1	8	yes	—
528	#127	C₂²×#99	1	4	yes	complemented
529	C₂×C₄²	#1913	1	2	yes	—
530	C₂×C₄²	#1913	1	2	yes	—
531	C₂×C₄²	C₂×#386	1	2	yes	—
532	C₂×C₄²	C₂×#386	1	1	yes	characteristic
533	C₂×C₄²	C₂×#386	1	2	yes	—
534	C₂×C₄²	C₂×#386	1	1	yes	characteristic
535	C₂×C₄²	C₂²×#99	1	2	yes	—
536	C₂×C₄²	C₂²×#99	1	2	yes	—
537	C₂×C₄²	C₂³×C₂²⋊C₄	1	1	yes	characteristic
538	C₂×C₄²	C₂²×#123	1	1	yes	characteristic
539	C₄×C₈	C₂²×#99	1	8	yes	complemented
540	C₂⁴	C₂×#1889	1	2	yes	—
541	C₂⁴	C₂×#1889	1	2	yes	—
542	C₂⁴	C₂×#1889	1	1	yes	characteristic
543	C₂⁴	C₂×#1889	1	1	yes	characteristic
544	C₂⁴	#33551	1	1	yes	characteristic
545	C₂⁴	#33551	1	1	yes	characteristic
546	C₂⁴	#33552	1	1	yes	characteristic
547	C₂⁴	#33552	1	1	yes	characteristic
548	C₂⁴	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
549	C₂×D₄	C₂×#1898	1	4	yes	—
550	C₂×D₄	C₂×#1898	1	4	yes	—
551	C₂²×C₄	C₂×#1889	1	2	yes	—
552	C₂²×C₄	C₂×#1889	1	2	yes	—
553	C₂²×C₄	C₂×#1889	1	1	yes	characteristic
554	C₂²×C₄	C₂×#1889	1	1	yes	characteristic
555	C₂²×C₄	C₂×#1898	1	1	yes	characteristic
556	C₂²×C₄	C₂×#1898	1	1	yes	characteristic
557	C₂²×C₄	C₂×#1913	1	1	yes	characteristic
558	C₂²×C₄	C₂×#1913	1	1	yes	characteristic
559	C₂²×C₄	#33551	1	1	yes	characteristic
560	C₂²×C₄	#33551	1	1	yes	characteristic
561	C₂²×C₄	#33552	1	1	yes	characteristic
562	C₂²×C₄	#33552	1	1	yes	characteristic
563	C₂²×C₄	C₂²×#386	1	1	yes	characteristic
564	C₂²×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
565	C₄²	C₂×#1898	1	2	yes	—
566	C₄²	C₂×#1913	1	4	yes	—
567	C₄²	C₂³×#99	1	2	yes	—
568	C₂³	#201664	1	1	yes	characteristic
569	C₂³	#201664	1	1	yes	characteristic
570	C₂³	#494600	1	1	yes	characteristic
571	C₂³	#494600	1	1	yes	characteristic
572	C₂³	#494600	1	1	yes	characteristic
573	C₂³	#494600	1	1	yes	characteristic
574	C₂³	#494600	1	1	yes	characteristic
575	C₂³	#494600	1	1	yes	characteristic
576	C₂³	C₂²×#1889	1	1	yes	characteristic
577	C₂³	C₂×#33551	1	1	yes	characteristic
578	C₂³	C₂×#33552	1	1	yes	characteristic
579	C₂×C₄	#201664	1	1	yes	characteristic
580	C₂×C₄	#201664	1	1	yes	characteristic
581	C₂×C₄	#494600	1	1	yes	characteristic
582	C₂×C₄	#494600	1	1	yes	characteristic
583	C₂×C₄	C₂²×#1898	1	1	yes	characteristic
584	C₂×C₄	C₂²×#1913	1	1	yes	characteristic
585	C₂×C₄	C₂×#33551	1	1	yes	characteristic
586	C₂×C₄	C₂×#33552	1	1	yes	characteristic
587	C₂²	#72210479	1	1	yes	characteristic
588	C₂²	#72210479	1	1	yes	characteristic
589	C₂²	#72210479	1	1	yes	characteristic
590	C₂²	#72210479	1	1	yes	characteristic
591	C₂²	C₂×#494600	1	1	yes	characteristic
592	C₂²	C₂×#494600	1	1	yes	characteristic
593	C₂²	C₂×#201664	1	1	yes	characteristic
594	C₂	#72216528	1	2	yes	direct product factor
595	C₂	C₂×#72210479	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72210479
commutator	C₂×#127	C₂⁴×C₄
Frattini, cosocle	C₂×#535	C₂⁵
prime radical	C₂×#72213685	C₂
LCS	G ▷ C₂×#127 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ C₂³×#535 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#72216516	C₂	semidirect
2. C₂×#72216511	C₂	semidirect
3. C₂×#72216523	C₂	semidirect
4. C₂×#72216507	C₂	semidirect
5. C₂×#72213409	C₂	semidirect
6. C₂×#72213452	C₂	semidirect
7. C₂×#72213429	C₂	semidirect
8. C₂×#72215536	C₂	semidirect
9. C₂×#72213443	C₂	semidirect
10. C₂×#152071	C₂²	semidirect
11. C₂×#94424	C₂²	semidirect
12. C₂×#94424	C₂²	semidirect
13. C₂×#152047	C₂²	semidirect
16. C₂×#94425	C₂²	semidirect
17. C₂×#94425	C₂²	semidirect
20. C₂×#246957	C₄	semidirect
22. C₂×#72210566	C₂	semidirect
23. C₂×#386041	C₂²	semidirect
24. C₂×#391879	C₂²	semidirect
25. C₂×#386039	C₂²	semidirect
30. C₂×#200876	C₄	semidirect
32. C₂×#6348967	C₄	semidirect
37. C₂×#375718	C₂²	semidirect
38. #72216528	C₂	direct (N × H)
39. #72216523	C₂²	semidirect
40. #72216516	C₂²	semidirect
41. #72216511	C₂²	semidirect
42. #72216507	C₂²	semidirect
43. #72215536	C₂²	semidirect
44. #72213685	C₄	semidirect
46. #72213452	C₂²	semidirect
47. #72213443	C₂²	semidirect
48. #72213429	C₂²	semidirect
49. #72213409	C₂²	semidirect
50. #72210566	C₂²	semidirect
52. C₂³×D₄²	C₂×C₄	semidirect
68. C₂×#52004	C₂×C₄	semidirect
69. C₂×#52004	C₂×C₄	semidirect
72. C₂²×#2380	C₂×C₄	semidirect
76. #6348967	C₂×C₄	semidirect
77. C₂×#18529	C₂×C₄	semidirect
78. C₂×#18524	C₂×C₄	semidirect
79. C₂×#17741	C₂×C₄	semidirect
84. #391879	C₂³	semidirect
85. #386041	C₂³	semidirect
86. #386039	C₂³	semidirect
87. #375718	C₂³	semidirect
100. #246957	C₂×C₄	semidirect
101. #246930	C₂×C₄	semidirect
103. #246927	C₂×C₄	semidirect
105. #200876	C₂×C₄	semidirect
106. C₂×#9428	C₂×C₄	semidirect
107. C₂×#9425	C₂×C₄	semidirect
108. #187811	C₂×C₄	semidirect
110. C₂×#8522	C₂×C₄	semidirect
112. #152071	C₂³	semidirect
113. #152047	C₂³	semidirect
118. #94425	C₂³	semidirect
119. #94425	C₂³	semidirect
120. #94424	C₂³	semidirect
121. #94424	C₂³	semidirect
127. C₂²×D₄²	C₂²×C₄	semidirect
195. #52004	C₂²×C₄	semidirect
198. C₂×#2380	C₂²×C₄	semidirect
204. #18529	C₂²×C₄	semidirect
205. #18524	C₂²×C₄	semidirect
206. #17741	C₂²×C₄	semidirect
222. #9428	C₂²×C₄	semidirect
223. #9425	C₂²×C₄	semidirect
225. #8522	C₂²×C₄	semidirect
338. #3151	#99	semidirect
342. C₂×#493	#99	semidirect
343. C₂×#492	#99	semidirect
416. #532	C₂×#99	semidirect
418. C₂×C₄⋊Q₈	C₂×#99	semidirect
427. C₂×#127	C₂×#99	semidirect
480. #493	C₂×#99	semidirect
481. #492	C₂×#99	semidirect
482. C₄×M₁₆	C₂×#99	semidirect
485. C₂×C₄×C₈	C₂×#99	semidirect
525. C₄⋊Q₈	C₂²×#99	semidirect
528. #127	C₂²×#99	semidirect
539. C₄×C₈	C₂²×#99	semidirect
594. C₂	#72216528	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×#72216516	2
C₂×#72216511	2
C₂×#72216523	2
C₂×#72216507	2
C₂×#72213409	2
C₂×#72213452	2
C₂×#72213429	2
C₂×#72215536	2
C₂×#72213443	2
C₂×#72213685	2
C₂×#72210566	2
#72216528	2

Subgroups — C2×#72216528 #72251458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72216528 #72251458