Subgroups — #72313192

Order 2048.

At least 9399 conjugacy classes of subgroups; only the 874 normal subgroups are shown.

The subgroup lattice has 664 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312760	C₂	1	1	yes	maximal, characteristic, complemented
2	#72312738	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312707	C₂	1	1	yes	maximal, characteristic, complemented
4	#72312558	C₂	1	1	yes	maximal, characteristic, complemented
5	#72312359	C₂	1	1	yes	maximal, characteristic, complemented
6	#72312292	C₂	1	1	yes	maximal, characteristic, complemented
7	#72312262	C₂	1	1	yes	maximal, characteristic, complemented
8	#72303810	C₂	1	1	yes	maximal, characteristic, complemented
9	#72303790	C₂	1	1	yes	maximal, characteristic, complemented
10	#72303729	C₂	1	1	yes	maximal, characteristic, complemented
11	#72303571	C₂	1	1	yes	maximal, characteristic, complemented
12	#72303321	C₂	1	1	yes	maximal, characteristic, complemented
13	#72280610	C₂	1	1	yes	maximal, characteristic, complemented
14	#72267981	C₂	1	1	yes	maximal, characteristic, complemented
15	#72256636	C₂	1	1	yes	maximal, characteristic
16	C₂×#63076	C₂²	1	1	yes	characteristic
17	C₂²×#3025	C₂²	1	1	yes	characteristic
18	C₂×#60387	C₂²	1	1	yes	characteristic
19	#10574202	C₂²	1	1	yes	characteristic
20	#7634872	C₂²	1	1	yes	characteristic
21	C₂×#33369	C₂²	1	1	yes	characteristic
22	C₂×#33358	C₂²	1	1	yes	characteristic
23	C₂²×#2120	C₂²	1	1	yes	characteristic
24	C₂²×#2120	C₂²	1	1	yes	characteristic
25	#6374140	C₂²	1	1	yes	characteristic
26	#6374112	C₂²	1	1	yes	characteristic
27	C₂²×#1891	C₂²	1	1	yes	characteristic
28	C₂×#20947	C₂²	1	1	yes	characteristic
29	#6371364	C₂²	1	1	yes	characteristic
30	C₂×#20860	C₂²	1	1	yes	characteristic
31	C₂×#20620	C₂²	1	1	yes	characteristic
32	C₂×#20612	C₂²	1	1	yes	characteristic
33	C₂×#20612	C₂²	1	1	yes	characteristic
34	#6370734	C₂²	1	1	yes	characteristic
35	C₂×#20401	C₂²	1	1	yes	characteristic
36	#498928	C₄	1	4	yes	complemented
37	#498927	C₄	1	4	yes	complemented
38	C₂×#12589	C₂²	1	1	yes	characteristic
39	C₂×#12589	C₂²	1	1	yes	characteristic
40	C₂×#12579	C₂²	1	1	yes	characteristic
41	C₂×#12579	C₂²	1	1	yes	characteristic
42	C₂×#12577	C₂²	1	1	yes	characteristic
43	C₂×#12577	C₂²	1	1	yes	characteristic
44	#492886	C₂²	1	1	yes	characteristic
45	#492886	C₂²	1	1	yes	characteristic
46	#355933	C₂²	1	1	yes	characteristic
47	#353434	C₂²	1	1	yes	characteristic
48	#353434	C₂²	1	1	yes	characteristic
49	#353430	C₂²	1	1	yes	characteristic
50	#353340	C₂²	1	1	yes	characteristic
51	#353340	C₂²	1	1	yes	characteristic
52	#353322	C₂²	1	1	yes	characteristic
53	C₂²×#585	C₂³	1	1	yes	characteristic
54	C₂×#3573	D₄	1	4	yes	complemented
55	C₂²×#568	D₄	1	1	yes	characteristic, complemented
56	C₂²×#568	C₂³	1	1	yes	characteristic
57	C₂²×#568	C₂³	1	1	yes	characteristic
58	C₂×#3449	C₂³	1	1	yes	characteristic
59	C₂²×#525	C₂³	1	1	yes	characteristic
60	C₂²×#458	C₂×C₄	1	2	yes	—
61	C₂²×B(4,𝔽₂)	C₂×C₄	1	2	yes	—
62	C₂×#3022	D₄	1	2	yes	—
63	C₂×#3021	D₄	1	2	yes	—
64	C₂×#3019	C₂×C₄	1	4	yes	—
65	C₂×#3018	D₄	1	4	yes	—
66	C₂×#2905	D₄	1	1	yes	characteristic, complemented
67	C₂²×#413	C₂³	1	1	yes	characteristic
68	C₂²×#413	C₂³	1	1	yes	characteristic
69	C₂²×#413	C₂³	1	1	yes	characteristic
70	C₂²×#410	C₂³	1	1	yes	characteristic
71	C₂²×#410	C₂³	1	1	yes	characteristic
72	C₂²×#410	C₂³	1	1	yes	characteristic
73	C₂×#2879	D₄	1	1	yes	characteristic, complemented
74	C₂×#2879	C₂³	1	1	yes	characteristic
75	C₂×#2879	C₂³	1	1	yes	characteristic
76	C₂×#2879	C₂³	1	1	yes	characteristic
77	C₂²×#407	D₄	1	1	yes	characteristic
78	C₂²×#407	D₄	1	1	yes	characteristic
79	C₂×#2873	D₄	1	1	yes	characteristic, complemented
80	#36955	C₂×C₄	1	2	yes	—
81	#36702	C₂×C₄	1	2	yes	—
82	#34044	C₂³	1	1	yes	characteristic
83	C₂×#2127	C₂×C₄	1	4	yes	—
84	C₂²×#352	C₂×C₄	1	2	yes	—
85	C₂²×AΣL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
86	C₂×#1884	D₄	1	4	yes	complemented
87	C₂×#1858	D₄	1	4	yes	complemented
88	C₂×#1794	D₄	1	4	yes	complemented
89	#20609	C₂×C₄	1	4	yes	—
90	C₂×#1757	D₄	1	4	yes	complemented
91	C₂²×#344	D₄	1	1	yes	characteristic, complemented
92	C₂²×#344	D₄	1	1	yes	characteristic, complemented
93	C₂²×#344	D₄	1	1	yes	characteristic, complemented
94	C₂²×#344	D₄	1	1	yes	characteristic, complemented
95	#20335	D₄	1	1	yes	characteristic
96	#20335	D₄	1	1	yes	characteristic
97	#12574	C₂×C₄	1	4	yes	—
98	#10764	D₄	1	4	yes	complemented
99	C₂×#585	C₂×D₄	1	2	yes	—
100	C₂×#585	C₂×D₄	1	2	yes	—
101	C₂×#585	C₂×D₄	1	2	yes	—
102	C₂×#585	C₂×D₄	1	2	yes	—
103	C₂×#585	#41	1	2	yes	—
104	C₂×#585	#41	1	2	yes	—
105	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
106	C₂³×#41	C₂×D₄	1	1	yes	characteristic
107	C₂³×#41	C₂⁴	1	1	yes	characteristic
108	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂×#568	C₂×D₄	1	2	yes	—
110	C₂×#568	C₂×D₄	1	2	yes	—
111	C₂×#568	C₂×D₄	1	2	yes	—
112	C₂²×#131	C₂×D₄	1	2	yes	—
113	C₂²×#131	C₂×D₄	1	2	yes	—
114	C₂²×#131	C₂×D₄	1	2	yes	—
115	C₂²×#131	#41	1	2	yes	—
116	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
117	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
118	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
119	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
120	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
121	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
122	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
123	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
124	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
125	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
126	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
127	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
128	C₂×#536	C₂×D₄	1	1	yes	characteristic
129	C₂×#536	#41	1	2	yes	—
130	C₂×#536	#41	1	1	yes	characteristic
131	C₂×#535	C₂²×C₄	1	2	yes	—
132	C₂×#535	C₂×D₄	1	1	yes	characteristic
133	C₂×#535	#41	1	1	yes	characteristic
134	C₂×#532	C₂×D₄	1	2	yes	—
135	C₂×#532	C₂×D₄	1	2	yes	—
136	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
137	C₂²×#123	#41	1	1	yes	characteristic
138	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
139	C₂²×#120	#41	1	1	yes	characteristic
140	C₂²×#117	C₂×D₄	1	1	yes	characteristic
141	C₂²×#117	C₂×D₄	1	1	yes	characteristic
142	C₂²×#117	C₂×D₄	1	1	yes	characteristic
143	C₂²×#117	C₂×D₄	1	1	yes	characteristic
144	C₂×#455	C₂²⋊C₄	1	8	yes	—
145	C₂×#453	C₂²⋊C₄	1	8	yes	—
146	C₂×#413	C₂×D₄	1	2	yes	—
147	C₂×#413	C₂×D₄	1	2	yes	—
148	C₂×#413	C₂×D₄	1	2	yes	—
149	C₂×#413	C₂×D₄	1	2	yes	—
150	C₂×#413	C₂×D₄	1	2	yes	—
151	C₂×#413	#41	1	2	yes	—
152	C₂×#413	#41	1	2	yes	—
153	C₂×#413	#41	1	2	yes	—
154	C₂²×#100	C₂²×C₄	1	2	yes	—
155	C₂×#410	C₂×D₄	1	2	yes	—
156	C₂×#410	C₂×D₄	1	2	yes	—
157	C₂×#410	C₂×D₄	1	2	yes	—
158	C₂×#410	C₂×D₄	1	2	yes	—
159	C₂×#410	#41	1	2	yes	—
160	C₂²×#99	C₂²×C₄	1	2	yes	—
161	C₂²×#99	C₂²×C₄	1	2	yes	—
162	C₂×#408	C₂×D₄	1	2	yes	—
163	C₂×#408	C₂×D₄	1	2	yes	—
164	#2308	C₂²×C₄	1	2	yes	—
165	#2308	#41	1	2	yes	—
166	C₂⁶	C₄×D₄	1	1	yes	characteristic
167	#585	#120	1	4	yes	—
168	#585	#121	1	4	yes	—
169	#585	#121	1	4	yes	—
170	#585	#123	1	4	yes	—
171	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	4	yes	—
172	C₂×H(5,𝔽₂)	C₄×D₄	1	4	yes	—
173	C₂×H(5,𝔽₂)	C₄×D₄	1	4	yes	—
174	C₂²×#41	C₄×D₄	1	2	yes	—
175	C₂²×#41	#120	1	2	yes	—
176	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
177	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
178	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
179	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
180	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
181	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
182	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
183	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
184	C₂²×#41	C₂×#41	1	1	yes	characteristic
185	C₂²×#41	C₂×#41	1	1	yes	characteristic
186	C₂²×#41	C₂×#41	1	1	yes	characteristic
187	C₂²×#41	C₂×#41	1	1	yes	characteristic
188	C₂²×#41	C₂×#41	1	1	yes	characteristic
189	C₂²×#41	C₂×#41	1	1	yes	characteristic
190	C₂²×#41	C₂×#41	1	1	yes	characteristic
191	C₂²×#41	C₂×#41	1	1	yes	characteristic
192	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂²×#41	2₋¹⁺⁴	1	1	yes	characteristic
200	C₂³×Q₈	#121	1	1	yes	characteristic
201	C₂³×Q₈	C₂×#41	1	1	yes	characteristic
202	C₂³×D₄	#121	1	1	yes	characteristic
203	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
204	C₂³×D₄	C₂×#41	1	1	yes	characteristic
205	C₂³×D₄	C₂×#41	1	1	yes	characteristic
206	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
207	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
208	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
209	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
210	#568	#120	1	4	yes	—
211	C₂²×M₁₆	#120	1	2	yes	—
212	C₂²×M₁₆	#120	1	1	yes	characteristic
213	C₂²×M₁₆	#120	1	1	yes	characteristic
214	C₂²×M₁₆	#120	1	2	yes	—
215	C₂²×M₁₆	#120	1	1	yes	characteristic
216	C₂²×M₁₆	#120	1	1	yes	characteristic
217	C₂²×M₁₆	#120	1	2	yes	—
218	C₂²×M₁₆	#121	1	2	yes	—
219	C₂²×M₁₆	#121	1	2	yes	—
220	C₂²×M₁₆	#121	1	2	yes	—
221	C₂²×M₁₆	#127	1	2	yes	—
222	C₂³×C₈	#120	1	2	yes	—
223	C₂³×C₈	#120	1	2	yes	—
224	C₂³×C₈	#120	1	2	yes	—
225	C₂³×C₈	#121	1	2	yes	—
226	C₂³×C₈	#121	1	2	yes	—
227	C₂³×C₈	#121	1	2	yes	—
228	C₂³×C₈	#127	1	2	yes	—
229	C₂×C₄⋊Q₈	#121	1	2	yes	—
230	C₂×C₄⋊Q₈	#123	1	2	yes	—
231	C₂×#127	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂×#127	C₄×D₄	1	2	yes	—
233	C₂×#124	C₂×C₂²⋊C₄	1	2	yes	—
234	C₂×#124	C₄×D₄	1	2	yes	—
235	C₂×#124	#121	1	2	yes	—
236	C₂×#124	#123	1	2	yes	—
237	C₂×#117	C₄×D₄	1	2	yes	—
238	C₂×#117	#121	1	1	yes	characteristic
239	C₂×#117	#121	1	1	yes	characteristic
240	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
241	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
242	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
244	C₂²×C₄²	#120	1	2	yes	—
245	#413	#120	1	4	yes	—
246	#413	#121	1	4	yes	—
247	#413	#121	1	4	yes	—
248	#413	#123	1	4	yes	—
249	C₂×#100	C₂×C₂²⋊C₄	1	4	yes	—
250	C₂×#100	C₄×D₄	1	4	yes	—
251	C₂×#100	C₄×D₄	1	4	yes	—
252	#410	#120	1	4	yes	—
253	C₂×#99	C₄×D₄	1	4	yes	—
254	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂⁵	#525	1	1	yes	characteristic
257	H(5,𝔽₂)	#386	1	8	yes	—
258	C₂×#41	C₂×#120	1	1	yes	characteristic
259	C₂×#41	C₂×#120	1	1	yes	characteristic
260	C₂×#41	C₂×#120	1	1	yes	characteristic
261	C₂×#41	C₂×#120	1	1	yes	characteristic
262	C₂×#41	C₂×#121	1	1	yes	characteristic
263	C₂×#41	C₂×#121	1	1	yes	characteristic
264	C₂×#41	C₂×#121	1	1	yes	characteristic
265	C₂×#41	C₂×#121	1	1	yes	characteristic
266	C₂×#41	C₂×#123	1	1	yes	characteristic
267	C₂×#41	C₂×#123	1	1	yes	characteristic
268	C₂×#41	#525	1	1	yes	characteristic
269	C₂×#41	#525	1	1	yes	characteristic
270	C₂×#41	#525	1	1	yes	characteristic
271	C₂×#41	#525	1	1	yes	characteristic
272	C₂×#41	#525	1	1	yes	characteristic
273	C₂×#41	#525	1	1	yes	characteristic
274	C₂×#41	#534	1	1	yes	characteristic
275	C₂×#41	#534	1	1	yes	characteristic
276	C₂×#41	#535	1	1	yes	characteristic
277	C₂×#41	#535	1	1	yes	characteristic
278	C₂×#41	#535	1	1	yes	characteristic
279	C₂×#41	#535	1	1	yes	characteristic
280	C₂×#41	#537	1	1	yes	characteristic
281	C₂×#41	#537	1	1	yes	characteristic
282	C₂×#41	#538	1	1	yes	characteristic
283	C₂×#41	#538	1	1	yes	characteristic
284	C₂×#41	#538	1	1	yes	characteristic
285	C₂×#41	#539	1	1	yes	characteristic
286	C₂×#41	#540	1	1	yes	characteristic
287	C₂×#41	#540	1	1	yes	characteristic
288	C₂×#41	#540	1	1	yes	characteristic
289	C₂×#41	#542	1	1	yes	characteristic
290	C₂×#41	D₄²	1	1	yes	characteristic
291	C₂×#41	D₄²	1	1	yes	characteristic
292	C₂×#41	D₄²	1	1	yes	characteristic
293	C₂×#41	D₄²	1	1	yes	characteristic
294	C₂×#41	D₄²	1	1	yes	characteristic
295	C₂×#41	#546	1	1	yes	characteristic
296	C₂×#41	#546	1	1	yes	characteristic
297	C₂×#41	#546	1	1	yes	characteristic
298	C₂×#41	#546	1	1	yes	characteristic
299	C₂×#41	#546	1	1	yes	characteristic
300	C₂×#41	#546	1	1	yes	characteristic
301	C₂×#41	#546	1	1	yes	characteristic
302	C₂×#41	#546	1	1	yes	characteristic
303	C₂×#41	#546	1	1	yes	characteristic
304	C₂×#41	#546	1	1	yes	characteristic
305	C₂×#41	#546	1	1	yes	characteristic
306	C₂×#41	#547	1	1	yes	characteristic
307	C₂×#41	#547	1	1	yes	characteristic
308	C₂×#41	#547	1	1	yes	characteristic
309	C₂×#41	#548	1	1	yes	characteristic
310	C₂×#41	#550	1	1	yes	characteristic
311	C₂×#41	#550	1	1	yes	characteristic
312	C₂×#41	#550	1	1	yes	characteristic
313	C₂×#41	#551	1	1	yes	characteristic
314	C₂×#41	#553	1	1	yes	characteristic
315	C₂×#41	#553	1	1	yes	characteristic
316	C₂×#41	#553	1	1	yes	characteristic
317	C₂×#41	#553	1	1	yes	characteristic
318	C₂×#41	#553	1	1	yes	characteristic
319	C₂×#41	#553	1	1	yes	characteristic
320	C₂×#41	#555	1	1	yes	characteristic
321	C₂×#41	#559	1	1	yes	characteristic
322	C₂²×Q₈	#386	1	2	yes	—
323	C₂²×Q₈	C₂×#121	1	1	yes	characteristic
324	C₂²×Q₈	C₂×#123	1	1	yes	characteristic
325	C₂²×Q₈	#525	1	1	yes	characteristic
326	C₂²×Q₈	#525	1	1	yes	characteristic
327	C₂²×Q₈	#525	1	1	yes	characteristic
328	C₂²×Q₈	#525	1	1	yes	characteristic
329	C₂²×Q₈	#525	1	1	yes	characteristic
330	C₂²×Q₈	#529	1	1	yes	characteristic
331	C₂²×Q₈	#538	1	1	yes	characteristic
332	C₂²×Q₈	#538	1	1	yes	characteristic
333	C₂²×Q₈	#538	1	1	yes	characteristic
334	C₂²×Q₈	#539	1	1	yes	characteristic
335	C₂²×D₄	#386	1	2	yes	—
336	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
337	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
338	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
339	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
340	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
341	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
342	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
343	C₂²×D₄	C₄×#41	1	1	yes	characteristic
344	C₂²×D₄	#518	1	1	yes	characteristic
345	C₂²×D₄	#518	1	1	yes	characteristic
346	C₂²×D₄	#518	1	1	yes	characteristic
347	C₂²×D₄	#520	1	1	yes	characteristic
348	C₂²×D₄	#525	1	1	yes	characteristic
349	C₂²×D₄	#525	1	1	yes	characteristic
350	C₂²×D₄	#525	1	1	yes	characteristic
351	C₂²×D₄	#529	1	1	yes	characteristic
352	C₂²×D₄	#529	1	1	yes	characteristic
353	C₂²×D₄	#532	1	1	yes	characteristic
354	C₂²×D₄	#532	1	1	yes	characteristic
355	C₂²×D₄	#532	1	1	yes	characteristic
356	C₂²×D₄	#546	1	1	yes	characteristic
357	C₂²×D₄	#546	1	1	yes	characteristic
358	C₂²×D₄	#546	1	1	yes	characteristic
359	C₂²×D₄	#546	1	1	yes	characteristic
360	C₂²×D₄	#546	1	1	yes	characteristic
361	C₂²×D₄	#546	1	1	yes	characteristic
362	C₂²×D₄	#546	1	1	yes	characteristic
363	C₂²×D₄	#547	1	1	yes	characteristic
364	C₂²×D₄	#548	1	1	yes	characteristic
365	C₂²×D₄	#550	1	1	yes	characteristic
366	C₂²×D₄	#551	1	1	yes	characteristic
367	C₂²×D₄	#551	1	1	yes	characteristic
368	C₂²×D₄	#551	1	1	yes	characteristic
369	C₂²×D₄	#553	1	1	yes	characteristic
370	C₂²×D₄	#553	1	1	yes	characteristic
371	C₂²×D₄	#559	1	1	yes	characteristic
372	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
373	C₂³×C₄	C₂×#120	1	1	yes	characteristic
374	C₂³×C₄	C₂×#120	1	1	yes	characteristic
375	C₂³×C₄	C₂×#120	1	1	yes	characteristic
376	C₂³×C₄	C₂×#120	1	1	yes	characteristic
377	C₂³×C₄	C₂×#120	1	1	yes	characteristic
378	C₂³×C₄	C₂×#121	1	1	yes	characteristic
379	C₂³×C₄	C₂×#121	1	1	yes	characteristic
380	C₂³×C₄	C₂×#121	1	1	yes	characteristic
381	C₂³×C₄	C₂×#121	1	1	yes	characteristic
382	C₂³×C₄	#525	1	1	yes	characteristic
383	C₂³×C₄	#525	1	1	yes	characteristic
384	C₂³×C₄	#525	1	1	yes	characteristic
385	C₂³×C₄	#525	1	1	yes	characteristic
386	C₂³×C₄	#525	1	1	yes	characteristic
387	C₂³×C₄	#525	1	1	yes	characteristic
388	C₂³×C₄	C₂×#127	1	1	yes	characteristic
389	C₂³×C₄	#532	1	1	yes	characteristic
390	C₂³×C₄	#532	1	1	yes	characteristic
391	C₂³×C₄	#532	1	1	yes	characteristic
392	C₂³×C₄	#534	1	1	yes	characteristic
393	C₂³×C₄	#534	1	1	yes	characteristic
394	C₂³×C₄	#535	1	1	yes	characteristic
395	C₂³×C₄	#537	1	1	yes	characteristic
396	C₂³×C₄	D₄²	1	1	yes	characteristic
397	C₂³×C₄	D₄²	1	1	yes	characteristic
398	C₂³×C₄	D₄²	1	1	yes	characteristic
399	C₂³×C₄	D₄²	1	1	yes	characteristic
400	C₂³×C₄	D₄²	1	1	yes	characteristic
401	C₂³×C₄	D₄²	1	1	yes	characteristic
402	C₂³×C₄	D₄²	1	1	yes	characteristic
403	C₂³×C₄	D₄²	1	1	yes	characteristic
404	C₂³×C₄	D₄²	1	1	yes	characteristic
405	C₂³×C₄	D₄²	1	1	yes	characteristic
406	C₂³×C₄	#546	1	1	yes	characteristic
407	C₂³×C₄	#546	1	1	yes	characteristic
408	C₂³×C₄	#546	1	1	yes	characteristic
409	C₂³×C₄	#547	1	1	yes	characteristic
410	C₂³×C₄	#548	1	1	yes	characteristic
411	C₂³×C₄	#550	1	1	yes	characteristic
412	C₂×M₁₆	B(4,𝔽₂)	1	2	yes	—
413	C₂×M₁₆	B(4,𝔽₂)	1	2	yes	—
414	C₂×M₁₆	B(4,𝔽₂)	1	2	yes	—
415	C₂×M₁₆	B(4,𝔽₂)	1	2	yes	—
416	C₂²×C₈	#392	1	2	yes	—
417	C₂²×C₈	#392	1	2	yes	—
418	C₂²×C₈	#392	1	2	yes	—
419	C₂²×C₈	#392	1	2	yes	—
420	C₂×C₄²	#386	1	2	yes	—
421	C₂×C₄²	#386	1	2	yes	—
422	#100	#386	1	8	yes	—
423	C₂⁴	C₄×#120	1	1	yes	characteristic
424	C₂⁴	C₄×#120	1	1	yes	characteristic
425	C₂⁴	C₄×#121	1	1	yes	characteristic
426	C₂⁴	C₄×#121	1	1	yes	characteristic
427	C₂⁴	C₄×#127	1	1	yes	characteristic
428	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
429	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
430	C₂⁴	#2358	1	1	yes	characteristic
431	C₂⁴	#2358	1	1	yes	characteristic
432	C₂⁴	#2358	1	1	yes	characteristic
433	C₂⁴	#2361	1	1	yes	characteristic
434	C₂⁴	#2361	1	1	yes	characteristic
435	C₂⁴	#2366	1	1	yes	characteristic
436	C₂⁴	#2388	1	1	yes	characteristic
437	C₂⁴	#2392	1	1	yes	characteristic
438	C₂⁴	#2404	1	1	yes	characteristic
439	C₂⁴	#2407	1	1	yes	characteristic
440	C₂⁴	#2534	1	1	yes	characteristic
441	C₂⁴	#2600	1	1	yes	characteristic
442	C₂×Q₈	#2424	1	1	yes	characteristic
443	C₂×Q₈	#2425	1	1	yes	characteristic
444	C₂×Q₈	#2443	1	1	yes	characteristic
445	C₂×Q₈	#2444	1	1	yes	characteristic
446	C₂×Q₈	#2456	1	1	yes	characteristic
447	C₂×Q₈	#2456	1	1	yes	characteristic
448	C₂×Q₈	#2459	1	1	yes	characteristic
449	C₂×Q₈	#2467	1	1	yes	characteristic
450	C₂×Q₈	#2630	1	1	yes	characteristic
451	C₂×Q₈	#2631	1	1	yes	characteristic
452	C₂×Q₈	#2635	1	1	yes	characteristic
453	C₂×Q₈	#2636	1	1	yes	characteristic
454	C₂×Q₈	#2847	1	1	yes	characteristic
455	C₂×Q₈	#2847	1	1	yes	characteristic
456	C₂×Q₈	#2849	1	1	yes	characteristic
457	C₂×Q₈	#2849	1	1	yes	characteristic
458	C₂×D₄	C₂×#386	1	1	yes	characteristic
459	C₂×D₄	C₂×#386	1	1	yes	characteristic
460	C₂×D₄	C₄×#120	1	1	yes	characteristic
461	C₂×D₄	C₄×#120	1	1	yes	characteristic
462	C₂×D₄	#2321	1	1	yes	characteristic
463	C₂×D₄	#2324	1	1	yes	characteristic
464	C₂×D₄	#2325	1	1	yes	characteristic
465	C₂×D₄	#2328	1	1	yes	characteristic
466	C₂×D₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
467	C₂×D₄	#2339	1	1	yes	characteristic
468	C₂×D₄	#2358	1	1	yes	characteristic
469	C₂×D₄	#2361	1	1	yes	characteristic
470	C₂×D₄	#2363	1	1	yes	characteristic
471	C₂×D₄	#2364	1	1	yes	characteristic
472	C₂×D₄	#2364	1	1	yes	characteristic
473	C₂×D₄	#2366	1	1	yes	characteristic
474	C₂×D₄	#2404	1	1	yes	characteristic
475	C₂×D₄	#2404	1	1	yes	characteristic
476	C₂×D₄	#2407	1	1	yes	characteristic
477	C₂×D₄	#2407	1	1	yes	characteristic
478	C₂×D₄	#2418	1	1	yes	characteristic
479	C₂×D₄	#2420	1	1	yes	characteristic
480	C₂×D₄	#2424	1	1	yes	characteristic
481	C₂×D₄	#2425	1	1	yes	characteristic
482	C₂×D₄	#2427	1	1	yes	characteristic
483	C₂×D₄	#2428	1	1	yes	characteristic
484	C₂×D₄	#2430	1	1	yes	characteristic
485	C₂×D₄	#2430	1	1	yes	characteristic
486	C₂×D₄	#2431	1	1	yes	characteristic
487	C₂×D₄	#2431	1	1	yes	characteristic
488	C₂×D₄	#2444	1	1	yes	characteristic
489	C₂×D₄	#2445	1	1	yes	characteristic
490	C₂×D₄	#2456	1	1	yes	characteristic
491	C₂×D₄	#2457	1	1	yes	characteristic
492	C₂×D₄	#2457	1	1	yes	characteristic
493	C₂×D₄	#2459	1	1	yes	characteristic
494	C₂×D₄	#2461	1	1	yes	characteristic
495	C₂×D₄	#2463	1	1	yes	characteristic
496	C₂×D₄	#2464	1	1	yes	characteristic
497	C₂×D₄	#2467	1	1	yes	characteristic
498	C₂×D₄	#2534	1	1	yes	characteristic
499	C₂×D₄	#2535	1	1	yes	characteristic
500	C₂×D₄	#2543	1	1	yes	characteristic
501	C₂×D₄	#2544	1	1	yes	characteristic
502	C₂×D₄	#2598	1	1	yes	characteristic
503	C₂×D₄	#2600	1	1	yes	characteristic
504	C₂×D₄	#2602	1	1	yes	characteristic
505	C₂×D₄	#2603	1	1	yes	characteristic
506	C₂²×C₄	C₂×#386	1	1	yes	characteristic
507	C₂²×C₄	C₄×#120	1	1	yes	characteristic
508	C₂²×C₄	C₄×#120	1	1	yes	characteristic
509	C₂²×C₄	C₄×#121	1	1	yes	characteristic
510	C₂²×C₄	#2321	1	1	yes	characteristic
511	C₂²×C₄	#2325	1	1	yes	characteristic
512	C₂²×C₄	C₂×#392	1	1	yes	characteristic
513	C₂²×C₄	#2388	1	1	yes	characteristic
514	C₂²×C₄	#2388	1	1	yes	characteristic
515	C₂²×C₄	#2392	1	1	yes	characteristic
516	C₂²×C₄	#2392	1	1	yes	characteristic
517	C₂²×C₄	#2397	1	1	yes	characteristic
518	C₂²×C₄	#2403	1	1	yes	characteristic
519	C₂²×C₄	#2403	1	1	yes	characteristic
520	C₂²×C₄	#2403	1	1	yes	characteristic
521	C₂²×C₄	#2403	1	1	yes	characteristic
522	C₂²×C₄	#2407	1	1	yes	characteristic
523	C₂²×C₄	#2407	1	1	yes	characteristic
524	C₂²×C₄	#2408	1	1	yes	characteristic
525	C₂²×C₄	#2408	1	1	yes	characteristic
526	C₂²×C₄	#2408	1	1	yes	characteristic
527	C₂²×C₄	#2408	1	1	yes	characteristic
528	C₂²×C₄	#2410	1	1	yes	characteristic
529	C₂²×C₄	#2410	1	1	yes	characteristic
530	C₂²×C₄	#2410	1	1	yes	characteristic
531	C₂²×C₄	#2410	1	1	yes	characteristic
532	C₂²×C₄	#2414	1	1	yes	characteristic
533	C₂²×C₄	#2414	1	1	yes	characteristic
534	C₂²×C₄	#2416	1	1	yes	characteristic
535	C₂²×C₄	#2416	1	1	yes	characteristic
536	C₂²×C₄	#2418	1	1	yes	characteristic
537	C₂²×C₄	#2418	1	1	yes	characteristic
538	C₂²×C₄	#2418	1	1	yes	characteristic
539	C₂²×C₄	#2422	1	1	yes	characteristic
540	C₂²×C₄	#2424	1	1	yes	characteristic
541	C₂²×C₄	#2424	1	1	yes	characteristic
542	C₂²×C₄	#2424	1	1	yes	characteristic
543	C₂²×C₄	#2424	1	1	yes	characteristic
544	C₂²×C₄	#2428	1	1	yes	characteristic
545	C₂²×C₄	#2428	1	1	yes	characteristic
546	C₂²×C₄	#2430	1	1	yes	characteristic
547	C₂²×C₄	#2431	1	1	yes	characteristic
548	C₂²×C₄	#2433	1	1	yes	characteristic
549	C₂²×C₄	#2433	1	1	yes	characteristic
550	C₂²×C₄	#2443	1	1	yes	characteristic
551	C₂²×C₄	#2445	1	1	yes	characteristic
552	C₂²×C₄	#2455	1	1	yes	characteristic
553	C₂²×C₄	#2455	1	1	yes	characteristic
554	C₂²×C₄	#2456	1	1	yes	characteristic
555	C₂²×C₄	#2456	1	1	yes	characteristic
556	C₂²×C₄	#2456	1	1	yes	characteristic
557	C₂²×C₄	#2456	1	1	yes	characteristic
558	C₂²×C₄	#2456	1	1	yes	characteristic
559	C₂²×C₄	#2461	1	1	yes	characteristic
560	C₂²×C₄	#2463	1	1	yes	characteristic
561	C₂²×C₄	#2464	1	1	yes	characteristic
562	C₂²×C₄	#2464	1	1	yes	characteristic
563	C₂²×C₄	#2464	1	1	yes	characteristic
564	C₂²×C₄	#2535	1	1	yes	characteristic
565	C₂²×C₄	#2537	1	1	yes	characteristic
566	C₂²×C₄	#2540	1	1	yes	characteristic
567	C₂²×C₄	#2541	1	1	yes	characteristic
568	C₂²×C₄	#2543	1	1	yes	characteristic
569	C₂²×C₄	#2602	1	1	yes	characteristic
570	C₂²×C₄	#2613	1	1	yes	characteristic
571	C₂²×C₄	#2615	1	1	yes	characteristic
572	C₂²×C₄	#2630	1	1	yes	characteristic
573	C₂²×C₄	#2635	1	1	yes	characteristic
574	C₂²×C₄	#2846	1	1	yes	characteristic
575	C₂²×C₄	#2847	1	1	yes	characteristic
576	C₂²×C₄	#2847	1	1	yes	characteristic
577	C₂²×C₄	#2849	1	1	yes	characteristic
578	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
579	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
580	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
581	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
582	C₂²×C₄	#3025	1	1	yes	characteristic
583	C₂²×C₄	#3025	1	1	yes	characteristic
584	C₂²×C₄	#3025	1	1	yes	characteristic
585	C₂²×C₄	#3025	1	1	yes	characteristic
586	C₂×C₈	#2021	1	4	yes	—
587	C₂×C₈	#2021	1	4	yes	—
588	C₂³	C₄×#392	1	1	yes	characteristic
589	C₂³	#13967	1	1	yes	characteristic
590	C₂³	#13967	1	1	yes	characteristic
591	C₂³	#13970	1	1	yes	characteristic
592	C₂³	#14023	1	1	yes	characteristic
593	C₂³	#14025	1	1	yes	characteristic
594	C₂³	#14171	1	1	yes	characteristic
595	C₂³	#14171	1	1	yes	characteristic
596	C₂³	#14175	1	1	yes	characteristic
597	C₂³	#14187	1	1	yes	characteristic
598	C₂³	#14189	1	1	yes	characteristic
599	C₂³	#14730	1	1	yes	characteristic
600	C₂³	#14732	1	1	yes	characteristic
601	C₂³	#14885	1	1	yes	characteristic
602	C₂³	#15159	1	1	yes	characteristic
603	C₂³	C₄×B(4,𝔽₂)	1	1	yes	characteristic
604	C₂³	C₄×B(4,𝔽₂)	1	1	yes	characteristic
605	C₂³	C₄×B(4,𝔽₂)	1	1	yes	characteristic
606	C₂³	C₄×B(4,𝔽₂)	1	1	yes	characteristic
607	C₂×C₄	#14023	1	1	yes	characteristic
608	C₂×C₄	#14187	1	1	yes	characteristic
609	C₂×C₄	#14299	1	1	yes	characteristic
610	C₂×C₄	#14694	1	1	yes	characteristic
611	C₂×C₄	#14715	1	1	yes	characteristic
612	C₂×C₄	#14721	1	1	yes	characteristic
613	C₂×C₄	#14721	1	1	yes	characteristic
614	C₂×C₄	#14722	1	1	yes	characteristic
615	C₂×C₄	#14722	1	1	yes	characteristic
616	C₂×C₄	#14730	1	1	yes	characteristic
617	C₂×C₄	#14732	1	1	yes	characteristic
618	C₂×C₄	#14795	1	1	yes	characteristic
619	C₂×C₄	#14796	1	1	yes	characteristic
620	C₂×C₄	#14806	1	1	yes	characteristic
621	C₂×C₄	#14811	1	1	yes	characteristic
622	C₂×C₄	#15123	1	1	yes	characteristic
623	C₂×C₄	C₂×#2021	1	1	yes	characteristic
624	C₂×C₄	C₂×#2021	1	1	yes	characteristic
625	C₂×C₄	#23443	1	1	yes	characteristic
626	C₂×C₄	#23443	1	1	yes	characteristic
627	C₂×C₄	#23595	1	1	yes	characteristic
628	C₂×C₄	#23595	1	1	yes	characteristic
629	C₂×C₄	#23595	1	1	yes	characteristic
630	C₂×C₄	#23595	1	1	yes	characteristic
631	C₂×C₄	#23900	1	1	yes	characteristic
632	C₂×C₄	#23900	1	1	yes	characteristic
633	C₂×C₄	#23900	1	1	yes	characteristic
634	C₂×C₄	#23900	1	1	yes	characteristic
635	C₂×C₄	#23902	1	1	yes	characteristic
636	C₂×C₄	#23902	1	1	yes	characteristic
637	C₂×C₄	#23902	1	1	yes	characteristic
638	C₂×C₄	#23902	1	1	yes	characteristic
639	C₂×C₄	#24140	1	1	yes	characteristic
640	C₂×C₄	#24140	1	1	yes	characteristic
641	C₂×C₄	#24140	1	1	yes	characteristic
642	C₂×C₄	#24140	1	1	yes	characteristic
643	C₂×C₄	#24260	1	1	yes	characteristic
644	C₂×C₄	#24260	1	1	yes	characteristic
645	C₂×C₄	#24260	1	1	yes	characteristic
646	C₂×C₄	#24260	1	1	yes	characteristic
647	C₂²	#154634	1	1	yes	characteristic
648	C₂²	C₄×#2021	1	1	yes	characteristic
649	C₂²	C₄×#2021	1	1	yes	characteristic
650	C₂²	#364513	1	1	yes	characteristic
651	C₂²	#364513	1	1	yes	characteristic
652	C₂²	#375865	1	1	yes	characteristic
653	C₂²	#375865	1	1	yes	characteristic
654	C₄	#394058	1	1	yes	characteristic
655	C₄	#394060	1	1	yes	characteristic
656	C₄	#394949	1	1	yes	characteristic
657	C₄	#394949	1	1	yes	characteristic
658	C₄	#395127	1	1	yes	characteristic
659	C₄	#395127	1	1	yes	characteristic
660	C₄	#411728	1	1	yes	characteristic
661	C₄	#411728	1	1	yes	characteristic
662	C₂	#72260737	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#394058
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂³×#41	C₂⁴
socle	C₂	#72260737
prime radical	#72256636	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#568 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72312760	C₂	semidirect
2. #72312738	C₂	semidirect
3. #72312707	C₂	semidirect
4. #72312558	C₂	semidirect
5. #72312359	C₂	semidirect
6. #72312292	C₂	semidirect
7. #72312262	C₂	semidirect
8. #72303810	C₂	semidirect
9. #72303790	C₂	semidirect
10. #72303729	C₂	semidirect
11. #72303571	C₂	semidirect
12. #72303321	C₂	semidirect
13. #72280610	C₂	semidirect
14. #72267981	C₂	semidirect
36. #498928	C₄	semidirect
37. #498927	C₄	semidirect
54. C₂×#3573	D₄	semidirect
55. C₂²×#568	D₄	semidirect
66. C₂×#2905	D₄	semidirect
73. C₂×#2879	D₄	semidirect
79. C₂×#2873	D₄	semidirect
86. C₂×#1884	D₄	semidirect
87. C₂×#1858	D₄	semidirect
88. C₂×#1794	D₄	semidirect
90. C₂×#1757	D₄	semidirect
91. C₂²×#344	D₄	semidirect
92. C₂²×#344	D₄	semidirect
93. C₂²×#344	D₄	semidirect
94. C₂²×#344	D₄	semidirect
98. #10764	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72312760	2
#72312738	2
#72312707	2
#72312558	2
#72312359	2
#72312292	2
#72312262	2
#72303810	2
#72303790	2
#72303729	2
#72303571	2
#72303321	2
#72280610	2
#72267981	2
#72256636	2