Subgroups — #72339852

Order 2048.

At least 6475 conjugacy classes of subgroups; only the 326 normal subgroups are shown.

The subgroup lattice has 306 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339848	C₂	1	1	yes	maximal, characteristic, complemented
2	#72339842	C₂	1	1	yes	maximal, characteristic, complemented
3	#72339820	C₂	1	1	yes	maximal, characteristic, complemented
4	#72339803	C₂	1	1	yes	maximal, characteristic
5	#72306804	C₂	1	1	yes	maximal, characteristic, complemented
6	#72306622	C₂	1	1	yes	maximal, characteristic, complemented
7	#72284046	C₂	1	2	yes	maximal, complemented
8	#72283810	C₂	1	2	yes	maximal, complemented
9	#72283622	C₂	1	2	yes	maximal, complemented
10	#72283381	C₂	1	2	yes	maximal, complemented
11	#72214702	C₂	1	1	yes	maximal, characteristic, complemented
12	#6382803	C₂²	1	1	yes	characteristic, complemented
13	#6363083	C₄	1	1	yes	characteristic, complemented
14	#6363080	C₄	1	1	yes	characteristic, complemented
15	#6363063	C₄	1	1	yes	characteristic, complemented
16	#6363053	C₄	1	1	yes	characteristic, complemented
17	#6363044	C₄	1	1	yes	characteristic, complemented
18	#6363023	C₄	1	1	yes	characteristic, complemented
19	#6363008	C₄	1	1	yes	characteristic, complemented
20	#6362952	C₄	1	1	yes	characteristic, complemented
21	#6361272	C₂²	1	1	yes	characteristic
22	#6359505	C₂²	1	1	yes	characteristic
23	#6359500	C₂²	1	1	yes	characteristic
24	#6358422	C₂²	1	1	yes	characteristic
25	#6357770	C₂²	1	1	yes	characteristic
26	#6348098	C₂²	1	1	yes	characteristic
27	#6348073	C₂²	1	1	yes	characteristic
28	#512608	C₂²	1	1	yes	characteristic, complemented
29	#512601	C₂²	1	1	yes	characteristic, complemented
30	#483304	C₂²	1	1	yes	characteristic, complemented
31	#153533	C₂²	1	2	yes	complemented
32	#153519	C₂²	1	2	yes	complemented
33	#153453	C₂²	1	2	yes	complemented
34	#153453	C₂²	1	2	yes	complemented
35	#153435	C₂²	1	2	yes	complemented
36	#153435	C₂²	1	2	yes	complemented
37	#152585	C₂²	1	2	yes	complemented
38	#152585	C₂²	1	2	yes	complemented
39	#152584	C₂²	1	2	yes	complemented
40	#152584	C₂²	1	2	yes	complemented
41	#137588	C₂²	1	2	yes	complemented
42	#137555	C₂²	1	2	yes	complemented
43	C₂×#3483	D₄	1	1	yes	characteristic
44	C₂×#3481	D₄	1	1	yes	characteristic
45	#60693	D₄	1	1	yes	characteristic
46	#60639	C₂³	1	1	yes	characteristic
47	#23196	C₂³	1	2	yes	—
48	#20323	C₂×C₄	1	1	yes	characteristic, complemented
49	#20314	C₂×C₄	1	1	yes	characteristic, complemented
50	#20294	C₂×C₄	1	1	yes	characteristic, complemented
51	#20289	C₂×C₄	1	1	yes	characteristic, complemented
52	#20157	C₂×C₄	1	1	yes	characteristic, complemented
53	#20150	C₂×C₄	1	1	yes	characteristic, complemented
54	#20138	C₂×C₄	1	1	yes	characteristic, complemented
55	#20131	C₂×C₄	1	1	yes	characteristic, complemented
56	#20117	C₂×C₄	1	1	yes	characteristic, complemented
57	#20084	C₂×C₄	1	1	yes	characteristic, complemented
58	#20053	C₂×C₄	1	1	yes	characteristic, complemented
59	#20040	C₂×C₄	1	1	yes	characteristic, complemented
60	#20017	C₂×C₄	1	1	yes	characteristic, complemented
61	#20013	C₂×C₄	1	1	yes	characteristic, complemented
62	#19987	C₂×C₄	1	1	yes	characteristic, complemented
63	SD₁₆²	C₂×C₄	1	1	yes	characteristic, complemented
64	#19750	C₂×C₄	1	1	yes	characteristic
65	#19750	D₄	1	1	yes	characteristic
66	#19749	C₂×C₄	1	1	yes	characteristic
67	#19749	D₄	1	1	yes	characteristic
68	#19746	C₂×C₄	1	1	yes	characteristic
69	#19746	D₄	1	1	yes	characteristic
70	#19745	C₂×C₄	1	1	yes	characteristic
71	#19745	D₄	1	1	yes	characteristic
72	#19616	C₂³	1	1	yes	characteristic
73	#19608	D₄	1	1	yes	characteristic
74	#19603	D₄	1	1	yes	characteristic
75	#19596	C₂³	1	1	yes	characteristic
76	#18549	D₄	1	1	yes	characteristic
77	#18549	D₄	1	1	yes	characteristic
78	#18548	C₂×C₄	1	1	yes	characteristic
79	#18548	C₂×C₄	1	1	yes	characteristic
80	#18547	C₂×C₄	1	1	yes	characteristic
81	#18547	D₄	1	1	yes	characteristic
82	#18546	C₂×C₄	1	1	yes	characteristic
83	#18546	D₄	1	1	yes	characteristic
84	#18528	C₂×C₄	1	1	yes	characteristic
85	#18528	C₂×C₄	1	1	yes	characteristic
86	#18527	D₄	1	1	yes	characteristic
87	#18527	D₄	1	1	yes	characteristic
88	#18526	C₂×C₄	1	1	yes	characteristic
89	#18526	D₄	1	1	yes	characteristic
90	#18524	C₂×C₄	1	1	yes	characteristic
91	#18524	D₄	1	1	yes	characteristic
92	#18510	D₄	1	1	yes	characteristic
93	#18510	D₄	1	1	yes	characteristic
94	#18510	C₂³	1	1	yes	characteristic
95	#18510	C₂³	1	1	yes	characteristic
96	#18480	D₄	1	1	yes	characteristic
97	#18480	D₄	1	1	yes	characteristic
98	#18480	C₂³	1	1	yes	characteristic
99	#18480	C₂³	1	1	yes	characteristic
100	#17738	D₄	1	1	yes	characteristic
101	#17738	D₄	1	1	yes	characteristic
102	C₂×#1455	D₄	1	1	yes	characteristic
103	#13589	C₂³	1	2	yes	—
104	#13583	C₂³	1	2	yes	—
105	#11920	C₂³	1	2	yes	—
106	#3483	C₂²⋊C₄	1	1	yes	characteristic
107	#3483	C₂²⋊C₄	1	1	yes	characteristic
108	#3483	C₂×D₄	1	1	yes	characteristic
109	#3483	C₂×D₄	1	1	yes	characteristic
110	#3481	C₂²⋊C₄	1	1	yes	characteristic
111	#3481	C₂²⋊C₄	1	1	yes	characteristic
112	#3481	C₂×D₄	1	1	yes	characteristic
113	#3481	C₂×D₄	1	1	yes	characteristic
114	C₂×#559	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×Q₈²	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×#555	C₂²⋊C₄	1	1	yes	characteristic
117	C₂×D₄²	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×#532	C₂×D₄	1	1	yes	characteristic
119	C₂×#532	C₂×D₄	1	1	yes	characteristic
120	#3185	C₂²⋊C₄	1	1	yes	characteristic
121	#3185	C₂²⋊C₄	1	1	yes	characteristic
122	#3184	C₂²⋊C₄	1	1	yes	characteristic
123	#3184	C₂²⋊C₄	1	1	yes	characteristic
124	#3160	C₂×D₄	1	1	yes	characteristic
125	#3152	C₂²×C₄	1	1	yes	characteristic
126	#3152	C₂×D₄	1	1	yes	characteristic
127	#3152	C₂×D₄	1	1	yes	characteristic
128	#3152	C₂×D₄	1	1	yes	characteristic
129	#3151	C₂²×C₄	1	1	yes	characteristic
130	#3151	C₂×D₄	1	1	yes	characteristic
131	#3151	C₂×D₄	1	1	yes	characteristic
132	#3151	C₂×D₄	1	1	yes	characteristic
133	#3147	C₂×D₄	1	1	yes	characteristic
134	#3147	C₂×D₄	1	1	yes	characteristic
135	#3147	C₂×D₄	1	1	yes	characteristic
136	#3147	C₂×D₄	1	1	yes	characteristic
137	#3147	C₂⁴	1	1	yes	characteristic
138	C₂×C₄×M₁₆	C₂×D₄	1	1	yes	characteristic
139	#1722	C₂²×C₄	1	1	yes	characteristic
140	#1722	C₂×D₄	1	1	yes	characteristic
141	#1721	C₂²×C₄	1	1	yes	characteristic
142	#1721	C₂×D₄	1	1	yes	characteristic
143	#1716	C₂²⋊C₄	1	1	yes	characteristic
144	#1716	C₂²⋊C₄	1	1	yes	characteristic
145	#1715	C₂²⋊C₄	1	1	yes	characteristic
146	#1715	C₂²⋊C₄	1	1	yes	characteristic
147	#1713	C₂²⋊C₄	1	1	yes	characteristic
148	#1712	C₂²⋊C₄	1	1	yes	characteristic
149	#1709	C₂²⋊C₄	1	1	yes	characteristic
150	#1709	C₂²⋊C₄	1	1	yes	characteristic
151	#1708	C₂²×C₄	1	1	yes	characteristic
152	#1708	C₂×D₄	1	1	yes	characteristic
153	#1705	C₂²×C₄	1	1	yes	characteristic
154	#1705	C₂×D₄	1	1	yes	characteristic
155	#1634	C₂²⋊C₄	1	1	yes	characteristic
156	#1634	C₂²⋊C₄	1	1	yes	characteristic
157	#1634	C₂×D₄	1	1	yes	characteristic
158	#1634	C₂×D₄	1	1	yes	characteristic
159	#1633	C₂²⋊C₄	1	1	yes	characteristic
160	#1633	C₂²⋊C₄	1	1	yes	characteristic
161	#1633	C₂×D₄	1	1	yes	characteristic
162	#1633	C₂×D₄	1	1	yes	characteristic
163	#1630	C₂²⋊C₄	1	1	yes	characteristic
164	#1630	C₂²⋊C₄	1	1	yes	characteristic
165	#1630	C₂²×C₄	1	1	yes	characteristic
166	#1630	C₂²×C₄	1	1	yes	characteristic
167	#1629	C₂²⋊C₄	1	1	yes	characteristic
168	#1629	C₂²⋊C₄	1	1	yes	characteristic
169	#1629	C₂²×C₄	1	1	yes	characteristic
170	#1629	C₂²×C₄	1	1	yes	characteristic
171	#1624	C₂²⋊C₄	1	1	yes	characteristic
172	#1624	C₂²⋊C₄	1	1	yes	characteristic
173	#1624	C₂²×C₄	1	1	yes	characteristic
174	#1624	C₂²×C₄	1	1	yes	characteristic
175	#1623	C₂²⋊C₄	1	1	yes	characteristic
176	#1623	C₂²⋊C₄	1	1	yes	characteristic
177	#1623	C₂²×C₄	1	1	yes	characteristic
178	#1623	C₂²×C₄	1	1	yes	characteristic
179	#1620	C₂²⋊C₄	1	1	yes	characteristic
180	#1620	C₂²⋊C₄	1	1	yes	characteristic
181	#1620	C₂×D₄	1	1	yes	characteristic
182	#1620	C₂×D₄	1	1	yes	characteristic
183	#1619	C₂²⋊C₄	1	1	yes	characteristic
184	#1619	C₂²⋊C₄	1	1	yes	characteristic
185	#1619	C₂×D₄	1	1	yes	characteristic
186	#1619	C₂×D₄	1	1	yes	characteristic
187	#1560	C₂²⋊C₄	1	1	yes	characteristic
188	#1560	C₂²⋊C₄	1	1	yes	characteristic
189	#1560	C₂²⋊C₄	1	1	yes	characteristic
190	#1560	C₂²⋊C₄	1	1	yes	characteristic
191	#1558	C₂²⋊C₄	1	1	yes	characteristic
192	#1558	C₂²⋊C₄	1	1	yes	characteristic
193	#1555	C₂²⋊C₄	1	1	yes	characteristic
194	#1555	C₂²⋊C₄	1	1	yes	characteristic
195	#1543	C₂×D₄	1	1	yes	characteristic
196	#1543	C₂×D₄	1	1	yes	characteristic
197	#1543	C₂×D₄	1	1	yes	characteristic
198	#1543	C₂×D₄	1	1	yes	characteristic
199	#1455	C₂²⋊C₄	1	1	yes	characteristic
200	#1455	C₂²⋊C₄	1	1	yes	characteristic
201	#1455	C₂²⋊C₄	1	1	yes	characteristic
202	#1455	C₂²⋊C₄	1	1	yes	characteristic
203	C₂×#322	C₂×D₄	1	1	yes	characteristic
204	C₂×C₈²	C₂×D₄	1	1	yes	characteristic
205	#559	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	#559	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	Q₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	Q₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	#555	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	#555	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	D₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	D₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	#532	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	#532	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	#532	#120	1	1	yes	characteristic
216	#532	#120	1	1	yes	characteristic
217	#532	#120	1	1	yes	characteristic
218	#532	#120	1	1	yes	characteristic
219	#532	C₂²×D₄	1	1	yes	characteristic
220	#532	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄⋊Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	#120	1	1	yes	characteristic
223	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×#127	#120	1	1	yes	characteristic
225	C₂²×C₄²	#120	1	1	yes	characteristic
226	#498	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	#498	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	#494	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	#494	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	#494	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	#494	#120	1	1	yes	characteristic
232	#494	C₂²×D₄	1	1	yes	characteristic
233	#493	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	#493	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	#493	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	#493	#120	1	1	yes	characteristic
237	#493	C₂²×D₄	1	1	yes	characteristic
238	#492	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	#492	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	#492	#120	1	1	yes	characteristic
241	#492	#120	1	1	yes	characteristic
242	#492	#120	1	1	yes	characteristic
243	#492	#120	1	1	yes	characteristic
244	#492	C₂³×C₄	1	1	yes	characteristic
245	#492	C₂²×D₄	1	1	yes	characteristic
246	C₄×M₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₄×M₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×M₁₆	#120	1	1	yes	characteristic
249	C₄×M₁₆	#120	1	1	yes	characteristic
250	C₂×C₄×C₈	#120	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×D₄	1	1	yes	characteristic
252	#333	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	#333	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	#333	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	#333	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	#331	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	#331	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	#331	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	#331	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	#322	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	#322	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂³×C₄	B(4,𝔽₂)	1	1	yes	characteristic
265	C₄⋊Q₈	#379	1	1	yes	characteristic
266	C₄⋊Q₈	#379	1	1	yes	characteristic
267	C₄⋊Q₈	C₂²×C₂²⋊C₄	1	1	yes	characteristic
268	C₄⋊Q₈	C₂×#120	1	1	yes	characteristic
269	#127	#379	1	1	yes	characteristic
270	#127	#379	1	1	yes	characteristic
271	#127	C₂²×C₂²⋊C₄	1	1	yes	characteristic
272	#127	C₂×#120	1	1	yes	characteristic
273	C₂×C₄²	#379	1	1	yes	characteristic
274	C₂×C₄²	#379	1	1	yes	characteristic
275	C₂×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄²	#458	1	1	yes	characteristic
277	C₂×C₄²	#458	1	1	yes	characteristic
278	C₂×C₄²	C₂×#120	1	1	yes	characteristic
279	C₄×C₈	#379	1	1	yes	characteristic
280	C₄×C₈	#379	1	1	yes	characteristic
281	C₄×C₈	C₂²×C₂²⋊C₄	1	1	yes	characteristic
282	C₄×C₈	C₂×#120	1	1	yes	characteristic
283	C₂⁴	#2196	1	1	yes	characteristic
284	C₂²×C₄	#1790	1	1	yes	characteristic
285	C₂²×C₄	#1790	1	1	yes	characteristic
286	C₂²×C₄	#2196	1	1	yes	characteristic
287	C₂²×C₄	#2196	1	1	yes	characteristic
288	C₂²×C₄	#2196	1	1	yes	characteristic
289	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
290	C₄²	#1790	1	1	yes	characteristic
291	C₄²	#1790	1	1	yes	characteristic
292	C₄²	C₂×#379	1	1	yes	characteristic
293	C₄²	C₂×#458	1	1	yes	characteristic
294	C₂³	#12759	1	1	yes	characteristic
295	C₂³	#12759	1	1	yes	characteristic
296	C₂³	C₂×#2196	1	1	yes	characteristic
297	C₂×C₄	#12759	1	1	yes	characteristic
298	C₂×C₄	#12759	1	1	yes	characteristic
299	C₂×C₄	C₂×#1790	1	1	yes	characteristic
300	C₂×C₄	C₂×#2196	1	1	yes	characteristic
301	C₂²	C₂×#12759	1	1	yes	characteristic
302	C₄	#512862	1	1	yes	characteristic
303	C₄	#512863	1	1	yes	characteristic
304	C₂	#72279353	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#512862
commutator	#492	C₂³×C₄
Frattini, cosocle	#3147	C₂⁴
socle	C₂	#72279353
prime radical	#72339803	C₂
LCS	G ▷ #492 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #3147 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339848	C₂	semidirect
2. #72339842	C₂	semidirect
3. #72339820	C₂	semidirect
5. #72306804	C₂	semidirect
6. #72306622	C₂	semidirect
7. #72284046	C₂	semidirect
8. #72283810	C₂	semidirect
9. #72283622	C₂	semidirect
10. #72283381	C₂	semidirect
11. #72214702	C₂	semidirect
12. #6382803	C₂²	semidirect
13. #6363083	C₄	semidirect
14. #6363080	C₄	semidirect
15. #6363063	C₄	semidirect
16. #6363053	C₄	semidirect
17. #6363044	C₄	semidirect
18. #6363023	C₄	semidirect
19. #6363008	C₄	semidirect
20. #6362952	C₄	semidirect
28. #512608	C₂²	semidirect
29. #512601	C₂²	semidirect
30. #483304	C₂²	semidirect
31. #153533	C₂²	semidirect
32. #153519	C₂²	semidirect
33. #153453	C₂²	semidirect
34. #153453	C₂²	semidirect
35. #153435	C₂²	semidirect
36. #153435	C₂²	semidirect
37. #152585	C₂²	semidirect
38. #152585	C₂²	semidirect
39. #152584	C₂²	semidirect
40. #152584	C₂²	semidirect
41. #137588	C₂²	semidirect
42. #137555	C₂²	semidirect
48. #20323	C₂×C₄	semidirect
49. #20314	C₂×C₄	semidirect
50. #20294	C₂×C₄	semidirect
51. #20289	C₂×C₄	semidirect
52. #20157	C₂×C₄	semidirect
53. #20150	C₂×C₄	semidirect
54. #20138	C₂×C₄	semidirect
55. #20131	C₂×C₄	semidirect
56. #20117	C₂×C₄	semidirect
57. #20084	C₂×C₄	semidirect
58. #20053	C₂×C₄	semidirect
59. #20040	C₂×C₄	semidirect
60. #20017	C₂×C₄	semidirect
61. #20013	C₂×C₄	semidirect
62. #19987	C₂×C₄	semidirect
63. SD₁₆²	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#72339848	2
#72339842	2
#72339820	2
#72339803	2
#72306804	2
#72306622	2
#72284046	2
#72283810	2
#72283622	2
#72283381	2
#72214702	2