Subgroups — #72255279

Order 3072.

At least 7664 conjugacy classes of subgroups; only the 273 normal subgroups are shown.

The subgroup lattice has 254 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72255276	C₂	1	1	yes	maximal, characteristic, complemented
2	#72255273	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255272	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#144386	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255270	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255269	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255260	C₂	1	1	yes	maximal, characteristic, complemented
8	#72255215	C₂	1	1	yes	maximal, characteristic, complemented
9	#72255133	C₂	1	1	yes	maximal, characteristic, complemented
10	#72255053	C₂	1	1	yes	maximal, characteristic, complemented
11	#72253585	C₂	1	1	yes	maximal, characteristic, complemented
12	#72253547	C₂	1	1	yes	maximal, characteristic, complemented
13	#72249302	C₂	1	1	yes	maximal, characteristic, complemented
14	#72248912	C₂	1	1	yes	maximal, characteristic, complemented
15	#72248661	C₂	1	1	yes	maximal, characteristic, complemented
16	D₃×#3484	C₂²	1	1	yes	characteristic
17	C₂²×#5715	C₂²	1	1	yes	characteristic
18	C₂×#82604	C₂²	1	1	yes	characteristic
19	#10828423	C₂²	1	1	yes	characteristic
20	#10828418	C₂²	1	1	yes	characteristic
21	#10826831	C₂²	1	1	yes	characteristic
22	#10826830	C₂²	1	1	yes	characteristic
23	#10826818	C₂²	1	1	yes	characteristic
24	#10826764	C₂²	1	1	yes	characteristic
25	D₃×#1783	C₂²	1	1	yes	characteristic
26	D₃×#1781	C₂²	1	1	yes	characteristic
27	D₃×#1781	C₂²	1	1	yes	characteristic
28	#10740388	C₂²	1	1	yes	characteristic, complemented
29	#10733626	C₂²	1	1	yes	characteristic, complemented
30	#10732891	C₂²	1	1	yes	characteristic, complemented
31	#10717052	C₂²	1	1	yes	characteristic
32	#10711426	C₂²	1	1	yes	characteristic, complemented
33	#10711423	C₂²	1	1	yes	characteristic, complemented
34	#10710675	C₂²	1	1	yes	characteristic, complemented
35	#10710675	C₂²	1	1	yes	characteristic, complemented
36	#10710673	C₂²	1	1	yes	characteristic, complemented
37	#10710647	C₂²	1	1	yes	characteristic, complemented
38	#10687759	C₂²	1	1	yes	characteristic
39	#10687758	C₂²	1	1	yes	characteristic
40	#10687757	C₂²	1	1	yes	characteristic
41	#10682460	C₂²	1	1	yes	characteristic, complemented
42	#10682459	C₂²	1	1	yes	characteristic, complemented
43	#10682458	C₂²	1	1	yes	characteristic, complemented
44	C₃×#23906	C₂²	1	1	yes	characteristic
45	C₃×#23900	C₂²	1	1	yes	characteristic
46	C₃×#15258	C₂²	1	1	yes	characteristic, complemented
47	C₃×#12693	C₂²	1	1	yes	characteristic, complemented
48	C₃×#12687	C₂²	1	1	yes	characteristic, complemented
49	C₃×#8534	C₂²	1	1	yes	characteristic
50	C₃×#7304	C₂²	1	1	yes	characteristic, complemented
51	#144386	D₃	1	1	yes	characteristic, complemented
52	C₂⁴×#67	D₄	1	1	yes	characteristic
53	C₂×D₃×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
54	C₂³×D₃×D₄	C₂³	1	1	yes	characteristic
55	C₂³×#237	D₄	1	1	yes	characteristic, complemented
56	C₃×#3484	C₂³	1	1	yes	characteristic
57	C₂²×C₃×#127	C₂³	1	1	yes	characteristic
58	C₂²×C₃×#120	C₂³	1	1	yes	characteristic
59	C₂²×#953	D₄	1	1	yes	characteristic
60	#87284	D₄	1	1	yes	characteristic, complemented
61	#87239	C₂³	1	1	yes	characteristic
62	C₂²×#940	D₄	1	1	yes	characteristic, complemented
63	C₂²×#936	C₂³	1	1	yes	characteristic
64	#85867	D₄	1	1	yes	characteristic, complemented
65	D₃×#518	D₄	1	1	yes	characteristic, complemented
66	C₂²×D₃×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
67	C₂²×#875	D₄	1	1	yes	characteristic, complemented
68	C₂²×#872	D₄	1	1	yes	characteristic, complemented
69	C₂×#5220	C₂³	1	1	yes	characteristic
70	C₂×#4955	D₄	1	1	yes	characteristic
71	C₂×#4955	D₄	1	1	yes	characteristic, complemented
72	C₂×#4958	D₄	1	1	yes	characteristic
73	C₂×D₃×#100	D₄	1	2	yes	complemented
74	C₂×D₃×#99	D₄	1	2	yes	complemented
75	C₃×#1783	C₂³	1	1	yes	characteristic
76	C₃×#1781	C₂³	1	1	yes	characteristic
77	C₃×#1781	C₂³	1	1	yes	characteristic
78	C₃×#1438	C₂³	1	1	yes	characteristic
79	#72131	C₂³	1	1	yes	characteristic
80	#72130	C₂³	1	1	yes	characteristic
81	#72130	C₂³	1	1	yes	characteristic
82	#72129	C₂³	1	1	yes	characteristic
83	#71550	D₄	1	1	yes	characteristic, complemented
84	#71550	D₄	1	1	yes	characteristic, complemented
85	#70079	D₄	1	2	yes	complemented
86	#70035	D₄	1	2	yes	complemented
87	#23906	C₂×D₃	1	1	yes	characteristic
88	#23900	C₂×D₃	1	1	yes	characteristic
89	#15258	C₂×D₃	1	1	yes	characteristic, complemented
90	#12693	C₂×D₃	1	1	yes	characteristic, complemented
91	#12687	C₂×D₃	1	1	yes	characteristic, complemented
92	#8534	C₂×D₃	1	1	yes	characteristic
93	#7304	C₂×D₃	1	1	yes	characteristic, complemented
94	C₂⁶×C₃	C₂×D₄	1	1	yes	characteristic
95	C₂⁵×D₃	C₂×D₄	1	1	yes	characteristic
96	C₂⁵×D₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₃×D₄	C₂⁴	1	1	yes	characteristic
100	C₂³×#67	C₂×D₄	1	1	yes	characteristic
101	C₂³×#67	C₂×D₄	1	1	yes	characteristic
102	C₂⁴×Dic₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#1004	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₃×D₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₃×D₄	#41	1	1	yes	characteristic
111	C₂²×D₃×D₄	#41	1	1	yes	characteristic
112	C₂³×D₁₂	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₄×D₃	C₂×D₄	1	1	yes	characteristic
114	C₂³×Dic₆	C₂×D₄	1	1	yes	characteristic
115	C₃×#518	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₃×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂²×#217	C₂×D₄	1	1	yes	characteristic
121	C₂²×#217	C₂×D₄	1	1	yes	characteristic
122	C₂²×#217	C₂×D₄	1	1	yes	characteristic
123	#5798	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×Dic₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×#100	C₂×D₄	1	2	yes	—
126	C₂×C₃×#99	C₂×D₄	1	2	yes	—
127	C₂×#834	C₂×D₄	1	2	yes	—
128	C₂×#833	C₂×D₄	1	2	yes	—
129	#3484	C₂²×D₃	1	1	yes	characteristic
130	C₂²×#127	C₂²×D₃	1	1	yes	characteristic
131	C₂²×#120	C₂²×D₃	1	1	yes	characteristic
132	#1783	C₂²×D₃	1	1	yes	characteristic
133	#1781	C₂²×D₃	1	1	yes	characteristic
134	#1781	C₂²×D₃	1	1	yes	characteristic
135	#1438	C₂²×D₃	1	1	yes	characteristic
136	C₂⁵×C₃	C₂²×D₄	1	1	yes	characteristic
137	C₂⁵×C₃	C₂²×D₄	1	1	yes	characteristic
138	C₂⁴×D₃	#120	1	1	yes	characteristic
139	C₂⁴×D₃	#120	1	1	yes	characteristic
140	C₂⁴×D₃	#120	1	1	yes	characteristic
141	C₂⁴×D₃	#120	1	1	yes	characteristic
142	C₂⁴×D₃	#120	1	1	yes	characteristic
143	C₂⁴×D₃	#121	1	1	yes	characteristic
144	C₂⁴×D₃	#121	1	1	yes	characteristic
145	C₂⁴×D₃	#121	1	1	yes	characteristic
146	C₂⁴×D₃	#121	1	1	yes	characteristic
147	C₂⁴×D₃	#127	1	1	yes	characteristic
148	C₂⁴×D₃	#127	1	1	yes	characteristic
149	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
151	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
152	C₂²×C₃×D₄	C₂×#41	1	1	yes	characteristic
153	C₂³×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂²×#67	#120	1	1	yes	characteristic
155	C₂²×#67	#120	1	1	yes	characteristic
156	C₂²×#67	#121	1	1	yes	characteristic
157	C₂²×#67	#121	1	1	yes	characteristic
158	C₂²×#67	#121	1	1	yes	characteristic
159	C₂²×#67	#121	1	1	yes	characteristic
160	C₂²×#67	#127	1	1	yes	characteristic
161	C₂²×#67	#127	1	1	yes	characteristic
162	C₂³×Dic₃	#120	1	1	yes	characteristic
163	C₂²×D₁₂	#120	1	1	yes	characteristic
164	C₂²×D₁₂	#120	1	1	yes	characteristic
165	C₂²×D₁₂	#121	1	1	yes	characteristic
166	C₂²×D₁₂	#121	1	1	yes	characteristic
167	C₂²×C₄×D₃	#120	1	1	yes	characteristic
168	C₂²×C₄×D₃	#120	1	1	yes	characteristic
169	C₂²×C₄×D₃	#121	1	1	yes	characteristic
170	C₂²×C₄×D₃	#121	1	1	yes	characteristic
171	C₂⁶	D₃×D₄	1	1	yes	characteristic
172	C₂×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
173	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
174	C₂³×D₄	C₂³×D₃	1	1	yes	characteristic
175	#518	D₃×D₄	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	D₃×D₄	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	D₃×D₄	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	D₃×D₄	1	1	yes	characteristic
179	C₂²×C₄²	D₃×D₄	1	1	yes	characteristic
180	C₂×#100	D₃×D₄	1	2	yes	—
181	C₂×#99	D₃×D₄	1	2	yes	—
182	C₂⁴×C₃	C₂×#120	1	1	yes	characteristic
183	C₂⁴×C₃	C₂×#120	1	1	yes	characteristic
184	C₂⁴×C₃	C₂×#121	1	1	yes	characteristic
185	C₂⁴×C₃	C₂×#121	1	1	yes	characteristic
186	C₂⁴×C₃	C₂×#127	1	1	yes	characteristic
187	C₂³×D₃	#392	1	1	yes	characteristic
188	C₂³×D₃	#392	1	1	yes	characteristic
189	C₂³×D₃	#392	1	1	yes	characteristic
190	C₂³×D₃	#392	1	1	yes	characteristic
191	C₂³×D₃	#392	1	1	yes	characteristic
192	C₂³×D₃	#392	1	1	yes	characteristic
193	C₂³×D₃	B(4,𝔽₂)	1	1	yes	characteristic
194	C₂³×D₃	B(4,𝔽₂)	1	1	yes	characteristic
195	C₂³×D₃	B(4,𝔽₂)	1	1	yes	characteristic
196	C₂³×D₃	B(4,𝔽₂)	1	1	yes	characteristic
197	C₂²×C₃×C₄	C₂×#120	1	1	yes	characteristic
198	C₂²×C₃×C₄	C₂×#121	1	1	yes	characteristic
199	C₂×#67	B(4,𝔽₂)	1	4	yes	—
200	C₂²×Dic₃	#392	1	1	yes	characteristic
201	C₂²×Dic₃	#392	1	1	yes	characteristic
202	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
203	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
204	C₂²×D₄	C₂×D₃×D₄	1	1	yes	characteristic
205	C₂²×D₄	C₂×D₃×D₄	1	1	yes	characteristic
206	C₂²×D₄	C₂×D₃×D₄	1	1	yes	characteristic
207	C₂²×D₄	D₃×#41	1	1	yes	characteristic
208	C₂³×C₄	C₂×D₃×D₄	1	1	yes	characteristic
209	C₂³×C₃	C₂×#392	1	1	yes	characteristic
210	C₂³×C₃	C₂×B(4,𝔽₂)	1	1	yes	characteristic
211	C₂³×C₃	C₂×B(4,𝔽₂)	1	1	yes	characteristic
212	C₂³×C₃	C₂×B(4,𝔽₂)	1	2	yes	—
213	C₂²×D₃	#2021	1	1	yes	characteristic
214	C₂²×D₃	#2021	1	1	yes	characteristic
215	C₂²×D₃	#2021	1	1	yes	characteristic
216	C₂²×D₃	#2021	1	1	yes	characteristic
217	C₂²×D₃	#2021	1	2	yes	—
218	C₂×C₃×C₄	#3026	1	2	yes	—
219	C₂×Dic₃	#2021	1	2	yes	—
220	C₂⁴	D₃×#120	1	1	yes	characteristic
221	C₂⁴	D₃×#120	1	1	yes	characteristic
222	C₂⁴	D₃×#121	1	1	yes	characteristic
223	C₂⁴	D₃×#121	1	1	yes	characteristic
224	C₂⁴	D₃×#127	1	1	yes	characteristic
225	C₂²×C₄	D₃×#120	1	1	yes	characteristic
226	C₂²×C₄	D₃×#121	1	1	yes	characteristic
227	C₂²×C₃	C₂×#2021	1	1	yes	characteristic
228	C₂²×C₃	C₂×#2021	1	1	yes	characteristic
229	C₂²×C₃	#23444	1	1	yes	characteristic
230	C₂²×C₃	#33551	1	1	yes	characteristic
231	C₂²×C₃	#33551	1	1	yes	characteristic
232	C₂²×C₃	#33551	1	1	yes	characteristic
233	C₂²×C₃	#33552	1	1	yes	characteristic
234	C₂³	D₃×#392	1	1	yes	characteristic
235	C₂³	D₃×B(4,𝔽₂)	1	1	yes	characteristic
236	C₂³	D₃×B(4,𝔽₂)	1	1	yes	characteristic
237	C₂³	D₃×B(4,𝔽₂)	1	2	yes	—
238	C₂×C₄	#80307	1	2	yes	—
239	C₂×C₃	#233541	1	1	yes	characteristic
240	C₂×C₃	#498927	1	1	yes	characteristic
241	C₂×C₃	#498931	1	1	yes	characteristic
242	C₂²	D₃×#2021	1	1	yes	characteristic
243	C₂²	D₃×#2021	1	1	yes	characteristic
244	C₂²	#10822434	1	1	yes	characteristic
245	C₂²	#10949241	1	1	yes	characteristic
246	C₂²	#10949241	1	1	yes	characteristic
247	C₂²	#10949247	1	1	yes	characteristic
248	C₂²	#10949254	1	1	yes	characteristic
249	C₃	#72213293	1	1	yes	characteristic, complemented, Sylow
250	C₂	#72248428	1	1	yes	characteristic
251	C₂	#72248608	1	1	yes	characteristic
252	C₂	#72248758	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#2021
commutator, cosocle	C₂³×C₃×D₄	C₂⁴
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₃×#144386	C₂
socle	C₂²×C₃	C₂×#2021
hypercenter	#144386	D₃
hypocenter	C₃	#72213293
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #144386

Complements

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

N	H ≅ G/N	type
1. #72255276	C₂	semidirect
2. #72255273	C₂	semidirect
3. #72255272	C₂	semidirect
4. C₃×#144386	C₂	semidirect
5. #72255270	C₂	semidirect
6. #72255269	C₂	semidirect
7. #72255260	C₂	semidirect
8. #72255215	C₂	semidirect
9. #72255133	C₂	semidirect
10. #72255053	C₂	semidirect
11. #72253585	C₂	semidirect
12. #72253547	C₂	semidirect
13. #72249302	C₂	semidirect
14. #72248912	C₂	semidirect
15. #72248661	C₂	semidirect
28. #10740388	C₂²	semidirect
29. #10733626	C₂²	semidirect
30. #10732891	C₂²	semidirect
32. #10711426	C₂²	semidirect
33. #10711423	C₂²	semidirect
34. #10710675	C₂²	semidirect
35. #10710675	C₂²	semidirect
36. #10710673	C₂²	semidirect
37. #10710647	C₂²	semidirect
41. #10682460	C₂²	semidirect
42. #10682459	C₂²	semidirect
43. #10682458	C₂²	semidirect
46. C₃×#15258	C₂²	semidirect
47. C₃×#12693	C₂²	semidirect
48. C₃×#12687	C₂²	semidirect
50. C₃×#7304	C₂²	semidirect
51. #144386	D₃	semidirect
53. C₂×D₃×H(5,𝔽₂)	D₄	semidirect
55. C₂³×#237	D₄	semidirect
60. #87284	D₄	semidirect
62. C₂²×#940	D₄	semidirect
64. #85867	D₄	semidirect
65. D₃×#518	D₄	semidirect
66. C₂²×D₃×C₂²⋊C₄	D₄	semidirect
67. C₂²×#875	D₄	semidirect
68. C₂²×#872	D₄	semidirect
71. C₂×#4955	D₄	semidirect
73. C₂×D₃×#100	D₄	semidirect
74. C₂×D₃×#99	D₄	semidirect
83. #71550	D₄	semidirect
84. #71550	D₄	semidirect
85. #70079	D₄	semidirect
86. #70035	D₄	semidirect
89. #15258	C₂×D₃	semidirect
90. #12693	C₂×D₃	semidirect
91. #12687	C₂×D₃	semidirect
93. #7304	C₂×D₃	semidirect
249. C₃	#72213293	semidirect

Maximal subgroups

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

subgroup	index
#72255276	2
#72255273	2
#72255272	2
C₃×#144386	2
#72255270	2
#72255269	2
#72255260	2
#72255215	2
#72255133	2
#72255053	2
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