Subgroups — #72231230

Order 2048.

At least 7363 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

The subgroup lattice has 268 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72231210	C₂	1	1	yes	maximal, characteristic, complemented
2	#72231195	C₂	1	1	yes	maximal, characteristic, complemented
3	#72231188	C₂	1	1	yes	maximal, characteristic, complemented
4	#72231177	C₂	1	1	yes	maximal, characteristic, complemented
5	#72231173	C₂	1	1	yes	maximal, characteristic, complemented
6	#72231165	C₂	1	1	yes	maximal, characteristic, complemented
7	#72231160	C₂	1	1	yes	maximal, characteristic, complemented
8	#72231156	C₂	1	1	yes	maximal, characteristic, complemented
9	#72231150	C₂	1	1	yes	maximal, characteristic, complemented
10	#72231143	C₂	1	1	yes	maximal, characteristic, complemented
11	#72231142	C₂	1	1	yes	maximal, characteristic, complemented
12	#72231140	C₂	1	1	yes	maximal, characteristic, complemented
13	#72231038	C₂	1	1	yes	maximal, characteristic, complemented
14	#72214821	C₂	1	1	yes	maximal, characteristic, complemented
15	#72214295	C₂	1	1	yes	maximal, characteristic, complemented
16	#10577301	C₂²	1	1	yes	characteristic, complemented
17	C₂×#45355	C₂²	1	1	yes	characteristic
18	C₂×D₄×#120	C₂²	1	1	yes	characteristic
19	#6491100	C₂²	1	1	yes	characteristic, complemented
20	#6454041	C₂²	1	1	yes	characteristic, complemented
21	#6346599	C₂²	1	1	yes	characteristic
22	#6346455	C₂²	1	1	yes	characteristic, complemented
23	#845307	C₂²	1	1	yes	characteristic
24	#719145	C₂²	1	1	yes	characteristic
25	#649038	C₂²	1	1	yes	characteristic
26	#609020	C₂²	1	1	yes	characteristic
27	#533071	C₂²	1	1	yes	characteristic
28	#496330	C₂²	1	1	yes	characteristic, complemented
29	#493995	C₂²	1	1	yes	characteristic, complemented
30	#482377	C₂²	1	1	yes	characteristic
31	#482357	C₂²	1	1	yes	characteristic
32	#481950	C₂²	1	1	yes	characteristic, complemented
33	#481950	C₂²	1	1	yes	characteristic, complemented
34	#480708	C₂²	1	1	yes	characteristic, complemented
35	#480376	C₂²	1	1	yes	characteristic, complemented
36	#201743	C₂²	1	1	yes	characteristic, complemented
37	#201386	C₂²	1	1	yes	characteristic
38	#194974	C₂²	1	1	yes	characteristic, complemented
39	#193892	C₂²	1	1	yes	characteristic, complemented
40	#186810	C₂²	1	1	yes	characteristic, complemented
41	#185755	C₂²	1	1	yes	characteristic, complemented
42	#185643	C₂²	1	1	yes	characteristic, complemented
43	#185643	C₂²	1	1	yes	characteristic, complemented
44	#185246	C₂²	1	1	yes	characteristic, complemented
45	#128198	C₂²	1	1	yes	characteristic, complemented
46	#128085	C₂²	1	1	yes	characteristic, complemented
47	#127207	C₂²	1	1	yes	characteristic, complemented
48	#127199	C₂²	1	1	yes	characteristic, complemented
49	#106758	C₂²	1	1	yes	characteristic, complemented
50	#106601	C₂²	1	1	yes	characteristic, complemented
51	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
52	#62773	C₂³	1	1	yes	characteristic
53	C₂²×D₄²	C₂³	1	1	yes	characteristic
54	C₂³×#127	D₄	1	1	yes	characteristic, complemented
55	#60230	C₂³	1	1	yes	characteristic
56	C₂×#2668	D₄	1	2	yes	complemented
57	C₂×#2847	D₄	1	1	yes	characteristic, complemented
58	C₂×#2541	D₄	1	1	yes	characteristic, complemented
59	C₂×#2535	D₄	1	1	yes	characteristic, complemented
60	C₂×#2433	D₄	1	1	yes	characteristic, complemented
61	C₂×#2433	D₄	1	1	yes	characteristic, complemented
62	C₂×#2408	C₂³	1	1	yes	characteristic
63	C₂×#2408	C₂³	1	1	yes	characteristic
64	C₂×#2403	D₄	1	2	yes	complemented
65	C₂×#2397	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
67	C₂×C₄×#120	C₂³	1	1	yes	characteristic
68	#20726	C₂³	1	1	yes	characteristic
69	#20722	C₂³	1	1	yes	characteristic
70	#20722	C₂³	1	1	yes	characteristic
71	#18196	D₄	1	2	yes	complemented
72	#18194	D₄	1	2	yes	complemented
73	#17364	C₂³	1	1	yes	characteristic
74	#17362	C₂³	1	1	yes	characteristic
75	#17362	C₂³	1	1	yes	characteristic
76	#15947	D₄	1	2	yes	—
77	#15420	D₄	1	1	yes	characteristic, complemented
78	#15406	D₄	1	1	yes	characteristic, complemented
79	#15284	D₄	1	1	yes	characteristic, complemented
80	#15267	D₄	1	1	yes	characteristic, complemented
81	#15266	D₄	1	2	yes	—
82	#14698	D₄	1	1	yes	characteristic, complemented
83	#14571	D₄	1	2	yes	complemented
84	#14272	D₄	1	1	yes	characteristic, complemented
85	#14181	D₄	1	2	yes	complemented
86	#13905	C₂³	1	1	yes	characteristic
87	#13902	D₄	1	1	yes	characteristic, complemented
88	#13785	D₄	1	1	yes	characteristic, complemented
89	C₄×#386	C₂³	1	1	yes	characteristic
90	#11873	D₄	1	4	yes	complemented
91	#9472	D₄	1	4	yes	complemented
92	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
94	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
97	#3501	C₂×D₄	1	1	yes	characteristic, complemented
98	#3496	C₂×D₄	1	1	yes	characteristic, complemented
99	C₂²×#127	C₂×D₄	1	1	yes	characteristic
100	C₂²×#121	C₂×D₄	1	1	yes	characteristic
101	C₂²×#121	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	2	yes	—
104	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
107	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
108	#2973	C₂×D₄	1	2	yes	complemented
109	#2408	C₂×D₄	1	2	yes	—
110	#2408	C₂×D₄	1	2	yes	—
111	#2408	C₂×D₄	1	2	yes	—
112	#2408	#41	1	2	yes	—
113	C₂×#399	C₂×D₄	1	1	yes	characteristic
114	C₂×#399	C₂×D₄	1	1	yes	characteristic
115	C₂×#394	C₂×D₄	1	2	yes	—
116	C₂×#392	C₂×D₄	1	1	yes	characteristic
117	C₂×#392	C₂×D₄	1	1	yes	characteristic
118	C₂×#392	C₂×D₄	1	2	yes	—
119	C₂×#386	C₂×D₄	1	1	yes	characteristic
120	C₂×#385	C₂×D₄	1	1	yes	characteristic
121	C₂×#382	C₂×D₄	1	1	yes	characteristic
122	C₂×#380	C₂×D₄	1	1	yes	characteristic
123	C₂×#380	C₂×D₄	1	1	yes	characteristic
124	C₂×#379	C₂×D₄	1	1	yes	characteristic
125	C₂×#379	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×#376	C₂×D₄	1	1	yes	characteristic
130	#1799	C₂×D₄	1	2	yes	complemented
131	#1799	C₂×D₄	1	2	yes	complemented
132	#1460	C₂×D₄	1	2	yes	complemented
133	#1459	C₂×D₄	1	2	yes	complemented
134	#1445	C₂×D₄	1	2	yes	—
135	#1438	C₂×D₄	1	2	yes	—
136	#1437	C₂×D₄	1	1	yes	characteristic
137	#1437	C₂×D₄	1	1	yes	characteristic
138	#1436	C₂×D₄	1	1	yes	characteristic
139	#1436	C₂×D₄	1	1	yes	characteristic
140	C₂⁶	#120	1	1	yes	characteristic
141	C₂⁶	#120	1	2	yes	—
142	C₂³×D₄	#120	1	1	yes	characteristic
143	C₂³×D₄	#120	1	2	yes	—
144	C₂³×D₄	#120	1	2	yes	—
145	C₂³×D₄	#120	1	1	yes	characteristic
146	C₂³×D₄	#120	1	1	yes	characteristic
147	C₂³×D₄	#120	1	1	yes	characteristic
148	C₂³×D₄	#121	1	2	yes	—
149	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂⁴×C₄	#120	1	1	yes	characteristic
151	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
165	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	#120	1	2	yes	—
167	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₄²	#120	1	1	yes	characteristic
170	C₂²×C₄²	#120	1	1	yes	characteristic
171	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
172	#394	#121	1	4	yes	—
173	#394	#127	1	4	yes	—
174	#392	#120	1	4	yes	—
175	#392	#121	1	4	yes	—
176	#380	#120	1	2	yes	—
177	#380	#121	1	2	yes	—
178	C₂×#95	#120	1	1	yes	characteristic
179	C₂×#95	#120	1	1	yes	characteristic
180	C₂×#95	#120	1	2	yes	—
181	C₂×#95	#120	1	1	yes	characteristic
182	C₂×#95	#120	1	2	yes	—
183	C₂×#95	#120	1	1	yes	characteristic
184	C₂×#95	#120	1	2	yes	—
185	C₂⁵	C₂×#120	1	1	yes	characteristic
186	C₂⁵	C₂×#120	1	1	yes	characteristic
187	C₂²×D₄	#392	1	4	yes	—
188	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
189	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
190	C₂²×D₄	D₄²	1	1	yes	characteristic
191	C₂²×D₄	D₄²	1	1	yes	characteristic
192	C₂²×D₄	D₄²	1	1	yes	characteristic
193	C₂²×D₄	#546	1	1	yes	characteristic
194	C₂³×C₄	C₂×#120	1	1	yes	characteristic
195	C₂³×C₄	C₂×#120	1	1	yes	characteristic
196	C₂³×C₄	C₂×#120	1	1	yes	characteristic
197	C₂³×C₄	C₂×#120	1	1	yes	characteristic
198	C₂³×C₄	C₂×#120	1	1	yes	characteristic
199	C₂³×C₄	C₂×#120	1	1	yes	characteristic
200	C₂³×C₄	C₂×#121	1	1	yes	characteristic
201	C₂³×C₄	#534	1	1	yes	characteristic
202	C₂³×C₄	#534	1	1	yes	characteristic
203	C₂³×C₄	D₄²	1	1	yes	characteristic
204	C₂³×C₄	D₄²	1	1	yes	characteristic
205	C₂³×C₄	D₄²	1	1	yes	characteristic
206	C₂³×C₄	#546	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×#121	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×#127	1	1	yes	characteristic
209	C₂×C₄⋊C₄	#535	1	1	yes	characteristic
210	C₂×C₄⋊C₄	#535	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
219	C₂×C₄²	D₄²	1	1	yes	characteristic
220	C₂×C₄²	D₄²	1	1	yes	characteristic
221	C₂×C₄²	D₄²	1	1	yes	characteristic
222	C₂×C₄²	#546	1	1	yes	characteristic
223	#95	#392	1	4	yes	—
224	C₂⁴	#2403	1	1	yes	characteristic
225	C₂⁴	#2403	1	1	yes	characteristic
226	C₂⁴	#2410	1	1	yes	characteristic
227	C₂⁴	#2410	1	1	yes	characteristic
228	C₂⁴	#2846	1	1	yes	characteristic
229	C₂×D₄	#2196	1	4	yes	—
230	C₂²×C₄	C₂×#392	1	1	yes	characteristic
231	C₂²×C₄	#2403	1	1	yes	characteristic
232	C₂²×C₄	#2403	1	1	yes	characteristic
233	C₂²×C₄	#2408	1	1	yes	characteristic
234	C₂²×C₄	#2408	1	1	yes	characteristic
235	C₂²×C₄	#2410	1	1	yes	characteristic
236	C₂²×C₄	#2410	1	1	yes	characteristic
237	C₂²×C₄	#2433	1	1	yes	characteristic
238	C₂²×C₄	#2433	1	1	yes	characteristic
239	C₂²×C₄	#2613	1	1	yes	characteristic
240	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
241	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
242	C₂²×C₄	#3026	1	1	yes	characteristic
243	C₂²×C₄	#3026	1	1	yes	characteristic
244	C₂³	#14715	1	1	yes	characteristic
245	C₂³	#23872	1	2	yes	—
246	C₂³	#23882	1	2	yes	—
247	C₂³	#32898	1	1	yes	characteristic
248	C₂³	#32898	1	1	yes	characteristic
249	C₂×C₄	#23900	1	1	yes	characteristic
250	C₂×C₄	#23903	1	1	yes	characteristic
251	C₂×C₄	#24404	1	1	yes	characteristic
252	C₂×C₄	#24404	1	1	yes	characteristic
253	C₂×C₄	C₂×#2196	1	2	yes	—
254	C₂×C₄	#33551	1	2	yes	—
255	C₂²	#394066	1	1	yes	characteristic
256	C₂²	#394098	1	1	yes	characteristic
257	C₂²	#394102	1	1	yes	characteristic
258	C₂²	#511771	1	1	yes	characteristic
259	C₂²	#511779	1	1	yes	characteristic
260	C₂²	#511819	1	1	yes	characteristic
261	C₂²	#511827	1	1	yes	characteristic
262	C₄	#499787	1	2	yes	—
263	C₄	#501156	1	2	yes	—
264	C₂	#72215601	1	1	yes	characteristic
265	C₂	#72231222	1	1	yes	characteristic
266	C₂	#72231228	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394066
commutator, Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
LCS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72231210	C₂	semidirect
2. #72231195	C₂	semidirect
3. #72231188	C₂	semidirect
4. #72231177	C₂	semidirect
5. #72231173	C₂	semidirect
6. #72231165	C₂	semidirect
7. #72231160	C₂	semidirect
8. #72231156	C₂	semidirect
9. #72231150	C₂	semidirect
10. #72231143	C₂	semidirect
11. #72231142	C₂	semidirect
12. #72231140	C₂	semidirect
13. #72231038	C₂	semidirect
14. #72214821	C₂	semidirect
15. #72214295	C₂	semidirect
16. #10577301	C₂²	semidirect
19. #6491100	C₂²	semidirect
20. #6454041	C₂²	semidirect
22. #6346455	C₂²	semidirect
28. #496330	C₂²	semidirect
29. #493995	C₂²	semidirect
32. #481950	C₂²	semidirect
33. #481950	C₂²	semidirect
34. #480708	C₂²	semidirect
35. #480376	C₂²	semidirect
36. #201743	C₂²	semidirect
38. #194974	C₂²	semidirect
39. #193892	C₂²	semidirect
40. #186810	C₂²	semidirect
41. #185755	C₂²	semidirect
42. #185643	C₂²	semidirect
43. #185643	C₂²	semidirect
44. #185246	C₂²	semidirect
45. #128198	C₂²	semidirect
46. #128085	C₂²	semidirect
47. #127207	C₂²	semidirect
48. #127199	C₂²	semidirect
49. #106758	C₂²	semidirect
50. #106601	C₂²	semidirect
51. C₂⁵×D₄	D₄	semidirect
54. C₂³×#127	D₄	semidirect
56. C₂×#2668	D₄	semidirect
57. C₂×#2847	D₄	semidirect
58. C₂×#2541	D₄	semidirect
59. C₂×#2535	D₄	semidirect
60. C₂×#2433	D₄	semidirect
61. C₂×#2433	D₄	semidirect
64. C₂×#2403	D₄	semidirect
66. C₂×D₄×C₂²⋊C₄	D₄	semidirect
71. #18196	D₄	semidirect
72. #18194	D₄	semidirect
77. #15420	D₄	semidirect
78. #15406	D₄	semidirect
79. #15284	D₄	semidirect
80. #15267	D₄	semidirect
82. #14698	D₄	semidirect
83. #14571	D₄	semidirect
84. #14272	D₄	semidirect
85. #14181	D₄	semidirect
87. #13902	D₄	semidirect
88. #13785	D₄	semidirect
90. #11873	D₄	semidirect
91. #9472	D₄	semidirect
97. #3501	C₂×D₄	semidirect
98. #3496	C₂×D₄	semidirect
108. #2973	C₂×D₄	semidirect
130. #1799	C₂×D₄	semidirect
131. #1799	C₂×D₄	semidirect
132. #1460	C₂×D₄	semidirect
133. #1459	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72231210	2
#72231195	2
#72231188	2
#72231177	2
#72231173	2
#72231165	2
#72231160	2
#72231156	2
#72231150	2
#72231143	2
#72231142	2
#72231140	2
#72231038	2
#72214821	2
#72214295	2