Subgroups — #72262735

Order 2048.

At least 7999 conjugacy classes of subgroups; only the 342 normal subgroups are shown.

The subgroup lattice has 258 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72256025	C₂	1	1	yes	maximal, characteristic, complemented
2	#72255980	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255874	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255863	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237473	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237467	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237373	C₂	1	1	yes	maximal, characteristic, complemented
8	#72236521	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236508	C₂	1	1	yes	maximal, characteristic, complemented
10	#72232592	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225654	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218855	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511790	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#34981	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	#6440460	C₂²	1	1	yes	characteristic
20	C₂×#23895	C₂²	1	1	yes	characteristic
21	#6437621	C₂²	1	1	yes	characteristic
22	#6437621	C₂²	1	1	yes	characteristic
23	#6437621	C₂²	1	1	yes	characteristic, complemented
24	C₂×#23900	C₂²	1	1	yes	characteristic
25	C₂×#23897	C₂²	1	1	yes	characteristic
26	#6374210	C₂²	1	1	yes	characteristic, complemented
27	C₂×#21746	C₂²	1	1	yes	characteristic
28	#493879	C₂²	1	1	yes	characteristic, complemented
29	#493879	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12707	C₂²	1	1	yes	characteristic
31	C₂×#12707	C₂²	1	1	yes	characteristic
32	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
33	C₂×#12705	C₂²	1	1	yes	characteristic
34	C₂×#12705	C₂²	1	1	yes	characteristic
35	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
36	C₂×#12687	C₂²	1	1	yes	characteristic
37	C₂×#12687	C₂²	1	1	yes	characteristic
38	#382707	C₂²	1	1	yes	characteristic, complemented
39	#359558	C₂²	1	1	yes	characteristic
40	#359549	C₂²	1	1	yes	characteristic
41	#359501	C₂²	1	1	yes	characteristic
42	#359442	C₂²	1	1	yes	characteristic, complemented
43	#188109	C₂²	1	1	yes	characteristic, complemented
44	#188049	C₂²	1	1	yes	characteristic, complemented
45	#146479	C₂²	1	1	yes	characteristic, complemented
46	#146475	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
51	#62817	D₄	1	4	yes	complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂²×#534	C₂³	1	1	yes	characteristic
56	C₂³×#120	D₄	1	1	yes	characteristic, complemented
57	C₂³×#120	C₂³	1	1	yes	characteristic
58	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
59	C₂²×B(4,𝔽₂)	D₄	1	2	yes	complemented
60	#34171	C₂³	1	1	yes	characteristic
61	C₂×#2315	D₄	1	1	yes	characteristic, complemented
62	C₂×#2308	D₄	1	1	yes	characteristic, complemented
63	C₂×#2308	C₂³	1	1	yes	characteristic
64	#23901	D₄	1	2	yes	complemented
65	#23901	D₄	1	2	yes	complemented
66	#23900	D₄	1	2	yes	complemented
67	#23900	D₄	1	2	yes	complemented
68	#21746	D₄	1	4	yes	complemented
69	C₂×#1889	D₄	1	2	yes	complemented
70	C₂×#1889	D₄	1	2	yes	complemented
71	C₂×#1889	D₄	1	2	yes	complemented
72	#20889	D₄	1	2	yes	complemented
73	#20889	D₄	1	2	yes	complemented
74	#20609	D₄	1	2	yes	complemented
75	#20609	D₄	1	2	yes	complemented
76	C₂×#1783	C₂³	1	1	yes	characteristic
77	C₂×#1783	C₂³	1	1	yes	characteristic
78	C₂×#1781	C₂³	1	1	yes	characteristic
79	C₂×#1781	C₂³	1	1	yes	characteristic
80	C₂×#1781	C₂³	1	1	yes	characteristic
81	C₂×#1781	C₂³	1	1	yes	characteristic
82	C₂×#1781	C₂³	1	1	yes	characteristic
83	C₂×#1781	C₂³	1	1	yes	characteristic
84	#12687	D₄	1	2	yes	complemented
85	#12687	D₄	1	2	yes	complemented
86	C₂⁷	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
92	#3484	C₂×D₄	1	2	yes	—
93	#3484	C₂×D₄	1	2	yes	—
94	#3484	C₂×D₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	C₂²×#123	C₂×D₄	1	1	yes	characteristic
97	C₂²×#123	C₂×D₄	1	1	yes	characteristic
98	C₂²×#120	C₂×D₄	1	2	yes	—
99	C₂²×#120	C₂×D₄	1	2	yes	—
100	C₂²×#120	C₂×D₄	1	1	yes	characteristic
101	C₂²×#120	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	2	yes	—
103	C₂²×#120	C₂×D₄	1	2	yes	—
104	C₂²×#120	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂²×#120	#41	1	1	yes	characteristic
107	C₂²×#120	#41	1	1	yes	characteristic
108	C₂×#518	C₂×D₄	1	1	yes	characteristic
109	C₂×#518	C₂×D₄	1	1	yes	characteristic
110	C₂×#518	C₂×D₄	1	1	yes	characteristic
111	C₂×#518	C₂×D₄	1	1	yes	characteristic
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
117	C₂²×#99	C₂×D₄	1	2	yes	—
118	C₂²×#99	C₂×D₄	1	2	yes	—
119	C₂²×#99	C₂×D₄	1	2	yes	—
120	C₂²×#99	C₂×D₄	1	2	yes	—
121	#2308	C₂×D₄	1	2	yes	—
122	#2308	C₂×D₄	1	2	yes	—
123	#1783	C₂×D₄	1	2	yes	—
124	#1781	C₂×D₄	1	2	yes	—
125	#1781	C₂×D₄	1	2	yes	—
126	#1781	C₂×D₄	1	2	yes	—
127	#1781	C₂×D₄	1	2	yes	—
128	#1781	C₂×D₄	1	2	yes	—
129	C₂⁶	#120	1	2	yes	—
130	C₂⁶	#120	1	2	yes	—
131	C₂⁶	#120	1	1	yes	characteristic
132	C₂⁶	#121	1	1	yes	characteristic
133	C₂⁶	#121	1	1	yes	characteristic
134	C₂⁶	C₂²×D₄	1	1	yes	characteristic
135	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×H(5,𝔽₂)	#120	1	2	yes	—
137	C₂×H(5,𝔽₂)	#120	1	2	yes	—
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
146	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
147	C₂³×D₄	C₂×#41	1	1	yes	characteristic
148	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂⁴×C₄	#120	1	1	yes	characteristic
151	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×#120	#120	1	2	yes	—
153	C₂×#120	#120	1	2	yes	—
154	C₂×#120	#121	1	2	yes	—
155	C₂×#120	#121	1	2	yes	—
156	C₂×#120	#121	1	2	yes	—
157	C₂×#120	#121	1	2	yes	—
158	C₂×#120	#127	1	2	yes	—
159	C₂×#120	#127	1	2	yes	—
160	#518	#120	1	2	yes	—
161	#518	#120	1	2	yes	—
162	C₂²×C₂²⋊C₄	#120	1	2	yes	—
163	C₂²×C₂²⋊C₄	#120	1	2	yes	—
164	C₂²×C₂²⋊C₄	#120	1	2	yes	—
165	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	#120	1	2	yes	—
168	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
170	C₂×#99	#120	1	4	yes	—
171	C₂×#99	#120	1	4	yes	—
172	C₂⁵	#392	1	2	yes	—
173	C₂⁵	#392	1	2	yes	—
174	C₂⁵	B(4,𝔽₂)	1	2	yes	—
175	C₂⁵	B(4,𝔽₂)	1	2	yes	—
176	C₂⁵	B(4,𝔽₂)	1	2	yes	—
177	C₂⁵	B(4,𝔽₂)	1	2	yes	—
178	C₂⁵	C₂×#120	1	1	yes	characteristic
179	C₂⁵	C₂×#120	1	1	yes	characteristic
180	C₂⁵	C₂×#120	1	1	yes	characteristic
181	C₂⁵	C₂×#120	1	1	yes	characteristic
182	C₂⁵	C₂×#121	1	1	yes	characteristic
183	C₂⁵	#534	1	1	yes	characteristic
184	C₂⁵	#534	1	1	yes	characteristic
185	C₂⁵	#534	1	1	yes	characteristic
186	C₂⁵	#535	1	1	yes	characteristic
187	C₂⁵	D₄²	1	1	yes	characteristic
188	C₂⁵	D₄²	1	1	yes	characteristic
189	C₂⁵	D₄²	1	1	yes	characteristic
190	C₂⁵	#546	1	1	yes	characteristic
191	C₂²×D₄	C₂×#120	1	1	yes	characteristic
192	C₂²×D₄	C₂×#120	1	1	yes	characteristic
193	C₂²×D₄	C₂×#120	1	1	yes	characteristic
194	C₂²×D₄	C₂×#120	1	1	yes	characteristic
195	C₂²×D₄	C₂×#120	1	1	yes	characteristic
196	C₂²×D₄	C₂×#121	1	1	yes	characteristic
197	C₂²×D₄	C₂×#121	1	1	yes	characteristic
198	C₂²×D₄	C₂×#127	1	1	yes	characteristic
199	C₂²×D₄	D₄²	1	1	yes	characteristic
200	C₂²×D₄	D₄²	1	1	yes	characteristic
201	C₂²×D₄	D₄²	1	1	yes	characteristic
202	C₂²×D₄	D₄²	1	1	yes	characteristic
203	C₂²×D₄	D₄²	1	1	yes	characteristic
204	C₂²×D₄	D₄²	1	1	yes	characteristic
205	C₂²×D₄	#546	1	1	yes	characteristic
206	C₂²×D₄	#546	1	1	yes	characteristic
207	C₂³×C₄	#534	1	1	yes	characteristic
208	C₂³×C₄	#535	1	1	yes	characteristic
209	C₂³×C₄	D₄²	1	1	yes	characteristic
210	C₂³×C₄	D₄²	1	1	yes	characteristic
211	C₂³×C₄	D₄²	1	1	yes	characteristic
212	C₂³×C₄	#546	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	#392	1	2	yes	—
214	C₂×C₂²⋊C₄	#392	1	2	yes	—
215	C₂⁴	#2021	1	2	yes	—
216	C₂⁴	#2021	1	2	yes	—
217	C₂⁴	#2021	1	2	yes	—
218	C₂⁴	#2021	1	2	yes	—
219	C₂⁴	C₂×#392	1	1	yes	characteristic
220	C₂⁴	#2403	1	1	yes	characteristic
221	C₂⁴	#2403	1	1	yes	characteristic
222	C₂⁴	#2403	1	1	yes	characteristic
223	C₂⁴	#2408	1	1	yes	characteristic
224	C₂⁴	#2410	1	1	yes	characteristic
225	C₂⁴	#2410	1	1	yes	characteristic
226	C₂⁴	#2410	1	1	yes	characteristic
227	C₂⁴	#2433	1	1	yes	characteristic
228	C₂⁴	#2613	1	1	yes	characteristic
229	C₂⁴	#2846	1	1	yes	characteristic
230	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
231	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
232	C₂⁴	#3025	1	2	yes	—
233	C₂⁴	#3027	1	2	yes	—
234	C₂²×C₄	#2403	1	1	yes	characteristic
235	C₂²×C₄	#2408	1	1	yes	characteristic
236	C₂²×C₄	#2410	1	1	yes	characteristic
237	C₂²×C₄	#2433	1	1	yes	characteristic
238	C₂³	#14715	1	1	yes	characteristic
239	C₂³	C₂×#2021	1	2	yes	—
240	C₂³	#23895	1	1	yes	characteristic
241	C₂³	#23895	1	1	yes	characteristic
242	C₂³	#23900	1	2	yes	—
243	C₂³	#23900	1	1	yes	characteristic
244	C₂³	#23900	1	1	yes	characteristic
245	C₂³	#23902	1	1	yes	characteristic
246	C₂³	#23904	1	1	yes	characteristic
247	C₂³	#24406	1	2	yes	—
248	C₂³	#32899	1	2	yes	—
249	C₂²	#245231	1	1	yes	characteristic
250	C₂²	#245231	1	1	yes	characteristic
251	C₂²	#245252	1	1	yes	characteristic
252	C₂²	#245252	1	1	yes	characteristic
253	C₂²	#394058	1	1	yes	characteristic
254	C₂²	#394064	1	1	yes	characteristic
255	C₂²	#394067	1	1	yes	characteristic
256	C₂	#72241299	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241299
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72256025	C₂	semidirect
2. #72255980	C₂	semidirect
3. #72255874	C₂	semidirect
4. #72255863	C₂	semidirect
5. #72237473	C₂	semidirect
6. #72237467	C₂	semidirect
7. #72237373	C₂	semidirect
8. #72236521	C₂	semidirect
9. #72236508	C₂	semidirect
10. #72232592	C₂	semidirect
11. #72225654	C₂	semidirect
12. #72218855	C₂	semidirect
13. #72218475	C₂	semidirect
14. C₂×#139731	C₂	semidirect
15. C₂×#511790	C₂	semidirect
23. #6437621	C₂²	semidirect
26. #6374210	C₂²	semidirect
28. #493879	C₂²	semidirect
29. #493879	C₂²	semidirect
32. C₂×#12707	C₂²	semidirect
35. C₂×#12705	C₂²	semidirect
38. #382707	C₂²	semidirect
42. #359442	C₂²	semidirect
43. #188109	C₂²	semidirect
44. #188049	C₂²	semidirect
45. #146479	C₂²	semidirect
46. #146475	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7310	C₂²	semidirect
51. #62817	D₄	semidirect
56. C₂³×#120	D₄	semidirect
58. C₂⁴×C₂²⋊C₄	D₄	semidirect
59. C₂²×B(4,𝔽₂)	D₄	semidirect
61. C₂×#2315	D₄	semidirect
62. C₂×#2308	D₄	semidirect
64. #23901	D₄	semidirect
65. #23901	D₄	semidirect
66. #23900	D₄	semidirect
67. #23900	D₄	semidirect
68. #21746	D₄	semidirect
69. C₂×#1889	D₄	semidirect
70. C₂×#1889	D₄	semidirect
71. C₂×#1889	D₄	semidirect
72. #20889	D₄	semidirect
73. #20889	D₄	semidirect
74. #20609	D₄	semidirect
75. #20609	D₄	semidirect
84. #12687	D₄	semidirect
85. #12687	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72256025	2
#72255980	2
#72255874	2
#72255863	2
#72237473	2
#72237467	2
#72237373	2
#72236521	2
#72236508	2
#72232592	2
#72225654	2
#72218855	2
#72218475	2
C₂×#139731	2
C₂×#511790	2