Subgroups — #72266321

Order 4096.

At least 13843 conjugacy classes of subgroups; only the 543 normal subgroups are shown.

The subgroup lattice has 256 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266320	C₂	1	4	yes	maximal, complemented
2	#72266319	C₂	1	4	yes	maximal, complemented
3	#72266289	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266288	C₂²	1	4	yes	complemented
5	#72266202	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266201	C₂²	1	4	yes	complemented
7	#72266068	C₂	1	1	yes	maximal, characteristic, complemented
8	#72236821	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236816	C₂²	1	2	yes	complemented
10	#72236812	C₂²	1	2	yes	complemented
11	#72236800	C₂	1	1	yes	maximal, characteristic
12	#72236754	C₂²	1	2	yes	—
13	#72236742	C₄	1	2	yes	complemented
14	C₂×#245229	C₄	1	2	yes	complemented
15	#72236034	C₂	1	1	yes	maximal, characteristic, complemented
16	#72235930	C₂²	1	1	yes	characteristic
17	#72235865	C₂²	1	2	yes	complemented
18	#72231822	C₂²	1	2	yes	complemented
19	#72226161	C₂	1	1	yes	maximal, characteristic, complemented
20	#72226038	C₂²	1	4	yes	—
21	#72225881	C₂²	1	1	yes	characteristic
22	#72225658	C₂²	1	4	yes	—
23	#72219377	C₂²	1	1	yes	characteristic
24	#72219377	C₂²	1	1	yes	characteristic
25	#72219307	C₂²	1	1	yes	characteristic
26	#72219307	C₂²	1	1	yes	characteristic
27	C₂×#7631286	C₂²	1	1	yes	characteristic
28	#72218846	C₄	1	2	yes	complemented
29	C₂×#245225	C₂²	1	2	yes	—
30	#72214832	C₄	1	2	yes	complemented
31	C₂⁴×H(5,𝔽₂)	D₄	1	1	yes	characteristic
32	C₂²×#3484	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂³×D₄²	C₂³	1	1	yes	characteristic
34	C₂×#35527	D₄	1	2	yes	—
35	C₂×#35512	C₂×C₄	1	2	yes	—
36	C₂×#35347	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×#35347	D₄	1	2	yes	—
38	C₂×#35346	C₂³	1	1	yes	characteristic
39	C₂×#35346	C₂³	1	2	yes	—
40	C₂×#34980	D₄	1	2	yes	—
41	C₂²×#2403	C₂×C₄	1	2	yes	—
42	C₂×#34171	D₄	1	1	yes	characteristic
43	C₂×#32898	C₂×C₄	1	4	yes	complemented
44	C₂×#21746	D₄	1	2	yes	complemented
45	#517411	C₂×C₄	1	1	yes	characteristic, complemented
46	#517411	D₄	1	2	yes	complemented
47	#515362	C₂×C₄	1	1	yes	characteristic, complemented
48	#515285	C₂³	1	1	yes	characteristic
49	#392658	C₂×C₄	1	4	yes	complemented
50	#392245	D₄	1	2	yes	complemented
51	#392109	C₂×C₄	1	2	yes	complemented
52	#392109	D₄	1	2	yes	complemented
53	#391709	D₄	1	8	yes	complemented
54	#382422	C₂³	1	2	yes	—
55	#382422	C₂³	1	2	yes	—
56	#382422	C₂³	1	2	yes	—
57	#374004	D₄	1	8	yes	complemented
58	C₂×#9495	C₂×C₄	1	4	yes	complemented
59	C₂×#9495	C₂×C₄	1	4	yes	complemented
60	C₂×#8547	C₂³	1	2	yes	—
61	C₂×#8542	C₂×C₄	1	2	yes	complemented
62	C₂×#8541	C₂³	1	2	yes	—
63	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
64	C₂³×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic, complemented
65	C₂³×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
66	C₂⁵×D₄	C₂²⋊C₄	1	2	yes	—
67	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
68	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
69	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
70	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
71	C₂³×#127	C₂×D₄	1	1	yes	characteristic
72	C₂³×#121	C₂²×C₄	1	1	yes	characteristic
73	C₂³×#121	C₂×D₄	1	1	yes	characteristic
74	C₂³×#121	#41	1	1	yes	characteristic
75	C₂³×#121	#41	1	1	yes	characteristic
76	C₂³×#120	C₂×D₄	1	1	yes	characteristic
77	C₂³×#120	C₂×D₄	1	1	yes	characteristic
78	C₂³×#120	#41	1	1	yes	characteristic
79	C₂³×#120	C₂⁴	1	1	yes	characteristic
80	C₂²×#518	C₂²×C₄	1	1	yes	characteristic, complemented
81	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₄×D₄	#41	1	1	yes	characteristic
83	#35346	C₂×D₄	1	4	yes	—
84	#35346	C₂×D₄	1	4	yes	—
85	#35346	#41	1	8	yes	—
86	C₂×#2405	C₂²⋊C₄	1	4	yes	—
87	C₂×#2403	C₂²⋊C₄	1	4	yes	—
88	C₂×#2403	C₂²×C₄	1	4	yes	—
89	C₂×#2403	C₂×D₄	1	4	yes	—
90	C₂×#2309	C₂²⋊C₄	1	2	yes	—
91	C₂×#2308	C₂²×C₄	1	1	yes	characteristic, complemented
92	C₂×#2308	C₂²×C₄	1	1	yes	characteristic, complemented
93	C₂×#2308	C₂×D₄	1	2	yes	—
94	C₂²×#379	C₂²⋊C₄	1	2	yes	—
95	C₂×#1819	C₂×D₄	1	4	yes	—
96	C₂×#1781	C₂²×C₄	1	4	yes	complemented
97	#14679	C₂×D₄	1	4	yes	—
98	#10965	C₂×D₄	1	8	yes	—
99	#10965	C₂×D₄	1	8	yes	—
100	C₂⁷	#99	1	2	yes	complemented
101	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂⁷	C₄×D₄	1	1	yes	characteristic
103	C₂⁷	#120	1	1	yes	characteristic
104	C₂⁷	C₂²×D₄	1	1	yes	characteristic
105	C₂²×H(5,𝔽₂)	#99	1	2	yes	complemented
106	C₂²×H(5,𝔽₂)	#99	1	4	yes	complemented
107	C₂²×H(5,𝔽₂)	#99	1	2	yes	complemented
108	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
109	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
110	C₂²×H(5,𝔽₂)	#123	1	4	yes	—
111	C₂⁴×D₄	#99	1	4	yes	complemented
112	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂⁴×D₄	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
116	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
117	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
118	C₂⁴×D₄	#120	1	1	yes	characteristic
119	C₂⁴×D₄	#120	1	1	yes	characteristic
120	C₂⁴×D₄	#121	1	1	yes	characteristic
121	C₂⁴×D₄	#121	1	1	yes	characteristic
122	C₂⁴×D₄	#121	1	1	yes	characteristic
123	C₂⁴×D₄	#121	1	1	yes	characteristic
124	C₂⁴×D₄	#121	1	1	yes	characteristic
125	C₂⁴×D₄	#123	1	1	yes	characteristic
126	C₂⁴×D₄	C₂³×C₄	1	1	yes	characteristic
127	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
129	C₂⁵×C₄	#99	1	2	yes	complemented
130	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
132	C₂⁵×C₄	#120	1	1	yes	characteristic
133	C₂⁵×C₄	#123	1	1	yes	characteristic
134	C₂²×#120	C₂²×D₄	1	4	yes	—
135	C₂²×#120	C₂×#41	1	4	yes	—
136	C₂³×C₄⋊C₄	C₄×D₄	1	1	yes	characteristic
137	C₂³×C₄⋊C₄	#123	1	1	yes	characteristic
138	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
140	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
141	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
142	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
143	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
145	C₂³×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
146	C₂³×C₄²	#121	1	1	yes	characteristic
147	#2403	C₄×D₄	1	8	yes	—
148	#2403	C₄×D₄	1	8	yes	—
149	#2403	#121	1	8	yes	—
150	#2403	#121	1	8	yes	—
151	#2308	#120	1	4	yes	—
152	#2308	#123	1	4	yes	—
153	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂×#379	#120	1	8	yes	complemented
155	C₂×#342	#120	1	8	yes	complemented
156	C₂⁶	#386	1	1	yes	characteristic
157	C₂⁶	#386	1	1	yes	characteristic
158	C₂⁶	#386	1	1	yes	characteristic
159	C₂⁶	#386	1	1	yes	characteristic
160	C₂⁶	C₂×#99	1	2	yes	—
161	C₂⁶	C₂×#99	1	2	yes	—
162	C₂⁶	C₂×#99	1	4	yes	—
163	C₂⁶	B(4,𝔽₂)	1	2	yes	—
164	C₂⁶	B(4,𝔽₂)	1	2	yes	—
165	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂⁶	C₂×#120	1	1	yes	characteristic
168	C₂⁶	C₂×#120	1	2	yes	—
169	C₂⁶	C₂×#121	1	1	yes	characteristic
170	C₂⁶	#525	1	2	yes	—
171	C₂³×D₄	#386	1	4	yes	—
172	C₂³×D₄	#386	1	4	yes	—
173	C₂³×D₄	C₂×#99	1	2	yes	—
174	C₂³×D₄	C₂×#99	1	2	yes	—
175	C₂³×D₄	C₂×#99	1	4	yes	—
176	C₂³×D₄	C₂×#99	1	4	yes	—
177	C₂³×D₄	C₂×C₄×D₄	1	4	yes	—
178	C₂³×D₄	C₂×#120	1	2	yes	—
179	C₂³×D₄	C₂×#121	1	4	yes	—
180	C₂³×D₄	C₂×#123	1	2	yes	—
181	C₂⁴×C₄	#386	1	1	yes	characteristic
182	C₂⁴×C₄	#386	1	1	yes	characteristic
183	C₂⁴×C₄	#386	1	1	yes	characteristic
184	C₂⁴×C₄	#386	1	1	yes	characteristic
185	C₂⁴×C₄	C₂×#99	1	4	yes	—
186	C₂⁴×C₄	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
188	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
190	C₂²×C₂²⋊C₄	#525	1	2	yes	—
191	C₂²×C₂²⋊C₄	#525	1	4	yes	—
192	C₂²×C₂²⋊C₄	#525	1	4	yes	—
193	#379	#386	1	8	yes	—
194	C₂⁵	#1889	1	1	yes	characteristic
195	C₂⁵	#1889	1	1	yes	characteristic
196	C₂⁵	#1889	1	2	yes	—
197	C₂⁵	#1898	1	1	yes	characteristic
198	C₂⁵	#1898	1	2	yes	—
199	C₂⁵	#1898	1	1	yes	characteristic
200	C₂⁵	#1898	1	1	yes	characteristic
201	C₂⁵	#1898	1	1	yes	characteristic
202	C₂⁵	#1898	1	4	yes	—
203	C₂⁵	#1898	1	4	yes	—
204	C₂⁵	#1913	1	1	yes	characteristic
205	C₂⁵	#1913	1	1	yes	characteristic
206	C₂⁵	#1913	1	2	yes	—
207	C₂⁵	C₂×#386	1	2	yes	—
208	C₂⁵	C₂×#386	1	1	yes	characteristic
209	C₂⁵	C₄×#120	1	2	yes	—
210	C₂⁵	#2846	1	1	yes	characteristic
211	C₂⁵	#2847	1	2	yes	—
212	C₂⁵	#2849	1	1	yes	characteristic
213	C₂⁵	C₂²×#99	1	2	yes	—
214	C₂⁵	C₂²×#99	1	2	yes	—
215	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
216	C₂⁵	#3026	1	2	yes	—
217	C₂³×C₄	#1898	1	2	yes	—
218	C₂³×C₄	#1898	1	4	yes	—
219	C₂³×C₄	#1898	1	4	yes	—
220	C₂³×C₄	C₄×#120	1	2	yes	—
221	C₂³×C₄	#2847	1	2	yes	—
222	C₂³×C₄	#2849	1	2	yes	—
223	C₂⁴	#9495	1	1	yes	characteristic
224	C₂⁴	#9495	1	1	yes	characteristic
225	C₂⁴	#9495	1	1	yes	characteristic
226	C₂⁴	#9495	1	1	yes	characteristic
227	C₂⁴	#9495	1	2	yes	—
228	C₂⁴	#9495	1	2	yes	—
229	C₂⁴	#14694	1	1	yes	characteristic
230	C₂⁴	C₂×#1889	1	1	yes	characteristic
231	C₂⁴	#20960	1	1	yes	characteristic
232	C₂⁴	C₂×#1898	1	1	yes	characteristic
233	C₂⁴	C₂×#1898	1	4	yes	—
234	C₂⁴	C₂×#1898	1	1	yes	characteristic
235	C₂⁴	C₂×#1913	1	1	yes	characteristic
236	C₂⁴	C₂×#1913	1	1	yes	characteristic
237	C₂⁴	#32898	1	2	yes	—
238	C₂⁴	#33020	1	2	yes	—
239	C₂³	#139750	1	2	yes	—
240	C₂³	C₂×#9495	1	1	yes	characteristic
241	C₂³	C₂×#9495	1	1	yes	characteristic
242	C₂³	#201687	1	1	yes	characteristic
243	C₂³	#201687	1	1	yes	characteristic
244	C₂³	#392658	1	1	yes	characteristic
245	C₂³	#392675	1	1	yes	characteristic
246	C₂³	#392699	1	1	yes	characteristic
247	C₂³	#511790	1	4	yes	—
248	C₂²	C₂×#139750	1	1	yes	characteristic
249	C₂²	#72218104	1	2	yes	—
250	C₂²	#72238349	1	1	yes	characteristic
251	C₂²	#72258052	1	1	yes	characteristic
252	C₂	#72258739	1	1	yes	characteristic
253	C₂	#72265802	1	1	yes	characteristic
254	C₂	#72265896	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72258052
commutator	C₂⁴×D₄	C₂³×C₄
Frattini, cosocle	C₂³×#120	C₂⁴
prime radical	#72236800	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266320	C₂	semidirect
2. #72266319	C₂	semidirect
3. #72266289	C₂	semidirect
4. #72266288	C₂²	semidirect
5. #72266202	C₂	semidirect
6. #72266201	C₂²	semidirect
7. #72266068	C₂	semidirect
8. #72236821	C₂	semidirect
9. #72236816	C₂²	semidirect
10. #72236812	C₂²	semidirect
13. #72236742	C₄	semidirect
14. C₂×#245229	C₄	semidirect
15. #72236034	C₂	semidirect
17. #72235865	C₂²	semidirect
18. #72231822	C₂²	semidirect
19. #72226161	C₂	semidirect
28. #72218846	C₄	semidirect
30. #72214832	C₄	semidirect
32. C₂²×#3484	C₂×C₄	semidirect
36. C₂×#35347	C₂×C₄	semidirect
43. C₂×#32898	C₂×C₄	semidirect
44. C₂×#21746	D₄	semidirect
45. #517411	C₂×C₄	semidirect
46. #517411	D₄	semidirect
47. #515362	C₂×C₄	semidirect
49. #392658	C₂×C₄	semidirect
50. #392245	D₄	semidirect
51. #392109	C₂×C₄	semidirect
52. #392109	D₄	semidirect
53. #391709	D₄	semidirect
57. #374004	D₄	semidirect
58. C₂×#9495	C₂×C₄	semidirect
59. C₂×#9495	C₂×C₄	semidirect
61. C₂×#8542	C₂×C₄	semidirect
64. C₂³×H(5,𝔽₂)	C₂²×C₄	semidirect
80. C₂²×#518	C₂²×C₄	semidirect
91. C₂×#2308	C₂²×C₄	semidirect
92. C₂×#2308	C₂²×C₄	semidirect
96. C₂×#1781	C₂²×C₄	semidirect
100. C₂⁷	#99	semidirect
105. C₂²×H(5,𝔽₂)	#99	semidirect
106. C₂²×H(5,𝔽₂)	#99	semidirect
107. C₂²×H(5,𝔽₂)	#99	semidirect
111. C₂⁴×D₄	#99	semidirect
129. C₂⁵×C₄	#99	semidirect
154. C₂×#379	#120	semidirect
155. C₂×#342	#120	semidirect

Maximal subgroups

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

subgroup	index
#72266320	2
#72266319	2
#72266289	2
#72266202	2
#72266068	2
#72236821	2
#72236800	2
#72236034	2
#72226161	2