Subgroups — #72315275

Order 2048.

At least 6500 conjugacy classes of subgroups; only the 308 normal subgroups are shown.

The subgroup lattice has 218 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314855	C₂	1	1	yes	maximal, characteristic, complemented
2	#72312214	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#139739	C₂	1	1	yes	maximal, characteristic, complemented
5	#72238976	C₂	1	1	yes	maximal, characteristic, complemented
6	#72225742	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#139748	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#493474	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219293	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219158	C₂	1	1	yes	maximal, characteristic, complemented
11	#72218391	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#144366	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#144382	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#347042	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×#23895	C₂²	1	1	yes	characteristic
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
22	C₂×#23897	C₂²	1	1	yes	characteristic
23	C₂×#21778	C₂²	1	1	yes	characteristic
24	C₂×#21776	C₂²	1	1	yes	characteristic
25	C₂×#15947	C₂²	1	1	yes	characteristic
26	C₂×#12707	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12705	C₂²	1	1	yes	characteristic
30	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
31	C₂×#12687	C₂²	1	1	yes	characteristic
32	C₂×#12687	C₂²	1	1	yes	characteristic
33	C₂×#12687	C₂²	1	1	yes	characteristic
34	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
35	#392030	C₂²	1	1	yes	characteristic, complemented
36	#382434	C₂²	1	1	yes	characteristic, complemented
37	#382422	C₂²	1	1	yes	characteristic, complemented
38	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8541	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7307	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	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂×#3026	D₄	1	2	yes	complemented
58	C₂×#2883	D₄	1	2	yes	complemented
59	C₂×#2846	D₄	1	2	yes	complemented
60	C₂×#1913	D₄	1	2	yes	complemented
61	C₂×#1898	D₄	1	2	yes	complemented
62	C₂×#1828	D₄	1	2	yes	complemented
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	C₂×#1783	C₂³	1	1	yes	characteristic
65	C₂×#1783	C₂³	1	1	yes	characteristic
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	C₂×#1781	C₂³	1	1	yes	characteristic
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1438	C₂³	1	1	yes	characteristic
73	#12705	D₄	1	4	yes	complemented
74	#12705	D₄	1	4	yes	complemented
75	#12687	D₄	1	4	yes	complemented
76	#8534	D₄	1	4	yes	complemented
77	C₂⁷	C₂×D₄	1	1	yes	characteristic
78	C₂⁷	#41	1	1	yes	characteristic
79	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
80	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
82	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
83	#3484	C₂×D₄	1	2	yes	—
84	#3484	C₂×D₄	1	2	yes	—
85	#3484	C₂×D₄	1	2	yes	—
86	#3484	C₂×D₄	1	2	yes	—
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	#41	1	2	yes	—
89	C₂×#534	C₂×D₄	1	2	yes	—
90	C₂²×#120	C₂×D₄	1	2	yes	—
91	C₂²×#120	C₂×D₄	1	2	yes	—
92	C₂²×#120	C₂×D₄	1	2	yes	—
93	C₂×#518	C₂×D₄	1	1	yes	characteristic
94	C₂×#518	C₂×D₄	1	1	yes	characteristic
95	C₂×#518	#41	1	1	yes	characteristic
96	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₂²⋊C₄	#41	1	1	yes	characteristic
98	C₂²×#99	C₂×D₄	1	2	yes	—
99	C₂²×#99	C₂×D₄	1	2	yes	—
100	C₂²×#99	C₂×D₄	1	2	yes	—
101	C₂×#379	C₂×D₄	1	2	yes	—
102	#1783	C₂×D₄	1	2	yes	—
103	#1783	C₂×D₄	1	2	yes	—
104	#1783	C₂×D₄	1	2	yes	—
105	#1781	C₂×D₄	1	2	yes	—
106	#1781	C₂×D₄	1	2	yes	—
107	#1781	C₂×D₄	1	2	yes	—
108	#1781	C₂×D₄	1	2	yes	—
109	#1438	C₂×D₄	1	2	yes	—
110	C₂⁶	#121	1	2	yes	complemented
111	C₂⁶	C₂²×D₄	1	1	yes	characteristic
112	C₂⁶	C₂×#41	1	1	yes	characteristic
113	C₂×H(5,𝔽₂)	#121	1	2	yes	—
114	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
115	C₂×H(5,𝔽₂)	#127	1	2	yes	—
116	C₂³×D₄	#120	1	2	yes	—
117	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
118	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
119	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
120	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂×#41	1	1	yes	characteristic
125	C₂³×D₄	C₂×#41	1	1	yes	characteristic
126	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
130	#534	#120	1	4	yes	—
131	#534	#121	1	4	yes	—
132	C₂×#120	#120	1	4	yes	—
133	C₂×#120	#121	1	4	yes	—
134	#518	#121	1	2	yes	—
135	#518	#121	1	2	yes	complemented
136	#518	#127	1	2	yes	—
137	C₂²×C₂²⋊C₄	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#121	1	2	yes	complemented
139	C₂×#99	#120	1	4	yes	—
140	C₂⁵	B(4,𝔽₂)	1	2	yes	—
141	C₂⁵	C₂×#120	1	1	yes	characteristic
142	C₂⁵	C₂×#120	1	1	yes	characteristic
143	C₂⁵	C₂×#121	1	1	yes	characteristic
144	C₂⁵	C₂×#121	1	1	yes	characteristic
145	C₂⁵	#525	1	1	yes	characteristic
146	C₂⁵	#525	1	1	yes	characteristic
147	C₂⁵	D₄²	1	1	yes	characteristic
148	C₂⁵	D₄²	1	1	yes	characteristic
149	C₂⁵	#546	1	1	yes	characteristic
150	C₂⁵	#546	1	1	yes	characteristic
151	C₂⁵	#546	1	1	yes	characteristic
152	C₂⁵	#546	1	1	yes	characteristic
153	C₂⁵	#546	1	1	yes	characteristic
154	C₂²×D₄	#392	1	4	yes	—
155	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
156	C₂²×D₄	C₂×#120	1	1	yes	characteristic
157	C₂²×D₄	C₂×#120	1	1	yes	characteristic
158	C₂²×D₄	C₂×#121	1	1	yes	characteristic
159	C₂²×D₄	C₂×#121	1	1	yes	characteristic
160	C₂²×D₄	C₂×#121	1	1	yes	characteristic
161	C₂²×D₄	#525	1	1	yes	characteristic
162	C₂²×D₄	C₂×#127	1	1	yes	characteristic
163	C₂²×D₄	#534	1	1	yes	characteristic
164	C₂²×D₄	#534	1	1	yes	characteristic
165	C₂²×D₄	#538	1	1	yes	characteristic
166	C₂²×D₄	D₄²	1	1	yes	characteristic
167	C₂²×D₄	D₄²	1	1	yes	characteristic
168	C₂²×D₄	D₄²	1	1	yes	characteristic
169	C₂²×D₄	D₄²	1	1	yes	characteristic
170	C₂²×D₄	#546	1	1	yes	characteristic
171	C₂²×D₄	#546	1	1	yes	characteristic
172	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
173	C₂³×C₄	#534	1	1	yes	characteristic
174	C₂³×C₄	#535	1	1	yes	characteristic
175	C₂³×C₄	#538	1	1	yes	characteristic
176	C₂³×C₄	D₄²	1	1	yes	characteristic
177	C₂³×C₄	D₄²	1	1	yes	characteristic
178	C₂³×C₄	#546	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	#392	1	4	yes	—
180	C₂⁴	#2021	1	4	yes	—
181	C₂⁴	C₂×#392	1	1	yes	characteristic
182	C₂⁴	#2403	1	1	yes	characteristic
183	C₂⁴	#2404	1	1	yes	characteristic
184	C₂⁴	#2410	1	1	yes	characteristic
185	C₂⁴	#2410	1	1	yes	characteristic
186	C₂⁴	#2410	1	1	yes	characteristic
187	C₂⁴	#2424	1	1	yes	characteristic
188	C₂⁴	#2424	1	1	yes	characteristic
189	C₂⁴	#2424	1	1	yes	characteristic
190	C₂⁴	#2539	1	1	yes	characteristic
191	C₂⁴	#2847	1	1	yes	characteristic
192	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
193	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
194	C₂²×C₄	#2021	1	4	yes	—
195	C₂²×C₄	#2410	1	1	yes	characteristic
196	C₂²×C₄	#2424	1	1	yes	characteristic
197	C₂²×C₄	#2433	1	1	yes	characteristic
198	C₂²×C₄	#2617	1	1	yes	characteristic
199	C₂³	#14716	1	1	yes	characteristic
200	C₂³	C₂×#2021	1	2	yes	—
201	C₂³	#23900	1	2	yes	—
202	C₂³	#23984	1	1	yes	characteristic
203	C₂³	#23984	1	1	yes	characteristic
204	C₂³	#23996	1	1	yes	characteristic
205	C₂³	#23996	1	1	yes	characteristic
206	C₂³	#23996	1	1	yes	characteristic
207	C₂³	#23996	1	1	yes	characteristic
208	C₂²	#251827	1	1	yes	characteristic
209	C₂²	#251854	1	1	yes	characteristic
210	C₂²	#251859	1	1	yes	characteristic
211	C₂²	#251859	1	1	yes	characteristic
212	C₂²	#394167	1	1	yes	characteristic
213	C₂²	#394273	1	1	yes	characteristic
214	C₂²	#394273	1	1	yes	characteristic
215	C₂	#72216344	1	2	yes	—
216	C₂	#72241287	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #72314855	C₂	semidirect
2. #72312214	C₂	semidirect
3. #72258334	C₂	semidirect
4. C₂×#139739	C₂	semidirect
5. #72238976	C₂	semidirect
6. #72225742	C₂	semidirect
7. C₂×#139748	C₂	semidirect
8. C₂×#493474	C₂	semidirect
9. #72219293	C₂	semidirect
10. #72219158	C₂	semidirect
11. #72218391	C₂	semidirect
12. C₂×#144366	C₂	semidirect
13. C₂×#144382	C₂	semidirect
14. C₂×#347042	C₂	semidirect
15. C₂×#511790	C₂	semidirect
21. C₂×#23900	C₂²	semidirect
30. C₂×#12705	C₂²	semidirect
34. C₂×#12687	C₂²	semidirect
35. #392030	C₂²	semidirect
36. #382434	C₂²	semidirect
37. #382422	C₂²	semidirect
38. C₂×#8547	C₂²	semidirect
42. C₂×#8453	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7304	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7314	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7307	C₂²	semidirect
51. #62817	D₄	semidirect
57. C₂×#3026	D₄	semidirect
58. C₂×#2883	D₄	semidirect
59. C₂×#2846	D₄	semidirect
60. C₂×#1913	D₄	semidirect
61. C₂×#1898	D₄	semidirect
62. C₂×#1828	D₄	semidirect
73. #12705	D₄	semidirect
74. #12705	D₄	semidirect
75. #12687	D₄	semidirect
76. #8534	D₄	semidirect
110. C₂⁶	#121	semidirect
114. C₂×H(5,𝔽₂)	#121	semidirect
135. #518	#121	semidirect
138. C₂²×C₂²⋊C₄	#121	semidirect

Maximal subgroups

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

subgroup	index
#72314855	2
#72312214	2
#72258334	2
C₂×#139739	2
#72238976	2
#72225742	2
C₂×#139748	2
C₂×#493474	2
#72219293	2
#72219158	2
#72218391	2
C₂×#144366	2
C₂×#144382	2
C₂×#347042	2
C₂×#511790	2