Subgroups — #72246758

Order 2048.

At least 7154 conjugacy classes of subgroups; only the 376 normal subgroups are shown.

The subgroup lattice has 245 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72246747	C₂	1	1	yes	maximal, characteristic, complemented
2	#72246453	C₂	1	1	yes	maximal, characteristic, complemented
3	#72246299	C₂	1	1	yes	maximal, characteristic, complemented
4	#72246292	C₂	1	1	yes	maximal, characteristic, complemented
5	#72246286	C₂	1	1	yes	maximal, characteristic, complemented
6	#72245877	C₂	1	1	yes	maximal, characteristic, complemented
7	#72242805	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238589	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235423	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219234	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219163	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219045	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511790	C₂	1	1	yes	maximal, characteristic
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	#7626243	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23897	C₂²	1	1	yes	characteristic
21	C₂×#21776	C₂²	1	1	yes	characteristic
22	#6371558	C₂²	1	1	yes	characteristic
23	#511790	C₄	1	8	yes	complemented
24	C₂×#12707	C₂²	1	1	yes	characteristic
25	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12687	C₂²	1	1	yes	characteristic
28	#373686	C₂²	1	1	yes	characteristic, complemented
29	#359552	C₂²	1	1	yes	characteristic
30	#359552	C₂²	1	1	yes	characteristic
31	#359549	C₂²	1	1	yes	characteristic
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic, complemented
34	#359501	C₂²	1	1	yes	characteristic
35	#359501	C₂²	1	1	yes	characteristic
36	#355096	C₂²	1	1	yes	characteristic, complemented
37	C₂×#8541	C₂²	1	1	yes	characteristic
38	C₂×#8534	C₂²	1	1	yes	characteristic
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
41	#154670	C₂²	1	1	yes	characteristic
42	#146479	C₂²	1	1	yes	characteristic, complemented
43	#146475	C₂²	1	1	yes	characteristic, complemented
44	#146457	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
49	#108222	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	#106132	C₂²	1	1	yes	characteristic, complemented
52	#62817	C₂×C₄	1	4	yes	complemented
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	#62652	D₄	1	1	yes	characteristic, complemented
58	C₂×#3026	D₄	1	2	yes	—
59	C₂×#2883	D₄	1	1	yes	characteristic, complemented
60	#34171	C₂³	1	1	yes	characteristic
61	#34171	C₂³	1	1	yes	characteristic
62	C₂²×#386	C₂³	1	1	yes	characteristic
63	#23900	C₂×C₄	1	4	yes	complemented
64	#23897	C₂×C₄	1	4	yes	complemented
65	#23895	C₂×C₄	1	4	yes	—
66	#23895	D₄	1	4	yes	—
67	C₂×#1913	D₄	1	2	yes	—
68	C₂×#1898	D₄	1	2	yes	complemented
69	#20609	D₄	1	2	yes	complemented
70	C₂×#1783	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	#20354	D₄	1	1	yes	characteristic, complemented
78	#20354	D₄	1	1	yes	characteristic, complemented
79	#14694	D₄	1	2	yes	complemented
80	#14679	D₄	1	2	yes	complemented
81	C₂×#1438	C₂³	1	1	yes	characteristic
82	#12707	C₂×C₄	1	4	yes	complemented
83	#12705	C₂×C₄	1	4	yes	complemented
84	#12687	C₂×C₄	1	4	yes	complemented
85	#10754	D₄	1	2	yes	complemented
86	#10735	D₄	1	2	yes	complemented
87	C₂⁷	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	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
92	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
93	#3484	C₂²×C₄	1	2	yes	—
94	#3484	C₂²×C₄	1	2	yes	—
95	#3484	C₂²×C₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	#3484	C₂×D₄	1	2	yes	—
98	#3484	#41	1	2	yes	—
99	#3484	#41	1	2	yes	—
100	#3484	#41	1	2	yes	—
101	C₂×#534	C₂×D₄	1	2	yes	—
102	C₂²×#120	C₂²×C₄	1	2	yes	—
103	C₂²×#120	C₂×D₄	1	2	yes	—
104	C₂²×#120	#41	1	2	yes	—
105	C₂×#518	C₂×D₄	1	1	yes	characteristic
106	C₂×#518	C₂×D₄	1	1	yes	characteristic
107	C₂×#518	C₂×D₄	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	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
114	#3026	C₂²⋊C₄	1	8	yes	complemented
115	C₂²×#99	C₂×D₄	1	2	yes	—
116	C₂²×#99	C₂×D₄	1	2	yes	—
117	#2308	C₂×D₄	1	2	yes	—
118	C₂×#379	C₂×D₄	1	2	yes	—
119	C₂×#379	C₂×D₄	1	2	yes	—
120	C₂×#379	C₂×D₄	1	2	yes	—
121	#1913	C₂²⋊C₄	1	8	yes	complemented
122	#1783	C₂²×C₄	1	2	yes	—
123	#1781	C₂²×C₄	1	2	yes	—
124	#1781	C₂²×C₄	1	2	yes	—
125	#1781	C₂×D₄	1	2	yes	—
126	C₂⁶	C₄×D₄	1	2	yes	complemented
127	C₂⁶	C₂²×D₄	1	1	yes	characteristic
128	C₂⁶	C₂²×D₄	1	1	yes	characteristic
129	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	complemented
130	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
131	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
132	C₂×H(5,𝔽₂)	#121	1	2	yes	—
133	C₂³×D₄	#120	1	2	yes	—
134	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
135	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
136	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
137	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂×#41	1	1	yes	characteristic
140	C₂³×D₄	C₂×#41	1	1	yes	characteristic
141	C₂³×D₄	C₂×#41	1	1	yes	characteristic
142	C₂³×D₄	C₂×#41	1	1	yes	characteristic
143	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
147	#534	C₂×C₂²⋊C₄	1	4	yes	—
148	#534	#123	1	4	yes	—
149	C₂×#120	C₄×D₄	1	4	yes	—
150	C₂×#120	#121	1	4	yes	—
151	#518	C₂×C₂²⋊C₄	1	2	yes	complemented
152	#518	C₄×D₄	1	2	yes	—
153	#518	C₄×D₄	1	2	yes	complemented
154	#518	#121	1	2	yes	—
155	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	complemented
156	C₂²×C₂²⋊C₄	#120	1	2	yes	—
157	C₂²×C₂²⋊C₄	#120	1	2	yes	—
158	C₂²×C₂²⋊C₄	#120	1	2	yes	—
159	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	complemented
160	C₂⁵	B(4,𝔽₂)	1	2	yes	—
161	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂⁵	C₂×#120	1	1	yes	characteristic
164	C₂⁵	C₂×#120	1	1	yes	characteristic
165	C₂⁵	C₂×#121	1	1	yes	characteristic
166	C₂⁵	#525	1	1	yes	characteristic
167	C₂⁵	D₄²	1	1	yes	characteristic
168	C₂⁵	D₄²	1	1	yes	characteristic
169	C₂⁵	D₄²	1	1	yes	characteristic
170	C₂⁵	D₄²	1	1	yes	characteristic
171	C₂⁵	#546	1	1	yes	characteristic
172	C₂⁵	#546	1	1	yes	characteristic
173	C₂⁵	#546	1	1	yes	characteristic
174	C₂²×D₄	#386	1	4	yes	—
175	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
176	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂²×D₄	#518	1	1	yes	characteristic
180	C₂²×D₄	C₂×#121	1	1	yes	characteristic
181	C₂²×D₄	C₂×#123	1	1	yes	characteristic
182	C₂²×D₄	#525	1	1	yes	characteristic
183	C₂²×D₄	#525	1	1	yes	characteristic
184	C₂²×D₄	#535	1	1	yes	characteristic
185	C₂²×D₄	#538	1	1	yes	characteristic
186	C₂²×D₄	D₄²	1	1	yes	characteristic
187	C₂²×D₄	#546	1	1	yes	characteristic
188	C₂²×D₄	#546	1	1	yes	characteristic
189	C₂²×D₄	#546	1	1	yes	characteristic
190	C₂²×D₄	#546	1	1	yes	characteristic
191	C₂²×D₄	#551	1	1	yes	characteristic
192	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
193	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
194	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
195	C₂³×C₄	#518	1	1	yes	characteristic
196	C₂³×C₄	#534	1	1	yes	characteristic
197	C₂³×C₄	#538	1	1	yes	characteristic
198	C₂³×C₄	D₄²	1	1	yes	characteristic
199	C₂³×C₄	#546	1	1	yes	characteristic
200	C₂³×C₄	#551	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	#386	1	4	yes	—
202	C₂⁴	#1889	1	4	yes	complemented
203	C₂⁴	C₂×#386	1	1	yes	characteristic
204	C₂⁴	C₄×#120	1	1	yes	characteristic
205	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
206	C₂⁴	#2358	1	1	yes	characteristic
207	C₂⁴	#2358	1	1	yes	characteristic
208	C₂⁴	#2403	1	1	yes	characteristic
209	C₂⁴	#2404	1	1	yes	characteristic
210	C₂⁴	#2408	1	1	yes	characteristic
211	C₂⁴	#2418	1	1	yes	characteristic
212	C₂⁴	#2424	1	1	yes	characteristic
213	C₂⁴	#2424	1	1	yes	characteristic
214	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
215	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
216	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
217	C₂⁴	#3025	1	2	yes	—
218	C₂²×C₄	#1889	1	4	yes	complemented
219	C₂²×C₄	#2321	1	1	yes	characteristic
220	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	#2418	1	1	yes	characteristic
222	C₂²×C₄	#2534	1	1	yes	characteristic
223	C₂³	#13967	1	1	yes	characteristic
224	C₂³	C₂×#1889	1	2	yes	—
225	C₂³	C₄×B(4,𝔽₂)	1	2	yes	—
226	C₂³	#23895	1	1	yes	characteristic
227	C₂³	#23896	1	1	yes	characteristic
228	C₂³	#23900	1	1	yes	characteristic
229	C₂³	#23900	1	1	yes	characteristic
230	C₂³	#23900	1	1	yes	characteristic
231	C₂³	#23902	1	1	yes	characteristic
232	C₂³	#23984	1	2	yes	—
233	C₂³	#24260	1	2	yes	—
234	C₂²	#245225	1	1	yes	characteristic
235	C₂²	#245227	1	1	yes	characteristic
236	C₂²	#245229	1	1	yes	characteristic
237	C₂²	#245231	1	1	yes	characteristic
238	C₂²	#364410	1	1	yes	characteristic
239	C₂²	#364500	1	1	yes	characteristic
240	C₂²	#364513	1	1	yes	characteristic
241	C₂	B(5,𝔽₂)	1	1	yes	characteristic
242	C₂	#72233445	1	1	yes	characteristic
243	C₂	#72237497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#245225
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂×#511790	C₂
LCS	G ▷ C₂³×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. #72246747	C₂	semidirect
2. #72246453	C₂	semidirect
3. #72246299	C₂	semidirect
4. #72246292	C₂	semidirect
5. #72246286	C₂	semidirect
6. #72245877	C₂	semidirect
7. #72242805	C₂	semidirect
8. #72238589	C₂	semidirect
9. #72235423	C₂	semidirect
10. #72219234	C₂	semidirect
11. #72219163	C₂	semidirect
12. #72219045	C₂	semidirect
13. #72218521	C₂	semidirect
14. C₂×#139743	C₂	semidirect
17. #7626243	C₂²	semidirect
23. #511790	C₄	semidirect
25. C₂×#12707	C₂²	semidirect
28. #373686	C₂²	semidirect
33. #359549	C₂²	semidirect
36. #355096	C₂²	semidirect
40. C₂×#8534	C₂²	semidirect
42. #146479	C₂²	semidirect
43. #146475	C₂²	semidirect
44. #146457	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7307	C₂²	semidirect
49. #108222	C₂²	semidirect
50. #107130	C₂²	semidirect
51. #106132	C₂²	semidirect
52. #62817	C₂×C₄	semidirect
57. #62652	D₄	semidirect
59. C₂×#2883	D₄	semidirect
63. #23900	C₂×C₄	semidirect
64. #23897	C₂×C₄	semidirect
68. C₂×#1898	D₄	semidirect
69. #20609	D₄	semidirect
77. #20354	D₄	semidirect
78. #20354	D₄	semidirect
79. #14694	D₄	semidirect
80. #14679	D₄	semidirect
82. #12707	C₂×C₄	semidirect
83. #12705	C₂×C₄	semidirect
84. #12687	C₂×C₄	semidirect
85. #10754	D₄	semidirect
86. #10735	D₄	semidirect
114. #3026	C₂²⋊C₄	semidirect
121. #1913	C₂²⋊C₄	semidirect
126. C₂⁶	C₄×D₄	semidirect
129. C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	semidirect
131. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
151. #518	C₂×C₂²⋊C₄	semidirect
153. #518	C₄×D₄	semidirect
155. C₂²×C₂²⋊C₄	C₄×D₄	semidirect
159. C₂×#99	C₂×C₂²⋊C₄	semidirect
202. C₂⁴	#1889	semidirect
218. C₂²×C₄	#1889	semidirect

Maximal subgroups

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

subgroup	index
#72246747	2
#72246453	2
#72246299	2
#72246292	2
#72246286	2
#72245877	2
#72242805	2
#72238589	2
#72235423	2
#72219234	2
#72219163	2
#72219045	2
#72218521	2
C₂×#139743	2
C₂×#511790	2