Subgroups — #72312503

Order 2048.

At least 6285 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

The subgroup lattice has 265 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312216	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310366	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309800	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266835	C₂	1	1	yes	maximal, characteristic
5	#72266378	C₂	1	1	yes	maximal, characteristic, complemented
6	#72256113	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255764	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251528	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251485	C₂	1	1	yes	maximal, characteristic, complemented
10	#72251422	C₂	1	1	yes	maximal, characteristic, complemented
11	#72246292	C₂	1	1	yes	maximal, characteristic, complemented
12	#72246288	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238611	C₂	1	1	yes	maximal, characteristic, complemented
14	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237467	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6437621	C₂²	1	1	yes	characteristic
18	#6437621	C₂²	1	1	yes	characteristic
19	C₂×#21778	C₂²	1	1	yes	characteristic
20	#6374128	C₂²	1	1	yes	characteristic, complemented
21	#6371558	C₂²	1	1	yes	characteristic
22	#763333	C₂²	1	1	yes	characteristic
23	C₂×#12705	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	#493370	C₂²	1	1	yes	characteristic
26	#493357	C₂²	1	1	yes	characteristic, complemented
27	#373637	C₂²	1	1	yes	characteristic, complemented
28	#359558	C₂²	1	1	yes	characteristic
29	#359558	C₂²	1	1	yes	characteristic
30	#359549	C₂²	1	1	yes	characteristic
31	#359549	C₂²	1	1	yes	characteristic
32	#359509	C₂²	1	1	yes	characteristic
33	#359509	C₂²	1	1	yes	characteristic
34	#359509	C₂²	1	1	yes	characteristic
35	#359442	C₂²	1	1	yes	characteristic, complemented
36	#355096	C₂²	1	1	yes	characteristic, complemented
37	#347057	C₄	1	2	yes	complemented
38	#347056	C₄	1	2	yes	complemented
39	#347047	C₄	1	2	yes	complemented
40	#347045	C₄	1	2	yes	complemented
41	#188109	C₂²	1	1	yes	characteristic, complemented
42	#188050	C₂²	1	1	yes	characteristic
43	C₂×#8534	C₂²	1	1	yes	characteristic
44	C₂×#8534	C₂²	1	1	yes	characteristic
45	#146479	C₂²	1	1	yes	characteristic, complemented
46	#146475	C₂²	1	1	yes	characteristic, complemented
47	#146475	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
51	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
52	#108222	C₂²	1	1	yes	characteristic, complemented
53	#107130	C₂²	1	1	yes	characteristic, complemented
54	#107130	C₂²	1	1	yes	characteristic, complemented
55	#62832	C₂×C₄	1	2	yes	—
56	#62824	C₂×C₄	1	2	yes	—
57	C₂×#3484	C₂³	1	1	yes	characteristic
58	C₂×#3484	C₂³	1	1	yes	characteristic
59	C₂²×#535	C₂³	1	1	yes	characteristic
60	C₂²×#534	C₂³	1	1	yes	characteristic
61	#62652	D₄	1	1	yes	characteristic, complemented
62	C₂×#2883	D₄	1	1	yes	characteristic, complemented
63	#34980	D₄	1	2	yes	—
64	#34171	C₂³	1	1	yes	characteristic
65	#34171	C₂³	1	1	yes	characteristic
66	#34171	C₂³	1	1	yes	characteristic
67	#23901	C₂×C₄	1	2	yes	—
68	#23901	C₂×C₄	1	2	yes	complemented
69	#23901	D₄	1	2	yes	—
70	#23900	C₂×C₄	1	2	yes	—
71	#23900	C₂×C₄	1	2	yes	complemented
72	#23900	D₄	1	2	yes	—
73	C₂×#1913	D₄	1	2	yes	complemented
74	C₂×#1898	D₄	1	2	yes	—
75	#20960	D₄	1	2	yes	complemented
76	#20616	D₄	1	2	yes	complemented
77	#20605	D₄	1	2	yes	complemented
78	C₂×#1783	C₂³	1	1	yes	characteristic
79	C₂×#1783	C₂³	1	1	yes	characteristic
80	C₂×#1783	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	C₂×#1781	C₂³	1	1	yes	characteristic
85	#20354	D₄	1	1	yes	characteristic, complemented
86	#20354	D₄	1	1	yes	characteristic, complemented
87	#15262	C₂×C₄	1	1	yes	characteristic, complemented
88	#15261	C₂×C₄	1	1	yes	characteristic, complemented
89	#15259	C₂×C₄	1	1	yes	characteristic, complemented
90	#15258	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂×#1438	C₂³	1	1	yes	characteristic
92	#10740	D₄	1	2	yes	complemented
93	#10716	D₄	1	2	yes	complemented
94	#8546	C₂×C₄	1	2	yes	complemented
95	#8544	C₂×C₄	1	2	yes	complemented
96	#8534	C₂×C₄	1	2	yes	—
97	#8534	C₂×C₄	1	2	yes	—
98	#8534	C₂×C₄	1	2	yes	complemented
99	#8534	C₂×C₄	1	2	yes	complemented
100	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
101	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
102	C₂³×#41	C₂×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
105	#3484	C₂²×C₄	1	2	yes	—
106	#3484	C₂²×C₄	1	2	yes	—
107	#3484	C₂×D₄	1	2	yes	—
108	#3484	#41	1	2	yes	—
109	#3484	#41	1	2	yes	—
110	#3484	#41	1	2	yes	—
111	C₂×#534	C₂×D₄	1	2	yes	—
112	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
113	C₂²×#127	#41	1	1	yes	characteristic
114	C₂²×#124	C₂²×C₄	1	1	yes	characteristic
115	C₂²×#124	#41	1	1	yes	characteristic
116	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
117	C₂²×#123	C₂×D₄	1	1	yes	characteristic
118	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
119	C₂²×#120	C₂×D₄	1	1	yes	characteristic
120	C₂²×#120	C₂×D₄	1	2	yes	—
121	C₂×#518	C₂×D₄	1	1	yes	characteristic
122	C₂×#518	C₂×D₄	1	1	yes	characteristic
123	C₂×#518	C₂×D₄	1	1	yes	characteristic
124	C₂×#518	C₂×D₄	1	1	yes	characteristic
125	C₂×#518	C₂×D₄	1	1	yes	characteristic
126	C₂×#518	C₂×D₄	1	1	yes	characteristic
127	C₂×#518	C₂×D₄	1	1	yes	characteristic
128	C₂×#518	C₂×D₄	1	1	yes	characteristic
129	C₂²×#117	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
131	C₂²×#100	C₂×D₄	1	2	yes	—
132	C₂²×#99	C₂×D₄	1	2	yes	—
133	C₂×#407	C₂×D₄	1	2	yes	—
134	#2849	C₂²⋊C₄	1	4	yes	complemented
135	#2846	C₂²⋊C₄	1	4	yes	complemented
136	#2309	C₂×D₄	1	2	yes	—
137	#2280	C₂×D₄	1	2	yes	—
138	C₂×#379	C₂×D₄	1	2	yes	—
139	#1898	C₂²⋊C₄	1	4	yes	complemented
140	#1898	C₂²⋊C₄	1	4	yes	complemented
141	#1781	C₂²×C₄	1	2	yes	—
142	#1781	C₂²×C₄	1	2	yes	—
143	#1781	C₂×D₄	1	2	yes	—
144	#1438	C₂²×C₄	1	1	yes	characteristic
145	#1438	C₂²×C₄	1	1	yes	characteristic
146	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂⁶	C₂²×D₄	1	1	yes	characteristic
148	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
150	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
151	C₂×H(5,𝔽₂)	#121	1	2	yes	—
152	C₂³×Q₈	C₄×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
154	C₂³×D₄	#120	1	2	yes	—
155	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
156	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
157	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
158	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
159	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
160	C₂³×D₄	C₂×#41	1	1	yes	characteristic
161	C₂³×D₄	C₂×#41	1	1	yes	characteristic
162	C₂³×D₄	C₂×#41	1	1	yes	characteristic
163	C₂³×D₄	C₂×#41	1	1	yes	characteristic
164	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
169	#534	C₄×D₄	1	4	yes	—
170	#534	#121	1	4	yes	—
171	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
172	C₂×#120	#123	1	4	yes	—
173	#518	C₄×D₄	1	2	yes	—
174	#518	C₄×D₄	1	2	yes	—
175	#518	#121	1	2	yes	—
176	C₂×#117	#120	1	2	yes	—
177	C₂×#117	#120	1	2	yes	—
178	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₂²⋊C₄	#120	1	2	yes	—
181	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
182	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
183	#379	C₂×C₂²⋊C₄	1	4	yes	—
184	C₂⁵	B(4,𝔽₂)	1	2	yes	—
185	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁵	#518	1	1	yes	characteristic
187	C₂⁵	C₂×#120	1	1	yes	characteristic
188	C₂⁵	C₂×#123	1	1	yes	characteristic
189	C₂⁵	#534	1	1	yes	characteristic
190	C₂⁵	#538	1	1	yes	characteristic
191	C₂⁵	D₄²	1	1	yes	characteristic
192	C₂⁵	D₄²	1	1	yes	characteristic
193	C₂⁵	D₄²	1	1	yes	characteristic
194	C₂⁵	#546	1	1	yes	characteristic
195	C₂⁵	#546	1	1	yes	characteristic
196	C₂⁵	#546	1	1	yes	characteristic
197	C₂⁵	#551	1	1	yes	characteristic
198	C₂²×D₄	#386	1	4	yes	—
199	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
200	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×D₄	#518	1	1	yes	characteristic
204	C₂²×D₄	C₂×#121	1	1	yes	characteristic
205	C₂²×D₄	C₂×#121	1	1	yes	characteristic
206	C₂²×D₄	#525	1	1	yes	characteristic
207	C₂²×D₄	#525	1	1	yes	characteristic
208	C₂²×D₄	#535	1	1	yes	characteristic
209	C₂²×D₄	#538	1	1	yes	characteristic
210	C₂²×D₄	D₄²	1	1	yes	characteristic
211	C₂²×D₄	#546	1	1	yes	characteristic
212	C₂²×D₄	#546	1	1	yes	characteristic
213	C₂²×D₄	#546	1	1	yes	characteristic
214	C₂²×D₄	#546	1	1	yes	characteristic
215	C₂²×D₄	#551	1	1	yes	characteristic
216	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
217	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂³×C₄	C₂×#120	1	1	yes	characteristic
219	C₂³×C₄	#525	1	1	yes	characteristic
220	C₂³×C₄	D₄²	1	1	yes	characteristic
221	C₂³×C₄	D₄²	1	1	yes	characteristic
222	C₂³×C₄	#546	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	#386	1	4	yes	—
224	C₂⁴	#1889	1	4	yes	—
225	C₂⁴	C₂×#386	1	1	yes	characteristic
226	C₂⁴	#2321	1	1	yes	characteristic
227	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
228	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
229	C₂⁴	#2358	1	1	yes	characteristic
230	C₂⁴	#2403	1	1	yes	characteristic
231	C₂⁴	#2404	1	1	yes	characteristic
232	C₂⁴	#2418	1	1	yes	characteristic
233	C₂⁴	#2418	1	1	yes	characteristic
234	C₂⁴	#2424	1	1	yes	characteristic
235	C₂⁴	#2534	1	1	yes	characteristic
236	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
237	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
238	C₂²×C₄	#1889	1	4	yes	—
239	C₂²×C₄	C₄×#120	1	1	yes	characteristic
240	C₂²×C₄	#2358	1	1	yes	characteristic
241	C₂²×C₄	#2408	1	1	yes	characteristic
242	C₂²×C₄	#2424	1	1	yes	characteristic
243	C₂²×C₄	#3026	1	2	yes	—
244	C₂²×C₄	#3027	1	2	yes	—
245	C₂³	#13967	1	1	yes	characteristic
246	C₂³	C₂×#1889	1	2	yes	—
247	C₂³	#23895	1	1	yes	characteristic
248	C₂³	#23896	1	1	yes	characteristic
249	C₂³	#23900	1	1	yes	characteristic
250	C₂³	#23900	1	1	yes	characteristic
251	C₂³	#23903	1	1	yes	characteristic
252	C₂³	#23904	1	1	yes	characteristic
253	C₂³	#23984	1	2	yes	—
254	C₂×C₄	#21482	1	2	yes	—
255	C₂×C₄	#24262	1	2	yes	—
256	C₂²	#245246	1	1	yes	characteristic
257	C₂²	#245247	1	1	yes	characteristic
258	C₂²	#245252	1	1	yes	characteristic
259	C₂²	#245254	1	1	yes	characteristic
260	C₂²	#364410	1	1	yes	characteristic
261	C₂²	#364500	1	1	yes	characteristic
262	C₂²	#364516	1	1	yes	characteristic
263	C₂	#72266002	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266002
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72266835	C₂
LCS	G ▷ C₂³×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. #72312216	C₂	semidirect
2. #72310366	C₂	semidirect
3. #72309800	C₂	semidirect
5. #72266378	C₂	semidirect
6. #72256113	C₂	semidirect
7. #72255764	C₂	semidirect
8. #72251528	C₂	semidirect
9. #72251485	C₂	semidirect
10. #72251422	C₂	semidirect
11. #72246292	C₂	semidirect
12. #72246288	C₂	semidirect
13. #72238611	C₂	semidirect
14. #72238343	C₂	semidirect
15. #72237467	C₂	semidirect
20. #6374128	C₂²	semidirect
26. #493357	C₂²	semidirect
27. #373637	C₂²	semidirect
35. #359442	C₂²	semidirect
36. #355096	C₂²	semidirect
37. #347057	C₄	semidirect
38. #347056	C₄	semidirect
39. #347047	C₄	semidirect
40. #347045	C₄	semidirect
41. #188109	C₂²	semidirect
45. #146479	C₂²	semidirect
46. #146475	C₂²	semidirect
47. #146475	C₂²	semidirect
48. C₂×#7304	C₂²	semidirect
49. C₂×#7316	C₂²	semidirect
50. C₂×#7314	C₂²	semidirect
51. C₂×#7311	C₂²	semidirect
52. #108222	C₂²	semidirect
53. #107130	C₂²	semidirect
54. #107130	C₂²	semidirect
61. #62652	D₄	semidirect
62. C₂×#2883	D₄	semidirect
68. #23901	C₂×C₄	semidirect
71. #23900	C₂×C₄	semidirect
73. C₂×#1913	D₄	semidirect
75. #20960	D₄	semidirect
76. #20616	D₄	semidirect
77. #20605	D₄	semidirect
85. #20354	D₄	semidirect
86. #20354	D₄	semidirect
87. #15262	C₂×C₄	semidirect
88. #15261	C₂×C₄	semidirect
89. #15259	C₂×C₄	semidirect
90. #15258	C₂×C₄	semidirect
92. #10740	D₄	semidirect
93. #10716	D₄	semidirect
94. #8546	C₂×C₄	semidirect
95. #8544	C₂×C₄	semidirect
98. #8534	C₂×C₄	semidirect
99. #8534	C₂×C₄	semidirect
134. #2849	C₂²⋊C₄	semidirect
135. #2846	C₂²⋊C₄	semidirect
139. #1898	C₂²⋊C₄	semidirect
140. #1898	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72312216	2
#72310366	2
#72309800	2
#72266835	2
#72266378	2
#72256113	2
#72255764	2
#72251528	2
#72251485	2
#72251422	2
#72246292	2
#72246288	2
#72238611	2
#72238343	2
#72237467	2