Subgroups — #72312258

Order 2048.

At least 6538 conjugacy classes of subgroups; only the 458 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	#72255980	C₂	1	2	yes	maximal, complemented
2	#72255874	C₂	1	2	yes	maximal, complemented
3	#72237638	C₂	1	2	yes	maximal, complemented
4	#72237374	C₂	1	2	yes	maximal, complemented
5	#72234661	C₂	1	1	yes	maximal, characteristic
6	#72225713	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219300	C₂	1	1	yes	maximal, characteristic, complemented
8	#72218381	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218313	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
11	#72210555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	C₂×#34980	C₂²	1	1	yes	characteristic
14	#6629258	C₄	1	2	yes	complemented
15	#6628892	C₄	1	2	yes	complemented
16	#6440460	C₂²	1	2	yes	—
17	#6437995	C₄	1	4	yes	complemented
18	#6437621	C₂²	1	2	yes	—
19	#6374251	C₂²	1	1	yes	characteristic, complemented
20	#6374128	C₂²	1	1	yes	characteristic, complemented
21	C₂²×#1781	C₂²	1	1	yes	characteristic
22	#493879	C₂²	1	2	yes	complemented
23	C₂×#12707	C₂²	1	2	yes	—
24	C₂×#12707	C₂²	1	2	yes	—
25	C₂×#12705	C₂²	1	2	yes	—
26	C₂×#12705	C₂²	1	2	yes	—
27	#493357	C₂²	1	2	yes	complemented
28	#392113	C₂²	1	1	yes	characteristic
29	#392109	C₂²	1	1	yes	characteristic
30	#392109	C₂²	1	1	yes	characteristic
31	#392109	C₂²	1	1	yes	characteristic
32	#382707	C₂²	1	1	yes	characteristic, complemented
33	#382707	C₂²	1	1	yes	characteristic, complemented
34	C₂×#7314	C₂²	1	2	yes	complemented
35	C₂×#7314	C₂²	1	2	yes	complemented
36	C₂×#7310	C₂²	1	2	yes	complemented
37	C₂×#7310	C₂²	1	2	yes	complemented
38	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂⁵×D₄	C₂³	1	1	yes	characteristic
40	C₂×#3484	C₂×C₄	1	2	yes	complemented
41	C₂×#3484	C₂³	1	2	yes	—
42	C₂²×#534	D₄	1	1	yes	characteristic
43	C₂²×#534	C₂³	1	1	yes	characteristic
44	C₂³×#120	D₄	1	1	yes	characteristic
45	C₂²×#518	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#3026	C₂×C₄	1	4	yes	—
47	C₂³×#99	C₂×C₄	1	2	yes	complemented
48	#34980	C₂×C₄	1	2	yes	—
49	#34980	C₂×C₄	1	2	yes	—
50	#34980	D₄	1	2	yes	—
51	#34980	D₄	1	2	yes	—
52	C₂×#2308	D₄	1	1	yes	characteristic
53	C₂×#2308	C₂³	1	1	yes	characteristic
54	C₂²×#379	D₄	1	1	yes	characteristic
55	#23903	D₄	1	8	yes	—
56	#23901	D₄	1	4	yes	—
57	#23900	D₄	1	4	yes	—
58	#20960	C₂×C₄	1	4	yes	—
59	#20889	C₂×C₄	1	4	yes	complemented
60	#20609	C₂×C₄	1	4	yes	complemented
61	C₂×#1783	C₂³	1	2	yes	—
62	C₂×#1781	C₂×C₄	1	2	yes	complemented
63	C₂×#1781	D₄	1	2	yes	complemented
64	C₂×#1781	D₄	1	2	yes	complemented
65	C₂×#1781	D₄	1	2	yes	complemented
66	C₂×#1781	D₄	1	2	yes	complemented
67	C₂×#1781	C₂³	1	2	yes	—
68	C₂×#1781	C₂³	1	2	yes	—
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×#1781	C₂³	1	1	yes	characteristic
73	C₂⁷	C₂×D₄	1	1	yes	characteristic
74	C₂⁷	#41	1	1	yes	characteristic
75	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
76	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
77	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
84	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
85	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	#41	1	1	yes	characteristic
87	C₂⁴×D₄	#41	1	1	yes	characteristic
88	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
89	C₂⁵×C₄	#41	1	1	yes	characteristic
90	#3484	C₂×D₄	1	4	yes	—
91	#3484	C₂×D₄	1	4	yes	—
92	C₂×#534	C₂²⋊C₄	1	2	yes	—
93	C₂×#534	C₂²⋊C₄	1	2	yes	—
94	C₂×#534	C₂²×C₄	1	2	yes	—
95	C₂×#534	C₂×D₄	1	2	yes	—
96	C₂×#534	C₂×D₄	1	4	yes	—
97	C₂²×#123	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂²⋊C₄	1	2	yes	—
99	C₂²×#120	C₂²⋊C₄	1	2	yes	—
100	C₂²×#120	C₂×D₄	1	2	yes	—
101	C₂×#518	C₂²×C₄	1	2	yes	—
102	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×#458	C₂²⋊C₄	1	8	yes	complemented
106	C₂×B(4,𝔽₂)	C₂²⋊C₄	1	8	yes	complemented
107	C₂²×#99	C₂²×C₄	1	4	yes	—
108	C₂²×#99	C₂×D₄	1	2	yes	—
109	C₂²×#99	C₂×D₄	1	2	yes	—
110	C₂²×#99	C₂×D₄	1	2	yes	—
111	C₂²×#99	C₂×D₄	1	2	yes	—
112	#2308	C₂²⋊C₄	1	2	yes	—
113	#2308	C₂²⋊C₄	1	2	yes	—
114	#2308	C₂²×C₄	1	2	yes	—
115	#2308	C₂×D₄	1	2	yes	—
116	C₂×#379	C₂²⋊C₄	1	2	yes	—
117	C₂×#379	C₂²⋊C₄	1	2	yes	—
118	#1890	C₂²⋊C₄	1	8	yes	complemented
119	#1889	C₂²⋊C₄	1	8	yes	complemented
120	#1781	C₂×D₄	1	4	yes	—
121	#1781	C₂×D₄	1	4	yes	—
122	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂⁶	C₄×D₄	1	1	yes	characteristic
125	C₂⁶	C₄×D₄	1	1	yes	characteristic
126	C₂⁶	#120	1	1	yes	characteristic
127	C₂⁶	#120	1	1	yes	characteristic
128	C₂⁶	#120	1	1	yes	characteristic
129	C₂⁶	#121	1	1	yes	characteristic
130	C₂⁶	#121	1	1	yes	characteristic
131	C₂⁶	#123	1	1	yes	characteristic
132	C₂⁶	#123	1	1	yes	characteristic
133	C₂⁶	#123	1	1	yes	characteristic
134	C₂⁶	C₂²×D₄	1	1	yes	characteristic
135	C₂⁶	C₂×#41	1	1	yes	characteristic
136	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
144	C₂³×D₄	#120	1	2	yes	—
145	C₂³×D₄	#120	1	1	yes	characteristic
146	C₂³×D₄	#120	1	1	yes	characteristic
147	C₂³×D₄	#121	1	1	yes	characteristic
148	C₂³×D₄	#121	1	1	yes	characteristic
149	C₂³×D₄	#121	1	1	yes	characteristic
150	C₂³×D₄	#121	1	1	yes	characteristic
151	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
152	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
154	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
155	C₂³×D₄	C₂²×D₄	1	2	yes	—
156	C₂³×D₄	C₂²×D₄	1	2	yes	—
157	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
158	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
159	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
160	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂⁴×C₄	#120	1	1	yes	characteristic
162	C₂⁴×C₄	#121	1	1	yes	characteristic
163	C₂⁴×C₄	#121	1	1	yes	characteristic
164	C₂⁴×C₄	#123	1	1	yes	characteristic
165	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
166	C₂×#123	C₂×C₂²⋊C₄	1	4	yes	—
167	C₂×#123	#120	1	4	yes	—
168	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
169	C₂×#120	#120	1	4	yes	—
170	#518	C₂×C₂²⋊C₄	1	4	yes	—
171	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂²×C₂²⋊C₄	#120	1	2	yes	—
175	C₂²×C₂²⋊C₄	#120	1	2	yes	—
176	C₂×#99	C₂×C₂²⋊C₄	1	8	yes	—
177	C₂⁵	#379	1	2	yes	—
178	C₂⁵	#386	1	1	yes	characteristic
179	C₂⁵	#386	1	1	yes	characteristic
180	C₂⁵	#386	1	1	yes	characteristic
181	C₂⁵	#386	1	1	yes	characteristic
182	C₂⁵	#386	1	1	yes	characteristic
183	C₂⁵	#386	1	1	yes	characteristic
184	C₂⁵	B(4,𝔽₂)	1	2	yes	—
185	C₂⁵	B(4,𝔽₂)	1	2	yes	—
186	C₂⁵	B(4,𝔽₂)	1	2	yes	—
187	C₂⁵	B(4,𝔽₂)	1	2	yes	—
188	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
189	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
190	C₂⁵	C₂×#120	1	1	yes	characteristic
191	C₂⁵	C₂×#120	1	1	yes	characteristic
192	C₂⁵	C₂×#120	1	1	yes	characteristic
193	C₂⁵	C₂×#121	1	1	yes	characteristic
194	C₂⁵	C₂×#123	1	1	yes	characteristic
195	C₂⁵	#534	1	1	yes	characteristic
196	C₂⁵	#534	1	1	yes	characteristic
197	C₂⁵	#546	1	2	yes	—
198	C₂⁵	#546	1	2	yes	—
199	C₂²×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
200	C₂²×D₄	#518	1	2	yes	—
201	C₂²×D₄	C₂×#120	1	2	yes	—
202	C₂²×D₄	#534	1	2	yes	—
203	C₂²×D₄	D₄²	1	2	yes	—
204	C₂²×D₄	D₄²	1	2	yes	—
205	C₂²×D₄	D₄²	1	2	yes	—
206	C₂²×D₄	D₄²	1	2	yes	—
207	C₂³×C₄	#379	1	2	yes	—
208	C₂³×C₄	#386	1	1	yes	characteristic
209	C₂³×C₄	#386	1	1	yes	characteristic
210	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂³×C₄	C₂×#121	1	1	yes	characteristic
212	C₂³×C₄	#546	1	2	yes	—
213	C₂³×C₄	#546	1	2	yes	—
214	C₂×C₂²⋊C₄	#379	1	4	yes	—
215	C₂⁴	#1889	1	1	yes	characteristic
216	C₂⁴	#1889	1	1	yes	characteristic
217	C₂⁴	#1889	1	1	yes	characteristic
218	C₂⁴	#1889	1	2	yes	—
219	C₂⁴	#1889	1	2	yes	—
220	C₂⁴	#1889	1	1	yes	characteristic
221	C₂⁴	#1913	1	2	yes	—
222	C₂⁴	C₂×#379	1	1	yes	characteristic
223	C₂⁴	C₂×#386	1	1	yes	characteristic
224	C₂⁴	#2308	1	1	yes	characteristic
225	C₂⁴	D₄×C₂²⋊C₄	1	2	yes	—
226	C₂⁴	#2403	1	1	yes	characteristic
227	C₂⁴	#2403	1	1	yes	characteristic
228	C₂⁴	#2405	1	1	yes	characteristic
229	C₂⁴	#2405	1	1	yes	characteristic
230	C₂⁴	#2410	1	2	yes	—
231	C₂⁴	#2883	1	2	yes	—
232	C₂⁴	#2883	1	2	yes	—
233	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
234	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
235	C₂²×C₄	#1913	1	2	yes	—
236	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₄	#2410	1	2	yes	—
238	C₂³	#13867	1	1	yes	characteristic
239	C₂³	#20585	1	2	yes	—
240	C₂³	C₂×#1889	1	1	yes	characteristic
241	C₂³	C₂×#1889	1	1	yes	characteristic
242	C₂³	#20984	1	1	yes	characteristic
243	C₂³	#20984	1	1	yes	characteristic
244	C₂³	C₂×#1913	1	1	yes	characteristic
245	C₂³	C₂×#1913	1	1	yes	characteristic
246	C₂³	#21749	1	2	yes	—
247	C₂³	#23900	1	2	yes	—
248	C₂³	#23900	1	2	yes	—
249	C₂²	#201678	1	1	yes	characteristic
250	C₂²	#201678	1	1	yes	characteristic
251	C₂²	#201678	1	1	yes	characteristic
252	C₂²	#201678	1	1	yes	characteristic
253	C₂²	#359434	1	1	yes	characteristic
254	C₂²	#359442	1	1	yes	characteristic
255	C₂²	#359501	1	1	yes	characteristic
256	C₂	#72255642	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72255642
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234661	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×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. #72255980	C₂	semidirect
2. #72255874	C₂	semidirect
3. #72237638	C₂	semidirect
4. #72237374	C₂	semidirect
6. #72225713	C₂	semidirect
7. #72219300	C₂	semidirect
8. #72218381	C₂	semidirect
9. #72218313	C₂	semidirect
10. #72218306	C₂	semidirect
11. #72210555	C₂	semidirect
14. #6629258	C₄	semidirect
15. #6628892	C₄	semidirect
17. #6437995	C₄	semidirect
19. #6374251	C₂²	semidirect
20. #6374128	C₂²	semidirect
22. #493879	C₂²	semidirect
27. #493357	C₂²	semidirect
32. #382707	C₂²	semidirect
33. #382707	C₂²	semidirect
34. C₂×#7314	C₂²	semidirect
35. C₂×#7314	C₂²	semidirect
36. C₂×#7310	C₂²	semidirect
37. C₂×#7310	C₂²	semidirect
38. C₂³×H(5,𝔽₂)	C₂×C₄	semidirect
40. C₂×#3484	C₂×C₄	semidirect
45. C₂²×#518	C₂×C₄	semidirect
47. C₂³×#99	C₂×C₄	semidirect
59. #20889	C₂×C₄	semidirect
60. #20609	C₂×C₄	semidirect
62. C₂×#1781	C₂×C₄	semidirect
63. C₂×#1781	D₄	semidirect
64. C₂×#1781	D₄	semidirect
65. C₂×#1781	D₄	semidirect
66. C₂×#1781	D₄	semidirect
105. C₂×#458	C₂²⋊C₄	semidirect
106. C₂×B(4,𝔽₂)	C₂²⋊C₄	semidirect
118. #1890	C₂²⋊C₄	semidirect
119. #1889	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72255980	2
#72255874	2
#72237638	2
#72237374	2
#72234661	2
#72225713	2
#72219300	2
#72218381	2
#72218313	2
#72218306	2
#72210555	2