Subgroups — #72266829

Order 2048.

At least 7479 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

The subgroup lattice has 248 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72256025	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251108	C₂	1	1	yes	maximal, characteristic, complemented
3	#72246450	C₂	1	1	yes	maximal, characteristic, complemented
4	#72246287	C₂	1	1	yes	maximal, characteristic, complemented
5	#72238351	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237632	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237601	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237473	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237472	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234693	C₂	1	1	yes	maximal, characteristic
11	#72234631	C₂	1	1	yes	maximal, characteristic, complemented
12	#72234613	C₂	1	1	yes	maximal, characteristic, complemented
13	#72226366	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#133839	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	C₂×#60399	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	#6440460	C₂²	1	1	yes	characteristic, complemented
20	#6440460	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	C₂×#23895	C₂²	1	1	yes	characteristic
23	#6437621	C₂²	1	1	yes	characteristic
24	#6437621	C₂²	1	1	yes	characteristic
25	#6374210	C₂²	1	1	yes	characteristic, complemented
26	C₂×#21776	C₂²	1	1	yes	characteristic
27	#763512	C₂²	1	1	yes	characteristic, complemented
28	#493879	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
31	C₂×#12705	C₂²	1	1	yes	characteristic
32	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
33	#493357	C₂²	1	1	yes	characteristic, complemented
34	#382434	C₂²	1	1	yes	characteristic, complemented
35	#359558	C₂²	1	1	yes	characteristic, complemented
36	#359549	C₂²	1	1	yes	characteristic
37	#359549	C₂²	1	1	yes	characteristic, complemented
38	#359442	C₂²	1	1	yes	characteristic, complemented
39	#190116	C₂²	1	1	yes	characteristic, complemented
40	#188049	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
43	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
44	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
45	#146479	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	#108222	C₂²	1	1	yes	characteristic, complemented
51	C₂×H(7,𝔽₂)	D₄	1	1	yes	characteristic
52	#62820	D₄	1	2	yes	—
53	#62817	D₄	1	2	yes	—
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂×#3484	C₂³	1	1	yes	characteristic
57	C₂×#3484	C₂³	1	1	yes	characteristic
58	C₂²×#534	C₂³	1	1	yes	characteristic
59	C₂×#3026	D₄	1	2	yes	—
60	C₂×#2883	D₄	1	1	yes	characteristic
61	C₂×#2883	D₄	1	1	yes	characteristic
62	C₂×#2883	D₄	1	1	yes	characteristic
63	#34992	D₄	1	2	yes	complemented
64	#34980	D₄	1	2	yes	—
65	#34171	C₂³	1	1	yes	characteristic
66	C₂×#2308	C₂³	1	1	yes	characteristic
67	#23901	D₄	1	2	yes	—
68	#23900	D₄	1	2	yes	—
69	#23895	D₄	1	4	yes	—
70	#21776	D₄	1	4	yes	—
71	C₂×#1913	D₄	1	2	yes	—
72	C₂×#1913	D₄	1	2	yes	complemented
73	C₂×#1898	D₄	1	2	yes	—
74	C₂×#1898	D₄	1	2	yes	complemented
75	#20960	D₄	1	2	yes	complemented
76	C₂×#1783	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	C₂×#1781	C₂³	1	1	yes	characteristic
79	C₂×#1781	C₂³	1	1	yes	characteristic
80	C₂×#1781	C₂³	1	1	yes	characteristic
81	C₂×#1781	C₂³	1	1	yes	characteristic
82	C₂×#1781	C₂³	1	1	yes	characteristic
83	C₂×#1438	C₂³	1	1	yes	characteristic
84	#8534	D₄	1	2	yes	complemented
85	#8534	D₄	1	2	yes	complemented
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	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	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
92	#3484	C₂×D₄	1	2	yes	—
93	#3484	C₂×D₄	1	2	yes	—
94	#3484	C₂×D₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	#3484	C₂×D₄	1	2	yes	—
98	#3484	C₂×D₄	1	2	yes	—
99	#3484	#41	1	2	yes	—
100	C₂×#534	C₂×D₄	1	2	yes	—
101	C₂×#534	C₂×D₄	1	2	yes	—
102	C₂²×#123	C₂×D₄	1	1	yes	characteristic
103	C₂²×#123	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂×D₄	1	2	yes	—
105	C₂²×#120	C₂×D₄	1	2	yes	—
106	C₂²×#120	C₂×D₄	1	1	yes	characteristic
107	C₂²×#120	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	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
113	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
114	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic, complemented
116	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic, complemented
117	C₂²×#99	C₂×D₄	1	2	yes	—
118	C₂²×#99	C₂×D₄	1	2	yes	complemented
119	C₂×#379	C₂×D₄	1	2	yes	—
120	C₂×#379	C₂×D₄	1	2	yes	complemented
121	#1781	C₂×D₄	1	2	yes	—
122	#1781	C₂×D₄	1	2	yes	—
123	#1781	C₂×D₄	1	2	yes	—
124	#1781	C₂×D₄	1	2	yes	—
125	#1781	C₂×D₄	1	2	yes	complemented
126	#1438	C₂×D₄	1	1	yes	characteristic, complemented
127	#1438	C₂×D₄	1	1	yes	characteristic, complemented
128	C₂⁶	#120	1	1	yes	characteristic
129	C₂⁶	C₂²×D₄	1	1	yes	characteristic
130	C₂⁶	C₂²×D₄	1	1	yes	characteristic
131	C₂×H(5,𝔽₂)	#120	1	2	yes	—
132	C₂×H(5,𝔽₂)	#120	1	2	yes	—
133	C₂×H(5,𝔽₂)	#120	1	2	yes	—
134	C₂×H(5,𝔽₂)	#121	1	2	yes	—
135	C₂³×D₄	#120	1	2	yes	—
136	C₂³×D₄	#120	1	2	yes	—
137	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂×#41	1	1	yes	characteristic
146	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂⁴×C₄	#120	1	1	yes	characteristic
150	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
151	#534	#121	1	4	yes	—
152	#534	#127	1	4	yes	—
153	C₂×#120	#120	1	4	yes	—
154	C₂×#120	#121	1	4	yes	—
155	#518	#120	1	2	yes	—
156	#518	#120	1	2	yes	—
157	#518	#120	1	2	yes	—
158	#518	#121	1	2	yes	—
159	C₂²×C₂²⋊C₄	#120	1	2	yes	—
160	C₂²×C₂²⋊C₄	#120	1	2	yes	—
161	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
162	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
163	C₂×#99	#120	1	4	yes	—
164	#379	#120	1	4	yes	—
165	C₂⁵	B(4,𝔽₂)	1	2	yes	—
166	C₂⁵	#458	1	2	yes	complemented
167	C₂⁵	C₂×#120	1	1	yes	characteristic
168	C₂⁵	C₂×#120	1	1	yes	characteristic
169	C₂⁵	C₂×#120	1	1	yes	characteristic
170	C₂⁵	C₂×#120	1	1	yes	characteristic
171	C₂⁵	C₂×#120	1	1	yes	characteristic
172	C₂⁵	C₂×#121	1	1	yes	characteristic
173	C₂⁵	D₄²	1	1	yes	characteristic
174	C₂⁵	D₄²	1	1	yes	characteristic
175	C₂⁵	D₄²	1	1	yes	characteristic
176	C₂⁵	D₄²	1	1	yes	characteristic
177	C₂⁵	D₄²	1	1	yes	characteristic
178	C₂⁵	D₄²	1	1	yes	characteristic
179	C₂⁵	#546	1	1	yes	characteristic
180	C₂²×D₄	#392	1	4	yes	—
181	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
182	C₂²×D₄	C₂×#120	1	1	yes	characteristic
183	C₂²×D₄	C₂×#120	1	1	yes	characteristic
184	C₂²×D₄	C₂×#120	1	1	yes	characteristic
185	C₂²×D₄	C₂×#120	1	1	yes	characteristic
186	C₂²×D₄	C₂×#121	1	1	yes	characteristic
187	C₂²×D₄	C₂×#121	1	1	yes	characteristic
188	C₂²×D₄	C₂×#127	1	1	yes	characteristic
189	C₂²×D₄	#534	1	1	yes	characteristic
190	C₂²×D₄	#534	1	1	yes	characteristic
191	C₂²×D₄	#535	1	1	yes	characteristic
192	C₂²×D₄	D₄²	1	1	yes	characteristic
193	C₂²×D₄	D₄²	1	1	yes	characteristic
194	C₂²×D₄	D₄²	1	1	yes	characteristic
195	C₂²×D₄	D₄²	1	1	yes	characteristic
196	C₂²×D₄	#546	1	1	yes	characteristic
197	C₂²×D₄	#546	1	1	yes	characteristic
198	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
199	C₂³×C₄	#458	1	2	yes	complemented
200	C₂³×C₄	#534	1	1	yes	characteristic
201	C₂³×C₄	#534	1	1	yes	characteristic
202	C₂³×C₄	#535	1	1	yes	characteristic
203	C₂³×C₄	D₄²	1	1	yes	characteristic
204	C₂³×C₄	D₄²	1	1	yes	characteristic
205	C₂³×C₄	#546	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	#392	1	4	yes	—
207	C₂⁴	#2021	1	4	yes	—
208	C₂⁴	C₂×#392	1	1	yes	characteristic
209	C₂⁴	#2403	1	1	yes	characteristic
210	C₂⁴	#2403	1	1	yes	characteristic
211	C₂⁴	#2403	1	1	yes	characteristic
212	C₂⁴	#2403	1	1	yes	characteristic
213	C₂⁴	#2408	1	1	yes	characteristic
214	C₂⁴	#2408	1	1	yes	characteristic
215	C₂⁴	#2410	1	1	yes	characteristic
216	C₂⁴	#2410	1	1	yes	characteristic
217	C₂⁴	#2433	1	1	yes	characteristic
218	C₂⁴	#2846	1	1	yes	characteristic
219	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
220	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
221	C₂⁴	C₂×#458	1	2	yes	—
222	C₂⁴	#3026	1	2	yes	—
223	C₂²×C₄	#2021	1	4	yes	—
224	C₂²×C₄	#2410	1	1	yes	characteristic
225	C₂²×C₄	#2410	1	1	yes	characteristic
226	C₂²×C₄	#2433	1	1	yes	characteristic
227	C₂²×C₄	#2613	1	1	yes	characteristic
228	C₂³	#14715	1	1	yes	characteristic
229	C₂³	C₂×#2021	1	2	yes	—
230	C₂³	#23895	1	1	yes	characteristic
231	C₂³	#23895	1	1	yes	characteristic
232	C₂³	#23900	1	2	yes	—
233	C₂³	#23900	1	1	yes	characteristic
234	C₂³	#23900	1	1	yes	characteristic
235	C₂³	#23901	1	1	yes	characteristic
236	C₂³	#23903	1	1	yes	characteristic
237	C₂³	#24405	1	2	yes	—
238	C₂³	#32900	1	2	yes	—
239	C₂²	#245226	1	1	yes	characteristic
240	C₂²	#245226	1	1	yes	characteristic
241	C₂²	#245246	1	1	yes	characteristic
242	C₂²	#245246	1	1	yes	characteristic
243	C₂²	#394058	1	1	yes	characteristic
244	C₂²	#394064	1	1	yes	characteristic
245	C₂²	#394068	1	1	yes	characteristic
246	C₂	#72241188	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241188
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234693	C₂
LCS	G ▷ C₂⁴×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. #72256025	C₂	semidirect
2. #72251108	C₂	semidirect
3. #72246450	C₂	semidirect
4. #72246287	C₂	semidirect
5. #72238351	C₂	semidirect
6. #72237632	C₂	semidirect
7. #72237601	C₂	semidirect
8. #72237473	C₂	semidirect
9. #72237472	C₂	semidirect
11. #72234631	C₂	semidirect
12. #72234613	C₂	semidirect
13. #72226366	C₂	semidirect
14. #72219160	C₂	semidirect
15. C₂×#133839	C₂	semidirect
19. #6440460	C₂²	semidirect
20. #6440460	C₂²	semidirect
25. #6374210	C₂²	semidirect
27. #763512	C₂²	semidirect
28. #493879	C₂²	semidirect
29. C₂×#12707	C₂²	semidirect
30. C₂×#12707	C₂²	semidirect
32. C₂×#12705	C₂²	semidirect
33. #493357	C₂²	semidirect
34. #382434	C₂²	semidirect
35. #359558	C₂²	semidirect
37. #359549	C₂²	semidirect
38. #359442	C₂²	semidirect
39. #190116	C₂²	semidirect
40. #188049	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. C₂×#8534	C₂²	semidirect
44. C₂×#8453	C₂²	semidirect
45. #146479	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. #108222	C₂²	semidirect
63. #34992	D₄	semidirect
72. C₂×#1913	D₄	semidirect
74. C₂×#1898	D₄	semidirect
75. #20960	D₄	semidirect
84. #8534	D₄	semidirect
85. #8534	D₄	semidirect
112. C₂×#518	C₂×D₄	semidirect
113. C₂×#518	C₂×D₄	semidirect
115. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
116. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
118. C₂²×#99	C₂×D₄	semidirect
120. C₂×#379	C₂×D₄	semidirect
125. #1781	C₂×D₄	semidirect
126. #1438	C₂×D₄	semidirect
127. #1438	C₂×D₄	semidirect
166. C₂⁵	#458	semidirect
199. C₂³×C₄	#458	semidirect

Maximal subgroups

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

subgroup	index
#72256025	2
#72251108	2
#72246450	2
#72246287	2
#72238351	2
#72237632	2
#72237601	2
#72237473	2
#72237472	2
#72234693	2
#72234631	2
#72234613	2
#72226366	2
#72219160	2
C₂×#133839	2