Finite Group Databasebeta

← C2×#72215683 #72267443

Subgroups — C2×#72215683 #72267443

Order 2048.

At least 6751 conjugacy classes of subgroups; only the 1040 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#133390C214yesmaximal, complemented
2C2×#133290C211yesmaximal, characteristic, complemented
3C2×#249319C211yesmaximal, characteristic, complemented
4C2×#133278C211yesmaximal, characteristic, complemented
5C2×#499787C214yesmaximal, complemented
6C2×#7670203C211yesmaximal, characteristic, complemented
7C2×#245225C211yesmaximal, characteristic, complemented
8C2×#133822C211yesmaximal, characteristic, complemented
9C2×#449475C211yesmaximal, characteristic, complemented
10#72215683C2116yesmaximal, direct product factor
11C23×B(4,𝔽2)C2212yescomplemented
12#7670203C2218yescomplemented
13C2×#58990C2214yes—
14C22×#2846C2211yescharacteristic
15C2×#35390C2211yescharacteristic
16C2×#35346C2211yescharacteristic
17C2×#35346C2211yescharacteristic, complemented
18C2×#32898C2212yescomplemented
19C2×#32888C2212yescomplemented
20#499787C22132yescomplemented
21C2×#13088C2214yescomplemented
22C2×#13086C2214yescomplemented
23C2×#12687C2214yescomplemented
24#449475C2218yescomplemented
25C2×#10991C2212yescomplemented
26C2×#10965C2211yescharacteristic, complemented
27#249319C2218yescomplemented
28#245225C2218yescomplemented
29C2×#9495C2212yescomplemented
30C2×#9463C2212yescomplemented
31C2×#8547C2211yescharacteristic, complemented
32C2×#8541C2211yescharacteristic, complemented
33#133822C2218yescomplemented
34#133390C22132yescomplemented
35#133290C2218yescomplemented
36#133278C2218yescomplemented
37C28D411yescharacteristic, complemented
38C23×H(5,𝔽2)D411yescharacteristic, complemented
39C23×H(5,𝔽2)C2311yescharacteristic, complemented
40C22×D42D414yescomplemented
41C22×#534D411yescharacteristic, complemented
42C23×#120C2312yes—
43C23×#120C2311yescharacteristic
44C22×B(4,𝔽2)C2318yescomplemented
45C23×#99C2312yescomplemented
46#58990D4116yescomplemented
47#58990C23116yes—
48C2×#2679D414yescomplemented
49C2×#2669D414yescomplemented
50C2×#2846D414yescomplemented
51C2×#2846C2314yes—
52#35390D414yescomplemented
53#35390C2314yes—
54#35346D414yescomplemented
55#35346C2314yes—
56#35346C2314yescomplemented
57C2×#2539D412yescomplemented
58C2×#2433C2312yes—
59C2×#2403D412yescomplemented
60C2×#2403C2312yes—
61C2×#2358D414yescomplemented
62C2×#2308D411yescharacteristic, complemented
63C2×#2308C2311yescharacteristic, complemented
64C22×#379D411yescharacteristic, complemented
65C22×#379D412yescomplemented
66C2×#2273D411yescharacteristic, complemented
67#32898C2318yescomplemented
68#32888C2318yescomplemented
69C2×#1783C2312yescomplemented
70C2×#1781C2312yescomplemented
71#13088C23116yescomplemented
72#13086C23116yescomplemented
73#12687C23116yescomplemented
74#10991C2318yescomplemented
75#10965C2314yescomplemented
76#9495C2318yescomplemented
77#9463C2318yescomplemented
78#8547C2314yescomplemented
79#8541C2314yescomplemented
80C27C2×D412yes—
81C27C2×D411yescharacteristic
82C27C2×D414yescomplemented
83C22×H(5,𝔽2)C2×D414yescomplemented
84C22×H(5,𝔽2)C2412yescomplemented
85C24×D4C2×D411yescharacteristic
86C24×D4C2×D411yescharacteristic
87C24×D4C2×D412yes—
88C24×D4C2411yescharacteristic
89C2×D42C2×D4116yescomplemented
90C2×#534C2×D414yescomplemented
91C22×#127C2×D414yes—
92C22×#121C2×D412yes—
93C22×#120C2×D414yes—
94C22×#120C2×D414yes—
95C22×#120C2×D418yes—
96C22×#120C2×D414yes—
97C22×#120C2×D414yes—
98C22×#120C2×D412yes—
99C22×#120C2×D412yes—
100C22×#120C2×D412yes—
101C22×#120C2414yes—
102C22×#120C2412yes—
103C23×C22⋊C4C2×D411yescharacteristic
104C23×C22⋊C4C2×D411yescharacteristic
105C23×C22⋊C4C2×D411yescharacteristic
106C23×C22⋊C4C2×D412yes—
107C22×#99C2414yescomplemented
108#2679C2×D4116yescomplemented
109#2669C2×D4116yescomplemented
110#2539C2×D418yescomplemented
111#2433C2×D418yes—
112#2433C2×D414yes—
113#2433C2414yes—
114#2403C2×D414yes—
115#2403C2×D418yes—
116#2403C2×D418yescomplemented
117#2403C2414yes—
118#2358C2×D4116yescomplemented
119#2308C2×D414yescomplemented
120#2308C2412yescomplemented
121C2×#390C2×D414yes—
122C2×#386C2×D414yes—
123C2×#379C2×D412yes—
124C2×#379C2×D414yescomplemented
125C2×#379C2×D418yescomplemented
126#2273C2×D414yescomplemented
127C2×C4×C22⋊C4C2×D412yes—
128#1783C2414yescomplemented
129#1781C2414yescomplemented
130C26#12011yescharacteristic
131C26#12011yescharacteristic
132C26#12012yes—
133C26#12012yes—
134C26#12014yes—
135C26#12012yes—
136C26#12012yescomplemented
137C26C22×D411yescharacteristic
138C26C22×D414yes—
139C26C22×D412yes—
140C26C22×D411yescharacteristic
141C23×D4#12012yes—
142C23×D4#12012yes—
143C23×D4#12012yescomplemented
144C23×D4#12014yescomplemented
145C23×D4C22×D412yes—
146C23×D4C22×D414yes—
147C23×D4C22×D412yes—
148C23×D4C22×D411yescharacteristic
149C23×D4C22×D411yescharacteristic
150C23×D4C22×D414yes—
151C23×D4C22×D414yes—
152C23×D4C22×D411yescharacteristic
153C23×D4C2511yescharacteristic
154C24×C4#12011yescharacteristic
155C24×C4#12011yescharacteristic
156C24×C4#12012yescomplemented
157C24×C4C22×D411yescharacteristic
158C2×#127#12018yescomplemented
159C2×#127C22×D418yes—
160C2×#121#12014yescomplemented
161C2×#121C22×D414yes—
162C2×#120#12018yes—
163C2×#120#12018yes—
164C2×#120#12018yescomplemented
165C2×#120C22×D418yes—
166C2×#120C22×D418yes—
167C2×#120C22×D418yes—
168C22×C4⋊C4#12012yescomplemented
169C22×C22⋊C4#12014yes—
170C22×C22⋊C4#12012yes—
171C22×C22⋊C4#12012yes—
172C22×C22⋊C4#12012yes—
173C22×C22⋊C4#12014yes—
174C22×C22⋊C4#12014yescomplemented
175C22×C22⋊C4C22×D412yes—
176C22×C22⋊C4C22×D412yes—
177C22×C22⋊C4C22×D412yes—
178C22×C22⋊C4C22×D414yes—
179C22×C42#12014yescomplemented
180#390#12018yescomplemented
181#390C22×D418yes—
182#386#12018yescomplemented
183#386C22×D418yes—
184#379#12014yescomplemented
185#379C22×D414yes—
186C4×C22⋊C4#12014yescomplemented
187C4×C22⋊C4C22×D414yes—
188C25B(4,𝔽2)12yes—
189C25B(4,𝔽2)12yes—
190C25B(4,𝔽2)18yescomplemented
191C25C2×#12012yes—
192C25C2×#12012yes—
193C25C2×#12011yescharacteristic
194C25C2×#12012yes—
195C25C2×#12012yes—
196C25C2×#12012yes—
197C25C2×#12014yes—
198C25C2×#12014yes—
199C25C2×#12012yes—
200C25C2×#12011yescharacteristic
201C25C2×#12011yescharacteristic
202C25C2×#12014yescomplemented
203C25C23×D411yescharacteristic
204C25C23×D411yescharacteristic
205C22×D4C2×#12014yes—
206C22×D4C2×#12014yes—
207C22×D4C2×#12014yes—
208C22×D4C2×#12014yescomplemented
209C22×D4C23×D414yes—
210C23×C4C2×#12012yes—
211C23×C4C2×#12012yes—
212C23×C4C2×#12012yes—
213C23×C4C2×#12011yescharacteristic
214C23×C4C2×#12012yes—
215C23×C4C2×#12014yescomplemented
216C23×C4C23×D411yescharacteristic
217C2×C4⋊C4C2×#12014yescomplemented
218C2×C22⋊C4C2×#12018yes—
219C2×C22⋊C4C2×#12018yescomplemented
220C2×C42C2×#12018yescomplemented
221C24#219618yescomplemented
222C24#284611yescharacteristic
223C24#284611yescharacteristic
224C24#284612yes—
225C24#284612yes—
226C24C2×B(4,𝔽2)11yescharacteristic
227C24C2×B(4,𝔽2)11yescharacteristic
228C24C2×B(4,𝔽2)14yes—
229C24C2×B(4,𝔽2)14yes—
230C24C2×B(4,𝔽2)18yescomplemented
231C24#302611yescharacteristic
232C24#302611yescharacteristic
233C24#302614yes—
234C24C22×#12011yescharacteristic
235C24C22×#12012yes—
236C24C22×#12012yes—
237C22×C4#284612yes—
238C22×C4C22×#12012yes—
239C23#3289811yescharacteristic
240C23#3289811yescharacteristic
241C23#3289811yescharacteristic
242C23#3289814yes—
243C23#3289814yes—
244C23#3289811yescharacteristic
245C23C2×#219614yes—
246C23C2×#219618yescomplemented
247C23C2×#284611yescharacteristic
248C23C22×B(4,𝔽2)11yescharacteristic
249C23C22×B(4,𝔽2)12yes—
250C23C2×#302611yescharacteristic
251C23C2×#302612yes—
252C22#34704212yes—
253C22#34704212yes—
254C22#51177112yes—
255C22#51177112yes—
256C22#51179012yes—
257C22C2×#3289812yes—
258C22C2×#3289811yescharacteristic
259C22C22×#219612yes—
260C2C2×#51179011yescharacteristic
261C2C2×#34704211yescharacteristic
262C2#7221568314yesdirect product factor
263C2C2×#51177111yescharacteristic
subgroupG/H
center, socleC23#32898
commutator, Frattini, cosocleC23×D4C25
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×#120 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#133390C2semidirect
2. C2×#133290C2semidirect
3. C2×#249319C2semidirect
4. C2×#133278C2semidirect
5. C2×#499787C2semidirect
6. C2×#7670203C2semidirect
7. C2×#245225C2semidirect
8. C2×#133822C2semidirect
9. C2×#449475C2semidirect
10. #72215683C2direct (N × H)
11. C23×B(4,𝔽2)C22semidirect
12. #7670203C22semidirect
17. C2×#35346C22semidirect
18. C2×#32898C22semidirect
19. C2×#32888C22semidirect
20. #499787C22semidirect
21. C2×#13088C22semidirect
22. C2×#13086C22semidirect
23. C2×#12687C22semidirect
24. #449475C22semidirect
25. C2×#10991C22semidirect
26. C2×#10965C22semidirect
27. #249319C22semidirect
28. #245225C22semidirect
29. C2×#9495C22semidirect
30. C2×#9463C22semidirect
31. C2×#8547C22semidirect
32. C2×#8541C22semidirect
33. #133822C22semidirect
34. #133390C22semidirect
35. #133290C22semidirect
36. #133278C22semidirect
37. C28D4semidirect
38. C23×H(5,𝔽2)D4semidirect
39. C23×H(5,𝔽2)C23semidirect
40. C22×D42D4semidirect
41. C22×#534D4semidirect
44. C22×B(4,𝔽2)C23semidirect
45. C23×#99C23semidirect
46. #58990D4semidirect
48. C2×#2679D4semidirect
49. C2×#2669D4semidirect
50. C2×#2846D4semidirect
52. #35390D4semidirect
54. #35346D4semidirect
56. #35346C23semidirect
57. C2×#2539D4semidirect
59. C2×#2403D4semidirect
61. C2×#2358D4semidirect
62. C2×#2308D4semidirect
63. C2×#2308C23semidirect
64. C22×#379D4semidirect
65. C22×#379D4semidirect
66. C2×#2273D4semidirect
67. #32898C23semidirect
68. #32888C23semidirect
69. C2×#1783C23semidirect
70. C2×#1781C23semidirect
71. #13088C23semidirect
72. #13086C23semidirect
73. #12687C23semidirect
74. #10991C23semidirect
75. #10965C23semidirect
76. #9495C23semidirect
77. #9463C23semidirect
78. #8547C23semidirect
79. #8541C23semidirect
82. C27C2×D4semidirect
83. C22×H(5,𝔽2)C2×D4semidirect
84. C22×H(5,𝔽2)C24semidirect
89. C2×D42C2×D4semidirect
90. C2×#534C2×D4semidirect
107. C22×#99C24semidirect
108. #2679C2×D4semidirect
109. #2669C2×D4semidirect
110. #2539C2×D4semidirect
116. #2403C2×D4semidirect
118. #2358C2×D4semidirect
119. #2308C2×D4semidirect
120. #2308C24semidirect
124. C2×#379C2×D4semidirect
125. C2×#379C2×D4semidirect
126. #2273C2×D4semidirect
128. #1783C24semidirect
129. #1781C24semidirect
136. C26#120semidirect
143. C23×D4#120semidirect
144. C23×D4#120semidirect
156. C24×C4#120semidirect
158. C2×#127#120semidirect
160. C2×#121#120semidirect
164. C2×#120#120semidirect
168. C22×C4⋊C4#120semidirect
174. C22×C22⋊C4#120semidirect
179. C22×C42#120semidirect
180. #390#120semidirect
182. #386#120semidirect
184. #379#120semidirect
186. C4×C22⋊C4#120semidirect
190. C25B(4,𝔽2)semidirect
202. C25C2×#120semidirect
208. C22×D4C2×#120semidirect
215. C23×C4C2×#120semidirect
217. C2×C4⋊C4C2×#120semidirect
219. C2×C22⋊C4C2×#120semidirect
220. C2×C42C2×#120semidirect
221. C24#2196semidirect
230. C24C2×B(4,𝔽2)semidirect
246. C23C2×#2196semidirect
262. C2#72215683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#1333902
C2×#1332902
C2×#2493192
C2×#1332782
C2×#4997872
C2×#76702032
C2×#2452252
C2×#1338222
C2×#4494752
#722156832