Finite Group Databasebeta

← C22×#72217027 #72344029

Subgroups — C22×#72217027 #72344029

Order 6144.

At least 9757 conjugacy classes of subgroups; only the 3651 normal subgroups are shown.

The subgroup lattice has 228 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#60399⋊C3C211yesmaximal, characteristic, complemented
2C23×#73026C212yesmaximal, complemented
3C22×#11714175C211yesmaximal, characteristic, complemented
4C22×#11714171C212yesmaximal, complemented
5C23×B(4,𝔽2)⋊C3C2212yescomplemented
6C22×#11714182C212yesmaximal, complemented
7C22×#11715828C211yesmaximal, characteristic, complemented
8C22×#11714174C211yesmaximal, characteristic, complemented
9C22×#87522C2211yescharacteristic, complemented
10C22×#73028C2212yescomplemented
11C22×#11714162C211yesmaximal, characteristic, complemented
12C22×#73025C2211yescharacteristic, complemented
13C22×#3025⋊C3C2211yescharacteristic, complemented
14C22×#11714184C211yesmaximal, characteristic, complemented
15C22×#3027⋊C3C2211yescharacteristic, complemented
16C22×#73031C2211yescharacteristic, complemented
17C22×#87520C2211yescharacteristic, complemented
18C22×#11714170C212yesmaximal, complemented
19C22×#73032C2212yescomplemented
20C22×#73027C2212yescomplemented
21C22×#73030C2211yescharacteristic, complemented
22C22×#11715839C211yesmaximal, characteristic, complemented
23C22×#87521C2211yescharacteristic, complemented
24C22×#87516C2212yescomplemented
25C22×#87515C2212yescomplemented
26C22×#3026⋊C3C2212yescomplemented
27C22×#87547C2211yescharacteristic, complemented
28C22×#87519C2211yescharacteristic, complemented
29C23×#5934C2212yescomplemented
30C2×#72217027C2148yesmaximal, direct product factor
31C2×#60399⋊C3C22124yescomplemented
32C2×#11714174C22124yescomplemented
33C2×#11714182C22148yescomplemented
34C2×#11714184C22124yescomplemented
35C2×#11714171C22148yescomplemented
36C2×#11714162C22124yescomplemented
37C2×#11714175C22124yescomplemented
38C22×#73026C2211yescharacteristic, complemented
39C22×#73026C2212yescomplemented
40C22×#73026C2212yescomplemented
41C22×#73026C2212yescomplemented
42C22×#73026C22148yescomplemented
43C22×#60399D311yescharacteristic, complemented
44C23×#5935C2212yescomplemented
45C2×#11714170C22148yescomplemented
46C2×#11715839C22124yescomplemented
47#72217027C221256yesdirect product factor
48C2×#11715828C22124yescomplemented
49#11715839C23164yescomplemented
50C2×#87521C23112yescomplemented
51C2×#87515C23124yescomplemented
52C2×#87519C23112yescomplemented
53C22×#5932C2311yescharacteristic, complemented
54C22×#5934C2311yescharacteristic
55C22×#5934C23124yescomplemented
56C22×#5934C2312yescomplemented
57#11715828C23164yescomplemented
58C2×#87522C23112yescomplemented
59C2×#87516C23124yescomplemented
60C2×#87520C23112yescomplemented
61C22×#5933C2311yescharacteristic, complemented
62C22×#5935C2311yescharacteristic
63C22×#5935C23124yescomplemented
64C22×#5935C2312yescomplemented
65#11714184C23164yescomplemented
66#11714182C231128yescomplemented
67#11714175C23164yescomplemented
68#11714174C23164yescomplemented
69#11714171C231128yescomplemented
70#11714170C231128yescomplemented
71C2×#73028C23124yescomplemented
72C2×#73032C23124yescomplemented
73C2×#73027C23124yescomplemented
74#11714162C23164yescomplemented
75C2×#73031C23112yescomplemented
76C2×#73025C23112yescomplemented
77C2×#73030C23112yescomplemented
78C2×#73026C23112yescomplemented
79C2×#73026C23124yescomplemented
80C2×#73026C23124yescomplemented
81C2×#73026C23124yescomplemented
82C2×#73026C231128yescomplemented
83C2×#87547C23112yescomplemented
84C22×#5950C2311yescharacteristic, complemented
85C23×#997C2312yescomplemented
86#60399⋊C3C23164yescomplemented
87C2×#3027⋊C3C23112yescomplemented
88C2×#3026⋊C3C23124yescomplemented
89C2×#3025⋊C3C23112yescomplemented
90C22×#458⋊C3C2311yescharacteristic, complemented
91C22×B(4,𝔽2)⋊C3C2311yescharacteristic
92C22×B(4,𝔽2)⋊C3C23124yescomplemented
93C22×B(4,𝔽2)⋊C3C2312yescomplemented
94C22×H(7,𝔽2)C2×D311yescharacteristic, complemented
95C22×#3027C2×D311yescharacteristic, complemented
96C22×#3025C2×D311yescharacteristic, complemented
97C2×#60399C2×D3124yescomplemented
98C22×#3026C2×D312yescomplemented
99C23×B(4,𝔽2)C2×D312yescomplemented
100#87547C24116yescomplemented
101C2×#5950C2416yescomplemented
102C22×#997C2411yescharacteristic
103C22×#997C24112yescomplemented
104#87522C24116yescomplemented
105#87521C24116yescomplemented
106#87520C24116yescomplemented
107#87519C24116yescomplemented
108#87516C24132yescomplemented
109#87515C24132yescomplemented
110C2×#5935C2416yes—
111C2×#5935C24132yescomplemented
112C2×#5935C24112yescomplemented
113C2×#5934C2416yes—
114C2×#5934C24132yescomplemented
115C2×#5934C24112yescomplemented
116C2×#5933C2416yescomplemented
117C2×#5932C2416yescomplemented
118#3027⋊C3C24116yescomplemented
119#3026⋊C3C24132yescomplemented
120#3025⋊C3C24116yescomplemented
121C2×#458⋊C3C2416yescomplemented
122C2×B(4,𝔽2)⋊C3C2416yes—
123C2×B(4,𝔽2)⋊C3C24132yescomplemented
124C2×B(4,𝔽2)⋊C3C24112yescomplemented
125#73032C24132yescomplemented
126#73031C24116yescomplemented
127#73030C24116yescomplemented
128#73028C24132yescomplemented
129#73027C24132yescomplemented
130#73026C24116yescomplemented
131#73026C24132yescomplemented
132#73026C24132yescomplemented
133#73026C24132yescomplemented
134#73025C24116yescomplemented
135C2×H(7,𝔽2)C22×D3112yescomplemented
136C22×#585C22×D311yescharacteristic, complemented
137C23×H(5,𝔽2)C22×D312yescomplemented
138C25×D4S411yescharacteristic, complemented
139#60399C22×D3164yescomplemented
140C2×#3027C22×D3112yescomplemented
141C2×#3026C22×D3124yescomplemented
142C2×#3025C22×D3112yescomplemented
143C22×#458C22×D311yescharacteristic, complemented
144C22×B(4,𝔽2)C22×D311yescharacteristic
145C22×B(4,𝔽2)C22×D312yescomplemented
146C22×B(4,𝔽2)C22×D3124yescomplemented
147#5950C2514yescomplemented
148C2×#997C2513yes—
149C2×#997C2518yescomplemented
150#5935C2514yes—
151#5935C2518yescomplemented
152#5934C2514yes—
153#5934C2518yescomplemented
154#5933C2514yescomplemented
155#5932C2514yescomplemented
156#458⋊C3C2514yescomplemented
157B(4,𝔽2)⋊C3C2514yes—
158B(4,𝔽2)⋊C3C2518yescomplemented
159C27C2×S412yescomplemented
160H(7,𝔽2)C23×D3116yescomplemented
161C2×#585C23×D316yescomplemented
162C22×H(5,𝔽2)C23×D311yescharacteristic
163C22×H(5,𝔽2)C23×D3112yescomplemented
164C24×D4C2×S411yescharacteristic
165C24×D4C2×S412yescomplemented
166C24×D4C2×S411yescharacteristic, complemented
167C24×D4C2×S4124yescomplemented
168C25×C4C2×S411yescharacteristic, complemented
169#3027C23×D3116yescomplemented
170#3026C23×D3132yescomplemented
171#3025C23×D3116yescomplemented
172C2×#458C23×D316yescomplemented
173C2×B(4,𝔽2)C23×D316yes—
174C2×B(4,𝔽2)C23×D3112yescomplemented
175C2×B(4,𝔽2)C23×D3132yescomplemented
176#997C2611yescharacteristic
177C26C22×S412yes—
178C26C22×S411yescharacteristic
179C26C22×S412yes—
180C26C22×S4124yescomplemented
181#585C24×D314yescomplemented
182C2×H(5,𝔽2)C24×D313yes—
183C2×H(5,𝔽2)C24×D318yescomplemented
184C23×D4C22×S4112yes—
185C23×D4C22×S4124yescomplemented
186C23×D4C22×S4112yescomplemented
187C23×D4C22×S4164yescomplemented
188C24×C4C22×S411yescharacteristic
189C24×C4C22×S411yescharacteristic
190C24×C4C22×S4112yescomplemented
191#458C24×D314yescomplemented
192B(4,𝔽2)C24×D314yes—
193B(4,𝔽2)C24×D318yescomplemented
194C25C23×S411yescharacteristic
195C25C23×S4112yes—
196C25C23×S416yes—
197C25C23×S4112yes—
198C25C23×S4132yescomplemented
199H(5,𝔽2)C25×D311yescharacteristic
200C22×D4#1020⋊C211yescharacteristic
201C22×D4C23×S4116yes—
202C22×D4C23×S4132yescomplemented
203C22×D4C23×S4116yescomplemented
204C23×C4C23×S416yes—
205C23×C4C23×S416yes—
206C23×C4C23×S4116yescomplemented
207C24C2×#1020⋊C212yes—
208C24C24×S413yes—
209C24C24×S418yes—
210C24C24×S414yes—
211C24C24×S418yes—
212C2×D4C2×#1020⋊C2112yes—
213C22×C4C2×#1020⋊C211yescharacteristic
214C22×C4C24×S414yes—
215C22×C4C24×S414yes—
216C23C22×#1020⋊C2112yes—
217C23C22×#1020⋊C211yescharacteristic
218C23C25×S411yescharacteristic
219D4C22×#1020⋊C2116yes—
220C2×C4C22×#1020⋊C216yes—
221C22#7221702714yesdirect product factor
222C22C23×#1020⋊C218yes—
223C22C23×#1020⋊C213yes—
224C4C23×#1020⋊C214yes—
225C2C2×#7221702716yesdirect product factor
226C2C24×#1020⋊C211yescharacteristic
subgroupG/H
center, socleC23C22×#1020⋊C2
commutator, cosocle, hypocenter#997C26
FrattiniC23C25×S4
FittingC22×#60399D3
hypercenterC22×D4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×#60399⋊C3C2semidirect
2. C23×#73026C2semidirect
3. C22×#11714175C2semidirect
4. C22×#11714171C2semidirect
5. C23×B(4,𝔽2)⋊C3C22semidirect
6. C22×#11714182C2semidirect
7. C22×#11715828C2semidirect
8. C22×#11714174C2semidirect
9. C22×#87522C22semidirect
10. C22×#73028C22semidirect
11. C22×#11714162C2semidirect
12. C22×#73025C22semidirect
13. C22×#3025⋊C3C22semidirect
14. C22×#11714184C2semidirect
15. C22×#3027⋊C3C22semidirect
16. C22×#73031C22semidirect
17. C22×#87520C22semidirect
18. C22×#11714170C2semidirect
19. C22×#73032C22semidirect
20. C22×#73027C22semidirect
21. C22×#73030C22semidirect
22. C22×#11715839C2semidirect
23. C22×#87521C22semidirect
24. C22×#87516C22semidirect
25. C22×#87515C22semidirect
26. C22×#3026⋊C3C22semidirect
27. C22×#87547C22semidirect
28. C22×#87519C22semidirect
29. C23×#5934C22semidirect
30. C2×#72217027C2direct (N × H)
31. C2×#60399⋊C3C22semidirect
32. C2×#11714174C22semidirect
33. C2×#11714182C22semidirect
34. C2×#11714184C22semidirect
35. C2×#11714171C22semidirect
36. C2×#11714162C22semidirect
37. C2×#11714175C22semidirect
38. C22×#73026C22semidirect
39. C22×#73026C22semidirect
40. C22×#73026C22semidirect
41. C22×#73026C22semidirect
42. C22×#73026C22semidirect
43. C22×#60399D3semidirect
44. C23×#5935C22semidirect
45. C2×#11714170C22semidirect
46. C2×#11715839C22semidirect
47. #72217027C22direct (N × H)
48. C2×#11715828C22semidirect
49. #11715839C23semidirect
50. C2×#87521C23semidirect
51. C2×#87515C23semidirect
52. C2×#87519C23semidirect
53. C22×#5932C23semidirect
55. C22×#5934C23semidirect
56. C22×#5934C23semidirect
57. #11715828C23semidirect
58. C2×#87522C23semidirect
59. C2×#87516C23semidirect
60. C2×#87520C23semidirect
61. C22×#5933C23semidirect
63. C22×#5935C23semidirect
64. C22×#5935C23semidirect
65. #11714184C23semidirect
66. #11714182C23semidirect
67. #11714175C23semidirect
68. #11714174C23semidirect
69. #11714171C23semidirect
70. #11714170C23semidirect
71. C2×#73028C23semidirect
72. C2×#73032C23semidirect
73. C2×#73027C23semidirect
74. #11714162C23semidirect
75. C2×#73031C23semidirect
76. C2×#73025C23semidirect
77. C2×#73030C23semidirect
78. C2×#73026C23semidirect
79. C2×#73026C23semidirect
80. C2×#73026C23semidirect
81. C2×#73026C23semidirect
82. C2×#73026C23semidirect
83. C2×#87547C23semidirect
84. C22×#5950C23semidirect
85. C23×#997C23semidirect
86. #60399⋊C3C23semidirect
87. C2×#3027⋊C3C23semidirect
88. C2×#3026⋊C3C23semidirect
89. C2×#3025⋊C3C23semidirect
90. C22×#458⋊C3C23semidirect
92. C22×B(4,𝔽2)⋊C3C23semidirect
93. C22×B(4,𝔽2)⋊C3C23semidirect
94. C22×H(7,𝔽2)C2×D3semidirect
95. C22×#3027C2×D3semidirect
96. C22×#3025C2×D3semidirect
97. C2×#60399C2×D3semidirect
98. C22×#3026C2×D3semidirect
99. C23×B(4,𝔽2)C2×D3semidirect
100. #87547C24semidirect
101. C2×#5950C24semidirect
103. C22×#997C24semidirect
104. #87522C24semidirect
105. #87521C24semidirect
106. #87520C24semidirect
107. #87519C24semidirect
108. #87516C24semidirect
109. #87515C24semidirect
111. C2×#5935C24semidirect
112. C2×#5935C24semidirect
114. C2×#5934C24semidirect
115. C2×#5934C24semidirect
116. C2×#5933C24semidirect
117. C2×#5932C24semidirect
118. #3027⋊C3C24semidirect
119. #3026⋊C3C24semidirect
120. #3025⋊C3C24semidirect
121. C2×#458⋊C3C24semidirect
123. C2×B(4,𝔽2)⋊C3C24semidirect
124. C2×B(4,𝔽2)⋊C3C24semidirect
125. #73032C24semidirect
126. #73031C24semidirect
127. #73030C24semidirect
128. #73028C24semidirect
129. #73027C24semidirect
130. #73026C24semidirect
131. #73026C24semidirect
132. #73026C24semidirect
133. #73026C24semidirect
134. #73025C24semidirect
135. C2×H(7,𝔽2)C22×D3semidirect
136. C22×#585C22×D3semidirect
137. C23×H(5,𝔽2)C22×D3semidirect
138. C25×D4S4semidirect
139. #60399C22×D3semidirect
140. C2×#3027C22×D3semidirect
141. C2×#3026C22×D3semidirect
142. C2×#3025C22×D3semidirect
143. C22×#458C22×D3semidirect
145. C22×B(4,𝔽2)C22×D3semidirect
146. C22×B(4,𝔽2)C22×D3semidirect
147. #5950C25semidirect
149. C2×#997C25semidirect
151. #5935C25semidirect
153. #5934C25semidirect
154. #5933C25semidirect
155. #5932C25semidirect
156. #458⋊C3C25semidirect
158. B(4,𝔽2)⋊C3C25semidirect
159. C27C2×S4semidirect
160. H(7,𝔽2)C23×D3semidirect
161. C2×#585C23×D3semidirect
163. C22×H(5,𝔽2)C23×D3semidirect
165. C24×D4C2×S4semidirect
166. C24×D4C2×S4semidirect
167. C24×D4C2×S4semidirect
168. C25×C4C2×S4semidirect
169. #3027C23×D3semidirect
170. #3026C23×D3semidirect
171. #3025C23×D3semidirect
172. C2×#458C23×D3semidirect
174. C2×B(4,𝔽2)C23×D3semidirect
175. C2×B(4,𝔽2)C23×D3semidirect
180. C26C22×S4semidirect
181. #585C24×D3semidirect
183. C2×H(5,𝔽2)C24×D3semidirect
185. C23×D4C22×S4semidirect
186. C23×D4C22×S4semidirect
187. C23×D4C22×S4semidirect
190. C24×C4C22×S4semidirect
191. #458C24×D3semidirect
193. B(4,𝔽2)C24×D3semidirect
198. C25C23×S4semidirect
202. C22×D4C23×S4semidirect
203. C22×D4C23×S4semidirect
206. C23×C4C23×S4semidirect
221. C22#72217027direct (N × H)
225. C2C2×#72217027direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#60399⋊C32
C23×#730262
C22×#117141752
C22×#117141712
C22×#117141822
C22×#117158282
C22×#117141742
C22×#117141622
C22×#117141842
C22×#117141702
C22×#117158392
C23×D4×S44
C22×#76228803
C2×#722170272