Finite Group Databasebeta

← #72240622

Subgroups — #72240622

Order 4096.

Only the 5179 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 316 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72210024C213yesmaximal, complemented
2C2×#8093108C2211yescharacteristic, complemented
3#72240547C2124yesmaximal, complemented
4#72240546C213yesmaximal, complemented
5C2×#357285C2211yescharacteristic, complemented
6#72240536C216yesmaximal, complemented
7C2×#391709C2213yescomplemented
8C2×#391709C2216yescomplemented
9#72240531C2213yescomplemented
10#72240529C2212yescomplemented
11#72240527C22124yescomplemented
12#72240520C2112yesmaximal, complemented
13C2×#417896C2216yescomplemented
14C2×#417896C22112yescomplemented
15C2×#417969C22112yescomplemented
16C2×#417969C22112yescomplemented
17C2×#417969C22112yescomplemented
18C2×#432839C22112yescomplemented
19#72240512C22124yescomplemented
20#72240512C22124yescomplemented
21#72240511C22112yescomplemented
22#72240511C22112yescomplemented
23#72240510C22124yescomplemented
24#72240509C22112yescomplemented
25#72240508C22124yescomplemented
26#72240508C22124yescomplemented
27C2×#72210025C213yesmaximal, complemented
28C2×#392135C22124yescomplemented
29C2×#392135C2213yescomplemented
30C2×#392135C2216yescomplemented
31C2×#392135C22112yescomplemented
32C2×H(9,𝔽2)C2211yescharacteristic, complemented
33#72240412C2112yesmaximal, complemented
34#72240407C22112yescomplemented
35#72240406C22112yescomplemented
36#72240406C22112yescomplemented
37#72240404C22124yescomplemented
38#72240404C22124yescomplemented
39#72240400C22124yescomplemented
40#72240398C22124yescomplemented
41#72240398C22124yescomplemented
42#72240193C22124yescomplemented
43#72240193C22124yescomplemented
44#72240192C22124yescomplemented
45#72240192C22124yescomplemented
46#72239046C2218yescomplemented
47#72238873C2214yescomplemented
48C2×#6628896C2213yes—
49C2×#6628896C22112yes—
50C2×#6628896C22124yes—
51C2×#6628896C2216yes—
52#72219166C2214yescomplemented
53C2×#449475C22112yescomplemented
54C2×#449475C2216yescomplemented
55C2×#449475C22112yescomplemented
56C22×H(7,𝔽2)C23112yes—
57C22×H(7,𝔽2)C2313yes—
58C23×B(4,𝔽2)C23112yes—
59C23×B(4,𝔽2)C23124yes—
60C23×B(4,𝔽2)C23112yes—
61C23×B(4,𝔽2)C23112yes—
62C22×#2846C23112yescomplemented
63C22×#2846C2313yescomplemented
64C2×#35367C2313yescomplemented
65C2×#35367C2316yescomplemented
66C2×#35346C2316yes—
67C2×#35346C2313yes—
68C2×#35346C23112yes—
69C2×#35346C23112yes—
70C2×#35346C23112yes—
71#6628896D4148yescomplemented
72C22×#2032C23124yescomplemented
73C2×#23468C23112yescomplemented
74C2×#20609C23112yes—
75C2×#20609C23124yes—
76C2×#20609C23124yes—
77C2×#20609C23124yes—
78C2×#20609C23124yes—
79C2×#20609C23124yes—
80C2×#20609C23124yes—
81C2×#20609C23112yes—
82C2×#20609C23112yes—
83C2×#20585C2316yes—
84C2×#20585C2313yes—
85C2×#20585C23112yes—
86C2×#20585C23112yes—
87C2×#20585C23112yes—
88C2×#20585C23124yes—
89C2×#20585C23124yes—
90C2×#20585C23124yes—
91C2×#20585C23112yes—
92C2×#20585C2316yes—
93#5955309C23124yescomplemented
94#515326C23112yescomplemented
95C2×#11172C23124yescomplemented
96C2×#11172C23124yescomplemented
97C2×#11172C23112yescomplemented
98C2×#11172C23112yescomplemented
99C2×#11172C23112yescomplemented
100C2×#11169C23124yescomplemented
101C2×#11169C23112yescomplemented
102C2×#11169C2316yescomplemented
103C2×#11089C23124yescomplemented
104C2×#11089C23124yescomplemented
105C2×#11089C23124yescomplemented
106C2×#11089C23124yescomplemented
107#357642C23112yescomplemented
108#357642C23124yescomplemented
109#357642C23112yescomplemented
110C2×#10991C23124yescomplemented
111C2×#10991C23112yescomplemented
112C2×#10991C23124yescomplemented
113C2×#10991C23124yescomplemented
114C2×#10991C23124yescomplemented
115C2×#10991C23124yescomplemented
116C2×#10991C23112yescomplemented
117C2×#10991C23124yescomplemented
118C2×#10991C23112yescomplemented
119C2×#10965C23112yescomplemented
120C2×#10965C23112yescomplemented
121C2×#10965C23124yescomplemented
122C2×#10965C23124yescomplemented
123C2×#10965C2316yescomplemented
124C2×#10965C23112yescomplemented
125C2×#10938C23112yescomplemented
126C2×#10938C23112yescomplemented
127C2×#10938C23124yescomplemented
128C2×#10938C23112yescomplemented
129C2×#10938C23124yescomplemented
130#123581C23124yescomplemented
131#123581C23124yescomplemented
132#123581C23124yescomplemented
133#123581C23124yescomplemented
134#123539C23124yescomplemented
135#123539C23124yescomplemented
136#123539C23124yescomplemented
137#123539C23124yescomplemented
138#123527C23124yescomplemented
139#123527C23124yescomplemented
140C28C2411yescharacteristic, complemented
141C2×H(7,𝔽2)C2×D4124yes—
142C23×H(5,𝔽2)C2414yes—
143C23×H(5,𝔽2)C2411yescharacteristic
144C23×H(5,𝔽2)C2416yes—
145C23×H(5,𝔽2)C24112yes—
146C23×H(5,𝔽2)C24112yes—
147C22×#543C2413yescomplemented
148C23×#120C24124yes—
149C23×#120C2413yes—
150C23×#120C2416yes—
151C23×#120C24112yes—
152C22×B(4,𝔽2)C2×D4196yes—
153C23×#99C24112yes—
154C23×#99C24124yes—
155C23×#99C24124yes—
156C23×#99C24124yes—
157C23×#99C24112yes—
158C23×#99C24112yes—
159C23×#99C24124yes—
160C23×#99C24124yes—
161C23×#99C24124yes—
162#35346C2×D4124yes—
163#35346C2×D4148yes—
164C2×#2315C24112yescomplemented
165C2×#2308C2413yes—
166C2×#2308C2416yes—
167C2×#2308C2416yes—
168C2×#2308C24112yes—
169C2×#2308C24112yes—
170C2×#2308C24112yes—
171C2×#2308C24124yes—
172C2×#2308C24124yes—
173C2×#2308C24112yes—
174C2×#2308C24124yes—
175C22×#379C2416yescomplemented
176C22×#379C2413yescomplemented
177C22×#379C24112yescomplemented
178C22×#379C24124yescomplemented
179C2×#2273C2411yescharacteristic, complemented
180C2×#2273C2412yescomplemented
181C2×#2273C2414yescomplemented
182#20609C2×D4196yes—
183#20585C2×D4124yes—
184#20585C2×D4148yes—
185C22×#342C24112yescomplemented
186C22×#342C24124yescomplemented
187C22×#342C24124yescomplemented
188C22×#342C24124yescomplemented
189C22×#342C24124yescomplemented
190C22×#342C24124yescomplemented
191C22×#342C24124yescomplemented
192C22×#342C24112yescomplemented
193C22×#342C24112yescomplemented
194#13794C2418yescomplemented
195C27C2513yes—
196C22×H(5,𝔽2)C22×D4112yes—
197C22×H(5,𝔽2)C22×D4124yes—
198C22×H(5,𝔽2)C22×D4148yes—
199C22×H(5,𝔽2)H(5,𝔽2)14yes—
200C22×H(5,𝔽2)H(5,𝔽2)148yes—
201C23×#41H(5,𝔽2)16yes—
202C23×#41H(5,𝔽2)124yes—
203C24×D4H(5,𝔽2)16yes—
204C24×D4H(5,𝔽2)124yes—
205C24×D4C2513yes—
206C24×D4C25112yes—
207C22×#120C22×D4112yes—
208C22×#120C22×D4124yes—
209C22×#120C22×D4148yes—
210C23×C22⋊C4C25112yes—
211C23×C22⋊C4C2513yes—
212C23×C22⋊C4C2516yes—
213C23×C22⋊C4C25124yes—
214C22×#99C22×D4148yes—
215C22×#99C22×D4196yes—
216#2308C22×D4112yes—
217#2308C22×D4124yes—
218#2308C22×D4124yes—
219#2308C22×D4148yes—
220C26H(3,𝔽4)12yes—
221C26H(3,𝔽4)18yescomplemented
222C26C23×D416yes—
223C26C2×H(5,𝔽2)13yes—
224C26C2×H(5,𝔽2)112yes—
225C26C2611yescharacteristic
226C22×#41#534124yes—
227C22×#41D42124yes—
228C23×Q8H(3,𝔽4)12yes—
229C23×Q8H(3,𝔽4)18yescomplemented
230C23×D4#534124yes—
231C23×D4D42124yes—
232C23×D4H(3,𝔽4)16yes—
233C23×D4H(3,𝔽4)124yescomplemented
234C23×D4C23×D416yes—
235C23×D4C23×D4112yes—
236C23×D4C23×D4124yes—
237C23×D4C2×H(5,𝔽2)16yes—
238C23×D4C2×H(5,𝔽2)124yes—
239C23×D4C2×H(5,𝔽2)112yes—
240C23×D4C2×H(5,𝔽2)148yes—
241C23×D4C2×H(5,𝔽2)148yes—
242C24×C4H(3,𝔽4)16yes—
243C24×C4H(3,𝔽4)124yescomplemented
244C24×C4C2×H(5,𝔽2)13yes—
245C24×C4C2×H(5,𝔽2)112yes—
246C22×C22⋊C4C23×D416yes—
247C22×C22⋊C4C23×D4124yes—
248C22×C22⋊C4C23×D4112yes—
249C25#267918yescomplemented
250C25C2×#53416yes—
251C25C2×D4216yes—
252C25#3484124yes—
253C25C2×H(3,𝔽4)124yes—
254C25C2×H(3,𝔽4)16yes—
255C25#3532124yes—
256C25C24×D413yes—
257C25C22×H(5,𝔽2)112yes—
258C25C22×H(5,𝔽2)112yes—
259C25C22×H(5,𝔽2)11yescharacteristic
260C25C22×H(5,𝔽2)13yes—
261C22×Q8#267918yescomplemented
262C22×D4#2679124yescomplemented
263C22×D4C2×#534124yes—
264C22×D4C2×D42124yes—
265C22×D4#3484196yes—
266C22×D4#3532196yes—
267C23×C4#2679124yescomplemented
268C23×C4C2×#53416yes—
269C23×C4C2×D4216yes—
270C23×C4#3484124yes—
271C23×C4C2×H(3,𝔽4)124yes—
272C23×C4C2×H(3,𝔽4)16yes—
273C23×C4#3532124yes—
274C24#45439148yes—
275C24C2×#2679112yes—
276C24C22×#53413yes—
277C24C22×H(3,𝔽4)11yescharacteristic
278C24C22×H(3,𝔽4)14yes—
279C24C22×D4213yes—
280C24C2×#3532112yes—
281C24C2×#3532124yes—
282C24C2×#3484112yes—
283C24C2×#3484124yes—
284C24#6281716yes—
285C24#62817112yes—
286C24#62817112yes—
287C24#62817112yes—
288C24#6281716yes—
289C24#62817112yes—
290C24#62817112yes—
291C22×C4#45439148yes—
292C22×C4C2×#2679112yes—
293C23C22×#267911yescharacteristic
294C23#764224416yes—
295C23#7642244112yes—
296C23#7670203112yes—
297C23#767020316yes—
298C23C2×#45439112yes—
299C23#7708349112yes—
300C23#7708349124yes—
301C23#809310813yes—
302C23#809310813yes—
303C23#8093108112yes—
304C23#809310814yes—
305C23#8108057124yes—
306C23#8108057112yes—
307C23#8108057112yes—
308C22#7220969911yescharacteristic
309C22#72210056112yes—
310C22#72210057112yes—
311C22#7221916413yes—
312C22#7221916613yes—
313C2#7221003012yes—
314C2#7221916811yescharacteristic
subgroupG/H
center, socleC22#72209699
commutator, Frattini, cosocleC26C26
LCSG ▷ C26 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C26 ◁ G
LECSG ▷ C26 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#72210024C2semidirect
2. C2×#8093108C22semidirect
3. #72240547C2semidirect
4. #72240546C2semidirect
5. C2×#357285C22semidirect
6. #72240536C2semidirect
7. C2×#391709C22semidirect
8. C2×#391709C22semidirect
9. #72240531C22semidirect
10. #72240529C22semidirect
11. #72240527C22semidirect
12. #72240520C2semidirect
13. C2×#417896C22semidirect
14. C2×#417896C22semidirect
15. C2×#417969C22semidirect
16. C2×#417969C22semidirect
17. C2×#417969C22semidirect
18. C2×#432839C22semidirect
19. #72240512C22semidirect
20. #72240512C22semidirect
21. #72240511C22semidirect
22. #72240511C22semidirect
23. #72240510C22semidirect
24. #72240509C22semidirect
25. #72240508C22semidirect
26. #72240508C22semidirect
27. C2×#72210025C2semidirect
28. C2×#392135C22semidirect
29. C2×#392135C22semidirect
30. C2×#392135C22semidirect
31. C2×#392135C22semidirect
32. C2×H(9,𝔽2)C22semidirect
33. #72240412C2semidirect
34. #72240407C22semidirect
35. #72240406C22semidirect
36. #72240406C22semidirect
37. #72240404C22semidirect
38. #72240404C22semidirect
39. #72240400C22semidirect
40. #72240398C22semidirect
41. #72240398C22semidirect
42. #72240193C22semidirect
43. #72240193C22semidirect
44. #72240192C22semidirect
45. #72240192C22semidirect
46. #72239046C22semidirect
47. #72238873C22semidirect
52. #72219166C22semidirect
53. C2×#449475C22semidirect
54. C2×#449475C22semidirect
55. C2×#449475C22semidirect
62. C22×#2846C23semidirect
63. C22×#2846C23semidirect
64. C2×#35367C23semidirect
65. C2×#35367C23semidirect
71. #6628896D4semidirect
72. C22×#2032C23semidirect
73. C2×#23468C23semidirect
93. #5955309C23semidirect
94. #515326C23semidirect
95. C2×#11172C23semidirect
96. C2×#11172C23semidirect
97. C2×#11172C23semidirect
98. C2×#11172C23semidirect
99. C2×#11172C23semidirect
100. C2×#11169C23semidirect
101. C2×#11169C23semidirect
102. C2×#11169C23semidirect
103. C2×#11089C23semidirect
104. C2×#11089C23semidirect
105. C2×#11089C23semidirect
106. C2×#11089C23semidirect
107. #357642C23semidirect
108. #357642C23semidirect
109. #357642C23semidirect
110. C2×#10991C23semidirect
111. C2×#10991C23semidirect
112. C2×#10991C23semidirect
113. C2×#10991C23semidirect
114. C2×#10991C23semidirect
115. C2×#10991C23semidirect
116. C2×#10991C23semidirect
117. C2×#10991C23semidirect
118. C2×#10991C23semidirect
119. C2×#10965C23semidirect
120. C2×#10965C23semidirect
121. C2×#10965C23semidirect
122. C2×#10965C23semidirect
123. C2×#10965C23semidirect
124. C2×#10965C23semidirect
125. C2×#10938C23semidirect
126. C2×#10938C23semidirect
127. C2×#10938C23semidirect
128. C2×#10938C23semidirect
129. C2×#10938C23semidirect
130. #123581C23semidirect
131. #123581C23semidirect
132. #123581C23semidirect
133. #123581C23semidirect
134. #123539C23semidirect
135. #123539C23semidirect
136. #123539C23semidirect
137. #123539C23semidirect
138. #123527C23semidirect
139. #123527C23semidirect
140. C28C24semidirect
147. C22×#543C24semidirect
164. C2×#2315C24semidirect
175. C22×#379C24semidirect
176. C22×#379C24semidirect
177. C22×#379C24semidirect
178. C22×#379C24semidirect
179. C2×#2273C24semidirect
180. C2×#2273C24semidirect
181. C2×#2273C24semidirect
185. C22×#342C24semidirect
186. C22×#342C24semidirect
187. C22×#342C24semidirect
188. C22×#342C24semidirect
189. C22×#342C24semidirect
190. C22×#342C24semidirect
191. C22×#342C24semidirect
192. C22×#342C24semidirect
193. C22×#342C24semidirect
194. #13794C24semidirect
221. C26H(3,𝔽4)semidirect
229. C23×Q8H(3,𝔽4)semidirect
233. C23×D4H(3,𝔽4)semidirect
243. C24×C4H(3,𝔽4)semidirect
249. C25#2679semidirect
261. C22×Q8#2679semidirect
262. C22×D4#2679semidirect
267. C23×C4#2679semidirect

Maximal subgroups

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

subgroupindex
C2×#722100242
#722405472
#722405462
#722405362
#722405202
C2×#722100252
#722404122