Finite Group Databasebeta

← #72262754

Subgroups — #72262754

Order 2048.

At least 6169 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

The subgroup lattice has 307 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72262682C211yesmaximal, characteristic
2#72262578C212yesmaximal, complemented
3#72262548C212yesmaximal, complemented
4#72262530C212yesmaximal, complemented
5#72262485C212yesmaximal, complemented
6#72238370C211yesmaximal, characteristic, complemented
7#72237989C211yesmaximal, characteristic, complemented
8#72237582C211yesmaximal, characteristic, complemented
9#72237033C211yesmaximal, characteristic, complemented
10#72234724C211yesmaximal, characteristic, complemented
11#72225985C211yesmaximal, characteristic, complemented
12#6374128C2211yescharacteristic, complemented
13#5118112C411yescharacteristic, complemented
14#5117977C411yescharacteristic, complemented
15#5082849C2211yescharacteristic
16#5082801C411yescharacteristic, complemented
17#4524223C2211yescharacteristic
18#4524032C411yescharacteristic, complemented
19#4444985C2211yescharacteristic
20#4444438C411yescharacteristic, complemented
21#4380237C2211yescharacteristic
22#4379699C2211yescharacteristic
23#2600715C2211yescharacteristic
24#2109240C411yescharacteristic, complemented
25#2032565C411yescharacteristic, complemented
26#648360C411yescharacteristic, complemented
27#616960C2211yescharacteristic
28#188122C2211yescharacteristic, complemented
29#188109C2211yescharacteristic, complemented
30#188109C2211yescharacteristic, complemented
31#139916C2212yescomplemented
32#139914C2212yescomplemented
33#139912C2212yes—
34#139912C2212yescomplemented
35#139717C2212yes—
36#139717C2212yescomplemented
37#139505C2212yescomplemented
38#139487C2212yescomplemented
39#135475C2212yescomplemented
40#133086C2212yescomplemented
41#119544C2212yescomplemented
42#112274C2212yescomplemented
43C22×#534C2311yescharacteristic
44C23×#120D411yescharacteristic
45C2×#2736C2×C411yescharacteristic, complemented
46C2×#2813D411yescharacteristic
47C2×#2846C2×C411yescharacteristic
48C2×#2846D411yescharacteristic
49C2×#2404D411yescharacteristic
50C2×#2431D411yescharacteristic
51C2×#2433D411yescharacteristic
52C2×#2403C2×C411yescharacteristic, complemented
53C2×#2615C2×C411yescharacteristic
54C2×#2661C2311yescharacteristic
55#21747C2312yes—
56#17392C2312yes—
57#17147C2×C411yescharacteristic, complemented
58#17105C2×C411yescharacteristic, complemented
59#16214C2×C411yescharacteristic, complemented
60#16208C2×C411yescharacteristic, complemented
61#15948C2311yescharacteristic
62#15948C2311yescharacteristic
63#15947D411yescharacteristic
64#15947D411yescharacteristic
65#15674C2×C411yescharacteristic
66#15668D411yescharacteristic
67#15668C2311yescharacteristic
68#15480C2×C411yescharacteristic, complemented
69#15467C2×C411yescharacteristic, complemented
70#15464D411yescharacteristic
71#15412D411yescharacteristic
72#15412C2311yescharacteristic
73#15340D411yescharacteristic
74#15310C2×C411yescharacteristic
75#15310D411yescharacteristic
76#15283C2×C411yescharacteristic
77#15267C2×C411yescharacteristic, complemented
78#15267C2×C411yescharacteristic, complemented
79#15261D411yescharacteristic
80#15259C2×C411yescharacteristic
81#15259D411yescharacteristic
82#15258C2×C411yescharacteristic
83#15258C2×C411yescharacteristic
84#15258D411yescharacteristic
85#15257C2×C411yescharacteristic
86#15257C2×C411yescharacteristic
87#15257D411yescharacteristic
88#15257D411yescharacteristic
89#14659D411yescharacteristic
90#14652C2×C411yescharacteristic, complemented
91#14649C2×C411yescharacteristic, complemented
92#14567C2×C411yescharacteristic, complemented
93#14533C2×C411yescharacteristic, complemented
94#14365C2×C411yescharacteristic, complemented
95#14335D411yescharacteristic
96#14290C2×C411yescharacteristic
97#14287C2311yescharacteristic
98#14255C2×C411yescharacteristic
99#14255D411yescharacteristic
100#14167D411yescharacteristic
101#13903D411yescharacteristic
102#13902D411yescharacteristic
103C22⋊C42C2×C411yescharacteristic, complemented
104#8553C2312yes—
105#8548C2312yes—
106C27C22⋊C411yescharacteristic
107C23×#41C22⋊C411yescharacteristic
108C24×D4C2×D411yescharacteristic
109C24×D4C2×D411yescharacteristic
110C22×#127C22⋊C411yescharacteristic
111C22×#127C22⋊C411yescharacteristic
112C22×#127C22⋊C411yescharacteristic
113C22×#124C22⋊C411yescharacteristic
114C22×#123C2×D411yescharacteristic
115C22×#123C2×D411yescharacteristic
116C22×#123C2×D411yescharacteristic
117C22×#123C2411yescharacteristic
118C22×#121C22×C411yescharacteristic
119C22×#121C2×D411yescharacteristic
120C22×#121C2×D411yescharacteristic
121C22×#121C2×D411yescharacteristic
122C22×#121C2×D411yescharacteristic
123C22×#120C22⋊C411yescharacteristic
124C22×#120C22⋊C411yescharacteristic
125C22×#120C22⋊C411yescharacteristic
126C22×#120C22⋊C411yescharacteristic
127C22×#120C22⋊C411yescharacteristic
128C22×#120C22⋊C411yescharacteristic
129C22×#120C22×C411yescharacteristic
130C22×#120C2×D411yescharacteristic
131C22×#120C2×D411yescharacteristic
132C22×#120C2×D411yescharacteristic
133C22×#120C2×D411yescharacteristic
134C22×#120C2×D411yescharacteristic
135C23×C22⋊C4C22⋊C411yescharacteristic
136C23×C22⋊C4C2×D411yescharacteristic
137C2×#400C22⋊C411yescharacteristic
138C2×#399C22⋊C411yescharacteristic
139C2×#398C2×D411yescharacteristic
140C2×#397C22×C411yescharacteristic
141C2×#394C22⋊C411yescharacteristic
142C2×#394C2×D411yescharacteristic
143C2×#393C22⋊C411yescharacteristic
144C2×#393C22⋊C411yescharacteristic
145C2×#393C22×C411yescharacteristic
146C2×#393C2×D411yescharacteristic
147C2×#392C22⋊C411yescharacteristic
148C2×#392C22⋊C411yescharacteristic
149C2×#392C22⋊C411yescharacteristic
150C2×#392C22×C411yescharacteristic
151C2×#392C22×C411yescharacteristic
152C2×#392C2×D411yescharacteristic
153C2×#391C2×D411yescharacteristic
154C2×#390C22⋊C411yescharacteristic
155C2×#390C22⋊C411yescharacteristic
156C2×#390C2×D411yescharacteristic
157C2×#381C22⋊C411yescharacteristic
158C2×#380C22⋊C411yescharacteristic
159C2×#380C2×D411yescharacteristic
160C2×#379C22⋊C411yescharacteristic
161C2×#379C22⋊C411yescharacteristic
162C2×#379C22×C411yescharacteristic
163C2×#379C22×C411yescharacteristic
164C2×#379C2×D411yescharacteristic
165C2×#379C2×D411yescharacteristic
166C2×C4×C22⋊C4C22⋊C411yescharacteristic
167C2×C4×C22⋊C4C2×D411yescharacteristic
168C2×C4×C22⋊C4C2×D411yescharacteristic
169#1445C2×D411yescharacteristic
170#1445C2×D411yescharacteristic
171#1440C22×C411yescharacteristic
172#1440C22×C411yescharacteristic
173#1440C2×D411yescharacteristic
174#1440C2×D411yescharacteristic
175#1438C22⋊C411yescharacteristic
176#1438C22⋊C411yescharacteristic
177#1438C22⋊C411yescharacteristic
178#1438C22⋊C411yescharacteristic
179#1438C22⋊C411yescharacteristic
180#1438C22⋊C411yescharacteristic
181#1438C22⋊C411yescharacteristic
182#1438C22⋊C411yescharacteristic
183#1438C22⋊C411yescharacteristic
184#1438C22⋊C411yescharacteristic
185#1438C22⋊C411yescharacteristic
186#1438C22⋊C411yescharacteristic
187#1438C2×D411yescharacteristic
188#1438C2×D411yescharacteristic
189#1437C22⋊C411yescharacteristic
190#1437C22⋊C411yescharacteristic
191#1437C22⋊C411yescharacteristic
192#1437C22⋊C411yescharacteristic
193#1437C22⋊C411yescharacteristic
194#1437C22⋊C411yescharacteristic
195#1437C22×C411yescharacteristic
196#1437C22×C411yescharacteristic
197#1437C22×C411yescharacteristic
198#1437C22×C411yescharacteristic
199#1437C2×D411yescharacteristic
200#1437C2×D411yescharacteristic
201#1436C22⋊C411yescharacteristic
202#1436C22⋊C411yescharacteristic
203#1436C2×D411yescharacteristic
204#1436C2×D411yescharacteristic
205C26C2×C22⋊C411yescharacteristic
206C26C2×C22⋊C411yescharacteristic
207C26C2×C22⋊C411yescharacteristic
208C26#12011yescharacteristic
209C26#12011yescharacteristic
210C23×Q8C2×C22⋊C411yescharacteristic
211C23×D4C2×C22⋊C411yescharacteristic
212C23×D4C2×C22⋊C411yescharacteristic
213C23×D4C2×C22⋊C411yescharacteristic
214C23×D4C2×C22⋊C411yescharacteristic
215C23×D4C2×C22⋊C411yescharacteristic
216C23×D4C2×C22⋊C411yescharacteristic
217C23×D4C2×C22⋊C411yescharacteristic
218C23×D4C2×C22⋊C411yescharacteristic
219C23×D4#12011yescharacteristic
220C23×D4#12011yescharacteristic
221C23×D4#12011yescharacteristic
222C23×D4#12011yescharacteristic
223C23×D4#12011yescharacteristic
224C23×D4C22×D411yescharacteristic
225C24×C4C2×C22⋊C411yescharacteristic
226C24×C4#12011yescharacteristic
227C24×C4#12011yescharacteristic
228C24×C4C22×D411yescharacteristic
229C22×C4⋊C4C2×C22⋊C411yescharacteristic
230C22×C4⋊C4C2×C22⋊C411yescharacteristic
231C22×C4⋊C4#12011yescharacteristic
232C22×C4⋊C4C22×D411yescharacteristic
233C22×C4⋊C4C22×D411yescharacteristic
234C22×C22⋊C4C2×C22⋊C411yescharacteristic
235C22×C22⋊C4C2×C22⋊C411yescharacteristic
236C22×C22⋊C4C2×C22⋊C411yescharacteristic
237C22×C22⋊C4C2×C22⋊C411yescharacteristic
238C22×C22⋊C4C2×C22⋊C411yescharacteristic
239C22×C22⋊C4C2×C22⋊C411yescharacteristic
240C22×C22⋊C4C2×C22⋊C411yescharacteristic
241C22×C22⋊C4C2×C22⋊C411yescharacteristic
242C22×C22⋊C4C2×C22⋊C411yescharacteristic
243C22×C22⋊C4#12011yescharacteristic
244C22×C22⋊C4#12011yescharacteristic
245C22×C22⋊C4#12011yescharacteristic
246C22×C22⋊C4#12011yescharacteristic
247C22×C22⋊C4#12011yescharacteristic
248C22×C22⋊C4#12011yescharacteristic
249C22×C22⋊C4C23×C411yescharacteristic
250C22×C22⋊C4C22×D411yescharacteristic
251C22×C22⋊C4C22×D411yescharacteristic
252C22×C42C2×C22⋊C411yescharacteristic
253C22×C42C2×C22⋊C411yescharacteristic
254C22×C42C2×C22⋊C411yescharacteristic
255C22×C42C2×C22⋊C411yescharacteristic
256C2×#95C2×C22⋊C411yescharacteristic
257C2×#95C2×C22⋊C411yescharacteristic
258C2×#95C2×C22⋊C411yescharacteristic
259C2×#95C2×C22⋊C411yescharacteristic
260C2×#95C2×C22⋊C411yescharacteristic
261C2×#95C2×C22⋊C411yescharacteristic
262C2×#95C2×C22⋊C411yescharacteristic
263C2×#95C2×C22⋊C411yescharacteristic
264C25#37911yescharacteristic
265C25#37911yescharacteristic
266C25#37911yescharacteristic
267C25#37911yescharacteristic
268C25#37911yescharacteristic
269C25B(4,𝔽2)11yescharacteristic
270C25B(4,𝔽2)11yescharacteristic
271C25#45811yescharacteristic
272C25#45811yescharacteristic
273C25C22×C22⋊C411yescharacteristic
274C25C2×#12011yescharacteristic
275C23×C4#37911yescharacteristic
276C23×C4#37911yescharacteristic
277C23×C4#37911yescharacteristic
278C23×C4C22×C22⋊C411yescharacteristic
279C23×C4C22×C22⋊C411yescharacteristic
280C23×C4C2×#12011yescharacteristic
281C23×C4C2×#12011yescharacteristic
282C23×C4C2×#12011yescharacteristic
283C24#179011yescharacteristic
284C24#179011yescharacteristic
285C24#179011yescharacteristic
286C24#179011yescharacteristic
287C24C2×#37911yescharacteristic
288C24C2×B(4,𝔽2)11yescharacteristic
289C24C2×#45811yescharacteristic
290C22×C4#288312yes—
291C22×C4#302612yes—
292C23#1275312yes—
293C23#2058511yescharacteristic
294C23C2×#179011yescharacteristic
295C23#2060911yescharacteristic
296C22#18793411yescharacteristic
297C22#49349211yescharacteristic
298C22#49350411yescharacteristic
299C22C2×#1275311yescharacteristic
300C22#49359411yescharacteristic
301C22#49364111yescharacteristic
302C22#49364711yescharacteristic
303C2#7224942611yescharacteristic
304C2#7226272411yescharacteristic
305C2#7226272511yescharacteristic
subgroupG/H
center, socleC22#187934
commutatorC22×C22⋊C4C23×C4
Frattini, cosocleC22×#123C24
prime radical#72262682C2
LCSG ▷ C22×C22⋊C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#534 ◁ G
LECSG ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #72262578C2semidirect
3. #72262548C2semidirect
4. #72262530C2semidirect
5. #72262485C2semidirect
6. #72238370C2semidirect
7. #72237989C2semidirect
8. #72237582C2semidirect
9. #72237033C2semidirect
10. #72234724C2semidirect
11. #72225985C2semidirect
12. #6374128C22semidirect
13. #5118112C4semidirect
14. #5117977C4semidirect
16. #5082801C4semidirect
18. #4524032C4semidirect
20. #4444438C4semidirect
24. #2109240C4semidirect
25. #2032565C4semidirect
26. #648360C4semidirect
28. #188122C22semidirect
29. #188109C22semidirect
30. #188109C22semidirect
31. #139916C22semidirect
32. #139914C22semidirect
34. #139912C22semidirect
36. #139717C22semidirect
37. #139505C22semidirect
38. #139487C22semidirect
39. #135475C22semidirect
40. #133086C22semidirect
41. #119544C22semidirect
42. #112274C22semidirect
45. C2×#2736C2×C4semidirect
52. C2×#2403C2×C4semidirect
57. #17147C2×C4semidirect
58. #17105C2×C4semidirect
59. #16214C2×C4semidirect
60. #16208C2×C4semidirect
68. #15480C2×C4semidirect
69. #15467C2×C4semidirect
77. #15267C2×C4semidirect
78. #15267C2×C4semidirect
90. #14652C2×C4semidirect
91. #14649C2×C4semidirect
92. #14567C2×C4semidirect
93. #14533C2×C4semidirect
94. #14365C2×C4semidirect
103. C22⋊C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
#722626822
#722625782
#722625482
#722625302
#722624852
#722383702
#722379892
#722375822
#722370332
#722347242
#722259852