Finite Group Databasebeta

← #37545322

Subgroups — #37545322

Order 23168.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153840C212yesmaximal, complemented
2#26041933C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic, complemented
4#26041652C211yesmaximal, characteristic, complemented
5#26041526C212yesmaximal, complemented
6#26041484C211yesmaximal, characteristic, complemented
7#26041474C211yesmaximal, characteristic, complemented
8#26041431C212yesmaximal, complemented
9#26041425C212yesmaximal, complemented
10#26041358C211yesmaximal, characteristic, complemented
11C4×D8×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2212yescomplemented
13C22×D724—21no—
14C2×D4×D181C2212yes—
15#19153840—28no—
16C2×#15668438C2212yescomplemented
17C2×#15668438C2212yescomplemented
18C2×#15668437C2212yes—
19C2×#15668434C2212yescomplemented
20C2×#15668432C2212yescomplemented
21#19153816C2211yescharacteristic, complemented
22#19153813C2211yescharacteristic
23#19153800—22no—
24#19153796C2212yescomplemented
25#19153791C2212yes—
26#19153790—21no—
27#19153790—21no—
28#19153786—22no—
29C4×D724C2211yescharacteristic
30C2×C4⋊Dic181—21no—
31C4⋊#13089678C2211yescharacteristic, complemented
32#19153739C2212yescomplemented
33#19153736C2212yescomplemented
34#19153728C2212yescomplemented
35C2×D8×C181C2211yescharacteristic
36C4×D4×C181C2211yescharacteristic
37C4×D4×C181C2211yescharacteristic, complemented
38#107×C181C2211yescharacteristic, complemented
39#102×C181C2211yescharacteristic
40#102×C181C2211yescharacteristic, complemented
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668448—24no—
44#15668448—42no—
45C2×Dic362C2312yes—
46C2×D724—22no—
47C2×D724—21no—
48C2×D724—21no—
49C2×D724—44no—
50C2×D724D411yescharacteristic
51C2×D724C2312yes—
52C2×C22⋊D181—22no—
53C2×C22⋊D181—22no—
54D4×D181—24no—
55D4×D181—42no—
56C22×Dic181—22no—
57C2×C4×D181C2312yes—
58#15668438—24no—
59#15668438—42no—
60#15668437—24no—
61#15668434—24no—
62#15668432—24no—
63C2×#13089678C2312yes—
64C4⋊Dic181—22no—
65C4⋊Dic181—21no—
66C4⋊Dic181—21no—
67C4⋊Dic181D411yescharacteristic
68#15668426—22no—
69#15668426—22no—
70#15668426D412yes—
71C2×D4×C181C2311yescharacteristic
72C2×D4×C181C2311yescharacteristic
73C22×C4×C181—21no—
74C22×C4×C181—21no—
75D8×C181D414yescomplemented
76C2×C8×C181C2311yescharacteristic
77C2×C8×C181C2311yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C4⋊C4×C181C2311yescharacteristic
80C22⋊C4×C181—21no—
81C22⋊C4×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—22no—
84C22×D181—44no—
85C22×D181—44no—
86C22×D181—82no—
87C22×D181C2×D412yes—
88Dic362—22no—
89Dic362—22no—
90D724—22no—
91D724—22no—
92D724—22no—
93D724—44no—
94D724—42no—
95D724—81no—
96C22⋊D181—44no—
97C22⋊D181—44no—
98C2×Dic181—22no—
99C2×Dic181—44no—
100C2×Dic181C2×D412yes—
101C4×D181—22no—
102C4×D181—22no—
103#13089678—22no—
104C23×C181—21no—
105C23×C181—21no—
106D4×C181—21no—
107D4×C181—21no—
108D4×C181C2×D412yes—
109D4×C181C2×D412yescomplemented
110C2×C4×C181—21no—
111C2×C4×C181—21no—
112C2×C4×C181—42no—
113C2×C4×C181—42no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2411yescharacteristic
117C8×C181—21no—
118C8×C181C2×D412yescomplemented
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—41no—
122C22×C181—41no—
123C22×C181—41no—
124C22×C181—41no—
125C22×C181C22×D411yescharacteristic
126C2×D181—22no—
127C2×D181—22no—
128C2×D181—22no—
129C2×D181—42no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—88no—
133C4×C181—21no—
134C4×C181—41no—
135C4×C181—41no—
136C4×C181AGL(1,ℤ/8)12yescomplemented
137C4×C181C22×D411yescharacteristic
138C4×C181H(5,𝔽2)11yescharacteristic
139Dic181—22no—
140Dic181—42no—
141C2×C181—42no—
142C2×C181—42no—
143C2×C181D4211yescharacteristic
144C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C181#57711yescharacteristic
146D181—42no—
147D181—82no—
148C181#327211yescharacteristic, complemented, Sylow
149#3272—1811nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1812no—
151#547—1811no—
152D42—1811no—
153#492—1811no—
154#468—1812no—
155#461—1811no—
156#459—1811no—
157#450—1812no—
158#449—1812no—
159C4×D8D18111yescharacteristic, complemented
160#436—1811no—
161C2×#41—1812no—
162C22×D4—3621no—
163C22×D4—1812no—
164AGL(1,ℤ/8)—3628no—
165C2×SD16—1812no—
166C2×SD16—1812no—
167C2×SD16—1812no—
168C2×D8—1812no—
169C2×D8C2×D18111yescharacteristic
170C2×M16—1812no—
171C4⋊Q8—1811no—
172#127—1811no—
173#123—3622no—
174C22⋊Q8—1812no—
175#121—3621no—
176#121—3621no—
177#121—1812no—
178#120—3622no—
179C4×D4—1811no—
180C4×D4C2×D18111yescharacteristic
181C4×D4C2×D18111yescharacteristic, complemented
182C2×C4⋊C4—3621no—
183#107C2×D18111yescharacteristic, complemented
184C4⋊C8—1811no—
185#103—1812no—
186#102—1812no—
187#102C2×D18111yescharacteristic
188#102C2×D18111yescharacteristic, complemented
189C22⋊C8—1812no—
190C4×C8C2×D18111yescharacteristic, complemented
191C24—3622no—
192#41—3624no—
193#41—7242no—
194C2×Q8—1812no—
195C2×D4—3622no—
196C2×D4—3624no—
197C2×D4—3622no—
198C2×D4—3622no—
199C2×D4—3621no—
200C2×D4—3621no—
201C2×D4—1811no—
202C2×D4—1812no—
203C2×D4—7244no—
204C2×D4—7242no—
205C2×D4C22×D18111yescharacteristic
206C2×D4C22×D18111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3622no—
210C22×C4—1812no—
211SD16—3624no—
212SD16—3624no—
213SD16—7242no—
214D8—3624no—
215D8D72414yescomplemented
216M16—3624no—
217C2×C8—1812no—
218C2×C8C22×D18111yescharacteristic
219C2×C8C22×D18111yescharacteristic
220C4⋊C4—3622no—
221C4⋊C4—3621no—
222C4⋊C4—3621no—
223C4⋊C4—1811no—
224C4⋊C4C22×D18111yescharacteristic
225C4⋊C4C22×D18111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3622no—
229C22⋊C4—3622no—
230C22⋊C4—1812no—
231C42C22×D18111yescharacteristic
232C23—21no—
233C23—21no—
234C23—3622no—
235C23—14482no—
236C23—1812no—
237C23—7244no—
238C23—7244no—
239Q8—3622no—
240Q8—3622no—
241D4—21no—
242D4—21no—
243D4—3622no—
244D4—3622no—
245D4—3622no—
246D4—14481no—
247D4—7244no—
248D4—7244no—
249D4—7244no—
250D4—7242no—
251D4C2×D72412yes—
252D4C2×D72412yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3622no—
258C2×C4—3622no—
259C2×C4—3622no—
260C2×C4—1812no—
261C2×C4—7244no—
262C2×C4D4×D18111yescharacteristic
263C2×C4D4×D18111yescharacteristic
264C2×C4C23×D18111yescharacteristic
265C8—21no—
266C8—3622no—
267C8C2×D72412yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3622no—
275C22—3622no—
276C22—3622no—
277C22—14488no—
278C22—7242no—
279C22—7242no—
280C22—7242no—
281C22C2×D4×D18111yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3622no—
286C4—7242no—
287C4#1915384012yescomplemented
288C4C22×D72411yescharacteristic
289C4#1915386211yescharacteristic
290C2—42no—
291C2—42no—
292C2—14482no—
293C2—7242no—
294C2D4×D72411yescharacteristic
295C2C2×#1915384011yescharacteristic
296C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×D8×C181C2
socleC22×C181C22×D4
hypercenterC4×D8D181
hypocenterC181#3272
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. #26041933C2semidirect
3. D4×D724C2semidirect
4. #26041652C2semidirect
5. #26041526C2semidirect
6. #26041484C2semidirect
7. #26041474C2semidirect
8. #26041431C2semidirect
9. #26041425C2semidirect
10. #26041358C2semidirect
11. C4×D8×C181C2semidirect
12. C2×#15668448C22semidirect
16. C2×#15668438C22semidirect
17. C2×#15668438C22semidirect
19. C2×#15668434C22semidirect
20. C2×#15668432C22semidirect
21. #19153816C22semidirect
24. #19153796C22semidirect
31. C4⋊#13089678C22semidirect
32. #19153739C22semidirect
33. #19153736C22semidirect
34. #19153728C22semidirect
37. C4×D4×C181C22semidirect
38. #107×C181C22semidirect
40. #102×C181C22semidirect
41. C4×C8×C181C22semidirect
75. D8×C181D4semidirect
109. D4×C181C2×D4semidirect
118. C8×C181C2×D4semidirect
136. C4×C181AGL(1,ℤ/8)semidirect
148. C181#3272semidirect
159. C4×D8D181semidirect
181. C4×D4C2×D181semidirect
183. #107C2×D181semidirect
188. #102C2×D181semidirect
190. C4×C8C2×D181semidirect
215. D8D724semidirect
252. D4C2×D724semidirect
267. C8C2×D724semidirect
287. C4#19153840semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
#260419332
D4×D7242
#260416522
#260415262
#260414842
#260414742
#260414312
#260414252
#260413582
C4×D8×C1812
#3272181