Finite Group Databasebeta

← #37545323

Subgroups — #37545323

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×#19153844C212yesmaximal, complemented
2#26041934C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic, complemented
4#26041652C211yesmaximal, characteristic, complemented
5#26041525C212yesmaximal, complemented
6#26041485C211yesmaximal, characteristic, complemented
7#26041475C211yesmaximal, characteristic, complemented
8#26041432C212yesmaximal, complemented
9#26041426C212yesmaximal, complemented
10C4×D1448C211yesmaximal, characteristic, complemented
11#436×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2212yescomplemented
13C22×D724C2212yes—
14C2×D4×D181—21no—
15#19153844—28no—
16C2×#15668438C2212yes—
17C2×#15668438C2212yescomplemented
18C2×#15668438C2212yescomplemented
19C2×D1448C2211yescharacteristic
20C2×D1448C2212yescomplemented
21#19153816C2211yescharacteristic, complemented
22#19153813C2211yescharacteristic
23#19153798—22no—
24#19153795C2212yescomplemented
25#19153791—21no—
26#19153790C2212yes—
27#19153787—21no—
28#19153786—22no—
29C4×D724C2211yescharacteristic
30C4×D724C2211yescharacteristic, complemented
31C4⋊C4×D181—21no—
32#19153750C2211yescharacteristic, complemented
33#19153741C2212yescomplemented
34#19153738C2211yescharacteristic
35#19153738C2211yescharacteristic, complemented
36#19153738C2212yescomplemented
37C2×M16×C181C2212yescomplemented
38C4×D4×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic, complemented
40C22⋊C8×C181C2212yescomplemented
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43#15668449—24no—
44#15668449—42no—
45C2×Dic362C2312yes—
46C2×D724—24no—
47C2×D724—22no—
48C2×D724—21no—
49C2×D724—21no—
50C2×D724—42no—
51C2×D724C2312yes—
52C2×D724C2311yescharacteristic
53C2×D724C2311yescharacteristic
54C2×C22⋊D181—22no—
55C2×C22⋊D181—22no—
56D4×D181—44no—
57C2×C4×D181—21no—
58C2×C4×D181—22no—
59C2×C4×D181—21no—
60#15668438—24no—
61#15668438—24no—
62#15668438—42no—
63D1448—24no—
64D1448D414yescomplemented
65C4⋊Dic181—21no—
66C4⋊Dic181—21no—
67C4⋊Dic181C2311yescharacteristic
68C4⋊Dic181C2311yescharacteristic
69#15668428—22no—
70C22⋊Dic181—22no—
71#15668426—21no—
72#15668426—21no—
73#15668426—22no—
74C2×D4×C181D411yescharacteristic
75C22×C4×C181C2312yes—
76M16×C181—24no—
77C2×C8×C181C2311yescharacteristic
78C2×C8×C181C2312yes—
79C2×C8×C181C2311yescharacteristic
80C4⋊C4×C181D411yescharacteristic
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—21no—
85C22×D181—22no—
86C22×D181—44no—
87C22×D181—44no—
88C22×D181—82no—
89Dic362—22no—
90Dic362—22no—
91D724—22no—
92D724—21no—
93D724—21no—
94D724—22no—
95D724—44no—
96D724—42no—
97D724—81no—
98D724C2×D412yes—
99D724C2×D412yescomplemented
100C22⋊D181—44no—
101C22⋊D181—44no—
102C2×Dic181—21no—
103C2×Dic181—21no—
104C2×Dic181—22no—
105C4×D181—44no—
106C4×D181—42no—
107C4×D181—42no—
108C23×C181C2×D412yes—
109D4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181—22no—
112C2×C4×C181C2×D411yescharacteristic
113C2×C4×C181C2×D412yes—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2411yescharacteristic
116C8×C181—22no—
117C8×C181—21no—
118C8×C181C2×D412yescomplemented
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—22no—
122C22×C181C22×D411yescharacteristic
123C2×D181—22no—
124C2×D181—22no—
125C2×D181—41no—
126C2×D181—41no—
127C2×D181—41no—
128C2×D181—41no—
129C2×D181—42no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—88no—
133C4×C181—22no—
134C4×C181—21no—
135C4×C181AGL(1,ℤ/8)12yescomplemented
136C4×C181C22×D411yescharacteristic
137C4×C181H(5,𝔽2)11yescharacteristic
138Dic181—42no—
139Dic181—41no—
140Dic181—41no—
141C2×C181—42no—
142C2×C181D4211yescharacteristic
143C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C181#57711yescharacteristic
145D181—42no—
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×D8—1811no—
160#436D18111yescharacteristic, complemented
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×D8—1811no—
170C2×M16C2×D18112yescomplemented
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×D4—1811no—
181C4×D4C2×D18111yescharacteristic
182C2×C4⋊C4—3621no—
183#107—1811no—
184C4⋊C8C2×D18111yescharacteristic, complemented
185#103—1812no—
186#102—1811no—
187#102—1811no—
188#102—1812no—
189C22⋊C8C2×D18112yescomplemented
190C4×C8C2×D18111yescharacteristic, complemented
191C24—3622no—
192#41—3624no—
193#41—7242no—
194C2×Q8—1812no—
195C2×D4—3624no—
196C2×D4—3622no—
197C2×D4—3622no—
198C2×D4—3621no—
199C2×D4—3622no—
200C2×D4—3621no—
201C2×D4—1812no—
202C2×D4—1811no—
203C2×D4—1811no—
204C2×D4—7242no—
205C2×D4—7244no—
206C2×D4D72411yescharacteristic
207C22×C4—3621no—
208C22×C4—3622no—
209C22×C4—3621no—
210C22×C4C22×D18112yes—
211SD16—3624no—
212SD16—3624no—
213SD16—7242no—
214D8—3624no—
215D8—1814no—
216M16—24no—
217C2×C8C22×D18111yescharacteristic
218C2×C8C22×D18112yes—
219C2×C8C22×D18111yescharacteristic
220C4⋊C4—3621no—
221C4⋊C4—3622no—
222C4⋊C4—3621no—
223C4⋊C4—1811no—
224C4⋊C4—1811no—
225C4⋊C4D72411yescharacteristic
226C22⋊C4—3622no—
227C22⋊C4—3621no—
228C22⋊C4—3621no—
229C22⋊C4—3622no—
230C22⋊C4D72412yes—
231C42C22×D18111yescharacteristic
232C23—3621no—
233C23—3621no—
234C23—3622no—
235C23—14482no—
236C23—7244no—
237C23—7244no—
238C23C2×D72412yes—
239Q8—3622no—
240Q8—3622no—
241D4—22no—
242D4—3622no—
243D4—3621no—
244D4—3621no—
245D4—3622no—
246D4—14481no—
247D4—1812no—
248D4—1812no—
249D4—7244no—
250D4—7244no—
251D4—7244no—
252D4—7242no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3621no—
256C2×C4—3621no—
257C2×C4—3622no—
258C2×C4—7244no—
259C2×C4—7242no—
260C2×C4—7242no—
261C2×C4C2×D72411yescharacteristic
262C2×C4C2×D72412yes—
263C2×C4C2×D72411yescharacteristic
264C2×C4C23×D18111yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D18112yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—3622no—
272C22—3622no—
273C22—14488no—
274C22—7242no—
275C22—7242no—
276C22—7242no—
277C22—7241no—
278C22—7241no—
279C22—7241no—
280C22—7241no—
281C22C22×D72411yescharacteristic
282C4—22no—
283C4—21no—
284C4—7242no—
285C4—7241no—
286C4—7241no—
287C4#1915384412yescomplemented
288C4C2×D4×D18111yescharacteristic
289C4#1915386211yescharacteristic
290C2—42no—
291C2—14482no—
292C2—7242no—
293C2—7242no—
294C2D4×D72411yescharacteristic
295C2C2×#1915384411yescharacteristic
296C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#436×C181C2
socleC22×C181C22×D4
hypercenter#436D181
hypocenterC181#3272
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. #26041934C2semidirect
3. D4×D724C2semidirect
4. #26041652C2semidirect
5. #26041525C2semidirect
6. #26041485C2semidirect
7. #26041475C2semidirect
8. #26041432C2semidirect
9. #26041426C2semidirect
10. C4×D1448C2semidirect
11. #436×C181C2semidirect
12. C2×#15668449C22semidirect
17. C2×#15668438C22semidirect
18. C2×#15668438C22semidirect
20. C2×D1448C22semidirect
21. #19153816C22semidirect
24. #19153795C22semidirect
30. C4×D724C22semidirect
32. #19153750C22semidirect
33. #19153741C22semidirect
35. #19153738C22semidirect
36. #19153738C22semidirect
37. C2×M16×C181C22semidirect
39. C4⋊C8×C181C22semidirect
40. C22⋊C8×C181C22semidirect
41. C4×C8×C181C22semidirect
64. D1448D4semidirect
99. D724C2×D4semidirect
118. C8×C181C2×D4semidirect
135. C4×C181AGL(1,ℤ/8)semidirect
148. C181#3272semidirect
160. #436D181semidirect
170. C2×M16C2×D181semidirect
184. C4⋊C8C2×D181semidirect
189. C22⋊C8C2×D181semidirect
190. C4×C8C2×D181semidirect
267. C8D4×D181semidirect
287. C4#19153844semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
#260419342
D4×D7242
#260416522
#260415252
#260414852
#260414752
#260414322
#260414262
C4×D14482
#436×C1812
#3272181