Finite Group Databasebeta

← #37544169

Subgroups — #37544169

Order 23168.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153842C211yesmaximal, characteristic, complemented
3C2×#19153841C211yesmaximal, characteristic, complemented
4#26041825C211yesmaximal, characteristic, complemented
5C2×#19153812C211yesmaximal, characteristic, complemented
6#26041667C212yesmaximal, complemented
7#26041665C212yesmaximal, complemented
8#26041654C212yesmaximal, complemented
9#26041650C212yesmaximal, complemented
10M16×Dic181C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×D724C2211yescharacteristic
14C22×C22⋊D181—22no—
15#19153844—24no—
16#19153842—24no—
17#19153841—24no—
18C2×#15668438C2212yescomplemented
19C2×#15668437C2212yescomplemented
20C2×#15668436C2212yescomplemented
21C2×D1448C2212yescomplemented
22C2×#15668434C2212yescomplemented
23C2×#15668434C2212yescomplemented
24C2×#15668433C2211yescharacteristic
25#19153814C2211yescharacteristic, complemented
26#19153812—22no—
27#19153812C2212yes—
28#19153812C2212yes—
29#19153812C2211yescharacteristic, complemented
30#19153805C2211yescharacteristic, complemented
31#19153803C2211yescharacteristic, complemented
32#19153788—22no—
33C4×C22⋊D181—22no—
34C2×C4×Dic181C2211yescharacteristic
35#19153726C2212yescomplemented
36C8×Dic181C2212yescomplemented
37C2×#41×C181C2211yescharacteristic
38C22×D4×C181C2211yescharacteristic
39AGL(1,ℤ/8)×C181—24no—
40C2×SD16×C181C2212yescomplemented
41C2×D8×C181C2212yescomplemented
42C2×M16×C181C2211yescharacteristic
43C23×D181—21no—
44#15668449—22no—
45#15668449—22no—
46C2×Dic362C2311yescharacteristic
47C2×D724—22no—
48C2×D724—22no—
49C2×D724—21no—
50C2×D724C2311yescharacteristic
51C2×D724C2312yes—
52C2×C22⋊D181—22no—
53C2×C22⋊D181—24no—
54C2×C22⋊D181—24no—
55C2×C22⋊D181—21no—
56C2×C22⋊D181—48no—
57C22×Dic181D412yes—
58C2×C4×D181—21no—
59#15668438—24no—
60#15668437—24no—
61#15668436—24no—
62D1448—24no—
63#15668434—24no—
64#15668434—24no—
65#15668433D414yescomplemented
66C2×#13089678C2312yes—
67#15668428—22no—
68C22⋊Dic181—22no—
69#15668426—22no—
70C4×Dic181D412yes—
71C4×Dic181C2311yescharacteristic
72C4×Dic181C2311yescharacteristic
73C24×C181—21no—
74#41×C181—22no—
75#41×C181—22no—
76C2×Q8×C181C2311yescharacteristic
77C2×D4×C181—21no—
78C2×D4×C181—22no—
79C2×D4×C181—22no—
80C2×D4×C181—21no—
81C2×D4×C181C2311yescharacteristic
82C2×D4×C181C2312yes—
83C22×C4×C181—21no—
84C22×C4×C181C2311yescharacteristic
85SD16×C181—24no—
86D8×C181—24no—
87M16×C181D414yescomplemented
88C2×C8×C181C2312yes—
89C22×D181—21no—
90C22×D181—22no—
91C22×D181—44no—
92C22×D181—42no—
93Dic362—21no—
94Dic362—21no—
95D724—21no—
96D724—22no—
97D724—21no—
98D724—22no—
99D724—42no—
100C22⋊D181—48no—
101C22⋊D181—42no—
102C22⋊D181—44no—
103C22⋊D181—48no—
104C2×Dic181—24no—
105C2×Dic181—21no—
106C2×Dic181C2×D412yes—
107C2×Dic181C2×D412yes—
108C4×D181—42no—
109#13089678C2×D414yescomplemented
110C23×C181—21no—
111C23×C181—22no—
112C23×C181—44no—
113C23×C181—42no—
114C23×C181C2×D411yescharacteristic
115Q8×C181—21no—
116Q8×C181—21no—
117D4×C181—21no—
118D4×C181—22no—
119D4×C181—22no—
120D4×C181—21no—
121D4×C181—42no—
122D4×C181—42no—
123C2×C4×C181—21no—
124C2×C4×C181—42no—
125C2×C4×C181C2×D411yescharacteristic
126C2×C4×C181C2×D412yes—
127C2×C4×C181C2×D412yes—
128C2×C4×C181C2411yescharacteristic
129C8×C181C2×D414yescomplemented
130C22×C181—22no—
131C22×C181—41no—
132C22×C181—42no—
133C22×C181—44no—
134C22×C181—42no—
135C22×C181—84no—
136C22×C181#12712yes—
137C22×C181C22×D411yescharacteristic
138C2×D181—41no—
139C2×D181—42no—
140C2×D181—44no—
141C2×D181—42no—
142C2×D181—84no—
143C4×C181—41no—
144C4×C181#12712yes—
145C4×C181C22×D411yescharacteristic
146C4×C181C22×D411yescharacteristic
147Dic181—22no—
148Dic181—41no—
149Dic181AGL(1,ℤ/8)14yescomplemented
150C2×C181—22no—
151C2×C181—81no—
152C2×C181—82no—
153C2×C181C2×#12711yescharacteristic
154C2×C181C2×AGL(1,ℤ/8)12yes—
155D181—81no—
156D181—82no—
157C181#315111yescharacteristic, complemented, Sylow
158#3151—1811nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1811no—
160C2×AGL(1,ℤ/8)—1811no—
161C2×AGL(1,ℤ/8)—1811no—
162C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
163#532—1811no—
164C2×#127—1811no—
165#496—1812no—
166#496—1812no—
167#493—1812no—
168#492—1812no—
169C4×M16—1811no—
170C2×#41—1811no—
171C2×#41C2×D18111yescharacteristic
172C22×D4—3622no—
173C22×D4—1811no—
174C22×D4C2×D18111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3624no—
177AGL(1,ℤ/8)—3624no—
178AGL(1,ℤ/8)—3624no—
179C2×SD16—1812no—
180C2×SD16—1812no—
181C2×SD16—1812no—
182C2×SD16C2×D18112yescomplemented
183C2×D8—1812no—
184C2×D8—1812no—
185C2×D8—1812no—
186C2×D8C2×D18112yescomplemented
187C2×M16—1811no—
188C2×M16C2×D18111yescharacteristic
189C4⋊Q8—1811no—
190#127—3622no—
191#127—1812no—
192#127—1812no—
193#127—1811no—
194#124—1811no—
195#124—1811no—
196#121—3622no—
197C4×D4—3622no—
198C2×C42—1811no—
199#97—1812no—
200C4×C8—1812no—
201C24—21no—
202C24—3621no—
203#41—22no—
204#41—22no—
205#41—3622no—
206#41—3622no—
207C2×Q8—1811no—
208C2×Q8C22×D18111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3621no—
214C2×D4—3622no—
215C2×D4—3624no—
216C2×D4—3624no—
217C2×D4—3622no—
218C2×D4—3621no—
219C2×D4—3622no—
220C2×D4—1811no—
221C2×D4—1812no—
222C2×D4—7248no—
223C2×D4C22×D18111yescharacteristic
224C2×D4C22×D18112yes—
225C22×C4—21no—
226C22×C4—3621no—
227C22×C4—1812no—
228C22×C4C22×D18111yescharacteristic
229SD16—24no—
230SD16—3624no—
231SD16—3624no—
232SD16—3624no—
233D8—24no—
234D8—3624no—
235D8—3624no—
236D8—3624no—
237M16—1814no—
238M16C22⋊D18114yescomplemented
239C2×C8—1812no—
240C2×C8C22×D18112yes—
241C4⋊C4—3622no—
242C22⋊C4—3622no—
243C22⋊C4—3622no—
244C42—1812no—
245C42—1811no—
246C42—1811no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3621no—
252C23—3622no—
253C23—7244no—
254C23—7242no—
255C23D4×D18111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3621no—
259Q8—3621no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3621no—
267D4—3622no—
268D4—3621no—
269D4—3622no—
270D4—7248no—
271D4—7248no—
272D4—7244no—
273D4—7242no—
274D4—7242no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3624no—
278C2×C4—3621no—
279C2×C4—1812no—
280C2×C4—1812no—
281C2×C4—7242no—
282C2×C4D4×D18111yescharacteristic
283C2×C4D4×D18112yes—
284C2×C4C2×C22⋊D18112yes—
285C2×C4C23×D18111yescharacteristic
286C8—1814no—
287C8C2×C22⋊D18114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—14484no—
295C22—7244no—
296C22—7242no—
297C22—7241no—
298C22—7242no—
299C22#1915381212yes—
300C22C2×D4×D18111yescharacteristic
301C4—41no—
302C4—3622no—
303C4—1814no—
304C4—7241no—
305C4#1915381212yes—
306C4C2×D4×D18111yescharacteristic
307C4C22×C22⋊D18111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—14482no—
312C2—14481no—
313C2C2×#1915381211yescharacteristic
314C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×AGL(1,ℤ/8)×C181C2
socleC22×C181C22×D4
hypercenterC2×AGL(1,ℤ/8)D181
hypocenterC181#3151
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153842C2semidirect
3. C2×#19153841C2semidirect
4. #26041825C2semidirect
5. C2×#19153812C2semidirect
6. #26041667C2semidirect
7. #26041665C2semidirect
8. #26041654C2semidirect
9. #26041650C2semidirect
10. M16×Dic181C2semidirect
11. C2×AGL(1,ℤ/8)×C181C2semidirect
18. C2×#15668438C22semidirect
19. C2×#15668437C22semidirect
20. C2×#15668436C22semidirect
21. C2×D1448C22semidirect
22. C2×#15668434C22semidirect
23. C2×#15668434C22semidirect
25. #19153814C22semidirect
29. #19153812C22semidirect
30. #19153805C22semidirect
31. #19153803C22semidirect
35. #19153726C22semidirect
36. C8×Dic181C22semidirect
40. C2×SD16×C181C22semidirect
41. C2×D8×C181C22semidirect
65. #15668433D4semidirect
87. M16×C181D4semidirect
109. #13089678C2×D4semidirect
129. C8×C181C2×D4semidirect
149. Dic181AGL(1,ℤ/8)semidirect
157. C181#3151semidirect
162. C2×AGL(1,ℤ/8)D181semidirect
182. C2×SD16C2×D181semidirect
186. C2×D8C2×D181semidirect
238. M16C22⋊D181semidirect
287. C8C2×C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538422
C2×#191538412
#260418252
C2×#191538122
#260416672
#260416652
#260416542
#260416502
M16×Dic1812
C2×AGL(1,ℤ/8)×C1812
#3151181