Finite Group Databasebeta

← #37544206

Subgroups — #37544206

Order 23168.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153836C211yesmaximal, characteristic, complemented
3C22×#15668434C211yesmaximal, characteristic, complemented
4#26041846C211yesmaximal, characteristic, complemented
5#26041843C211yesmaximal, characteristic, complemented
6#26041667C212yesmaximal, complemented
7#26041665C212yesmaximal, complemented
8#26041660C212yesmaximal, complemented
9#26041654C212yesmaximal, complemented
10#26041237C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×D724C2211yescharacteristic
14#19153844—24no—
15#19153836—24no—
16C2×#15668439C2211yescharacteristic, complemented
17C2×#15668438C2212yescomplemented
18C2×#15668437C2211yescharacteristic, complemented
19C2×#15668436C2211yescharacteristic, complemented
20C2×D1448C2212yescomplemented
21C2×#15668434—24no—
22C2×#15668434C2212yes—
23C2×#15668434C2212yes—
24C2×#15668434C2211yescharacteristic, complemented
25C22×#13089678C2211yescharacteristic
26#19153812C2212yescomplemented
27#19153812C2212yescomplemented
28#19153805C2212yescomplemented
29#19153803C2212yescomplemented
30#19153790—22no—
31#19153787—22no—
32#19153785—22no—
33#19153784—22no—
34#19153775C2211yescharacteristic
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×D724C2312yes—
51C2×D724C2311yescharacteristic
52C2×C22⋊D181—24no—
53C2×C22⋊D181—24no—
54C2×C22⋊D181—21no—
55C2×C4×D181—21no—
56#15668439—22no—
57#15668438—24no—
58#15668437—22no—
59#15668436—22no—
60D1448—24no—
61#15668434—24no—
62#15668434—24no—
63#15668434—22no—
64C2×#13089678D414yescomplemented
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C4⋊Dic181D412yes—
68C22⋊Dic181—24no—
69C22⋊Dic181D412yes—
70#15668426—24no—
71C4×Dic181C2312yes—
72C24×C181—21no—
73#41×C181—22no—
74#41×C181—22no—
75C2×Q8×C181C2311yescharacteristic
76C2×D4×C181—21no—
77C2×D4×C181—22no—
78C2×D4×C181—22no—
79C2×D4×C181—21no—
80C2×D4×C181C2311yescharacteristic
81C2×D4×C181C2312yes—
82C22×C4×C181—21no—
83C22×C4×C181C2311yescharacteristic
84SD16×C181—24no—
85D8×C181—24no—
86M16×C181D414yescomplemented
87C2×C8×C181C2312yes—
88C22×D181—22no—
89C22×D181—21no—
90C22×D181—42no—
91C22×D181—82no—
92Dic362—21no—
93Dic362—21no—
94D724—22no—
95D724—21no—
96D724—21no—
97D724—22no—
98D724—42no—
99C22⋊D181—42no—
100C22⋊D181—84no—
101C22⋊D181—84no—
102C2×Dic181—21no—
103C2×Dic181C2×D414yes—
104C4×D181—42no—
105#13089678C2×D412yes—
106#13089678C2×D412yes—
107C23×C181—21no—
108C23×C181—22no—
109C23×C181—42no—
110C23×C181—82no—
111C23×C181C2×D411yescharacteristic
112Q8×C181—21no—
113Q8×C181—21no—
114D4×C181—21no—
115D4×C181—22no—
116D4×C181—22no—
117D4×C181—21no—
118D4×C181—42no—
119D4×C181—42no—
120C2×C4×C181—21no—
121C2×C4×C181—42no—
122C2×C4×C181C2×D411yescharacteristic
123C2×C4×C181C2×D412yes—
124C2×C4×C181C2×D412yes—
125C2×C4×C181C2411yescharacteristic
126C8×C181C2×D414yes—
127C22×C181—22no—
128C22×C181—41no—
129C22×C181—42no—
130C22×C181—44no—
131C22×C181—42no—
132C22×C181—84no—
133C22×C181#12712yes—
134C22×C181C22×D411yescharacteristic
135C2×D181—41no—
136C2×D181—42no—
137C2×D181—42no—
138C2×D181—44no—
139C2×D181—84no—
140C4×C181—41no—
141C4×C181#12712yes—
142C4×C181C22×D411yescharacteristic
143C4×C181C22×D411yescharacteristic
144Dic181—24no—
145Dic181—41no—
146C2×C181—22no—
147C2×C181—81no—
148C2×C181—82no—
149C2×C181C2×#12711yescharacteristic
150C2×C181#57612yes—
151D181—81no—
152D181—82no—
153C181#315511yescharacteristic, complemented, Sylow
154#3155—1811nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1811no—
156C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
157C2×#135—1811no—
158C22×D8—1811no—
159#535—1811no—
160#535—1811no—
161#496—1812no—
162#496—1812no—
163#495—1812no—
164#493—1812no—
165#405—1811no—
166C2×#41—1811no—
167C2×#41C2×D18111yescharacteristic
168C22×D4—1811no—
169C22×D4C2×D18111yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—3624no—
172#135—3624no—
173C2×Q16—1811no—
174C2×SD16—1811no—
175C2×SD16—1811no—
176C2×SD16—1812no—
177C2×SD16C2×D18112yescomplemented
178C2×D8—3624no—
179C2×D8—1812no—
180C2×D8—1812no—
181C2×D8—1811no—
182C2×D8—1812no—
183C2×D8C2×D18112yescomplemented
184C2×M16C2×D18111yescharacteristic
185C22×C8—1811no—
186#127—1812no—
187#127—1812no—
188#124—1812no—
189#124—1812no—
190#121—3622no—
191#121—3622no—
192#120—3622no—
193#120—3622no—
194#117—1811no—
195#97—1812no—
196C4×C8—1812no—
197C24—21no—
198C24—3621no—
199#41—22no—
200#41—22no—
201#41—3622no—
202#41—3622no—
203C2×Q8—1811no—
204C2×Q8C22×D18111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3621no—
210C2×D4—3624no—
211C2×D4—3624no—
212C2×D4—3622no—
213C2×D4—3621no—
214C2×D4—3622no—
215C2×D4—1812no—
216C2×D4—1811no—
217C2×D4C22×D18112yes—
218C2×D4C22×D18111yescharacteristic
219C22×C4—21no—
220C22×C4—3621no—
221C22×C4C22×D18111yescharacteristic
222Q16—3622no—
223SD16—24no—
224SD16—3622no—
225SD16—3622no—
226SD16—3624no—
227D8—24no—
228D8—3624no—
229D8—3624no—
230D8—3622no—
231D8—3624no—
232M16C22⋊D18114yescomplemented
233C2×C8—1814no—
234C2×C8—1811no—
235C2×C8—1811no—
236C2×C8C22×D18112yes—
237C4⋊C4—1812no—
238C22⋊C4—3624no—
239C22⋊C4—3624no—
240C22⋊C4—1812no—
241C42—1812no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—3622no—
247C23—3621no—
248C23—14482no—
249C23—7242no—
250C23D4×D18111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3621no—
254Q8—3621no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3622no—
262D4—3621no—
263D4—3621no—
264D4—3622no—
265D4—14484no—
266D4—14484no—
267D4—7242no—
268D4—7242no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3621no—
272C2×C4—1814no—
273C2×C4—7242no—
274C2×C4D4×D18111yescharacteristic
275C2×C4D4×D18112yes—
276C2×C4C2×C22⋊D18112yes—
277C2×C4C23×D18111yescharacteristic
278C8—1812no—
279C8—1812no—
280C8C2×C22⋊D18114yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—14484no—
288C22—7241no—
289C22—7242no—
290C22—7242no—
291C22—7244no—
292C22#1915381212yes—
293C22C2×D4×D18111yescharacteristic
294C4—41no—
295C4—3624no—
296C4—7241no—
297C4#1915381212yes—
298C4C2×D4×D18111yescharacteristic
299C4C22×C22⋊D18111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—14482no—
304C2—14481no—
305C2C2×#1915381211yescharacteristic
306C2#2604208312yes—
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#3155
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×#19153836C2semidirect
3. C22×#15668434C2semidirect
4. #26041846C2semidirect
5. #26041843C2semidirect
6. #26041667C2semidirect
7. #26041665C2semidirect
8. #26041660C2semidirect
9. #26041654C2semidirect
10. #26041237C2semidirect
11. C2×AGL(1,ℤ/8)×C181C2semidirect
16. C2×#15668439C22semidirect
17. C2×#15668438C22semidirect
18. C2×#15668437C22semidirect
19. C2×#15668436C22semidirect
20. C2×D1448C22semidirect
24. C2×#15668434C22semidirect
26. #19153812C22semidirect
27. #19153812C22semidirect
28. #19153805C22semidirect
29. #19153803C22semidirect
35. #19153726C22semidirect
36. C8×Dic181C22semidirect
40. C2×SD16×C181C22semidirect
41. C2×D8×C181C22semidirect
64. C2×#13089678D4semidirect
86. M16×C181D4semidirect
153. C181#3155semidirect
156. C2×AGL(1,ℤ/8)D181semidirect
177. C2×SD16C2×D181semidirect
183. C2×D8C2×D181semidirect
232. M16C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538362
C22×#156684342
#260418462
#260418432
#260416672
#260416652
#260416602
#260416542
#260412372
C2×AGL(1,ℤ/8)×C1812
#3155181