Finite Group Databasebeta

← #37542945

Subgroups — #37542945

Order 23168.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D181C211yesmaximal, characteristic, complemented
2C2×#19153840C212yesmaximal, complemented
3#26041789C211yesmaximal, characteristic, complemented
4#26041432C211yesmaximal, characteristic, complemented
5#26041431C212yesmaximal, complemented
6#26041430C211yesmaximal, characteristic, complemented
7#26041412C211yesmaximal, characteristic, complemented
8#26041411C212yesmaximal, complemented
9#26041282C212yesmaximal, complemented
10#26041241C211yesmaximal, characteristic, complemented
11#447×C181C211yesmaximal, characteristic, complemented
12C24×D181—21no—
13C2×#15668448C2211yescharacteristic, complemented
14C22×D724C2211yescharacteristic
15C22×C22⋊D181—21no—
16C2×D4×D181—28no—
17C2×D4×D181—21no—
18C2×D4×D181—21no—
19C2×D4×D181C2212yes—
20C2×D4×D181C2212yes—
21C22×C4×D181C2211yescharacteristic
22#19153840—28no—
23C2×#15668438C2212yescomplemented
24C2×#15668437C2212yescomplemented
25C2×#15668434C2212yescomplemented
26C2×#15668432C2212yescomplemented
27#19153798—21no—
28#19153796C2211yescharacteristic, complemented
29#19153795C2211yescharacteristic, complemented
30#19153787C2211yescharacteristic, complemented
31#19153784—21no—
32D4×Dic181—21no—
33#19153776—21no—
34#19153774C2211yescharacteristic, complemented
35#19153738C2212yescomplemented
36#19153737C2212yescomplemented
37#19153736C2212yescomplemented
38C22⋊#13089678C2211yescharacteristic, complemented
39#19153728C2212yescomplemented
40C22×D4×C181C2211yescharacteristic
41C2×D8×C181C2212yescomplemented
42#121×C181C2211yescharacteristic, complemented
43#102×C181C2212yescomplemented
44C22⋊C8×C181C2211yescharacteristic, complemented
45C23×D181—28no—
46C23×D181—22no—
47C23×D181—22no—
48C23×D181—21no—
49C23×D181—44no—
50C23×D181—44no—
51C23×D181D411yescharacteristic
52#15668448—24no—
53C2×Dic362C2311yescharacteristic, complemented
54C2×D724—24no—
55C2×D724—21no—
56C2×D724C2312yes—
57C2×C22⋊D181—28no—
58C2×C22⋊D181—21no—
59C2×C22⋊D181—21no—
60C2×C22⋊D181—22no—
61C2×C22⋊D181—22no—
62D4×D181—28no—
63D4×D181—28no—
64D4×D181—24no—
65D4×D181—24no—
66D4×D181—24no—
67D4×D181D418yescomplemented
68C22×Dic181—21no—
69C22×Dic181D411yescharacteristic
70C2×C4×D181—24no—
71C2×C4×D181D411yescharacteristic
72C2×C4×D181D411yescharacteristic
73C2×C4×D181C2311yescharacteristic
74C2×C4×D181C2311yescharacteristic
75#15668438—24no—
76#15668437—24no—
77#15668434—24no—
78#15668432—24no—
79C2×#13089678C2312yescomplemented
80C4⋊Dic181—21no—
81C4⋊Dic181C2311yescharacteristic, complemented
82#15668428—21no—
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C24×C181—21no—
89C2×D4×C181—24no—
90C2×D4×C181—21no—
91C2×D4×C181—21no—
92C2×D4×C181C2312yes—
93C2×D4×C181C2311yescharacteristic, complemented
94C22×C4×C181C2311yescharacteristic
95D8×C181—24no—
96C2×C8×C181C2312yescomplemented
97C4⋊C4×C181C2311yescharacteristic, complemented
98C22⋊C4×C181—21no—
99C22×D181—22no—
100C22×D181—22no—
101C22×D181—22no—
102C22×D181—28no—
103C22×D181—24no—
104C22×D181—22no—
105C22×D181—28no—
106C22×D181—432no—
107C22×D181—44no—
108C22×D181—44no—
109C22×D181—44no—
110C22×D181—42no—
111C22×D181—42no—
112C22×D181—44no—
113C22×D181C2×D411yescharacteristic
114C22×D181C2×D411yescharacteristic
115Dic362—22no—
116D724—24no—
117D724—22no—
118D724C2×D414yes—
119C22⋊D181—28no—
120C22⋊D181—28no—
121C22⋊D181—42no—
122C22⋊D181—42no—
123C2×Dic181—22no—
124C2×Dic181—21no—
125C2×Dic181—21no—
126C2×Dic181—42no—
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—22no—
130C4×D181—22no—
131C4×D181—22no—
132C4×D181C2×D414yes—
133#13089678—22no—
134C23×C181—24no—
135C23×C181—22no—
136C23×C181—21no—
137C23×C181—42no—
138C23×C181—42no—
139C23×C181C2×D411yescharacteristic
140D4×C181—24no—
141D4×C181—22no—
142D4×C181—22no—
143D4×C181—42no—
144D4×C181C2×D414yes—
145C2×C4×C181—22no—
146C2×C4×C181—21no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C8×C181—22no—
150C22×C181—21no—
151C22×C181—21no—
152C22×C181—24no—
153C22×C181—48no—
154C22×C181—42no—
155C22×C181—42no—
156C22×C181—41no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181AGL(1,ℤ/8)12yescomplemented
160C22×C181C22×D411yescharacteristic
161C2×D181—28no—
162C2×D181—21no—
163C2×D181—21no—
164C2×D181—21no—
165C2×D181—21no—
166C2×D181—24no—
167C2×D181—416no—
168C2×D181—416no—
169C2×D181—48no—
170C2×D181—42no—
171C2×D181—42no—
172C2×D181#12012yes—
173C2×D181AGL(1,ℤ/8)12yescomplemented
174C4×C181—21no—
175C4×C181—41no—
176C4×C181C22×D412yes—
177Dic181—21no—
178Dic181—41no—
179Dic181—41no—
180Dic181#12012yes—
181C2×C181—22no—
182C2×C181—44no—
183C2×C181—81no—
184C2×C181C2×#12011yescharacteristic
185C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
187D181—22no—
188D181—22no—
189D181—44no—
190C181#300311yescharacteristic, complemented, Sylow
191#3003—1811nomaximal, Sylow
192C23×D4—1811no—
193C2×AGL(1,ℤ/8)—1812no—
194#525—1811no—
195#450—1811no—
196#450—1812no—
197#450—1811no—
198#447—1812no—
199#447—1811no—
200#447D18111yescharacteristic, complemented
201#418—1812no—
202#407—1811no—
203C25—3621no—
204C2×#41—1811no—
205C22×D4—3628no—
206C22×D4—3621no—
207C22×D4—3621no—
208C22×D4—3621no—
209C22×D4—1812no—
210C22×D4—1812no—
211C22×D4—1811no—
212C22×D4C2×D18111yescharacteristic
213C23×C4—1811no—
214AGL(1,ℤ/8)—3628no—
215C2×SD16—1812no—
216C2×SD16—1812no—
217C2×D8—1812no—
218C2×D8C2×D18112yescomplemented
219C2×M16—1812no—
220#123—3621no—
221C22⋊Q8—1811no—
222C22⋊Q8—1811no—
223#121—1811no—
224#121C2×D18111yescharacteristic, complemented
225#120—3621no—
226C4×D4—3621no—
227C4×D4—3621no—
228#117—1811no—
229#102—1812no—
230#102—1812no—
231#102—1812no—
232#102C2×D18112yescomplemented
233C22⋊C8—1811no—
234C22⋊C8—1812no—
235C22⋊C8C2×D18111yescharacteristic, complemented
236C24—21no—
237C24—3628no—
238C24—3622no—
239C24—3622no—
240C24—3621no—
241C24—1811no—
242C24—7244no—
243C24—7244no—
244#41—3624no—
245C2×Q8—1811no—
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—3624no—
250C2×D4—3628no—
251C2×D4—3628no—
252C2×D4—3624no—
253C2×D4—3628no—
254C2×D4—3624no—
255C2×D4—3621no—
256C2×D4—3621no—
257C2×D4—3624no—
258C2×D4—3622no—
259C2×D4—3622no—
260C2×D4—3621no—
261C2×D4—1818no—
262C2×D4—1812no—
263C2×D4C22×D18112yes—
264C2×D4C22×D18111yescharacteristic, complemented
265C22×C4—3624no—
266C22×C4—3621no—
267C22×C4—1811no—
268C22×C4—1811no—
269C22×C4—1811no—
270C22×C4—1811no—
271C22×C4—1811no—
272C22×C4C22×D18111yescharacteristic
273SD16—3624no—
274SD16—3624no—
275D8—24no—
276D8—3624no—
277M16—3624no—
278C2×C8—1812no—
279C2×C8C22×D18112yescomplemented
280C4⋊C4—3621no—
281C4⋊C4—3621no—
282C4⋊C4—1811no—
283C4⋊C4C22×D18111yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C42—3621no—
290C23—24no—
291C23—22no—
292C23—21no—
293C23—42no—
294C23—42no—
295C23—3622no—
296C23—3622no—
297C23—3624no—
298C23—3628no—
299C23—3628no—
300C23—3622no—
301C23—3622no—
302C23—1811no—
303C23—1811no—
304C23—7242no—
305C23—72432no—
306C23—7244no—
307C23—7244no—
308C23—7244no—
309C23—7244no—
310C23—7242no—
311C23D4×D18111yescharacteristic
312Q8—3622no—
313D4—24no—
314D4—22no—
315D4—22no—
316D4—42no—
317D4—3624no—
318D4—3628no—
319D4—3628no—
320D4—3622no—
321D4—1814no—
322D4—7242no—
323D4—7242no—
324D4D4×D18114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—3622no—
328C2×C4—3622no—
329C2×C4—3622no—
330C2×C4—3622no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—1811no—
334C2×C4—1811no—
335C2×C4—1814no—
336C2×C4—7242no—
337C2×C4D4×D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—22no—
340C8—3622no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—3628no—
351C22—3621no—
352C22—3621no—
353C22—3621no—
354C22—3621no—
355C22—3624no—
356C22—1812no—
357C22—1812no—
358C22—7248no—
359C22—72416no—
360C22—72416no—
361C22—7242no—
362C22—7242no—
363C22#1915384012yescomplemented
364C22C2×D4×D18111yescharacteristic
365C4—21no—
366C4—41no—
367C4—3621no—
368C4—1812no—
369C4—7241no—
370C4—7241no—
371C4C2×D4×D18112yes—
372C2—22no—
373C2—44no—
374C2—81no—
375C2—3622no—
376C2—3622no—
377C2—7244no—
378C2#120×D18111yescharacteristic
379C2AGL(1,ℤ/8)×D18111yescharacteristic
380C2C2×#1915384011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3003
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D181C2semidirect
2. C2×#19153840C2semidirect
3. #26041789C2semidirect
4. #26041432C2semidirect
5. #26041431C2semidirect
6. #26041430C2semidirect
7. #26041412C2semidirect
8. #26041411C2semidirect
9. #26041282C2semidirect
10. #26041241C2semidirect
11. #447×C181C2semidirect
13. C2×#15668448C22semidirect
23. C2×#15668438C22semidirect
24. C2×#15668437C22semidirect
25. C2×#15668434C22semidirect
26. C2×#15668432C22semidirect
28. #19153796C22semidirect
29. #19153795C22semidirect
30. #19153787C22semidirect
34. #19153774C22semidirect
35. #19153738C22semidirect
36. #19153737C22semidirect
37. #19153736C22semidirect
38. C22⋊#13089678C22semidirect
39. #19153728C22semidirect
41. C2×D8×C181C22semidirect
42. #121×C181C22semidirect
43. #102×C181C22semidirect
44. C22⋊C8×C181C22semidirect
53. C2×Dic362C23semidirect
67. D4×D181D4semidirect
79. C2×#13089678C23semidirect
81. C4⋊Dic181C23semidirect
93. C2×D4×C181C23semidirect
96. C2×C8×C181C23semidirect
97. C4⋊C4×C181C23semidirect
159. C22×C181AGL(1,ℤ/8)semidirect
173. C2×D181AGL(1,ℤ/8)semidirect
190. C181#3003semidirect
200. #447D181semidirect
218. C2×D8C2×D181semidirect
224. #121C2×D181semidirect
232. #102C2×D181semidirect
235. C22⋊C8C2×D181semidirect
264. C2×D4C22×D181semidirect
279. C2×C8C22×D181semidirect
283. C4⋊C4C22×D181semidirect
363. C22#19153840semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1812
C2×#191538402
#260417892
#260414322
#260414312
#260414302
#260414122
#260414112
#260412822
#260412412
#447×C1812
#3003181