Finite Group Databasebeta

← #37542968

Subgroups — #37542968

Order 23168.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15668448C211yesmaximal, characteristic, complemented
2C2×#19153840C212yesmaximal, complemented
3#121×D181C211yesmaximal, characteristic, complemented
4#26041425C212yesmaximal, complemented
5#26041421C212yesmaximal, complemented
6#26041420C211yesmaximal, characteristic, complemented
7#26041417C211yesmaximal, characteristic, complemented
8#26041413C211yesmaximal, characteristic, complemented
9#26041277C212yesmaximal, complemented
10#26041241C211yesmaximal, characteristic, complemented
11#447×C181C211yesmaximal, characteristic, complemented
12C2×#15668448—28no—
13C2×#15668448—21no—
14C2×#15668448—21no—
15C2×#15668448C2212yes—
16C2×#15668448C2212yescomplemented
17C22×Dic362C2211yescharacteristic, complemented
18C22×C22⋊D181—21no—
19C2×D4×D181—21no—
20C2×D4×D181C2211yescharacteristic, complemented
21C23×Dic181—21no—
22C22×C4×D181C2211yescharacteristic
23#19153840—28no—
24C2×#15668438C2212yescomplemented
25C2×#15668437C2212yescomplemented
26C2×#15668434C2212yescomplemented
27C2×#15668432C2212yescomplemented
28#19153791C2211yescharacteristic, complemented
29#19153790C2211yescharacteristic
30#19153789—21no—
31#19153788—21no—
32#19153787C2211yescharacteristic, complemented
33C4⋊C4×D181C2211yescharacteristic, complemented
34C22⋊C4×D181—21no—
35#19153741C2212yescomplemented
36#19153739C2212yescomplemented
37#19153737C2212yescomplemented
38C22⋊#13089678C2211yescharacteristic, complemented
39#19153728C2212yescomplemented
40C22×D4×C181C2211yescharacteristic, complemented
41C2×D8×C181C2212yescomplemented
42#121×C181C2211yescharacteristic
43#102×C181C2212yescomplemented
44C22⋊C8×C181C2211yescharacteristic, complemented
45C23×D181—21no—
46C23×D181D411yescharacteristic
47#15668448—28no—
48#15668448—28no—
49#15668448—24no—
50#15668448—24no—
51#15668448—24no—
52#15668448D418yescomplemented
53C2×Dic362—24no—
54C2×Dic362—21no—
55C2×Dic362C2312yes—
56C2×D724—21no—
57C2×D724C2311yescharacteristic
58C2×C22⋊D181—28no—
59C2×C22⋊D181—22no—
60C2×C22⋊D181—22no—
61C2×C22⋊D181—21no—
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64D4×D181—24no—
65D4×D181—44no—
66C22×Dic181—28no—
67C22×Dic181—22no—
68C22×Dic181—22no—
69C22×Dic181—21no—
70C22×Dic181D411yescharacteristic
71C2×C4×D181—24no—
72C2×C4×D181—21no—
73C2×C4×D181D411yescharacteristic
74C2×C4×D181D411yescharacteristic
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2311yescharacteristic
77#15668438—24no—
78#15668437—24no—
79#15668434—24no—
80#15668432—24no—
81C2×#13089678C2312yes—
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84C22⋊Dic181—21no—
85#15668426—21no—
86#15668426—21no—
87C24×C181—21no—
88C2×D4×C181—24no—
89C2×D4×C181—21no—
90C2×D4×C181—21no—
91C2×D4×C181C2311yescharacteristic
92C2×D4×C181C2312yes—
93C22×C4×C181C2311yescharacteristic
94D8×C181—24no—
95C2×C8×C181C2312yes—
96C4⋊C4×C181C2311yescharacteristic
97C22⋊C4×C181—21no—
98C22×D181—22no—
99C22×D181—22no—
100C22×D181—21no—
101C22×D181—41no—
102C22×D181—42no—
103C22×D181—42no—
104C22×D181—42no—
105C22×D181C2×D411yescharacteristic
106C22×D181C2×D411yescharacteristic
107Dic362—24no—
108Dic362—22no—
109Dic362C2×D414yescomplemented
110D724—22no—
111D724—42no—
112C22⋊D181—28no—
113C22⋊D181—28no—
114C22⋊D181—42no—
115C22⋊D181—42no—
116C22⋊D181—42no—
117C2×Dic181—24no—
118C2×Dic181—28no—
119C2×Dic181—22no—
120C2×Dic181—22no—
121C2×Dic181—28no—
122C2×Dic181—21no—
123C2×Dic181C2×D411yescharacteristic
124C2×Dic181C2×D411yescharacteristic
125C4×D181—22no—
126C4×D181—22no—
127C4×D181—22no—
128C4×D181—42no—
129C4×D181C2×D414yes—
130#13089678—22no—
131C23×C181—24no—
132C23×C181—22no—
133C23×C181—21no—
134C23×C181—42no—
135C23×C181—42no—
136C23×C181C2×D411yescharacteristic
137D4×C181—24no—
138D4×C181—22no—
139D4×C181—22no—
140D4×C181—42no—
141D4×C181C2×D414yescomplemented
142C2×C4×C181—22no—
143C2×C4×C181—21no—
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—22no—
147C22×C181—21no—
148C22×C181—21no—
149C22×C181—24no—
150C22×C181—48no—
151C22×C181—42no—
152C22×C181—42no—
153C22×C181—41no—
154C22×C181—41no—
155C22×C181—41no—
156C22×C181AGL(1,ℤ/8)12yescomplemented
157C22×C181C22×D411yescharacteristic
158C2×D181—21no—
159C2×D181—21no—
160C2×D181—21no—
161C2×D181—21no—
162C2×D181—21no—
163C2×D181—44no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—84no—
168C2×D181#12012yes—
169C4×C181—21no—
170C4×C181—41no—
171C4×C181C22×D412yes—
172Dic181—21no—
173Dic181—24no—
174Dic181—41no—
175Dic181#12012yes—
176C2×C181—22no—
177C2×C181—44no—
178C2×C181—81no—
179C2×C181C2×#12011yescharacteristic
180C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C181#57511yescharacteristic
182D181—22no—
183D181—41no—
184D181—81no—
185C181#300611yescharacteristic, complemented, Sylow
186#3006—1811nomaximal, Sylow
187C22×#41—1811no—
188C2×AGL(1,ℤ/8)—1812no—
189C2×#121—1811no—
190#449—1812no—
191#449—1812no—
192#448—1811no—
193#448—1811no—
194#447—1811no—
195#447D18111yescharacteristic, complemented
196#417—1812no—
197#407—1811no—
198C2×#41—3628no—
199C2×#41—3621no—
200C2×#41—3621no—
201C2×#41—1812no—
202C2×#41—1812no—
203C22×Q8—1811no—
204C22×D4—3621no—
205C22×D4—3621no—
206C22×D4—1811no—
207C22×D4C2×D18111yescharacteristic, complemented
208C23×C4—3621no—
209C23×C4—1811no—
210AGL(1,ℤ/8)—3628no—
211C2×SD16—1812no—
212C2×SD16—1812no—
213C2×D8—1812no—
214C2×D8C2×D18112yescomplemented
215C2×M16—1812no—
216#121—3621no—
217#121—3621no—
218#121—1811no—
219#121—1811no—
220#121—1811no—
221#121C2×D18111yescharacteristic
222C2×C4⋊C4—1811no—
223C2×C22⋊C4—3621no—
224#103—1812no—
225#103—1812no—
226#102—1812no—
227#102C2×D18112yescomplemented
228C22⋊C8—1811no—
229C22⋊C8—1812no—
230C22⋊C8C2×D18111yescharacteristic, complemented
231C24—21no—
232C24—3621no—
233C24—1811no—
234#41—3628no—
235#41—3628no—
236#41—3624no—
237#41—3624no—
238#41—3624no—
239#41—1818no—
240C2×Q8—3624no—
241C2×Q8—3621no—
242C2×Q8—1812no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—3621no—
247C2×D4—3628no—
248C2×D4—3622no—
249C2×D4—3622no—
250C2×D4—3621no—
251C2×D4—3621no—
252C2×D4—3624no—
253C2×D4—3621no—
254C2×D4—1811no—
255C2×D4—7244no—
256C2×D4C22×D18111yescharacteristic
257C2×D4C22×D18112yes—
258C22×C4—3624no—
259C22×C4—3621no—
260C22×C4—3628no—
261C22×C4—3621no—
262C22×C4—3622no—
263C22×C4—3622no—
264C22×C4—1811no—
265C22×C4—1811no—
266C22×C4—1811no—
267C22×C4—1811no—
268C22×C4—1811no—
269C22×C4C22×D18111yescharacteristic
270SD16—3624no—
271SD16—3624no—
272D8—24no—
273D8—3624no—
274M16—3624no—
275C2×C8—1812no—
276C2×C8C22×D18112yes—
277C4⋊C4—3621no—
278C4⋊C4—1811no—
279C4⋊C4C22×D18111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C23—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—3622no—
290C23—3622no—
291C23—3621no—
292C23—1811no—
293C23—1811no—
294C23—7241no—
295C23—7242no—
296C23—7242no—
297C23—7242no—
298C23D4×D18111yescharacteristic
299Q8—3624no—
300Q8—3622no—
301Q8—1814no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—3628no—
307D4—3628no—
308D4—3622no—
309D4—7242no—
310D4—7242no—
311D4—7242no—
312D4—7242no—
313D4D4×D18114yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—3622no—
317C2×C4—3622no—
318C2×C4—3624no—
319C2×C4—3628no—
320C2×C4—3622no—
321C2×C4—3622no—
322C2×C4—3628no—
323C2×C4—3622no—
324C2×C4—3621no—
325C2×C4—1811no—
326C2×C4—1811no—
327C2×C4—1814no—
328C2×C4—7242no—
329C2×C4D4×D18111yescharacteristic
330C2×C4C23×D18111yescharacteristic
331C8—22no—
332C8—3622no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3621no—
343C22—3621no—
344C22—3621no—
345C22—3621no—
346C22—3621no—
347C22—14484no—
348C22—1812no—
349C22—7244no—
350C22—7241no—
351C22—7241no—
352C22—7241no—
353C22#1915384012yescomplemented
354C22C2×D4×D18111yescharacteristic
355C4—21no—
356C4—41no—
357C4—3621no—
358C4—3624no—
359C4—1812no—
360C4—7241no—
361C4C2×D4×D18112yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—3622no—
366C2—14481no—
367C2—7241no—
368C2#120×D18111yescharacteristic
369C2C2×#1915384011yescharacteristic
370C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3006
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×#15668448C2semidirect
2. C2×#19153840C2semidirect
3. #121×D181C2semidirect
4. #26041425C2semidirect
5. #26041421C2semidirect
6. #26041420C2semidirect
7. #26041417C2semidirect
8. #26041413C2semidirect
9. #26041277C2semidirect
10. #26041241C2semidirect
11. #447×C181C2semidirect
16. C2×#15668448C22semidirect
17. C22×Dic362C22semidirect
20. C2×D4×D181C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668437C22semidirect
26. C2×#15668434C22semidirect
27. C2×#15668432C22semidirect
28. #19153791C22semidirect
32. #19153787C22semidirect
33. C4⋊C4×D181C22semidirect
35. #19153741C22semidirect
36. #19153739C22semidirect
37. #19153737C22semidirect
38. C22⋊#13089678C22semidirect
39. #19153728C22semidirect
40. C22×D4×C181C22semidirect
41. C2×D8×C181C22semidirect
43. #102×C181C22semidirect
44. C22⋊C8×C181C22semidirect
52. #15668448D4semidirect
109. Dic362C2×D4semidirect
141. D4×C181C2×D4semidirect
156. C22×C181AGL(1,ℤ/8)semidirect
185. C181#3006semidirect
195. #447D181semidirect
207. C22×D4C2×D181semidirect
214. C2×D8C2×D181semidirect
227. #102C2×D181semidirect
230. C22⋊C8C2×D181semidirect
313. D4D4×D181semidirect
353. C22#19153840semidirect

Maximal subgroups

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

subgroupindex
C22×#156684482
C2×#191538402
#121×D1812
#260414252
#260414212
#260414202
#260414172
#260414132
#260412772
#260412412
#447×C1812
#3006181