Finite Group Databasebeta

← #26370065

Subgroups — #26370065

Order 12032.

1380 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1380 conjugacy classes of subgroups — too many to draw. Only its 211 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×#15689559C212yesmaximal, complemented
2#20252969C211yesmaximal, characteristic, complemented
3#20245317C214yesmaximal, complemented
4#20245300C214yesmaximal, complemented
5#20245104C212yesmaximal, complemented
6#20244999C211yesmaximal, characteristic, complemented
7C23×D8×C47C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic, complemented
9#15689634—21no—
10#15689559—28no—
11C2×#13093968C2214yescomplemented
12C2×#13093965C2214yescomplemented
13C2×#13093964C2212yescomplemented
14#15689364C2212yescomplemented
15#15689359C2212yescomplemented
16#15689352—22no—
17#15689161C2211yescharacteristic, complemented
18#15689120—21no—
19#15688924C2218yescomplemented
20C22⋊(C47⋊C16)C2214yescomplemented
21C23×D4×C47—21no—
22C22×D8×C47—28no—
23C22×D8×C47—22no—
24C22×D8×C47—21no—
25C22×D8×C47C2214yes—
26C22×D8×C47C2212yes—
27C23×C8×C47C2211yescharacteristic
28C2×#10625350—21no—
29#13094046—24no—
30C2×Dic188C2311yescharacteristic, complemented
31C2×#10625325—21no—
32C2×D376C2311yescharacteristic, complemented
33#13094007—41no—
34#13094003—22no—
35#13093998—22no—
36#13093993—41no—
37C4×C22⋊D47—42no—
38#13093983—21no—
39#13093968—28no—
40#13093965—28no—
41#13093964—24no—
42C2×C47⋊C16C2314yescomplemented
43#13093958C2312yescomplemented
44#13093956—21no—
45#13093949—22no—
46#13093946—22no—
47C25×C47—41no—
48C22×D4×C47—28no—
49C22×D4×C47—24no—
50C22×D4×C47—22no—
51C22×D4×C47—44no—
52C22×D4×C47—42no—
53C22×D4×C47—41no—
54C23×C4×C47D411yescharacteristic
55C2×D8×C47—28no—
56C2×D8×C47—216no—
57C2×D8×C47—216no—
58C2×D8×C47—24no—
59C2×D8×C47—22no—
60C2×D8×C47—24no—
61C2×D8×C47D418yescomplemented
62C2×D8×C47C2314yes—
63C22×C8×C47—24no—
64C22×C8×C47D412yes—
65C22×C8×C47D411yescharacteristic
66C22×C8×C47C2312yes—
67C22×C8×C47C2311yescharacteristic
68C24×C47—48no—
69C24×C47—44no—
70C24×C47—42no—
71C24×C47—88no—
72C24×C47D811yescharacteristic
73C2×D4×C47—216no—
74C2×D4×C47—28no—
75C2×D4×C47—24no—
76C2×D4×C47—48no—
77C2×D4×C47—48no—
78C2×D4×C47—416no—
79C2×D4×C47—44no—
80C2×D4×C47—42no—
81C2×D4×C47—44no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47D812yes—
85C22×C4×C47D811yescharacteristic
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×D411yescharacteristic
88C2×C22⋊D47—42no—
89C2×C22⋊D47—41no—
90#10625350—42no—
91#10625350—42no—
92C2×D188—21no—
93C2×C4×D47—41no—
94C2×Dic94—21no—
95D8×C47—216no—
96D8×C47—28no—
97D8×C47—24no—
98D8×C47C2×D418yes—
99C2×C8×C47—22no—
100C2×C8×C47—24no—
101C2×C8×C47—24no—
102C2×C8×C47C2×D412yes—
103C2×C8×C47C2×D411yescharacteristic
104C2×C8×C47C2×D414yes—
105C2×C8×C47C2411yescharacteristic
106C22⋊Dic47—41no—
107C22⋊Dic47—42no—
108(C2×D47)⋊C4—42no—
109C4⋊Dic47—22no—
110Dic47⋊C4—42no—
111C4×Dic47—41no—
112Dic188—22no—
113D376—22no—
114#10625325—42no—
115C47⋊C16—24no—
116C23×C47—22no—
117C23×C47—24no—
118C23×C47—416no—
119C23×C47—44no—
120C23×C47—48no—
121C23×C47—832no—
122C23×C47—88no—
123C23×C47—88no—
124C23×C47—88no—
125C23×C47C2×D811yescharacteristic
126C23×C47C2×D812yes—
127C22×D47—41no—
128D4×C47—28no—
129D4×C47—416no—
130D4×C47—44no—
131D4×C47—48no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—24no—
136C2×C4×C47#12012yes—
137C2×C4×C47C2×D812yes—
138C2×C4×C47C2×D811yescharacteristic
139C2×C4×C47AGL(1,ℤ/8)12yes—
140C2×C4×C47C22×D411yescharacteristic
141C22⋊D47—82no—
142C22⋊D47—84no—
143C2×Dic47—42no—
144C2×Dic47—41no—
145D188—41no—
146D188—41no—
147C4×D47—82no—
148Dic94—41no—
149Dic94—41no—
150C8×C47—22no—
151C8×C47#12012yes—
152C8×C47C22×D412yes—
153#33517—471nomaximal, Sylow
154C22×C47—28no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—22no—
158C22×C47—48no—
159C22×C47—816no—
160C22×C47—832no—
161C22×C47—88no—
162C22×C47#44712yes—
163C22×C47#50914yescomplemented
164C22×C47C22×D811yescharacteristic
165C2×D47—82no—
166C2×D47—81no—
167C4×C47—22no—
168C4×C47#44712yes—
169C4×C47C2×#12011yescharacteristic
170C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
171Dic47—82no—
172Dic47—81no—
173C23×D8D4711yescharacteristic, complemented
174C2×#509—472no—
175#3050—471no—
176#2187—474no—
177#2184—474no—
178#2140—472no—
179#2112—471no—
180C2×C47—22no—
181C2×C47—24no—
182C2×C47—88no—
183C2×C47C2×#44711yescharacteristic
184C2×C47C2×#50912yes—
185D47—161no—
186C23×D4—21no—
187C2×#135—471no—
188C22×D8—28no—
189C22×D8—21no—
190C22×D8—22no—
191C22×D8C2×D4714yes—
192C22×D8C2×D4712yes—
193C23×C8C2×D4711yescharacteristic
194#525—941no—
195#509—948no—
196C2×SD32—474no—
197C2×D16—474no—
198C2×M32—472no—
199#467—472no—
200#466—472no—
201#465—942no—
202#427—471no—
203#416—941no—
204#357—478no—
205C22⋊C16—474no—
206C47#3351711yescharacteristic, complemented, Sylow
207C25—41no—
208C2×#41—941no—
209C22×D4—28no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—41no—
215C23×C4C22⋊D4711yescharacteristic
216#135—944no—
217C2×Q16—471no—
218C2×SD16—941no—
219C2×D8—28no—
220C2×D8—216no—
221C2×D8—216no—
222C2×D8—24no—
223C2×D8—22no—
224C2×D8—24no—
225C2×D8—471no—
226C2×D8C22⋊D4718yescomplemented
227C2×D8C22×D4714yes—
228C22×C8—24no—
229C22×C8C22⋊D4711yescharacteristic
230C22×C8C22⋊D4712yes—
231C22×C8C22×D4712yes—
232C22×C8C22×D4711yescharacteristic
233#123—1881no—
234C22⋊Q8—942no—
235#121—942no—
236#120—1881no—
237C4×D4—1882no—
238#117—941no—
239SD32—948no—
240D16—948no—
241M32—944no—
242C2×C16—474no—
243#107—472no—
244#106—941no—
245#103—942no—
246#102—942no—
247C24—48no—
248C24—44no—
249C24—42no—
250C24—88no—
251C24#1062533411yescharacteristic
252#41—1882no—
253#41—1882no—
254C2×Q8—941no—
255C2×D4—216no—
256C2×D4—28no—
257C2×D4—24no—
258C2×D4—48no—
259C2×D4—416no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—941no—
265C2×D4—1882no—
266C2×D4—1881no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—1881no—
270C22×C4#1062533412yes—
271C22×C4#1062533411yescharacteristic
272C22×C4C2×C22⋊D4712yes—
273C22×C4C2×C22⋊D4711yescharacteristic
274Q16—942no—
275SD16—1882no—
276D8—216no—
277D8—28no—
278D8—24no—
279D8—942no—
280D8C2×C22⋊D4718yes—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8C2×C22⋊D4712yes—
285C2×C8C2×C22⋊D4711yescharacteristic
286C2×C8C2×C22⋊D4714yes—
287C2×C8C23×D4711yescharacteristic
288C4⋊C4—942no—
289C4⋊C4—1882no—
290C22⋊C4—1881no—
291C22⋊C4—1882no—
292C22⋊C4—1882no—
293C42—1881no—
294C16—944no—
295C23—22no—
296C23—24no—
297C23—416no—
298C23—48no—
299C23—44no—
300C23—832no—
301C23—88no—
302C23—88no—
303C23—88no—
304C23—1881no—
305C23C2×#1062533411yescharacteristic
306C23C2×#1062533412yes—
307Q8—1881no—
308Q8—1881no—
309D4—28no—
310D4—416no—
311D4—48no—
312D4—44no—
313D4—3762no—
314D4—3764no—
315D4—1881no—
316D4—1881no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—3762no—
322C2×C4—1882no—
323C2×C4—1881no—
324C2×C4#1309399312yes—
325C2×C4C2×#1062533412yes—
326C2×C4C2×#1062533411yescharacteristic
327C2×C4#1309405012yes—
328C2×C4C22×C22⋊D4711yescharacteristic
329C8—22no—
330C8#1309399312yes—
331C8C22×C22⋊D4712yes—
332C22—28no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—48no—
337C22—816no—
338C22—832no—
339C22—88no—
340C22—3761no—
341C22—3762no—
342C22#1568925712yes—
343C22#1568955914yescomplemented
344C22C22×#1062533411yescharacteristic
345C4—22no—
346C4—3761no—
347C4—3762no—
348C4#1568925712yes—
349C4C2×#1309399311yescharacteristic
350C4C2×#1309405011yescharacteristic
351C2—22no—
352C2—24no—
353C2—88no—
354C2—7521no—
355C2C2×#1568925711yescharacteristic
356C2C2×#1568955912yes—
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC23×D8×C47C2
socleC22×C47C22×D8
hypercenterC23×D8D47
hypocenterC47#33517
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15689559C2semidirect
2. #20252969C2semidirect
3. #20245317C2semidirect
4. #20245300C2semidirect
5. #20245104C2semidirect
6. #20244999C2semidirect
7. C23×D8×C47C2semidirect
8. C2×#13094046C22semidirect
11. C2×#13093968C22semidirect
12. C2×#13093965C22semidirect
13. C2×#13093964C22semidirect
14. #15689364C22semidirect
15. #15689359C22semidirect
17. #15689161C22semidirect
19. #15688924C22semidirect
20. C22⋊(C47⋊C16)C22semidirect
30. C2×Dic188C23semidirect
32. C2×D376C23semidirect
42. C2×C47⋊C16C23semidirect
43. #13093958C23semidirect
61. C2×D8×C47D4semidirect
163. C22×C47#509semidirect
173. C23×D8D47semidirect
206. C47#33517semidirect
226. C2×D8C22⋊D47semidirect
343. C22#15689559semidirect

Maximal subgroups

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

subgroupindex
C2×#156895592
#202529692
#202453172
#202453002
#202451042
#202449992
C23×D8×C472
#3351747