Finite Group Databasebeta

← #26373101

Subgroups — #26373101

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255736C211yesmaximal, characteristic, complemented
2#20245611C211yesmaximal, characteristic, complemented
3#20245566C212yesmaximal, complemented
4#20245525C211yesmaximal, characteristic, complemented
5#20245472C212yesmaximal, complemented
6#20245420C211yesmaximal, characteristic, complemented
7#20245300C212yesmaximal, complemented
8C4×#13093965C211yesmaximal, characteristic, complemented
9D4×C47⋊C16C211yesmaximal, characteristic, complemented
10C2×#15688924C212yesmaximal, complemented
11D4×D8×C47C211yesmaximal, characteristic, complemented
12#15689777—21no—
13C2×#13093965C2211yescharacteristic
14C2×#13093965C2212yescomplemented
15C22×C47⋊C16C2212yescomplemented
16#15689528C2211yescharacteristic, complemented
17#15689437—22no—
18#15689404—21no—
19#15689359C2212yescomplemented
20C4×D376C2211yescharacteristic, complemented
21C2×#13093958C2212yescomplemented
22#15688948C2212yescomplemented
23#15688948C2211yescharacteristic, complemented
24C4⋊(C47⋊C16)C2211yescharacteristic, complemented
25#15688924C2212yes—
26#15688924C2211yescharacteristic
27#15688924C2212yescomplemented
28#15688924C2212yescomplemented
29#15688924C2212yescomplemented
30#15688923C2211yescharacteristic, complemented
31C22⋊(C47⋊C16)C2212yescomplemented
32C4×C47⋊C16C2211yescharacteristic, complemented
33C22×D8×C47C2212yes—
34D42×C47—21no—
35C47×#493C2211yescharacteristic
36C47×#466C2212yes—
37C47×#459—21no—
38C47×#447—22no—
39C4×D8×C47C2211yescharacteristic
40C8×D4×C47C2211yescharacteristic
41C2×#10625352—41no—
42C2×D376C2311yescharacteristic
43#13094024—21no—
44#13094008—42no—
45#13094004—41no—
46#13093998—22no—
47C4×D188—21no—
48C2×C4⋊Dic47—22no—
49#13093965—22no—
50#13093965—42no—
51C2×C47⋊C16—42no—
52C2×C47⋊C16C2311yescharacteristic
53C2×C47⋊C16C2312yes—
54C2×C47⋊C16C2311yescharacteristic
55#13093958—21no—
56#13093958—22no—
57#13093958C2312yes—
58#13093958C2311yescharacteristic
59#13093946—22no—
60#13093946—21no—
61C22×D4×C47—22no—
62C22×D4×C47—41no—
63C2×D8×C47—24no—
64C2×D8×C47—22no—
65C2×D8×C47—21no—
66C2×D8×C47—42no—
67C2×D8×C47C2312yes—
68C2×D8×C47C2311yescharacteristic
69C22×C8×C47C2312yes—
70#127×C47—21no—
71#121×C47—22no—
72#121×C47—41no—
73#120×C47—42no—
74C4×D4×C47—21no—
75C4×D4×C47D411yescharacteristic
76#107×C47C2311yescharacteristic
77C4⋊C8×C47D411yescharacteristic
78#102×C47—22no—
79#102×C47—21no—
80C22⋊C8×C47D412yes—
81C4×C8×C47C2311yescharacteristic
82C24×C47—42no—
83C2×D4×C47—21no—
84C2×D4×C47—22no—
85C2×D4×C47—44no—
86C2×D4×C47—42no—
87C2×D4×C47—42no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—82no—
91C2×D4×C47—84no—
92C2×D4×C47D811yescharacteristic
93C22×C4×C47—41no—
94C22×C4×C47C2×D412yes—
95C2×C22⋊D47—42no—
96C22×Dic47—42no—
97#10625352—84no—
98C2×D188—21no—
99C2×C4×D47—41no—
100C2×Dic94—41no—
101D8×C47—22no—
102D8×C47—21no—
103D8×C47—22no—
104D8×C47—44no—
105D8×C47—81no—
106D8×C47#4112yescomplemented
107C2×C8×C47—22no—
108C2×C8×C47—22no—
109C2×C8×C47C2×D412yes—
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47C2411yescharacteristic
112C4⋊C4×C47—21no—
113C4⋊C4×C47D811yescharacteristic
114C22⋊C4×C47—41no—
115C22⋊C4×C47—42no—
116C22⋊C4×C47D812yes—
117C42×C47C2×D411yescharacteristic
118(C2×D47)⋊C4—41no—
119(C2×D47)⋊C4—42no—
120C4⋊Dic47—21no—
121C4⋊Dic47—22no—
122C4⋊Dic47—42no—
123C4⋊Dic47—41no—
124D376—21no—
125D376—21no—
126C47⋊C16—21no—
127C47⋊C16—21no—
128C47⋊C16—22no—
129C23×C47—41no—
130C23×C47—42no—
131C23×C47—84no—
132C23×C47—84no—
133C23×C47—162no—
134C23×C47C2×D812yes—
135C22×D47—41no—
136D4×C47—22no—
137D4×C47—42no—
138D4×C47—41no—
139D4×C47—42no—
140D4×C47—84no—
141D4×C47—84no—
142D4×C47—82no—
143D4×C47—161no—
144D4×C47D1614yescomplemented
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47—41no—
148C2×C4×C47—82no—
149C2×C4×C47C2×D811yescharacteristic
150C2×C4×C47C2×D812yes—
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47C22×D411yescharacteristic
153C22⋊D47—84no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—84no—
157D188—41no—
158D188—41no—
159C4×D47—82no—
160Dic94—82no—
161C8×C47—22no—
162C8×C47—21no—
163C8×C47C2×#4111yescharacteristic
164C8×C47H(5,𝔽2)11yescharacteristic
165#33861—471nomaximal, Sylow
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—42no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—82no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47—168no—
175C22×C47C2×D1614yes—
176C22×C47C22×D811yescharacteristic
177C2×D47—81no—
178C2×D47—81no—
179C2×D47—81no—
180C4×C47—21no—
181C4×C47—22no—
182C4×C47—81no—
183C4×C47C2×D1612yes—
184C4×C47#54611yescharacteristic
185C4×C47#57711yescharacteristic
186Dic47—82no—
187Dic47—81no—
188#3334—471no—
189D4×D8D4711yescharacteristic, complemented
190#2240—471no—
191#2232—472no—
192#2224—471no—
193#2215—472no—
194#2206—471no—
195#2184—472no—
196C4×D16—471no—
197D4×C16—471no—
198C2×#357—472no—
199C2×C47—24no—
200C2×C47—82no—
201C2×C47—162no—
202C2×C47#329411yescharacteristic
203C2×C47C22×D1611yescharacteristic
204C2×C47#341611yescharacteristic
205D47—161no—
206C22×D8C2×D4712yes—
207#547—941no—
208D42—21no—
209C2×D16—471no—
210C2×D16—472no—
211C22×C16—472no—
212#500—471no—
213#493C2×D4711yescharacteristic
214#480—942no—
215#474—941no—
216#466—472no—
217#466C2×D4712yes—
218#459—21no—
219#447—22no—
220C4×D8—471no—
221C4×D8C2×D4711yescharacteristic
222C8×D4C2×D4711yescharacteristic
223C2×#107—472no—
224#366—471no—
225#366—472no—
226C4⋊C16—471no—
227#357—471no—
228#357—472no—
229#357—472no—
230#357—472no—
231#357—472no—
232#357—471no—
233C22⋊C16—472no—
234C4×C16—471no—
235C47#3386111yescharacteristic, complemented, Sylow
236C2×#41—1881no—
237C22×D4—22no—
238C22×D4—41no—
239C2×D8—24no—
240C2×D8—22no—
241C2×D8—21no—
242C2×D8—42no—
243C2×D8—471no—
244C2×D8C22×D4712yes—
245C2×D8C22×D4711yescharacteristic
246C22×C8C22×D4712yes—
247C4⋊Q8—941no—
248#127—21no—
249#123—1882no—
250C22⋊Q8—1881no—
251#121—22no—
252#121—41no—
253#121—942no—
254#120—42no—
255C4×D4—21no—
256C4×D4—941no—
257C4×D4C22⋊D4711yescharacteristic
258C2×C4⋊C4—942no—
259D16—942no—
260D16—1882no—
261C2×C16—471no—
262C2×C16—471no—
263C2×C16—472no—
264C2×C16—1882no—
265#107—471no—
266#107—472no—
267#107—942no—
268#107—941no—
269#107C22×D4711yescharacteristic
270C4⋊C8C22⋊D4711yescharacteristic
271#102—21no—
272#102—22no—
273#102—941no—
274#102—942no—
275C22⋊C8C22⋊D4712yes—
276C4×C8C22×D4711yescharacteristic
277C24—42no—
278#41—3764no—
279C2×Q8—1881no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4—941no—
290C2×D4—1882no—
291C2×D4#1062533411yescharacteristic
292C22×C4—41no—
293C22×C4—1882no—
294C22×C4—1881no—
295C22×C4C2×C22⋊D4712yes—
296D8—22no—
297D8—21no—
298D8—22no—
299D8—44no—
300D8—81no—
301D8—941no—
302D8—941no—
303D8#1062535212yescomplemented
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×C22⋊D4711yescharacteristic
307C2×C8C2×C22⋊D4712yes—
308C2×C8C23×D4711yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—941no—
311C4⋊C4—942no—
312C4⋊C4—1882no—
313C4⋊C4—1881no—
314C4⋊C4#1062533411yescharacteristic
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—1881no—
318C22⋊C4—1882no—
319C22⋊C4#1062533412yes—
320C42C2×C22⋊D4711yescharacteristic
321C16—941no—
322C16—941no—
323C16—942no—
324C23—41no—
325C23—42no—
326C23—84no—
327C23—84no—
328C23—162no—
329C23—1881no—
330C23C2×#1062533412yes—
331Q8—3762no—
332D4—22no—
333D4—42no—
334D4—41no—
335D4—42no—
336D4—84no—
337D4—84no—
338D4—82no—
339D4—161no—
340D4—3764no—
341D4—1881no—
342D4—1881no—
343D4#1309396514yescomplemented
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—3764no—
349C2×C4—3762no—
350C2×C4—1881no—
351C2×C4—1882no—
352C2×C4C2×#1062533411yescharacteristic
353C2×C4C2×#1062533412yes—
354C2×C4C2×#1062533411yescharacteristic
355C2×C4C22×C22⋊D4711yescharacteristic
356C8—22no—
357C8—21no—
358C8C2×#1062535211yescharacteristic
359C8#1309406911yescharacteristic
360C22—22no—
361C22—22no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—168no—
369C22—3761no—
370C22—3761no—
371C22—3761no—
372C22C2×#1309396514yes—
373C22C22×#1062533411yescharacteristic
374C4—21no—
375C4—22no—
376C4—81no—
377C4—3761no—
378C4—3762no—
379C4C2×#1309396512yes—
380C4#1568976711yescharacteristic
381C4#1568993411yescharacteristic
382C2—24no—
383C2—82no—
384C2—162no—
385C2—7521no—
386C2#2025520111yescharacteristic
387C2C22×#1309396511yescharacteristic
388C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingD4×D8×C47C2
socleC22×C47C22×D8
hypercenterD4×D8D47
hypocenterC47#33861
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #20255736C2semidirect
2. #20245611C2semidirect
3. #20245566C2semidirect
4. #20245525C2semidirect
5. #20245472C2semidirect
6. #20245420C2semidirect
7. #20245300C2semidirect
8. C4×#13093965C2semidirect
9. D4×C47⋊C16C2semidirect
10. C2×#15688924C2semidirect
11. D4×D8×C47C2semidirect
14. C2×#13093965C22semidirect
15. C22×C47⋊C16C22semidirect
16. #15689528C22semidirect
19. #15689359C22semidirect
20. C4×D376C22semidirect
21. C2×#13093958C22semidirect
22. #15688948C22semidirect
23. #15688948C22semidirect
24. C4⋊(C47⋊C16)C22semidirect
27. #15688924C22semidirect
28. #15688924C22semidirect
29. #15688924C22semidirect
30. #15688923C22semidirect
31. C22⋊(C47⋊C16)C22semidirect
32. C4×C47⋊C16C22semidirect
106. D8×C47#41semidirect
144. D4×C47D16semidirect
189. D4×D8D47semidirect
235. C47#33861semidirect
303. D8#10625352semidirect
343. D4#13093965semidirect

Maximal subgroups

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

subgroupindex
#202557362
#202456112
#202455662
#202455252
#202454722
#202454202
#202453002
C4×#130939652
D4×C47⋊C162
C2×#156889242
D4×D8×C472
#3386147