Finite Group Databasebeta

← #26201501

Subgroups — #26201501

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic, complemented
2#20252848C211yesmaximal, characteristic
3#20252411C211yesmaximal, characteristic, complemented
4#20242794C214yesmaximal, complemented
5#20242760C214yesmaximal, complemented
6#20242159C211yesmaximal, characteristic, complemented
7#20242069C211yesmaximal, characteristic, complemented
8#20242037C211yesmaximal, characteristic, complemented
9C47×#3227C211yesmaximal, characteristic, complemented
10C2×#13094049—21no—
11C2×C4×D188C2211yescharacteristic
12#15689410—22no—
13#15689410C2214yes—
14#15689335—22no—
15#15689325C2212yes—
16#15689315C2212yes—
17C8×D188C2214yes—
18#15689188—22no—
19#15689176C2211yescharacteristic
20C2×#13093955—21no—
21#15689147C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23#15689124—21no—
24C2×#13093944—21no—
25C22⋊C8×D47—22no—
26#15688874C2214yescomplemented
27#15688872C2214yescomplemented
28#15688869C2212yescomplemented
29#15688867C2212yescomplemented
30C47×#532C2211yescharacteristic
31C2×C4⋊Q8×C47C2211yescharacteristic
32C47×#498C2212yescomplemented
33C47×#486C2212yescomplemented
34C47×#481—22no—
35C47×#476—22no—
36C47×#431C2211yescharacteristic
37C22×D188D411yescharacteristic
38C22×C4×D47—21no—
39#13094049—44no—
40C2×#10625336—22no—
41C2×#10625334—22no—
42C2×#10625329—21no—
43C22×C47⋊C8—21no—
44C2×C8×D47—24no—
45C4×D188—21no—
46C4×D188—21no—
47C4×D188D412yes—
48C4×D188C2312yes—
49C2×C4⋊Dic47D411yescharacteristic
50C2×(C2×D47)⋊C4—21no—
51#13093955—24no—
52C4⋊(C47⋊C8)—21no—
53C4⋊(C47⋊C8)—21no—
54C4⋊(C47⋊C8)C2312yes—
55C4⋊(C47⋊C8)C2312yes—
56C4⋊(C47⋊C8)C2312yes—
57#13093948—22no—
58#13093945—22no—
59#13093944—24no—
60C22⋊(C47⋊C8)—21no—
61(C2×D47)⋊C8—24no—
62C4×C47⋊C8—21no—
63C2×#41×C47—21no—
64C22×Q8×C47—21no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47C2312yes—
69C4⋊Q8×C47C2311yescharacteristic
70#127×C47C2311yescharacteristic
71#124×C47—21no—
72#121×C47—21no—
73#121×C47—21no—
74C4×D4×C47—21no—
75C4×D4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C42×C47C2311yescharacteristic
79#106×C47—24no—
80C4⋊C8×C47D412yes—
81#102×C47—24no—
82C22⋊C8×C47D412yes—
83C4×C8×C47C2312yes—
84C23×D47—21no—
85#41×C47—42no—
86#41×C47—42no—
87C2×Q8×C47—21no—
88C2×Q8×C47—22no—
89C2×Q8×C47—41no—
90C2×Q8×C47—82no—
91C2×D4×C47—21no—
92C2×D4×C47—21no—
93C2×D4×C47—21no—
94C2×D4×C47—41no—
95C22×C4×C47—41no—
96C22×C4×C47—41no—
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C22×Dic47—21no—
101C2×D188—21no—
102C2×D188—22no—
103C2×D188SD1614yes—
104C2×D188C2×D412yes—
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—82no—
108C2×C8×C47C2×D414yes—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C42×C47C2×D411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47#4111yescharacteristic
120C42×C47C2411yescharacteristic
121#10625336—44no—
122#10625334—44no—
123(C2×D47)⋊C4—22no—
124(C2×D47)⋊C4—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47C2×D412yes—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—42no—
132C8×D47—84no—
133C23×C47—41no—
134C23×C47#12111yescharacteristic
135C22×D47—22no—
136C22×D47—24no—
137C22×D47—81no—
138C22×D47—81no—
139Q8×C47—41no—
140Q8×C47—41no—
141Q8×C47—82no—
142Q8×C47—82no—
143D4×C47—41no—
144D4×C47—41no—
145D4×C47—42no—
146D4×C47—81no—
147D4×C47—82no—
148C2×C4×C47—22no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
152C2×C4×C47—82no—
153C2×C4×C47—82no—
154C2×C4×C47#12111yescharacteristic
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47C2×SD1612yes—
158C2×C4×C47#13512yes—
159C2×C4×C47C22×D411yescharacteristic
160C2×C4×C47C22×D411yescharacteristic
161C2×C4×C47C2×#4111yescharacteristic
162C2×Dic47—22no—
163C2×Dic47—81no—
164D188—22no—
165D188—24no—
166D188C2×SD1614yes—
167C4×D47—44no—
168C4×D47—82no—
169C8×C47—24no—
170C47⋊C8—42no—
171C47⋊C8—42no—
172C47⋊C8—42no—
173#19391—471nomaximal, Sylow
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—81no—
177C22×C47—81no—
178C22×C47—81no—
179C22×C47#46512yes—
180C22×C47C2×#12111yescharacteristic
181C2×D47—24no—
182C2×D47—42no—
183C2×D47—42no—
184C2×D47—84no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—81no—
189C4×C47—82no—
190C4×C47#46512yes—
191C4×C47C22×SD1611yescharacteristic
192C4×C47C2×#13511yescharacteristic
193C4×C47#57712yes—
194Dic47—42no—
195#3227D4711yescharacteristic, complemented
196C2×#476—471no—
197#3036—471no—
198#2972—471no—
199#1677—474no—
200#1671—474no—
201#1570—471no—
202#1548—471no—
203#1540—471no—
204C2×C47—22no—
205C2×C47—161no—
206C2×C47#304211yescharacteristic
207C2×C47C2×#46511yescharacteristic
208C2×C47#306811yescharacteristic
209D47—44no—
210C2×AGL(1,ℤ/8)—941no—
211#532C2×D4711yescharacteristic
212C2×C4⋊Q8C2×D4711yescharacteristic
213C2×C4×D4—471no—
214#498C2×D4712yescomplemented
215#486C2×D4712yescomplemented
216#481—22no—
217#476—22no—
218#476—474no—
219#476—942no—
220#463—942no—
221#461—472no—
222#459—472no—
223C8×D4—474no—
224C8×D4—942no—
225#431—471no—
226#431C2×D4711yescharacteristic
227C2×#106—941no—
228#423—471no—
229C2×C4⋊C8—471no—
230#417—941no—
231C2×#102—941no—
232C2×C22⋊C8—942no—
233#334—474no—
234#333—474no—
235#332—472no—
236#331—472no—
237C47#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—21no—
240C22×D4—471no—
241C23×C4—941no—
242AGL(1,ℤ/8)—1884no—
243C2×SD16—942no—
244C2×D8—942no—
245C2×M16—941no—
246C22×C8—941no—
247C22×C8—944no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22×D4712yes—
252C4⋊Q8C22×D4711yescharacteristic
253#127C22×D4711yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—472no—
260C4×D4—472no—
261C4×D4—941no—
262C4×D4—941no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—471no—
266C2×C22⋊C4—941no—
267C2×C42C22×D4711yescharacteristic
268#106—24no—
269#106—944no—
270C4⋊C8—472no—
271C4⋊C8—472no—
272C4⋊C8—472no—
273C4⋊C8—941no—
274C4⋊C8—941no—
275C4⋊C8C22⋊D4712yes—
276#103—942no—
277#102—24no—
278#102—944no—
279#102—942no—
280C22⋊C8—941no—
281C22⋊C8—944no—
282C22⋊C8C22⋊D4712yes—
283C4×C8—941no—
284C4×C8C22×D4712yes—
285C24—941no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—474no—
297C2×D4—472no—
298C2×D4—941no—
299C2×D4—942no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—941no—
303C22×C4—942no—
304C22×C4—942no—
305C22×C4—3762no—
306C22×C4D4×D4711yescharacteristic
307C22×C4#1062535211yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309SD16—1884no—
310D8—1884no—
311M16—1882no—
312C2×C8—942no—
313C2×C8—942no—
314C2×C8—942no—
315C2×C8—3764no—
316C2×C8—1882no—
317C2×C8C2×C22⋊D4714yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—472no—
325C4⋊C4—941no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—942no—
329C22⋊C4—942no—
330C42D4×D4711yescharacteristic
331C42#1062535211yescharacteristic
332C42C2×C22⋊D4711yescharacteristic
333C42C23×D4711yescharacteristic
334C23—41no—
335C23—944no—
336C23—942no—
337C23—3761no—
338C23—3761no—
339C23#1309399511yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—82no—
343Q8—82no—
344D4—41no—
345D4—41no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—474no—
350D4—944no—
351D4—942no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—942no—
359C2×C4—3762no—
360C2×C4—3761no—
361C2×C4—1884no—
362C2×C4#1309399511yescharacteristic
363C2×C4#1309399511yescharacteristic
364C2×C4#1309399511yescharacteristic
365C2×C4SD16×D4712yes—
366C2×C4#1309404512yes—
367C2×C4C2×D4×D4711yescharacteristic
368C2×C4C22×C22⋊D4711yescharacteristic
369C2×C4C2×#1062535211yescharacteristic
370C8—24no—
371C8—1882no—
372C8—1882no—
373C8—1882no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—944no—
380C22—3764no—
381C22—1882no—
382C22—1882no—
383C22#1568934912yes—
384C22C2×#1309399511yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—82no—
390C4—1882no—
391C4#1568934912yes—
392C4C2×SD16×D4711yescharacteristic
393C4C2×#1309404511yescharacteristic
394C4#1568993112yes—
395C2—22no—
396C2—161no—
397C2—1884no—
398C2#2025289611yescharacteristic
399C2C2×#1568934911yescharacteristic
400C2#2025311211yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3227C2
socleC22×C47C2×#121
prime radical#20252848C2
hypercenter#3227D47
hypocenterC47#19391
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C2×#15689410C2semidirect
3. #20252411C2semidirect
4. #20242794C2semidirect
5. #20242760C2semidirect
6. #20242159C2semidirect
7. #20242069C2semidirect
8. #20242037C2semidirect
9. C47×#3227C2semidirect
26. #15688874C22semidirect
27. #15688872C22semidirect
28. #15688869C22semidirect
29. #15688867C22semidirect
32. C47×#498C22semidirect
33. C47×#486C22semidirect
195. #3227D47semidirect
214. #498C2×D47semidirect
215. #486C2×D47semidirect
237. C47#19391semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
#202528482
#202524112
#202427942
#202427602
#202421592
#202420692
#202420372
C47×#32272
#1939147