Finite Group Databasebeta

← #26201502

Subgroups — #26201502

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
1#20254516C211yesmaximal, characteristic, complemented
2#20252852C211yesmaximal, characteristic
3#20252412C211yesmaximal, characteristic, complemented
4#20242799C214yesmaximal, complemented
5#20242765C214yesmaximal, complemented
6#20242158C211yesmaximal, characteristic, complemented
7#20242069C211yesmaximal, characteristic, complemented
8#20242038C211yesmaximal, characteristic, complemented
9C2×C47×#476C211yesmaximal, characteristic, complemented
10C2×#13094050—21no—
11#15689673C2211yescharacteristic
12#15689519C2212yescomplemented
13#15689471C2212yescomplemented
14#15689441—22no—
15#15689410—22no—
16#15689338—22no—
17#15689328C2212yes—
18#15689318C2212yes—
19D4×C47⋊C8—22no—
20D4×C47⋊C8C2214yes—
21#15689176C2211yescharacteristic
22#15689176C2211yescharacteristic
23#15689147C2211yescharacteristic
24#15689122—21no—
25C2×C22⋊(C47⋊C8)—22no—
26#15688873C2214yescomplemented
27#15688872C2214yescomplemented
28#15688870C2212yescomplemented
29#15688866C2212yescomplemented
30C2×C4⋊Q8×C47C2211yescharacteristic
31C2×C4×D4×C47C2211yescharacteristic
32C47×#476—22no—
33C47×#476C2214yes—
34C2×#106×C47—21no—
35C2×C4⋊C8×C47C2211yescharacteristic
36C2×#102×C47—21no—
37C2×#10625350—21no—
38#13094050—44no—
39C2×#10625335—22no—
40C2×#10625334—22no—
41C2×#10625329—21no—
42C22×C47⋊C8—24no—
43#13094024C2311yescharacteristic
44#13094021C2311yescharacteristic
45#13094012—21no—
46#13093998—21no—
47#13093998—21no—
48C4×D188—21no—
49C4×D188—21no—
50#13093955—24no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)D412yes—
53C4⋊(C47⋊C8)C2312yes—
54C4⋊(C47⋊C8)C2312yes—
55#13093947—22no—
56#13093944—22no—
57#13093944—24no—
58C22⋊(C47⋊C8)—21no—
59C22⋊(C47⋊C8)—24no—
60C22⋊(C47⋊C8)D412yes—
61C4×C47⋊C8—21no—
62C4×C47⋊C8C2312yes—
63C22×Q8×C47—21no—
64C22×D4×C47D411yescharacteristic
65C23×C4×C47—21no—
66C22×C8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47—21no—
69C4⋊Q8×C47—21no—
70C4⋊Q8×C47C2312yes—
71C4×D4×C47—21no—
72C4×D4×C47—21no—
73C4×D4×C47D412yes—
74C4×D4×C47C2312yes—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47D411yescharacteristic
78C2×C22⋊C4×C47—21no—
79C2×C42×C47C2311yescharacteristic
80#106×C47—24no—
81C4⋊C8×C47—21no—
82C4⋊C8×C47C2312yes—
83#102×C47—24no—
84C24×C47—21no—
85C2×Q8×C47—22no—
86C2×Q8×C47—41no—
87C2×Q8×C47—82no—
88C2×D4×C47—21no—
89C2×D4×C47—22no—
90C2×D4×C47SD1614yes—
91C2×D4×C47C2×D412yes—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—41no—
96C22×C4×C47—82no—
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C2×C22⋊D47—41no—
101#10625350—42no—
102#10625350—42no—
103C2×D188—21no—
104C2×D188—21no—
105C2×D188—21no—
106C2×C4×D47—41no—
107C2×Dic94—21no—
108C2×C8×C47—22no—
109C2×C8×C47—42no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C4⋊C4×C47C2×D412yes—
116C22⋊C4×C47—22no—
117C22⋊C4×C47—22no—
118C42×C47C2×D411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47C2411yescharacteristic
122#10625335—44no—
123#10625334—44no—
124(C2×D47)⋊C4—41no—
125(C2×D47)⋊C4—41no—
126C4⋊Dic47—21no—
127C4⋊Dic47—21no—
128#10625329—42no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—84no—
132C2×C47⋊C8C2×D414yes—
133C23×C47—22no—
134C23×C47—24no—
135C23×C47—81no—
136C23×C47—81no—
137C23×C47#12111yescharacteristic
138C22×D47—41no—
139Q8×C47—82no—
140Q8×C47—82no—
141D4×C47—22no—
142D4×C47—24no—
143D4×C47C2×SD1614yes—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—44no—
148C2×C4×C47—42no—
149C2×C4×C47—82no—
150C2×C4×C47—81no—
151C2×C4×C47—82no—
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47#12111yescharacteristic
155C2×C4×C47C2×SD1612yes—
156C2×C4×C47#13512yes—
157C2×C4×C47C22×D411yescharacteristic
158C2×C4×C47C22×D411yescharacteristic
159C2×C4×C47C2×#4111yescharacteristic
160C22⋊D47—82no—
161C2×Dic47—41no—
162D188—41no—
163D188—41no—
164D188—42no—
165D188—81no—
166C4×D47—82no—
167Dic94—41no—
168Dic94—41no—
169C8×C47—42no—
170C47⋊C8—24no—
171C47⋊C8—42no—
172C47⋊C8—42no—
173#19391—471nomaximal, Sylow
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—24no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—84no—
180C22×C47#46512yes—
181C22×C47C2×#12111yescharacteristic
182C2×D47—81no—
183C2×D47—81no—
184C2×D47—81no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—42no—
189C4×C47—82no—
190C4×C47#46512yes—
191C4×C47C22×SD1611yescharacteristic
192C4×C47C2×#13511yescharacteristic
193C4×C47#57712yes—
194Dic47—81no—
195#3227—471no—
196C2×#476D4711yescharacteristic, complemented
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—44no—
206C2×C47#304211yescharacteristic
207C2×C47C2×#46511yescharacteristic
208C2×C47#306811yescharacteristic
209D47—161no—
210C2×AGL(1,ℤ/8)—941no—
211#532—471no—
212C2×C4⋊Q8C2×D4711yescharacteristic
213C2×C4×D4C2×D4711yescharacteristic
214#498—472no—
215#486—472no—
216#481—942no—
217#476—22no—
218#476—942no—
219#476C2×D4714yes—
220#463—942no—
221#461—472no—
222#459—472no—
223C8×D4—474no—
224C8×D4—942no—
225#431—471no—
226#431—471no—
227C2×#106—21no—
228#423—471no—
229C2×C4⋊C8C2×D4711yescharacteristic
230#417—941no—
231C2×#102—21no—
232C2×C22⋊C8—942no—
233#334—474no—
234#333—474no—
235#332—472no—
236#331—472no—
237C47#1939111yescharacteristic, complemented, Sylow
238C2×#41—941no—
239C22×Q8—21no—
240C22×D4C22⋊D4711yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—1884no—
243C2×SD16—942no—
244C2×D8—942no—
245C2×M16—941no—
246C22×C8—21no—
247C22×C8—944no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—471no—
252C4⋊Q8C22×D4712yes—
253#127—471no—
254#124—941no—
255#121—941no—
256#121—941no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—941no—
260C4×D4—941no—
261C4×D4C22⋊D4712yes—
262C4×D4C22×D4712yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D4711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D4711yescharacteristic
268#106—24no—
269#106—944no—
270C4⋊C8—21no—
271C4⋊C8—472no—
272C4⋊C8—472no—
273C4⋊C8—472no—
274C4⋊C8—941no—
275C4⋊C8C22×D4712yes—
276#103—942no—
277#102—24no—
278#102—944no—
279#102—942no—
280C22⋊C8—472no—
281C22⋊C8—944no—
282C22⋊C8—941no—
283C4×C8—472no—
284C4×C8—941no—
285C24—21no—
286#41—1882no—
287#41—1882no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—941no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—941no—
295C2×D4—941no—
296C2×D4—941no—
297C2×D4—1881no—
298C2×D4#1062533614yes—
299C2×D4C2×C22⋊D4712yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—1881no—
306C22×C4D4×D4711yescharacteristic
307C22×C4#1062535211yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309SD16—1884no—
310D8—1884no—
311M16—1882no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—474no—
315C2×C8—942no—
316C2×C8—942no—
317C2×C8—3764no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—941no—
324C4⋊C4—941no—
325C4⋊C4C2×C22⋊D4712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—1881no—
329C22⋊C4—1881no—
330C42D4×D4711yescharacteristic
331C42#1062535211yescharacteristic
332C42C2×C22⋊D4711yescharacteristic
333C42C23×D4711yescharacteristic
334C23—22no—
335C23—24no—
336C23—81no—
337C23—81no—
338C23—1881no—
339C23#1309399611yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—1881no—
343Q8—1881no—
344D4—24no—
345D4—22no—
346D4—3761no—
347D4—3762no—
348D4—1881no—
349D4—1881no—
350D4—1882no—
351D4C2×#1062533614yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—3762no—
361C2×C4—1881no—
362C2×C4#1309399611yescharacteristic
363C2×C4#1309399611yescharacteristic
364C2×C4#1309399611yescharacteristic
365C2×C4C2×#1062533612yes—
366C2×C4#1309404412yes—
367C2×C4C2×D4×D4711yescharacteristic
368C2×C4C22×C22⋊D4711yescharacteristic
369C2×C4C2×#1062535211yescharacteristic
370C8—42no—
371C8—944no—
372C8—1882no—
373C8—1882no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—3761no—
381C22—3761no—
382C22—3761no—
383C22#1568935412yes—
384C22C2×#1309399611yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4—3761no—
391C4#1568935412yes—
392C4C22×#1062533611yescharacteristic
393C4C2×#1309404411yescharacteristic
394C4#1568992912yes—
395C2—22no—
396C2—44no—
397C2—7521no—
398C2#2025289711yescharacteristic
399C2C2×#1568935411yescharacteristic
400C2#2025311311yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#476C2
socleC22×C47C2×#121
prime radical#20252852C2
hypercenterC2×#476D47
hypocenterC47#19391
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #20254516C2semidirect
3. #20252412C2semidirect
4. #20242799C2semidirect
5. #20242765C2semidirect
6. #20242158C2semidirect
7. #20242069C2semidirect
8. #20242038C2semidirect
9. C2×C47×#476C2semidirect
12. #15689519C22semidirect
13. #15689471C22semidirect
26. #15688873C22semidirect
27. #15688872C22semidirect
28. #15688870C22semidirect
29. #15688866C22semidirect
196. C2×#476D47semidirect
237. C47#19391semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202528522
#202524122
#202427992
#202427652
#202421582
#202420692
#202420382
C2×C47×#4762
#1939147