Finite Group Databasebeta

← #26201507

Subgroups — #26201507

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#20254513C211yesmaximal, characteristic, complemented
2C2×#15689413C211yesmaximal, characteristic, complemented
3#20252412C211yesmaximal, characteristic, complemented
4#20242795C214yesmaximal, complemented
5#20242761C214yesmaximal, complemented
6#20242160C211yesmaximal, characteristic, complemented
7#20242070C211yesmaximal, characteristic, complemented
8#20242037C211yesmaximal, characteristic, complemented
9C47×#3036C211yesmaximal, characteristic
10C2×#13094024C2211yescharacteristic
11#15689519C2212yescomplemented
12#15689469C2212yescomplemented
13#15689439—22no—
14#15689413—22no—
15#15689413—22no—
16#15689413C2214yes—
17D4×C47⋊C8—22no—
18D4×C47⋊C8C2214yes—
19#15689176C2211yescharacteristic
20#15689176C2211yescharacteristic
21C2×#13093955—21no—
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×#13093945—21no—
24C2×C22⋊(C47⋊C8)—22no—
25#15688873C2214yescomplemented
26#15688871C2214yescomplemented
27#15688869C2212yescomplemented
28#15688867C2212yescomplemented
29C2×AGL(1,ℤ/8)×C47—21no—
30C47×#532C2211yescharacteristic
31C2×C4×D4×C47C2211yescharacteristic
32C47×#463—22no—
33C47×#461C2212yes—
34C47×#459C2212yes—
35C47×#423C2211yescharacteristic
36C47×#417—21no—
37C22×Dic94—21no—
38C22×C47⋊C8—21no—
39C22×C47⋊C8—24no—
40#13094024—21no—
41#13094024—21no—
42#13094024—21no—
43#13094024C2312yes—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46#13093955—24no—
47#13093955—24no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)D412yes—
50C4⋊(C47⋊C8)C2312yes—
51C4⋊(C47⋊C8)C2312yes—
52#13093945—24no—
53#13093945—24no—
54C22⋊(C47⋊C8)—21no—
55C22⋊(C47⋊C8)—24no—
56C22⋊(C47⋊C8)D412yes—
57C4×C47⋊C8—21no—
58C4×C47⋊C8C2312yes—
59C2×#41×C47—21no—
60C22×D4×C47D411yescharacteristic
61C23×C4×C47—21no—
62AGL(1,ℤ/8)×C47—44no—
63C2×SD16×C47—22no—
64C2×D8×C47—22no—
65C2×M16×C47—21no—
66C4⋊Q8×C47C2311yescharacteristic
67#127×C47C2311yescharacteristic
68#124×C47—21no—
69#121×C47—21no—
70#121×C47—21no—
71C4×D4×C47—21no—
72C4×D4×C47—21no—
73C4×D4×C47—21no—
74C4×D4×C47—21no—
75C4×D4×C47D412yes—
76C4×D4×C47C2312yes—
77C2×C4⋊C4×C47D411yescharacteristic
78C2×C22⋊C4×C47—21no—
79C2×C42×C47C2311yescharacteristic
80C4⋊C8×C47—21no—
81C4⋊C8×C47C2312yes—
82#103×C47—22no—
83#102×C47—22no—
84C24×C47—21no—
85#41×C47—42no—
86#41×C47—42no—
87C2×Q8×C47—21no—
88C2×D4×C47—21no—
89C2×D4×C47—22no—
90C2×D4×C47—21no—
91C2×D4×C47—21no—
92C2×D4×C47—21no—
93C2×D4×C47—41no—
94C2×D4×C47SD1614yes—
95C2×D4×C47C2×D412yes—
96C22×C4×C47—21no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—41no—
100C22×C4×C47—82no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×Dic47—41no—
105C2×Dic94—22no—
106C2×Dic94—41no—
107C2×Dic94—82no—
108SD16×C47—44no—
109D8×C47—44no—
110M16×C47—42no—
111C2×C8×C47—22no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47C2×D412yes—
116C22⋊C4×C47—22no—
117C22⋊C4×C47—22no—
118C22⋊C4×C47—41no—
119C22⋊C4×C47—41no—
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C42×C47C2411yescharacteristic
124C4⋊Dic47—22no—
125C4⋊Dic47—22no—
126C4⋊Dic47—41no—
127C4⋊Dic47—41no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131C2×C47⋊C8—84no—
132C2×C47⋊C8C2×D414yes—
133C23×C47—22no—
134C23×C47—24no—
135C23×C47—41no—
136C23×C47—81no—
137C23×C47—81no—
138C23×C47#12111yescharacteristic
139Q8×C47—41no—
140Q8×C47—41no—
141D4×C47—22no—
142D4×C47—24no—
143D4×C47—41no—
144D4×C47—41no—
145D4×C47—42no—
146D4×C47—81no—
147D4×C47—82no—
148D4×C47C2×SD1614yes—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—41no—
152C2×C4×C47—44no—
153C2×C4×C47—41no—
154C2×C4×C47—82no—
155C2×C4×C47—81no—
156C2×C4×C47—82no—
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47#12111yescharacteristic
159C2×C4×C47#12111yescharacteristic
160C2×C4×C47C2×SD1612yes—
161C2×C4×C47#13512yes—
162C2×C4×C47C22×D411yescharacteristic
163C2×C4×C47C22×D411yescharacteristic
164C2×C4×C47C2×#4111yescharacteristic
165C2×Dic47—42no—
166C2×Dic47—82no—
167Dic94—82no—
168Dic94—82no—
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—81no—
181C22×C47—81no—
182C22×C47—81no—
183C22×C47#46512yes—
184C22×C47C2×#12111yescharacteristic
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—42no—
189C4×C47—81no—
190C4×C47#46512yes—
191C4×C47C22×SD1611yescharacteristic
192C4×C47C2×#13511yescharacteristic
193C4×C47#57712yes—
194Dic47—82no—
195#3227—471no—
196C2×#476—471no—
197#3036D4711yescharacteristic
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—161no—
207C2×C47#304211yescharacteristic
208C2×C47C2×#46511yescharacteristic
209C2×C47#306811yescharacteristic
210C2×AGL(1,ℤ/8)—21no—
211#532C2×D4711yescharacteristic
212C2×C4⋊Q8—471no—
213C2×C4×D4C2×D4711yescharacteristic
214#498—472no—
215#486—472no—
216#481—942no—
217#476—474no—
218#476—942no—
219#476—942no—
220#463—22no—
221#461C2×D4712yes—
222#459C2×D4712yes—
223C8×D4—474no—
224C8×D4—942no—
225#431—471no—
226#431—471no—
227C2×#106—941no—
228#423C2×D4711yescharacteristic
229C2×C4⋊C8—471no—
230#417—21no—
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—941no—
240C22×D4C22⋊D4711yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—44no—
243C2×SD16—22no—
244C2×D8—22no—
245C2×M16—21no—
246C22×C8—944no—
247C22×C8—941no—
248C4⋊Q8—472no—
249C4⋊Q8—941no—
250C4⋊Q8—941no—
251C4⋊Q8—941no—
252C4⋊Q8C22×D4711yescharacteristic
253#127C22×D4711yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4C22⋊D4712yes—
262C4×D4C22×D4712yes—
263C2×C4⋊C4—941no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4C22⋊D4711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D4711yescharacteristic
268#106—944no—
269#106—944no—
270C4⋊C8—21no—
271C4⋊C8—472no—
272C4⋊C8—472no—
273C4⋊C8—472no—
274C4⋊C8—941no—
275C4⋊C8C22×D4712yes—
276#103—22no—
277#102—22no—
278#102—944no—
279#102—944no—
280C22⋊C8—472no—
281C22⋊C8—944no—
282C22⋊C8—941no—
283C4×C8—472no—
284C4×C8—941no—
285C24—21no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—942no—
290C2×Q8—3762no—
291C2×Q8—1881no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4#1062533514yes—
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—44no—
310D8—44no—
311M16—42no—
312C2×C8—22no—
313C2×C8—474no—
314C2×C8—942no—
315C2×C8—942no—
316C2×C8—3764no—
317C2×C8—1882no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—942no—
322C4⋊C4—942no—
323C4⋊C4—1881no—
324C4⋊C4—1881no—
325C4⋊C4C2×C22⋊D4712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42D4×D4711yescharacteristic
331C42#1062535211yescharacteristic
332C42C2×C22⋊D4711yescharacteristic
333C42C23×D4711yescharacteristic
334C23—22no—
335C23—24no—
336C23—41no—
337C23—81no—
338C23—81no—
339C23#1309399611yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—3762no—
343Q8—3762no—
344D4—24no—
345D4—22no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—82no—
351D4C2×#1062533514yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—3762no—
361C2×C4—1882no—
362C2×C4#1309399611yescharacteristic
363C2×C4#1309399611yescharacteristic
364C2×C4#1309399611yescharacteristic
365C2×C4C2×#1062533512yes—
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—81no—
381C22—81no—
382C22—81no—
383C22#1568935012yes—
384C22C2×#1309399611yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—81no—
390C4—3762no—
391C4#1568935012yes—
392C4C22×#1062533511yescharacteristic
393C4C2×#1309404411yescharacteristic
394C4#1568993312yes—
395C2—22no—
396C2—44no—
397C2—161no—
398C2#2025290211yescharacteristic
399C2C2×#1568935011yescharacteristic
400C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC47×#3036C2
socleC22×C47C2×#121
hypercenter#3036D47
hypocenterC47#19391
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20254513C2semidirect
2. C2×#15689413C2semidirect
3. #20252412C2semidirect
4. #20242795C2semidirect
5. #20242761C2semidirect
6. #20242160C2semidirect
7. #20242070C2semidirect
8. #20242037C2semidirect
11. #15689519C22semidirect
12. #15689469C22semidirect
25. #15688873C22semidirect
26. #15688871C22semidirect
27. #15688869C22semidirect
28. #15688867C22semidirect
237. C47#19391semidirect

Maximal subgroups

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

subgroupindex
#202545132
C2×#156894132
#202524122
#202427952
#202427612
#202421602
#202420702
#202420372
C47×#30362
#1939147