Finite Group Databasebeta

← #27080197

Subgroups — #27080197

Order 12032.

2722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094032C212yesmaximal, complemented
2#20256721C211yesmaximal, characteristic
3#20252105C214yesmaximal, complemented
4#20252099C214yesmaximal, complemented
5#20252090C218yesmaximal, complemented
6#20252081C218yesmaximal, complemented
7#20251832C212yesmaximal, complemented
8C2×#15689079C211yesmaximal, characteristic, complemented
9C47×#2905C211yesmaximal, characteristic, complemented
10C22×#10625350C2212yes—
11C2×#13094032—216no—
12C2×#13094032C2214yes—
13C2×#13094032C2214yes—
14C2×#13094032C2214yescomplemented
15C2×#13094032C2214yescomplemented
16C22×#10625329C2211yescharacteristic
17C22×#10625324C2214yescomplemented
18C23×C47⋊C8C2211yescharacteristic
19C22×C8×D47C2214yescomplemented
20#15689704C414yes—
21#15689699C414yes—
22#15689692C414yes—
23#15689685C414yes—
24#15689673C2214yes—
25#15689635C2218yes—
26#15689200C22116yescomplemented
27#15689193C22132yescomplemented
28#15689188C22116yescomplemented
29#15689091C2218yescomplemented
30#15689089C2218yes—
31#15689086C2218yescomplemented
32#15689079C2214yescomplemented
33#15689079C2214yescomplemented
34C2×#13093935C2212yescomplemented
35C2×C4×C47⋊C8C2212yescomplemented
36C22×M16×C47C2212yes—
37C2×#117×C47C2211yescharacteristic
38C47×#424C2218yescomplemented
39C47×#423C2214yescomplemented
40C2×#10625350—28no—
41C2×#10625350—28no—
42C2×#10625350C2×C414yes—
43C2×#10625350C2×C414yes—
44C2×#10625350C2314yes—
45C2×#10625350C2314yes—
46C22×Dic94C2×C412yes—
47C22×D188C2×C412yes—
48C22×C22⋊D47C2×C414yes—
49C22×C4×D47C2314yes—
50#13094032—216no—
51#13094032—216no—
52#13094032—216no—
53#13094032—216no—
54C2×#10625329—24no—
55C2×#10625329C2314yes—
56C2×#10625324—216no—
57C2×#10625324C2318yes—
58C2×#10625324C2318yes—
59C22×C47⋊C8D418yescomplemented
60C22×C47⋊C8C2312yes—
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2312yes—
64C22×C47⋊C8C2312yes—
65C2×C8×D47—216no—
66C2×C8×D47C2318yes—
67C2×C8×D47C2318yes—
68#13094024C2×C414yes—
69#13094021C2×C414yes—
70#13094012C2×C418yes—
71#13094008C2×C4116yes—
72#13094004C2×C4116yes—
73#13093999C2×C4116yes—
74#13093998C2×C4116yes—
75#13093994C2×C4116yes—
76C4×D188C23116yes—
77(C2×D47)⋊C8C23132yes—
78#13093935C2318yes—
79C4×C47⋊C8C2314yes—
80C4×C47⋊C8C2314yes—
81C23×C4×C47C2311yescharacteristic
82C2×M16×C47—28no—
83C2×M16×C47C2314yes—
84C2×M16×C47C2314yes—
85C22×C8×C47C2314yes—
86#117×C47C2×C414yes—
87#117×C47C2314yes—
88C2×C4⋊C4×C47C2×C412yes—
89C2×C22⋊C4×C47C2×C412yes—
90C2×C42×C47C2312yes—
91C4⋊C8×C47C23116yes—
92C24×C47C22×C411yescharacteristic
93C23×D47C22×C414yes—
94C22×C4×C47C22×C414yes—
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C22×C412yes—
97C22×C4×C47C2×D414yes—
98C22×C4×C47#4114yes—
99C22×C4×C47C2411yescharacteristic
100C22×C4×C47C2412yes—
101C2×C22⋊D47—216no—
102C2×C22⋊D47C22×C418yes—
103C2×C22⋊D47C22×C418yes—
104C22×Dic47C22×C414yes—
105#10625350—216no—
106#10625350—216no—
107#10625350—28no—
108#10625350—28no—
109#10625350—28no—
110#10625350—28no—
111C2×D188—28no—
112C2×D188C22×C414yes—
113C2×D188C22×C414yes—
114C2×D188C22×C4116yes—
115C2×C4×D47—28no—
116C2×C4×D47—28no—
117C2×C4×D47C22×C418yes—
118C2×C4×D47C2418yes—
119C2×Dic94—28no—
120C2×Dic94C22×C414yes—
121C2×Dic94C22×C414yes—
122M16×C47—28no—
123M16×C47—28no—
124C2×C8×C47—28no—
125C2×C8×C47C2418yes—
126C4⋊C4×C47C22×C418yes—
127C22⋊C4×C47C22×C418yes—
128C42×C47C22×C414yes—
129C42×C47C2414yes—
130(C2×D47)⋊C4C22×C4132yes—
131C4⋊Dic47C22×C4116yes—
132#10625329—28no—
133C2×C47⋊C8—24no—
134C2×C47⋊C8C2×D418yes—
135C2×C47⋊C8C2×D418yes—
136C2×C47⋊C8C2×D414yes—
137C2×C47⋊C8C2×D414yes—
138C2×C47⋊C8C2414yes—
139C2×C47⋊C8C2411yescharacteristic
140C2×C47⋊C8C2411yescharacteristic
141C2×C47⋊C8C2412yes—
142#10625324—216no—
143#10625324—216no—
144C8×D47—216no—
145C8×D47—216no—
146C23×C47—42no—
147C23×C47C4×D414yes—
148C23×C47C23×C412yes—
149C23×C47C23×C411yescharacteristic
150C22×D47—28no—
151C22×D47—48no—
152C22×D47C23×C418yes—
153C2×C4×C47—44no—
154C2×C4×C47C4×D418yes—
155C2×C4×C47C4×D414yes—
156C2×C4×C47C23×C418yes—
157C2×C4×C47C23×C411yescharacteristic
158C2×C4×C47C23×C412yes—
159C2×C4×C47C22×D414yes—
160C2×C4×C47C22×D412yes—
161C2×C4×C47C2×#4112yes—
162C2×C4×C47C2×#4114yes—
163C2×C4×C47C2511yescharacteristic
164C22⋊D47—216no—
165C22⋊D47—216no—
166C2×Dic47—28no—
167C2×Dic47C23×C418yes—
168D188—28no—
169D188—28no—
170D188—416no—
171C4×D47—216no—
172C4×D47—28no—
173C4×D47—28no—
174Dic94—28no—
175Dic94—28no—
176C8×C47—28no—
177C47⋊C8—24no—
178C47⋊C8C22×D412yes—
179C47⋊C8C22×D414yes—
180C47⋊C8C22×D412yes—
181#60140—471nomaximal, Sylow
182C22×C47—22no—
183C22×C47—24no—
184C22×C47—44no—
185C22×C47C2×C4×D414yes—
186C22×C47C2×C4×D412yes—
187C22×C47C24×C411yescharacteristic
188C2×D47—28no—
189C2×D47—216no—
190C2×D47—416no—
191C4×C47—28no—
192C4×C47C2×C4×D412yes—
193C4×C47C2×C4×D414yes—
194C4×C47C23×D411yescharacteristic
195C4×C47C22×#4111yescharacteristic
196Dic47—28no—
197C22×#131—472no—
198#3449—471no—
199#2932—474no—
200#2931—474no—
201#2930—478no—
202#2929—478no—
203#2905D4711yescharacteristic, complemented
204#2879—472no—
205C2×#405—471no—
206C2×C47—24no—
207C2×C47—22no—
208C2×C47C22×C4×D411yescharacteristic
209C2×C47#357312yes—
210D47—48no—
211C22×#41—472no—
212C2×#131—474no—
213C2×#131—474no—
214C2×#131—474no—
215C2×#131—474no—
216C2×#131—9416no—
217C22×M16—471no—
218C22×M16—474no—
219C22×M16C2×D4712yes—
220C23×C8—471no—
221C23×C8—474no—
222#537—474no—
223#536—474no—
224#535—474no—
225#534—474no—
226#532—474no—
227#525—478no—
228C2×#117C2×D4711yescharacteristic
229#436—4716no—
230#435—4732no—
231C8×D4—4716no—
232#424C2×D4718yescomplemented
233#423C2×D4714yescomplemented
234#408—478no—
235#408—478no—
236#407—478no—
237#405—474no—
238#405—474no—
239C2×#97—472no—
240C2×C4×C8—472no—
241C47#6014011yescharacteristic, complemented, Sylow
242C2×#41—474no—
243C2×#41—474no—
244C2×#41—474no—
245C2×#41—474no—
246C2×#41—948no—
247C2×#41—948no—
248C22×Q8—472no—
249C22×D4—474no—
250C22×D4—472no—
251C23×C4—474no—
252C23×C4C22×D4711yescharacteristic
253#131—9416no—
254#131—9416no—
255#131—9416no—
256#131—9416no—
257C2×M16—28no—
258C2×M16—474no—
259C2×M16—478no—
260C2×M16—478no—
261C2×M16—9416no—
262C2×M16—944no—
263C2×M16C22×D4714yes—
264C2×M16C22×D4714yes—
265C22×C8—478no—
266C22×C8—472no—
267C22×C8—472no—
268C22×C8—471no—
269C22×C8—471no—
270C22×C8—472no—
271C22×C8—478no—
272C22×C8—478no—
273C22×C8—9416no—
274C22×C8C22×D4714yes—
275C4⋊Q8—474no—
276#127—474no—
277#124—478no—
278#123—4716no—
279C22⋊Q8—4716no—
280#121—4716no—
281#121—4716no—
282#120—4716no—
283C4×D4—4716no—
284#117C4×D4714yes—
285#117C22×D4714yes—
286C2×C4⋊C4C4×D4712yes—
287C2×C22⋊C4C4×D4712yes—
288C2×C42C22×D4712yes—
289C4⋊C8C22×D47116yes—
290C22⋊C8—4732no—
291#97—478no—
292C4×C8—474no—
293C4×C8—474no—
294C24—474no—
295C24C2×C4×D4711yescharacteristic
296#41—9416no—
297#41—9416no—
298#41—948no—
299#41—948no—
300#41—948no—
301#41—948no—
302C2×Q8—474no—
303C2×Q8—474no—
304C2×Q8—948no—
305C2×D4—478no—
306C2×D4—478no—
307C2×D4—474no—
308C2×D4—474no—
309C2×D4—4716no—
310C2×D4—9416no—
311C2×D4—948no—
312C22×C4—474no—
313C22×C4—478no—
314C22×C4—478no—
315C22×C4—948no—
316C22×C4—948no—
317C22×C4C2×C4×D4714yes—
318C22×C4C2×C4×D4711yescharacteristic
319C22×C4C2×C4×D4712yes—
320C22×C4D4×D4714yes—
321C22×C4#1062535414yes—
322C22×C4C23×D4711yescharacteristic
323C22×C4C23×D4712yes—
324M16—28no—
325M16—28no—
326M16—9416no—
327M16—9416no—
328M16—948no—
329C2×C8—28no—
330C2×C8—478no—
331C2×C8—478no—
332C2×C8—474no—
333C2×C8—474no—
334C2×C8—471no—
335C2×C8—471no—
336C2×C8—472no—
337C2×C8—474no—
338C2×C8—9416no—
339C2×C8—9416no—
340C2×C8—944no—
341C2×C8C23×D4718yes—
342C4⋊C4—4716no—
343C4⋊C4C2×C4×D4718yes—
344C22⋊C4—4732no—
345C22⋊C4C2×C4×D4718yes—
346C42C2×C4×D4714yes—
347C42C23×D4714yes—
348C23—42no—
349C23—478no—
350C23—948no—
351C23—1888no—
352C23#1309398614yes—
353C23C22×C4×D4712yes—
354C23C22×C4×D4711yescharacteristic
355Q8—948no—
356Q8—948no—
357D4—9416no—
358D4—9416no—
359D4—948no—
360D4—948no—
361D4—18816no—
362C2×C4—44no—
363C2×C4—478no—
364C2×C4—948no—
365C2×C4—9416no—
366C2×C4—948no—
367C2×C4—948no—
368C2×C4#1309398618yes—
369C2×C4#1309398614yes—
370C2×C4C22×C4×D4718yes—
371C2×C4C22×C4×D4711yescharacteristic
372C2×C4C22×C4×D4712yes—
373C2×C4C2×D4×D4714yes—
374C2×C4C2×D4×D4712yes—
375C2×C4C2×#1062535412yes—
376C2×C4C2×#1062535414yes—
377C2×C4C24×D4711yescharacteristic
378C8—28no—
379C8—472no—
380C8—474no—
381C8—472no—
382C8—944no—
383C22—22no—
384C22—24no—
385C22—44no—
386C22—948no—
387C22—9416no—
388C22—18816no—
389C22C2×#1309398614yes—
390C22C2×#1309398612yes—
391C22C23×C4×D4711yescharacteristic
392C4—28no—
393C4—948no—
394C4C2×#1309398614yes—
395C4C2×#1309398612yes—
396C4C22×D4×D4711yescharacteristic
397C4C22×#1062535411yescharacteristic
398C2—24no—
399C2—22no—
400C2—1888no—
401C2C22×#1309398611yescharacteristic
402C2#2025810112yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2905C2
prime radical#20256721C2
cosocleC2×C4×C47C25
hypercenter#2905D47
hypocenterC47#60140
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C22×#13094032C2semidirect
3. #20252105C2semidirect
4. #20252099C2semidirect
5. #20252090C2semidirect
6. #20252081C2semidirect
7. #20251832C2semidirect
8. C2×#15689079C2semidirect
9. C47×#2905C2semidirect
14. C2×#13094032C22semidirect
15. C2×#13094032C22semidirect
17. C22×#10625324C22semidirect
19. C22×C8×D47C22semidirect
26. #15689200C22semidirect
27. #15689193C22semidirect
28. #15689188C22semidirect
29. #15689091C22semidirect
31. #15689086C22semidirect
32. #15689079C22semidirect
33. #15689079C22semidirect
34. C2×#13093935C22semidirect
35. C2×C4×C47⋊C8C22semidirect
38. C47×#424C22semidirect
39. C47×#423C22semidirect
59. C22×C47⋊C8D4semidirect
203. #2905D47semidirect
232. #424C2×D47semidirect
233. #423C2×D47semidirect
241. C47#60140semidirect

Maximal subgroups

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

subgroupindex
C22×#130940322
#202567212
#202521052
#202520992
#202520902
#202520812
#202518322
C2×#156890792
C47×#29052
#6014047