Finite Group Databasebeta

← #26370667

Subgroups — #26370667

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256522C212yesmaximal, complemented
2C2×#15689241C211yesmaximal, characteristic, complemented
3#20245466C212yesmaximal, complemented
4#20245461C212yesmaximal, complemented
5#20245454C214yesmaximal, complemented
6C2×#15688944C211yesmaximal, characteristic, complemented
7#20245137C212yesmaximal, complemented
8C47×#3147C211yesmaximal, characteristic, complemented
9C2×#13094046C2211yescharacteristic
10C2×#13094030—21no—
11#15689564C2214yes—
12#15689563—22no—
13#15689559C2214yes—
14#15689558—22no—
15#15689553—24no—
16C2×#13093964C2212yes—
17#15689241—21no—
18#15689241—21no—
19#15689241C2212yes—
20#15689241C2212yes—
21C2×#13093960C2211yescharacteristic
22C2×#13093952—21no—
23#15688944C2212yescomplemented
24#15688944C2212yescomplemented
25#15688939C2214yescomplemented
26#15688936C2214yescomplemented
27C2×#135×C47—21no—
28C2×#135×C47C2212yes—
29C47×#532—21no—
30C47×#495—21no—
31C47×#495C2212yescomplemented
32C47×#494C2211yescharacteristic, complemented
33C47×#493C2211yescharacteristic, complemented
34C47×#492—21no—
35C2×C4×C8×C47C2211yescharacteristic
36C2×#10625350—21no—
37#13094046—21no—
38#13094046—21no—
39#13094046D412yescomplemented
40#13094046C2312yes—
41C2×Dic188D411yescharacteristic, complemented
42C2×#10625325—21no—
43C2×D376D411yescharacteristic, complemented
44#13094030—22no—
45#13094030—22no—
46#13094030—41no—
47#13094030—41no—
48C2×#10625329—21no—
49C2×#10625324—41no—
50C2×C8×D47—41no—
51#13093970—24no—
52#13093968—24no—
53#13093967—24no—
54#13093965—24no—
55#13093964—22no—
56#13093964C2314yes—
57C2×C47⋊C16—22no—
58#13093960—21no—
59#13093960C2312yes—
60#13093952—22no—
61#13093952—22no—
62C2×#41×C47—22no—
63#135×C47—22no—
64#135×C47—22no—
65#135×C47—22no—
66#135×C47—22no—
67#135×C47—41no—
68#135×C47—41no—
69#135×C47—41no—
70#135×C47—41no—
71C2×Q16×C47—21no—
72C2×Q16×C47C2312yes—
73C2×SD16×C47—22no—
74C2×SD16×C47—22no—
75C2×D8×C47—21no—
76C2×D8×C47C2312yes—
77C22×C8×C47D411yescharacteristic
78C22×C8×C47C2311yescharacteristic
79C4⋊Q8×C47—21no—
80#127×C47—21no—
81#124×C47—22no—
82#121×C47—42no—
83C4×D4×C47—42no—
84C2×C42×C47D411yescharacteristic
85C4×C8×C47D411yescharacteristic
86C4×C8×C47D411yescharacteristic
87C4×C8×C47C2311yescharacteristic
88C4×C8×C47C2311yescharacteristic
89#41×C47—42no—
90#41×C47—42no—
91#41×C47—42no—
92#41×C47—42no—
93C2×Q8×C47—22no—
94C2×D4×C47—22no—
95C2×D4×C47—42no—
96C2×D4×C47—42no—
97C22×C4×C47—21no—
98C22×C4×C47—42no—
99C22×C4×C47C2×D411yescharacteristic
100C2×C22⋊D47—41no—
101#10625350—22no—
102#10625350—22no—
103#10625350—41no—
104#10625350—41no—
105C2×D188—21no—
106C2×C4×D47—41no—
107C2×Dic94—21no—
108Q16×C47—22no—
109Q16×C47—22no—
110Q16×C47—41no—
111Q16×C47—41no—
112SD16×C47—42no—
113SD16×C47—42no—
114SD16×C47—42no—
115SD16×C47—42no—
116D8×C47—22no—
117D8×C47—22no—
118D8×C47—41no—
119D8×C47—41no—
120C2×C8×C47D811yescharacteristic
121C2×C8×C47D811yescharacteristic
122C2×C8×C47D811yescharacteristic
123C2×C8×C47D811yescharacteristic
124C2×C8×C47C2×D411yescharacteristic
125C2×C8×C47C2×D411yescharacteristic
126C2×C8×C47C2×D411yescharacteristic
127C2×C8×C47C2×D411yescharacteristic
128C2×C8×C47C2×D411yescharacteristic
129C2×C8×C47#4111yescharacteristic
130C2×C8×C47#4111yescharacteristic
131C2×C8×C47C2411yescharacteristic
132C4⋊C4×C47—42no—
133C22⋊C4×C47—44no—
134C42×C47—21no—
135C42×C47C2×D411yescharacteristic
136C42×C47C2×D411yescharacteristic
137#10625329—22no—
138#10625329—41no—
139C2×C47⋊C8—41no—
140Dic188—21no—
141Dic188C2×D412yes—
142D376—21no—
143D376C2×D412yes—
144#10625325—22no—
145#10625325—22no—
146#10625324—42no—
147#10625324—42no—
148C8×D47—42no—
149C8×D47—42no—
150C47⋊C16—24no—
151C23×C47—42no—
152C23×C47AGL(1,ℤ/8)11yescharacteristic
153C22×D47—41no—
154Q8×C47—42no—
155Q8×C47—42no—
156D4×C47—42no—
157D4×C47—42no—
158D4×C47—84no—
159D4×C47—82no—
160D4×C47—82no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—82no—
166C2×C4×C47—82no—
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47C2×D811yescharacteristic
170C2×C4×C47C2×D811yescharacteristic
171C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C47C22×D411yescharacteristic
173C22⋊D47—42no—
174C22⋊D47—42no—
175C2×Dic47—41no—
176D188—22no—
177D188—41no—
178C4×D47—42no—
179C4×D47—42no—
180Dic94—22no—
181Dic94—41no—
182C8×C47#12111yescharacteristic
183C8×C47#12111yescharacteristic
184C8×C47C2×D811yescharacteristic
185C8×C47C2×D811yescharacteristic
186C8×C47C2×D811yescharacteristic
187C8×C47C2×D811yescharacteristic
188C8×C47C22×D411yescharacteristic
189C8×C47C2×#4111yescharacteristic
190C47⋊C8—42no—
191#33594—471nomaximal, Sylow
192C22×C47—41no—
193C22×C47—84no—
194C22×C47—82no—
195C22×C47#45911yescharacteristic
196C22×C47#45911yescharacteristic
197C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
198C2×D47—42no—
199C2×D47—82no—
200C4×C47—22no—
201C4×C47—41no—
202C4×C47—82no—
203C4×C47#45911yescharacteristic
204C4×C47#45911yescharacteristic
205C4×C47C2×#12111yescharacteristic
206C4×C47C22×D811yescharacteristic
207Dic47—42no—
208#3414—472no—
209#3147D4711yescharacteristic, complemented
210C2×#443—471no—
211#2214—472no—
212#2213—472no—
213#2212—474no—
214C2×#364—471no—
215#2148—472no—
216C2×C47—21no—
217C2×C47—21no—
218C2×C47—21no—
219C2×C47—162no—
220C2×C47C2×#45911yescharacteristic
221D47—82no—
222C2×#135—21no—
223C2×#135—471no—
224C2×#135C2×D4712yes—
225C2×#131—941no—
226#532—21no—
227#510—474no—
228#510—942no—
229#509—474no—
230#509—942no—
231#508—944no—
232C2×M32—472no—
233#495—21no—
234#495C2×D4712yescomplemented
235#494C2×D4711yescharacteristic, complemented
236#493C2×D4711yescharacteristic, complemented
237#492—21no—
238#443—472no—
239#443—472no—
240#443—941no—
241#443—941no—
242C2×#108—471no—
243C2×#104—941no—
244C2×C4×C8C2×D4711yescharacteristic
245#364—472no—
246#364—472no—
247#362—474no—
248#361—474no—
249C47#3359411yescharacteristic, complemented, Sylow
250C2×#41—22no—
251C2×#41—941no—
252#135—22no—
253#135—22no—
254#135—22no—
255#135—22no—
256#135—41no—
257#135—41no—
258#135—41no—
259#135—41no—
260#135—472no—
261#135—472no—
262#135—941no—
263#135—941no—
264C2×Q16—21no—
265C2×Q16—471no—
266C2×Q16C22×D4712yes—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×SD16—941no—
270C2×D8—21no—
271C2×D8—471no—
272C2×D8C22×D4712yes—
273#131—942no—
274#131—942no—
275#131—1881no—
276#131—1881no—
277C2×M16—941no—
278C2×M16—1881no—
279C22×C8—1881no—
280C22×C8C22⋊D4711yescharacteristic
281C22×C8C22×D4711yescharacteristic
282C4⋊Q8—21no—
283#127—21no—
284#124—22no—
285#121—42no—
286C4×D4—42no—
287C2×C42C22⋊D4711yescharacteristic
288Q32—944no—
289SD32—944no—
290SD32—944no—
291D16—944no—
292M32—474no—
293M32—942no—
294C2×C16—942no—
295#108—472no—
296#108—941no—
297#104—942no—
298#104—942no—
299C4×C8C22⋊D4711yescharacteristic
300C4×C8C22⋊D4711yescharacteristic
301C4×C8C22×D4711yescharacteristic
302C4×C8C22×D4711yescharacteristic
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—942no—
308#41—942no—
309#41—1881no—
310#41—1881no—
311C2×Q8—22no—
312C2×Q8—941no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—941no—
317C2×D4—1881no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—1881no—
321C22×C4C2×C22⋊D4711yescharacteristic
322Q16—22no—
323Q16—22no—
324Q16—41no—
325Q16—41no—
326Q16—472no—
327Q16—941no—
328SD16—42no—
329SD16—42no—
330SD16—42no—
331SD16—42no—
332SD16—942no—
333SD16—942no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8—472no—
339D8—941no—
340M16—942no—
341M16—1881no—
342M16—1882no—
343M16—1882no—
344C2×C8—1881no—
345C2×C8—1882no—
346C2×C8—1882no—
347C2×C8#1062533411yescharacteristic
348C2×C8#1062533411yescharacteristic
349C2×C8#1062533411yescharacteristic
350C2×C8#1062533411yescharacteristic
351C2×C8D4×D4711yescharacteristic
352C2×C8D4×D4711yescharacteristic
353C2×C8#1062535211yescharacteristic
354C2×C8#1062535211yescharacteristic
355C2×C8C2×C22⋊D4711yescharacteristic
356C2×C8C2×C22⋊D4711yescharacteristic
357C2×C8C2×C22⋊D4711yescharacteristic
358C2×C8C23×D4711yescharacteristic
359C4⋊C4—42no—
360C22⋊C4—44no—
361C42—21no—
362C42C2×C22⋊D4711yescharacteristic
363C42C2×C22⋊D4711yescharacteristic
364C16—944no—
365C23—42no—
366C23—1881no—
367C23#1309405011yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—942no—
371Q8—1881no—
372D4—42no—
373D4—42no—
374D4—84no—
375D4—82no—
376D4—82no—
377D4—942no—
378D4—1882no—
379D4—1882no—
380D4—1881no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—1881no—
388C2×C4—1882no—
389C2×C4—1882no—
390C2×C4#1309399511yescharacteristic
391C2×C4#1309399511yescharacteristic
392C2×C4C2×#1062533411yescharacteristic
393C2×C4C2×#1062533411yescharacteristic
394C2×C4#1309405011yescharacteristic
395C2×C4C22×C22⋊D4711yescharacteristic
396C8—1882no—
397C8#1309399511yescharacteristic
398C8#1309399511yescharacteristic
399C8C2×#1062533411yescharacteristic
400C8C2×#1062533411yescharacteristic
401C8C2×#1062533411yescharacteristic
402C8C2×#1062533411yescharacteristic
403C8C2×D4×D4711yescharacteristic
404C8C2×#1062535211yescharacteristic
405C22—41no—
406C22—84no—
407C22—82no—
408C22—3762no—
409C22—1882no—
410C22#1568931511yescharacteristic
411C22#1568931511yescharacteristic
412C22C2×#1309405011yescharacteristic
413C4—22no—
414C4—41no—
415C4—82no—
416C4—1882no—
417C4#1568931511yescharacteristic
418C4#1568931511yescharacteristic
419C4C2×#1309399511yescharacteristic
420C4C22×#1062533411yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—162no—
425C2—3762no—
426C2C2×#1568931511yescharacteristic
subgroupG/H
centerC4#15689315
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3147C2
socleC2×C47C2×#459
hypercenter#3147D47
hypocenterC47#33594
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #20256522C2semidirect
2. C2×#15689241C2semidirect
3. #20245466C2semidirect
4. #20245461C2semidirect
5. #20245454C2semidirect
6. C2×#15688944C2semidirect
7. #20245137C2semidirect
8. C47×#3147C2semidirect
23. #15688944C22semidirect
24. #15688944C22semidirect
25. #15688939C22semidirect
26. #15688936C22semidirect
31. C47×#495C22semidirect
32. C47×#494C22semidirect
33. C47×#493C22semidirect
39. #13094046D4semidirect
41. C2×Dic188D4semidirect
43. C2×D376D4semidirect
209. #3147D47semidirect
234. #495C2×D47semidirect
235. #494C2×D47semidirect
236. #493C2×D47semidirect
249. C47#33594semidirect

Maximal subgroups

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

subgroupindex
#202565222
C2×#156892412
#202454662
#202454612
#202454542
C2×#156889442
#202451372
C47×#31472
#3359447