Finite Group Databasebeta

← #26201528

Subgroups — #26201528

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253475C211yesmaximal, characteristic
2#20253392C211yesmaximal, characteristic, complemented
3#20252411C211yesmaximal, characteristic, complemented
4#20242822C212yesmaximal, complemented
5#20242794C212yesmaximal, complemented
6#20242784C212yesmaximal, complemented
7#20242760C212yesmaximal, complemented
8#20242168C211yesmaximal, characteristic, complemented
9#20242098C211yesmaximal, characteristic, complemented
10#20242050C211yesmaximal, characteristic, complemented
11C47×#3228C211yesmaximal, characteristic, complemented
12C2×C4×D188C2211yescharacteristic
13#15689451—21no—
14#15689439—21no—
15#15689410C2212yes—
16#15689410C2212yes—
17#15689325C2212yes—
18#15689315C2212yes—
19#15689273—21no—
20#15689256—21no—
21#15689233—22no—
22C4×#10625336—22no—
23C8×D188C2212yes—
24C8×D188C2212yes—
25#15689188—22no—
26#15689180C2211yescharacteristic
27#15689176C2211yescharacteristic
28#15689176C2211yescharacteristic
29C22⋊C8×D47—22no—
30#15688876C2212yescomplemented
31#15688874C2212yescomplemented
32#15688872C2212yescomplemented
33#15688872C2212yescomplemented
34#15688869C2212yescomplemented
35#15688867C2212yescomplemented
36C47×#533C2211yescharacteristic
37C47×#532C2211yescharacteristic
38C47×#500C2211yescharacteristic, complemented
39C47×#498C2211yescharacteristic, complemented
40C47×#486C2211yescharacteristic, complemented
41C47×#486C2211yescharacteristic, complemented
42C47×#482—22no—
43C47×#478—22no—
44C47×#431C2211yescharacteristic
45C22×D188D411yescharacteristic
46C22×C4×D47—21no—
47C2×#10625336—22no—
48C2×#10625334—22no—
49C2×C8×D47—22no—
50C2×C8×D47—22no—
51C4×D188—21no—
52C4×D188—21no—
53C4×D188D412yes—
54C4×D188C2312yes—
55C2×C4⋊Dic47D411yescharacteristic
56C2×(C2×D47)⋊C4—21no—
57#13093955—22no—
58#13093955—22no—
59C4⋊(C47⋊C8)C2312yes—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093948—22no—
63#13093945—22no—
64#13093944—22no—
65#13093944—22no—
66C22⋊(C47⋊C8)—21no—
67C22⋊(C47⋊C8)—21no—
68C22⋊(C47⋊C8)—21no—
69(C2×D47)⋊C8—22no—
70(C2×D47)⋊C8—22no—
71#13093935—21no—
72C4×C47⋊C8—21no—
73C4×C47⋊C8—21no—
74C2×#41×C47—21no—
75C4⋊Q8×C47C2311yescharacteristic
76C4⋊Q8×C47C2311yescharacteristic
77C4⋊Q8×C47C2311yescharacteristic
78#127×C47C2311yescharacteristic
79#125×C47—21no—
80#124×C47—21no—
81C22⋊Q8×C47—21no—
82C22⋊Q8×C47—21no—
83#121×C47—21no—
84#121×C47—21no—
85C4×Q8×C47—21no—
86C4×Q8×C47—21no—
87C4×D4×C47—21no—
88C4×D4×C47—21no—
89#117×C47—21no—
90C2×C42×C47C2311yescharacteristic
91#107×C47—22no—
92#106×C47—22no—
93C4⋊C8×C47D412yes—
94#102×C47—22no—
95#102×C47—22no—
96C22⋊C8×C47D412yes—
97C4×C8×C47C2311yescharacteristic
98C4×C8×C47C2311yescharacteristic
99C23×D47—21no—
100#41×C47—82no—
101C2×Q8×C47—21no—
102C2×Q8×C47—21no—
103C2×Q8×C47—21no—
104C2×D4×C47—21no—
105C2×D4×C47—21no—
106C2×D4×C47—21no—
107C2×D4×C47—41no—
108C22×C4×C47—41no—
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×Dic47—21no—
113C2×D188—21no—
114C2×D188—24no—
115C2×D188C2×D412yes—
116C2×C4×D47—22no—
117C2×C4×D47—22no—
118C2×C4×D47—82no—
119C2×C8×C47C2×D412yes—
120C2×C8×C47C2×D412yes—
121C4⋊C4×C47—21no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—21no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47—41no—
128C4⋊C4×C47—41no—
129C22⋊C4×C47—41no—
130C22⋊C4×C47—41no—
131C22⋊C4×C47—41no—
132C42×C47—41no—
133C42×C47C2×D411yescharacteristic
134C42×C47C2×D411yescharacteristic
135C42×C47#4111yescharacteristic
136C42×C47C2411yescharacteristic
137#10625336—44no—
138#10625334—44no—
139(C2×D47)⋊C4—22no—
140(C2×D47)⋊C4—22no—
141C4⋊Dic47—21no—
142C4⋊Dic47C2×D412yes—
143C2×C47⋊C8—22no—
144C2×C47⋊C8—22no—
145C2×C47⋊C8—22no—
146C8×D47—82no—
147C8×D47—82no—
148C23×C47—41no—
149C23×C47#12111yescharacteristic
150C22×D47—22no—
151C22×D47—24no—
152C22×D47—81no—
153C22×D47—81no—
154Q8×C47—42no—
155Q8×C47—81no—
156Q8×C47—81no—
157D4×C47—42no—
158D4×C47—42no—
159D4×C47—81no—
160D4×C47—82no—
161C2×C4×C47—22no—
162C2×C4×C47—22no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—82no—
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47#12111yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×C4×C47C22×D411yescharacteristic
172C2×C4×C47C22×D411yescharacteristic
173C2×C4×C47C2×#4111yescharacteristic
174C2×Dic47—22no—
175C2×Dic47—81no—
176D188—24no—
177D188—22no—
178D188#13514yes—
179C4×D47—42no—
180C4×D47—42no—
181C4×D47—82no—
182C8×C47—22no—
183C8×C47—22no—
184C47⋊C8—42no—
185C47⋊C8—42no—
186C47⋊C8—42no—
187#19394—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—81no—
192C22×C47—81no—
193C22×C47—81no—
194C22×C47C2×#12111yescharacteristic
195C2×D47—24no—
196C2×D47—42no—
197C2×D47—42no—
198C2×D47—84no—
199C4×C47—21no—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—21no—
203C4×C47—81no—
204C4×C47—81no—
205C4×C47—81no—
206C4×C47C2×#13512yes—
207C4×C47#57712yes—
208Dic47—42no—
209#3228D4711yescharacteristic, complemented
210#3109—471no—
211#3101—471no—
212#2972—471no—
213#1681—472no—
214#1677—472no—
215#1675—472no—
216#1671—472no—
217#1572—471no—
218#1554—471no—
219#1543—471no—
220C2×C47—41no—
221C2×C47—161no—
222C2×C47#304211yescharacteristic
223C2×C47#305011yescharacteristic
224C2×C47#306811yescharacteristic
225D47—44no—
226#533C2×D4711yescharacteristic
227#532C2×D4711yescharacteristic
228C2×C4×D4—471no—
229#500C2×D4711yescharacteristic, complemented
230#498C2×D4711yescharacteristic, complemented
231#486C2×D4711yescharacteristic, complemented
232#486C2×D4711yescharacteristic, complemented
233#483—941no—
234#482—22no—
235#481—941no—
236#478—22no—
237#476—472no—
238#476—472no—
239#461—472no—
240#459—472no—
241#450—941no—
242#447—941no—
243#442—942no—
244C4×SD16—942no—
245C8×D4—472no—
246C8×D4—472no—
247C8×D4—942no—
248#432—471no—
249#431—471no—
250#431—471no—
251#431C2×D4711yescharacteristic
252C2×C22⋊C8—942no—
253#335—472no—
254#334—472no—
255#333—472no—
256#333—472no—
257#332—472no—
258#331—472no—
259C47#1939411yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C22×D4—471no—
262C23×C4—941no—
263C2×SD16—942no—
264C2×D8—942no—
265C22×C8—942no—
266C22×C8—942no—
267C4⋊Q8C22×D4711yescharacteristic
268C4⋊Q8C22×D4711yescharacteristic
269C4⋊Q8C22×D4711yescharacteristic
270#127C22×D4711yescharacteristic
271#125—21no—
272#124—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—21no—
276#121—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—472no—
282C4×D4—472no—
283C4×D4—941no—
284C4×D4—941no—
285#117—21no—
286C2×C4⋊C4—471no—
287C2×C22⋊C4—941no—
288C2×C42C22×D4711yescharacteristic
289#107—22no—
290#106—22no—
291#106—942no—
292#106—942no—
293C4⋊C8—472no—
294C4⋊C8—472no—
295C4⋊C8—472no—
296C4⋊C8C22⋊D4712yes—
297#103—942no—
298#102—22no—
299#102—22no—
300#102—942no—
301#102—942no—
302#102—942no—
303C22⋊C8—941no—
304C22⋊C8—941no—
305C22⋊C8—941no—
306C22⋊C8—942no—
307C22⋊C8—942no—
308C22⋊C8C22⋊D4712yes—
309#97—941no—
310C4×C8—941no—
311C4×C8—941no—
312C4×C8C22×D4711yescharacteristic
313C4×C8C22×D4711yescharacteristic
314C24—941no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—472no—
324C2×D4—944no—
325C2×D4—941no—
326C22×C4—41no—
327C22×C4—941no—
328C22×C4—942no—
329C22×C4—942no—
330C22×C4—3762no—
331C22×C4D4×D4711yescharacteristic
332C22×C4#1062535211yescharacteristic
333C22×C4C2×C22⋊D4711yescharacteristic
334SD16—1884no—
335D8—1884no—
336C2×C8—942no—
337C2×C8—942no—
338C2×C8—942no—
339C2×C8—3762no—
340C2×C8—3762no—
341C2×C8C2×C22⋊D4712yes—
342C2×C8C2×C22⋊D4712yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—472no—
352C4⋊C4—941no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—942no—
357C22⋊C4—942no—
358C42—41no—
359C42D4×D4711yescharacteristic
360C42#1062535211yescharacteristic
361C42C2×C22⋊D4711yescharacteristic
362C42C23×D4711yescharacteristic
363C23—41no—
364C23—944no—
365C23—942no—
366C23—3761no—
367C23—3761no—
368C23#1309399511yescharacteristic
369Q8—42no—
370Q8—81no—
371Q8—81no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—82no—
376D4—474no—
377D4—944no—
378D4—942no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—942no—
387C2×C4—3762no—
388C2×C4—3761no—
389C2×C4—1882no—
390C2×C4—1882no—
391C2×C4#1309399511yescharacteristic
392C2×C4#1309399511yescharacteristic
393C2×C4#1309399511yescharacteristic
394C2×C4C2×D4×D4711yescharacteristic
395C2×C4C22×C22⋊D4711yescharacteristic
396C2×C4C2×#1062535211yescharacteristic
397C8—22no—
398C8—22no—
399C8—1882no—
400C8—1882no—
401C8—1882no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—944no—
409C22—3764no—
410C22—1882no—
411C22—1882no—
412C22C2×#1309399511yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—1882no—
421C4#135×D4712yes—
422C4#1568993112yes—
423C2—41no—
424C2—161no—
425C2—1884no—
426C2#2025289611yescharacteristic
427C2#2025296511yescharacteristic
428C2#2025311211yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3228C2
socleC22×C47C2×#121
prime radical#20253475C2
hypercenter#3228D47
hypocenterC47#19394
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
2. #20253392C2semidirect
3. #20252411C2semidirect
4. #20242822C2semidirect
5. #20242794C2semidirect
6. #20242784C2semidirect
7. #20242760C2semidirect
8. #20242168C2semidirect
9. #20242098C2semidirect
10. #20242050C2semidirect
11. C47×#3228C2semidirect
30. #15688876C22semidirect
31. #15688874C22semidirect
32. #15688872C22semidirect
33. #15688872C22semidirect
34. #15688869C22semidirect
35. #15688867C22semidirect
38. C47×#500C22semidirect
39. C47×#498C22semidirect
40. C47×#486C22semidirect
41. C47×#486C22semidirect
209. #3228D47semidirect
229. #500C2×D47semidirect
230. #498C2×D47semidirect
231. #486C2×D47semidirect
232. #486C2×D47semidirect
259. C47#19394semidirect

Maximal subgroups

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

subgroupindex
#202534752
#202533922
#202524112
#202428222
#202427942
#202427842
#202427602
#202421682
#202420982
#202420502
C47×#32282
#1939447