Finite Group Databasebeta

← #26200079

Subgroups — #26200079

Order 12032.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253483C211yesmaximal, characteristic, complemented
2#20253475C211yesmaximal, characteristic
3C2×C8×D188C211yesmaximal, characteristic, complemented
4#20242836C212yesmaximal, complemented
5#20242822C212yesmaximal, complemented
6#20242760C212yesmaximal, complemented
7#20242726C212yesmaximal, complemented
8#20242151C211yesmaximal, characteristic, complemented
9#20242088C211yesmaximal, characteristic, complemented
10#20242083C211yesmaximal, characteristic, complemented
11C47×#3167C211yesmaximal, characteristic, complemented
12C22×C8×D47—21no—
13C2×C4×D188C2211yescharacteristic
14#15689456—21no—
15#15689439—21no—
16#15689410C2212yes—
17#15689400C2212yes—
18#15689325C2212yes—
19#15689315C2212yes—
20#15689273—21no—
21#15689256—21no—
22#15689233—22no—
23#15689221—22no—
24C8×D188—21no—
25C8×D188—21no—
26C8×D188C2212yes—
27C8×D188C2212yes—
28#15689180C2211yescharacteristic
29#15689180C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31C2×(C2×D47)⋊C8—21no—
32#15688876C2212yescomplemented
33#15688874C2212yescomplemented
34#15688872C2212yescomplemented
35#15688869C2212yescomplemented
36#15688869C2212yescomplemented
37#15688867C2212yescomplemented
38C47×#533C2211yescharacteristic
39C47×#532C2211yescharacteristic
40C47×#498C2211yescharacteristic, complemented
41C47×#492C2211yescharacteristic, complemented
42C47×#487C2211yescharacteristic, complemented
43C47×#486C2211yescharacteristic, complemented
44C47×#465—21no—
45C47×#465—21no—
46C4×SD16×C47—21no—
47C4×SD16×C47—21no—
48C2×C4×C8×C47C2211yescharacteristic
49C22×D188D411yescharacteristic
50C22×C4×D47—21no—
51C2×#10625336—22no—
52C2×#10625334—22no—
53C22×C47⋊C8—21no—
54C2×C8×D47—22no—
55C2×C8×D47—22no—
56C2×C8×D47—44no—
57C4×D188—21no—
58C4×D188—21no—
59C4×D188D412yes—
60C4×D188C2312yes—
61C2×C4⋊Dic47D411yescharacteristic
62C2×(C2×D47)⋊C4—21no—
63#13093957—22no—
64#13093955—22no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)C2312yes—
67C4⋊(C47⋊C8)C2312yes—
68C4⋊(C47⋊C8)C2312yes—
69#13093948—22no—
70#13093945—22no—
71#13093944—22no—
72#13093944—22no—
73C22⋊(C47⋊C8)—21no—
74C22⋊(C47⋊C8)—21no—
75(C2×D47)⋊C8—22no—
76(C2×D47)⋊C8—22no—
77#13093935—21no—
78#13093935—21no—
79C2×#41×C47—21no—
80C2×SD16×C47—21no—
81C2×SD16×C47—21no—
82C22×C8×C47D411yescharacteristic
83C22×C8×C47D411yescharacteristic
84C4⋊Q8×C47C2311yescharacteristic
85C4⋊Q8×C47C2311yescharacteristic
86C4⋊Q8×C47C2311yescharacteristic
87#127×C47C2311yescharacteristic
88#125×C47—21no—
89#124×C47—21no—
90C22⋊Q8×C47—21no—
91C22⋊Q8×C47—21no—
92#121×C47—21no—
93#121×C47—21no—
94C4×Q8×C47—21no—
95C4×Q8×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47—21no—
98#117×C47—21no—
99C2×C42×C47C2311yescharacteristic
100#106×C47—21no—
101#106×C47—21no—
102#103×C47—21no—
103#103×C47—21no—
104#102×C47—21no—
105#102×C47—21no—
106C4×C8×C47D411yescharacteristic
107C4×C8×C47D411yescharacteristic
108C4×C8×C47C2311yescharacteristic
109C4×C8×C47C2311yescharacteristic
110C23×D47—21no—
111#41×C47—82no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—41no—
119C22×C4×C47—41no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47#4111yescharacteristic
123C22×Dic47—21no—
124C2×D188—21no—
125C2×D188—24no—
126C2×D188C2×D412yes—
127C2×C4×D47—22no—
128C2×C4×D47—22no—
129C2×C4×D47—42no—
130C2×C4×D47—42no—
131SD16×C47—42no—
132SD16×C47—42no—
133C2×C8×C47—22no—
134C2×C8×C47—41no—
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×D411yescharacteristic
137C2×C8×C47C2×D411yescharacteristic
138C2×C8×C47C2×D411yescharacteristic
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—41no—
146C4⋊C4×C47—41no—
147C22⋊C4×C47—41no—
148C22⋊C4×C47—41no—
149C22⋊C4×C47—41no—
150C42×C47—41no—
151C42×C47C2×D411yescharacteristic
152C42×C47C2×D411yescharacteristic
153C42×C47#4111yescharacteristic
154C42×C47C2411yescharacteristic
155#10625336—44no—
156#10625334—44no—
157(C2×D47)⋊C4—22no—
158(C2×D47)⋊C4—22no—
159C4⋊Dic47—21no—
160C4⋊Dic47C2×D412yes—
161C2×C47⋊C8—22no—
162C2×C47⋊C8—22no—
163C2×C47⋊C8—22no—
164C2×C47⋊C8—42no—
165C8×D47—44no—
166C8×D47—44no—
167C23×C47—41no—
168C23×C47#12111yescharacteristic
169C22×D47—24no—
170C22×D47—22no—
171C22×D47—42no—
172C22×D47—42no—
173Q8×C47—42no—
174Q8×C47—41no—
175Q8×C47—41no—
176Q8×C47—81no—
177D4×C47—42no—
178D4×C47—41no—
179D4×C47—41no—
180D4×C47—81no—
181D4×C47—82no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—21no—
185C2×C4×C47—21no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—82no—
190C2×C4×C47#12111yescharacteristic
191C2×C4×C47#12111yescharacteristic
192C2×C4×C47#12111yescharacteristic
193C2×C4×C47C22×D411yescharacteristic
194C2×C4×C47C22×D411yescharacteristic
195C2×C4×C47C2×#4111yescharacteristic
196C2×Dic47—22no—
197C2×Dic47—42no—
198D188—24no—
199D188—22no—
200D188#13514yes—
201C4×D47—44no—
202C4×D47—42no—
203C4×D47—42no—
204C8×C47—21no—
205C8×C47—21no—
206C8×C47—21no—
207C8×C47AGL(1,ℤ/8)12yes—
208C47⋊C8—42no—
209C47⋊C8—42no—
210C47⋊C8—42no—
211#19257—471nomaximal, Sylow
212C22×C47—21no—
213C22×C47—21no—
214C22×C47—81no—
215C22×C47—81no—
216C22×C47—81no—
217C22×C47#47112yes—
218C22×C47C2×#12111yescharacteristic
219C2×D47—24no—
220C2×D47—48no—
221C2×D47—42no—
222C2×D47—42no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—21no—
226C4×C47—21no—
227C4×C47—81no—
228C4×C47—81no—
229C4×C47—81no—
230C4×C47C2×#13512yes—
231C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C47#57511yescharacteristic
233Dic47—42no—
234#3167D4711yescharacteristic, complemented
235#3110—471no—
236#3109—471no—
237C2×C8×D4—471no—
238#1683—472no—
239#1681—472no—
240#1671—472no—
241#1665—472no—
242#1568—471no—
243#1552—471no—
244#1551—471no—
245C2×C47—22no—
246C2×C47—161no—
247C2×C47#303611yescharacteristic
248C2×C47#305011yescharacteristic
249C2×C47C2×#47111yescharacteristic
250D47—44no—
251C23×C8—941no—
252#533C2×D4711yescharacteristic
253#532C2×D4711yescharacteristic
254C2×C4×D4—471no—
255#498C2×D4711yescharacteristic, complemented
256#492C2×D4711yescharacteristic, complemented
257#487C2×D4711yescharacteristic, complemented
258#486C2×D4711yescharacteristic, complemented
259#484—941no—
260#481—941no—
261#476—472no—
262#474—472no—
263#465—21no—
264#465—21no—
265#461—472no—
266#459—472no—
267#450—941no—
268#447—941no—
269#442—942no—
270#440—942no—
271C4×SD16—21no—
272C4×SD16—21no—
273C8×D4—472no—
274C8×D4—472no—
275C8×D4—941no—
276C8×D4—941no—
277#432—471no—
278#432—471no—
279C2×C4⋊C8—471no—
280C2×C22⋊C8—941no—
281C2×C4×C8C2×D4711yescharacteristic
282#335—472no—
283#334—472no—
284#333—472no—
285#332—472no—
286#332—472no—
287#331—472no—
288C47#1925711yescharacteristic, complemented, Sylow
289C2×#41—21no—
290C22×D4—471no—
291C23×C4—941no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—942no—
295C2×D8—942no—
296C22×C8—941no—
297C22×C8—942no—
298C22×C8—942no—
299C22×C8—1884no—
300C22×C8C22⋊D4711yescharacteristic
301C22×C8C22⋊D4711yescharacteristic
302C4⋊Q8C22×D4711yescharacteristic
303C4⋊Q8C22×D4711yescharacteristic
304C4⋊Q8C22×D4711yescharacteristic
305#127C22×D4711yescharacteristic
306#125—21no—
307#124—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—21no—
311#121—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—472no—
317C4×D4—472no—
318C4×D4—941no—
319C4×D4—941no—
320#117—21no—
321C2×C4⋊C4—471no—
322C2×C22⋊C4—941no—
323C2×C42C22×D4711yescharacteristic
324#107—942no—
325#106—21no—
326#106—21no—
327#106—942no—
328C4⋊C8—472no—
329C4⋊C8—472no—
330C4⋊C8—472no—
331C4⋊C8—941no—
332#103—21no—
333#103—21no—
334#103—942no—
335#102—21no—
336#102—21no—
337#102—942no—
338#102—942no—
339#102—942no—
340C22⋊C8—941no—
341C22⋊C8—941no—
342C22⋊C8—942no—
343C22⋊C8—942no—
344#97—941no—
345#97—941no—
346C4×C8C22⋊D4711yescharacteristic
347C4×C8C22⋊D4711yescharacteristic
348C4×C8C22×D4711yescharacteristic
349C4×C8C22×D4711yescharacteristic
350C24—941no—
351#41—82no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—472no—
360C2×D4—944no—
361C2×D4—941no—
362C22×C4—41no—
363C22×C4—941no—
364C22×C4—942no—
365C22×C4—942no—
366C22×C4—1882no—
367C22×C4—1882no—
368C22×C4D4×D4711yescharacteristic
369C22×C4#1062535211yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371SD16—42no—
372SD16—42no—
373SD16—1884no—
374D8—1884no—
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—942no—
378C2×C8—942no—
379C2×C8—942no—
380C2×C8—1884no—
381C2×C8—1882no—
382C2×C8—1884no—
383C2×C8C2×C22⋊D4711yescharacteristic
384C2×C8C2×C22⋊D4711yescharacteristic
385C2×C8C2×C22⋊D4711yescharacteristic
386C2×C8C2×C22⋊D4711yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—472no—
396C4⋊C4—941no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—942no—
401C22⋊C4—942no—
402C42—41no—
403C42D4×D4711yescharacteristic
404C42#1062535211yescharacteristic
405C42C2×C22⋊D4711yescharacteristic
406C42C23×D4711yescharacteristic
407C23—41no—
408C23—944no—
409C23—942no—
410C23—1882no—
411C23—1882no—
412C23#1309399511yescharacteristic
413Q8—42no—
414Q8—41no—
415Q8—41no—
416Q8—81no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—81no—
421D4—82no—
422D4—474no—
423D4—944no—
424D4—942no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—942no—
434C2×C4—1884no—
435C2×C4—1882no—
436C2×C4—1882no—
437C2×C4—1882no—
438C2×C4#1309399511yescharacteristic
439C2×C4#1309399511yescharacteristic
440C2×C4#1309399511yescharacteristic
441C2×C4C2×D4×D4711yescharacteristic
442C2×C4C22×C22⋊D4711yescharacteristic
443C2×C4C2×#1062535211yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—1882no—
448C8—1882no—
449C8—1882no—
450C8#1309404912yes—
451C22—21no—
452C22—21no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—944no—
457C22—1888no—
458C22—1882no—
459C22—1882no—
460C22#1568938312yes—
461C22C2×#1309399511yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—1882no—
470C4#135×D4712yes—
471C4C2×#1309404911yescharacteristic
472C4#1568992111yescharacteristic
473C2—22no—
474C2—161no—
475C2—1884no—
476C2#2025284811yescharacteristic
477C2#2025296511yescharacteristic
478C2C2×#1568938311yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3167C2
socleC22×C47C2×#121
prime radical#20253475C2
hypercenter#3167D47
hypocenterC47#19257
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20253483C2semidirect
3. C2×C8×D188C2semidirect
4. #20242836C2semidirect
5. #20242822C2semidirect
6. #20242760C2semidirect
7. #20242726C2semidirect
8. #20242151C2semidirect
9. #20242088C2semidirect
10. #20242083C2semidirect
11. C47×#3167C2semidirect
32. #15688876C22semidirect
33. #15688874C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
36. #15688869C22semidirect
37. #15688867C22semidirect
40. C47×#498C22semidirect
41. C47×#492C22semidirect
42. C47×#487C22semidirect
43. C47×#486C22semidirect
234. #3167D47semidirect
255. #498C2×D47semidirect
256. #492C2×D47semidirect
257. #487C2×D47semidirect
258. #486C2×D47semidirect
288. C47#19257semidirect

Maximal subgroups

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

subgroupindex
#202534832
#202534752
C2×C8×D1882
#202428362
#202428222
#202427602
#202427262
#202421512
#202420882
#202420832
C47×#31672
#1925747