Finite Group Databasebeta

← #26200052

Subgroups — #26200052

Order 12032.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253181C211yesmaximal, characteristic, complemented
2#20252848C211yesmaximal, characteristic
3C2×C8×D188C211yesmaximal, characteristic, complemented
4#20242808C212yesmaximal, complemented
5#20242794C212yesmaximal, complemented
6#20242760C212yesmaximal, complemented
7#20242726C212yesmaximal, complemented
8#20242144C211yesmaximal, characteristic, complemented
9#20242078C211yesmaximal, characteristic, complemented
10#20242065C211yesmaximal, characteristic, complemented
11C47×#3167C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13C22×C8×D47—21no—
14C2×C4×D188C2211yescharacteristic
15#15689420—22no—
16#15689410C2212yes—
17#15689400C2212yes—
18#15689335—22no—
19#15689325C2212yes—
20#15689315C2212yes—
21C8×D188—21no—
22C8×D188—21no—
23C8×D188C2212yes—
24C8×D188C2212yes—
25#15689166—21no—
26#15689147C2211yescharacteristic
27#15689147C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29#15689127—21no—
30#15689124—21no—
31C2×(C2×D47)⋊C8—21no—
32#15688874C2212yescomplemented
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—
51#13094049—44no—
52C2×#10625336—22no—
53C2×#10625334—22no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C22×C47⋊C8—21no—
57C2×C8×D47—22no—
58C2×C8×D47—22no—
59C2×C8×D47—44no—
60C4×D188—21no—
61C4×D188—21no—
62C4×D188D412yes—
63C4×D188C2312yes—
64C2×C4⋊Dic47D411yescharacteristic
65C2×(C2×D47)⋊C4—21no—
66#13093957—22no—
67#13093955—22no—
68C4⋊(C47⋊C8)—21no—
69C4⋊(C47⋊C8)—21no—
70C4⋊(C47⋊C8)—21no—
71C4⋊(C47⋊C8)C2312yes—
72C4⋊(C47⋊C8)C2312yes—
73C4⋊(C47⋊C8)C2312yes—
74#13093948—22no—
75#13093945—22no—
76#13093944—22no—
77#13093944—22no—
78(C2×D47)⋊C8—22no—
79(C2×D47)⋊C8—22no—
80C2×#41×C47—21no—
81C2×SD16×C47—21no—
82C2×SD16×C47—21no—
83C22×C8×C47D411yescharacteristic
84C22×C8×C47D411yescharacteristic
85C4⋊Q8×C47C2311yescharacteristic
86C4⋊Q8×C47C2311yescharacteristic
87C4⋊Q8×C47C2311yescharacteristic
88#127×C47C2311yescharacteristic
89#125×C47—21no—
90#124×C47—21no—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93#121×C47—21no—
94#121×C47—21no—
95C4×Q8×C47—21no—
96C4×Q8×C47—21no—
97C4×D4×C47—21no—
98C4×D4×C47—21no—
99#117×C47—21no—
100C2×C42×C47C2311yescharacteristic
101#106×C47—21no—
102#106×C47—21no—
103#103×C47—21no—
104#103×C47—21no—
105#102×C47—21no—
106#102×C47—21no—
107C4×C8×C47D411yescharacteristic
108C4×C8×C47D411yescharacteristic
109C4×C8×C47C2311yescharacteristic
110C4×C8×C47C2311yescharacteristic
111C23×D47—21no—
112#41×C47—42no—
113#41×C47—42no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—41no—
121C22×C4×C47—41no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C22×Dic47—21no—
126C2×D188—21no—
127C2×D188—22no—
128C2×D188SD1614yescomplemented
129C2×D188C2×D412yes—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—42no—
133C2×C4×D47—42no—
134SD16×C47—42no—
135SD16×C47—42no—
136C2×C8×C47—22no—
137C2×C8×C47—41no—
138C2×C8×C47C2×D411yescharacteristic
139C2×C8×C47C2×D411yescharacteristic
140C2×C8×C47C2×D411yescharacteristic
141C2×C8×C47C2×D411yescharacteristic
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—41no—
149C4⋊C4×C47—41no—
150C22⋊C4×C47—41no—
151C22⋊C4×C47—41no—
152C22⋊C4×C47—41no—
153C42×C47—41no—
154C42×C47C2×D411yescharacteristic
155C42×C47C2×D411yescharacteristic
156C42×C47#4111yescharacteristic
157C42×C47C2411yescharacteristic
158#10625336—44no—
159#10625334—44no—
160(C2×D47)⋊C4—22no—
161(C2×D47)⋊C4—22no—
162C4⋊Dic47—21no—
163C4⋊Dic47C2×D412yes—
164#10625329—42no—
165#10625329—42no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—42no—
170C8×D47—44no—
171C8×D47—44no—
172C23×C47—41no—
173C23×C47#12111yescharacteristic
174C22×D47—22no—
175C22×D47—24no—
176C22×D47—42no—
177C22×D47—42no—
178Q8×C47—41no—
179Q8×C47—41no—
180Q8×C47—41no—
181Q8×C47—41no—
182Q8×C47—81no—
183D4×C47—41no—
184D4×C47—41no—
185D4×C47—41no—
186D4×C47—41no—
187D4×C47—81no—
188D4×C47—82no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—82no—
196C2×C4×C47#12111yescharacteristic
197C2×C4×C47#12111yescharacteristic
198C2×C4×C47#12111yescharacteristic
199C2×C4×C47C2×SD1612yes—
200C2×C4×C47#13512yes—
201C2×C4×C47C22×D411yescharacteristic
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47C2×#4111yescharacteristic
204C2×Dic47—22no—
205C2×Dic47—42no—
206D188—22no—
207D188—24no—
208D188C2×SD1614yes—
209C4×D47—44no—
210C4×D47—42no—
211C4×D47—42no—
212C8×C47—21no—
213C8×C47—21no—
214C8×C47—21no—
215C8×C47AGL(1,ℤ/8)12yescomplemented
216C47⋊C8—42no—
217C47⋊C8—42no—
218C47⋊C8—42no—
219#19254—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—81no—
223C22×C47—81no—
224C22×C47—81no—
225C22×C47#46512yes—
226C22×C47C2×#12111yescharacteristic
227C2×D47—24no—
228C2×D47—48no—
229C2×D47—42no—
230C2×D47—42no—
231C4×C47—21no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—81no—
235C4×C47—81no—
236C4×C47—81no—
237C4×C47#46512yes—
238C4×C47C22×SD1611yescharacteristic
239C4×C47C2×#13511yescharacteristic
240C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C47#57511yescharacteristic
242Dic47—42no—
243#3167D4711yescharacteristic, complemented
244#3077—471no—
245#3036—471no—
246C2×C8×D4—471no—
247#1679—472no—
248#1677—472no—
249#1671—472no—
250#1665—472no—
251#1566—471no—
252#1550—471no—
253#1547—471no—
254C2×C47—22no—
255C2×C47—161no—
256C2×C47#303611yescharacteristic
257C2×C47C2×#46511yescharacteristic
258C2×C47#306711yescharacteristic
259D47—44no—
260C2×AGL(1,ℤ/8)—941no—
261C23×C8—941no—
262#533C2×D4711yescharacteristic
263#532C2×D4711yescharacteristic
264C2×C4×D4—471no—
265#498C2×D4711yescharacteristic, complemented
266#492C2×D4711yescharacteristic, complemented
267#487C2×D4711yescharacteristic, complemented
268#486C2×D4711yescharacteristic, complemented
269#478—942no—
270#476—472no—
271#474—472no—
272#465—21no—
273#465—21no—
274#463—942no—
275#461—472no—
276#459—472no—
277C4×SD16—21no—
278C4×SD16—21no—
279C8×D4—472no—
280C8×D4—472no—
281C8×D4—941no—
282C8×D4—941no—
283#428—941no—
284#423—471no—
285#423—471no—
286C2×C4⋊C8—471no—
287#418—941no—
288#417—941no—
289C2×C22⋊C8—941no—
290C2×C4×C8C2×D4711yescharacteristic
291#334—472no—
292#334—472no—
293#333—472no—
294#332—472no—
295#332—472no—
296#331—472no—
297C47#1925411yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C22×D4—471no—
300C23×C4—941no—
301AGL(1,ℤ/8)—1884no—
302C2×SD16—21no—
303C2×SD16—21no—
304C2×SD16—942no—
305C2×D8—942no—
306C2×M16—941no—
307C2×M16—941no—
308C22×C8—941no—
309C22×C8—942no—
310C22×C8—942no—
311C22×C8—1884no—
312C22×C8C22⋊D4711yescharacteristic
313C22×C8C22⋊D4711yescharacteristic
314C4⋊Q8C22×D4711yescharacteristic
315C4⋊Q8C22×D4711yescharacteristic
316C4⋊Q8C22×D4711yescharacteristic
317#127C22×D4711yescharacteristic
318#125—21no—
319#124—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—472no—
329C4×D4—472no—
330C4×D4—941no—
331C4×D4—941no—
332#117—21no—
333C2×C4⋊C4—471no—
334C2×C22⋊C4—941no—
335C2×C42C22×D4711yescharacteristic
336#107—942no—
337#106—21no—
338#106—21no—
339#106—942no—
340C4⋊C8—472no—
341C4⋊C8—472no—
342C4⋊C8—472no—
343C4⋊C8—941no—
344C4⋊C8—941no—
345C4⋊C8—941no—
346#103—21no—
347#103—21no—
348#103—942no—
349#102—21no—
350#102—21no—
351#102—942no—
352#102—942no—
353#102—942no—
354C22⋊C8—942no—
355C22⋊C8—942no—
356C4×C8C22⋊D4711yescharacteristic
357C4×C8C22⋊D4711yescharacteristic
358C4×C8C22×D4711yescharacteristic
359C4×C8C22×D4711yescharacteristic
360C24—941no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—474no—
371C2×D4—472no—
372C2×D4—941no—
373C2×D4—942no—
374C22×C4—41no—
375C22×C4—941no—
376C22×C4—942no—
377C22×C4—942no—
378C22×C4—1882no—
379C22×C4—1882no—
380C22×C4D4×D4711yescharacteristic
381C22×C4#1062535211yescharacteristic
382C22×C4C2×C22⋊D4711yescharacteristic
383SD16—42no—
384SD16—42no—
385SD16—1884no—
386D8—1884no—
387M16—1882no—
388M16—1882no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—942no—
392C2×C8—942no—
393C2×C8—942no—
394C2×C8—1884no—
395C2×C8—1882no—
396C2×C8—1884no—
397C2×C8C2×C22⋊D4711yescharacteristic
398C2×C8C2×C22⋊D4711yescharacteristic
399C2×C8C2×C22⋊D4711yescharacteristic
400C2×C8C2×C22⋊D4711yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—472no—
410C4⋊C4—941no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—942no—
415C22⋊C4—942no—
416C42—41no—
417C42D4×D4711yescharacteristic
418C42#1062535211yescharacteristic
419C42C2×C22⋊D4711yescharacteristic
420C42C23×D4711yescharacteristic
421C23—41no—
422C23—944no—
423C23—942no—
424C23—1882no—
425C23—1882no—
426C23#1309399511yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438D4—474no—
439D4—944no—
440D4—942no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—942no—
449C2×C4—1884no—
450C2×C4—1882no—
451C2×C4—1882no—
452C2×C4—1882no—
453C2×C4#1309399511yescharacteristic
454C2×C4#1309399511yescharacteristic
455C2×C4#1309399511yescharacteristic
456C2×C4SD16×D4712yes—
457C2×C4#1309404512yes—
458C2×C4C2×D4×D4711yescharacteristic
459C2×C4C22×C22⋊D4711yescharacteristic
460C2×C4C2×#1062535211yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—1882no—
465C8—1882no—
466C8—1882no—
467C8#1309404912yescomplemented
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—944no—
474C22—1888no—
475C22—1882no—
476C22—1882no—
477C22#1568934912yes—
478C22C2×#1309399511yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—1882no—
486C4#1568934912yes—
487C4C2×SD16×D4711yescharacteristic
488C4C2×#1309404511yescharacteristic
489C4C2×#1309404911yescharacteristic
490C4#1568992111yescharacteristic
491C2—22no—
492C2—161no—
493C2—1884no—
494C2#2025284811yescharacteristic
495C2C2×#1568934911yescharacteristic
496C2#2025310411yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3167C2
socleC22×C47C2×#121
prime radical#20252848C2
hypercenter#3167D47
hypocenterC47#19254
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. #20253181C2semidirect
3. C2×C8×D188C2semidirect
4. #20242808C2semidirect
5. #20242794C2semidirect
6. #20242760C2semidirect
7. #20242726C2semidirect
8. #20242144C2semidirect
9. #20242078C2semidirect
10. #20242065C2semidirect
11. C47×#3167C2semidirect
32. #15688874C22semidirect
33. #15688874C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
36. #15688869C22semidirect
37. #15688867C22semidirect
40. C47×#498C22semidirect
41. C47×#492C22semidirect
42. C47×#487C22semidirect
43. C47×#486C22semidirect
128. C2×D188SD16semidirect
215. C8×C47AGL(1,ℤ/8)semidirect
243. #3167D47semidirect
265. #498C2×D47semidirect
266. #492C2×D47semidirect
267. #487C2×D47semidirect
268. #486C2×D47semidirect
297. C47#19254semidirect
467. C8#13094049semidirect

Maximal subgroups

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

subgroupindex
#202531812
#202528482
C2×C8×D1882
#202428082
#202427942
#202427602
#202427262
#202421442
#202420782
#202420652
C47×#31672
#1925447