Finite Group Databasebeta

← #26199136

Subgroups — #26199136

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253841C211yesmaximal, characteristic
2C2×#15689323C211yesmaximal, characteristic, complemented
3Q8×#10625329C211yesmaximal, characteristic, complemented
4#20242870C212yesmaximal, complemented
5#20242862C212yesmaximal, complemented
6#20242746C212yesmaximal, complemented
7#20242738C212yesmaximal, complemented
8#20242141C211yesmaximal, characteristic, complemented
9#20242065C211yesmaximal, characteristic, complemented
10#20242055C211yesmaximal, characteristic, complemented
11C47×#3103C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13C2×#13094049—21no—
14C22×#10625336—21no—
15C2×#13094021C2211yescharacteristic
16#15689513—22no—
17#15689499C2212yes—
18#15689497C2212yes—
19#15689323—21no—
20#15689323—21no—
21#15689323C2212yes—
22#15689323C2212yes—
23#15689254—22no—
24#15689254C2212yescomplemented
25Q8×C47⋊C8C2212yescomplemented
26#15689147—21no—
27#15689147C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093944—21no—
30C4×#10625329—21no—
31C4×#10625329C2211yescharacteristic
32#15688875C2212yescomplemented
33#15688873C2212yescomplemented
34#15688869C2212yescomplemented
35#15688867C2212yescomplemented
36#15688866C2212yes—
37#15688866C2212yes—
38C47×#532C2211yescharacteristic
39C2×C4×Q8×C47C2211yescharacteristic
40C47×#481—21no—
41C47×#475C2211yescharacteristic, complemented
42C47×#475C2211yescharacteristic, complemented
43C47×#460C2211yescharacteristic, complemented
44C47×#460C2211yescharacteristic, complemented
45C47×#448—21no—
46C4×SD16×C47—21no—
47C4×SD16×C47—21no—
48C47×#431C2211yescharacteristic
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52#13094049—44no—
53#13094049—44no—
54C2×#10625336—22no—
55C2×#10625336—22no—
56C2×#10625336—22no—
57C2×#10625336—22no—
58C2×#10625336—44no—
59C2×#10625334—22no—
60C2×#10625334—22no—
61C2×#10625329—21no—
62C2×#10625329D411yescharacteristic
63C2×#10625329D411yescharacteristic
64C22×C47⋊C8—21no—
65#13094021—21no—
66#13094021—21no—
67#13094021—21no—
68#13094021C2312yes—
69C4⋊(C47⋊C8)—21no—
70C4⋊(C47⋊C8)—21no—
71C4⋊(C47⋊C8)—22no—
72C4⋊(C47⋊C8)—22no—
73C4⋊(C47⋊C8)C2312yes—
74C4⋊(C47⋊C8)C2312yes—
75#13093944—22no—
76#13093944—22no—
77#13093935—22no—
78#13093935D412yes—
79C4×C47⋊C8—22no—
80C4×C47⋊C8C2312yes—
81C2×#41×C47—21no—
82C22×Q8×C47D411yescharacteristic
83C2×SD16×C47—21no—
84C2×SD16×C47—21no—
85C4⋊Q8×C47C2311yescharacteristic
86#127×C47C2311yescharacteristic
87#124×C47—21no—
88#121×C47—21no—
89#121×C47—21no—
90C4×Q8×C47—21no—
91C4×Q8×C47—21no—
92C4×Q8×C47D411yescharacteristic
93C4×Q8×C47D411yescharacteristic
94C4×Q8×C47C2311yescharacteristic
95C4×Q8×C47C2311yescharacteristic
96C4×D4×C47—21no—
97C4×D4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C42×C47—21no—
101C2×C42×C47C2311yescharacteristic
102#106×C47—21no—
103#106×C47—21no—
104C4⋊C8×C47C2311yescharacteristic
105C4⋊C8×C47C2311yescharacteristic
106#103×C47—21no—
107#103×C47—21no—
108#102×C47—21no—
109#102×C47—21no—
110C22⋊C8×C47—21no—
111C4×C8×C47—21no—
112C23×D47—41no—
113#41×C47—42no—
114#41×C47—42no—
115C2×Q8×C47—21no—
116C2×Q8×C47—22no—
117C2×Q8×C47—21no—
118C2×Q8×C47—41no—
119C2×Q8×C47C2×D411yescharacteristic
120C2×Q8×C47C2×D411yescharacteristic
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—41no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—41no—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47#4111yescharacteristic
131C2×D188—22no—
132C2×D188—22no—
133C2×D188—22no—
134C2×D188—42no—
135C2×D188—42no—
136C2×D188—42no—
137C2×D188—42no—
138C2×D188—41no—
139C2×D188—41no—
140C2×D188—41no—
141C2×D188—82no—
142SD16×C47—42no—
143SD16×C47—42no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47C2×D411yescharacteristic
154C4⋊C4×C47C2×D411yescharacteristic
155C22⋊C4×C47—41no—
156C22⋊C4×C47—41no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47C2×D411yescharacteristic
162C42×C47C2×D411yescharacteristic
163C42×C47#4111yescharacteristic
164C42×C47C2411yescharacteristic
165#10625336—44no—
166#10625336—44no—
167#10625336—44no—
168#10625336—44no—
169#10625334—44no—
170#10625334—44no—
171#10625329—22no—
172#10625329—42no—
173#10625329SD1614yescomplemented
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—42no—
177C2×C47⋊C8C2×D412yes—
178C2×C47⋊C8C2×D412yes—
179C23×C47—41no—
180C23×C47#12111yescharacteristic
181C22×D47—42no—
182C22×D47—82no—
183C22×D47—82no—
184C22×D47—82no—
185Q8×C47—21no—
186Q8×C47—21no—
187Q8×C47—21no—
188Q8×C47—42no—
189Q8×C47—41no—
190Q8×C47—41no—
191Q8×C47AGL(1,ℤ/8)12yescomplemented
192D4×C47—41no—
193D4×C47—41no—
194D4×C47—41no—
195D4×C47—41no—
196D4×C47—81no—
197D4×C47—82no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—81no—
208C2×C4×C47—82no—
209C2×C4×C47#12111yescharacteristic
210C2×C4×C47#12111yescharacteristic
211C2×C4×C47#12111yescharacteristic
212C2×C4×C47C2×SD1612yes—
213C2×C4×C47AGL(1,ℤ/8)12yes—
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217D188—42no—
218D188—42no—
219D188—42no—
220D188—42no—
221D188—82no—
222D188—82no—
223D188—82no—
224D188—82no—
225C8×C47—41no—
226C8×C47—41no—
227C47⋊C8—22no—
228C47⋊C8—42no—
229C47⋊C8—42no—
230C47⋊C8C2×SD1614yescomplemented
231#19170—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—81no—
235C22×C47—81no—
236C22×C47—81no—
237C22×C47#46012yes—
238C22×C47C2×#12111yescharacteristic
239C2×D47—82no—
240C2×D47—82no—
241C2×D47—82no—
242C2×D47—164no—
243C4×C47—21no—
244C4×C47—22no—
245C4×C47—21no—
246C4×C47—21no—
247C4×C47—41no—
248C4×C47—41no—
249C4×C47—41no—
250C4×C47—81no—
251C4×C47#46012yes—
252C4×C47C22×SD1611yescharacteristic
253C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C47#57511yescharacteristic
256#3151—471no—
257#3103D4711yescharacteristic, complemented
258C2×#460—471no—
259Q8×M16—471no—
260#1688—472no—
261#1686—472no—
262#1668—472no—
263#1666—472no—
264#1565—471no—
265#1547—471no—
266#1545—471no—
267C2×C47—22no—
268C2×C47—161no—
269C2×C47C2×#46011yescharacteristic
270C2×C47#305511yescharacteristic
271C2×C47#306811yescharacteristic
272D47—162no—
273C2×AGL(1,ℤ/8)—941no—
274C2×AGL(1,ℤ/8)—941no—
275C22×SD16—941no—
276#532C2×D4711yescharacteristic
277C2×#127—471no—
278C2×C4×Q8C2×D4711yescharacteristic
279#496—942no—
280#493—472no—
281#492—472no—
282#481—21no—
283#475C2×D4711yescharacteristic, complemented
284#475C2×D4711yescharacteristic, complemented
285#460—472no—
286#460—472no—
287#460—941no—
288#460—941no—
289#460C2×D4711yescharacteristic, complemented
290#460C2×D4711yescharacteristic, complemented
291#448—21no—
292#446—472no—
293#446—942no—
294C8×Q8—472no—
295C4×SD16—21no—
296C4×SD16—21no—
297#431C2×D4711yescharacteristic
298#423—471no—
299#423—941no—
300C2×C4⋊C8—471no—
301C2×#102—941no—
302C4×M16—471no—
303C4×M16—941no—
304#335—472no—
305#334—472no—
306#332—472no—
307#331—472no—
308#331—472no—
309#331—472no—
310C47#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8C22⋊D4711yescharacteristic
313C22×D4—941no—
314C22×D4—941no—
315C22×D4—941no—
316AGL(1,ℤ/8)—1884no—
317AGL(1,ℤ/8)—1884no—
318C2×SD16—21no—
319C2×SD16—21no—
320C2×SD16—942no—
321C2×SD16—942no—
322C2×SD16—942no—
323C2×SD16—942no—
324C2×SD16—1884no—
325C2×D8—942no—
326C2×D8—942no—
327C2×M16—471no—
328C2×M16—471no—
329C2×M16—941no—
330C22×C8—941no—
331C4⋊Q8C22×D4711yescharacteristic
332#127—472no—
333#127—941no—
334#127—941no—
335#127—941no—
336#127C22×D4711yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D4711yescharacteristic
343C4×Q8C22⋊D4711yescharacteristic
344C4×Q8C22×D4711yescharacteristic
345C4×Q8C22×D4711yescharacteristic
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D4711yescharacteristic
350C2×C42—21no—
351C2×C42C22×D4711yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—472no—
355C4⋊C8—472no—
356C4⋊C8—942no—
357C4⋊C8—942no—
358C4⋊C8—941no—
359C4⋊C8—941no—
360C4⋊C8C22×D4711yescharacteristic
361C4⋊C8C22×D4711yescharacteristic
362#103—21no—
363#103—21no—
364#102—21no—
365#102—21no—
366#102—942no—
367#102—942no—
368C22⋊C8—21no—
369#97—472no—
370#97—942no—
371C4×C8—21no—
372C4×C8—472no—
373C4×C8—942no—
374C24—1881no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8C2×C22⋊D4711yescharacteristic
382C2×Q8C2×C22⋊D4711yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—942no—
388C2×D4—942no—
389C2×D4—942no—
390C2×D4—3762no—
391C2×D4—1882no—
392C2×D4—1882no—
393C2×D4—1881no—
394C2×D4—1881no—
395C2×D4—1881no—
396C2×D4—1882no—
397C2×D4—1882no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4D4×D4711yescharacteristic
402C22×C4#1062535211yescharacteristic
403C22×C4C2×C22⋊D4711yescharacteristic
404SD16—42no—
405SD16—42no—
406SD16—1884no—
407SD16—1884no—
408SD16—1884no—
409SD16—1884no—
410D8—1884no—
411D8—1884no—
412M16—474no—
413M16—942no—
414M16—1882no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—472no—
418C2×C8—472no—
419C2×C8—942no—
420C2×C8—942no—
421C2×C8—1882no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C2×C22⋊D4711yescharacteristic
430C4⋊C4C2×C22⋊D4711yescharacteristic
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D4711yescharacteristic
438C42#1062535211yescharacteristic
439C42C2×C22⋊D4711yescharacteristic
440C42C23×D4711yescharacteristic
441C23—41no—
442C23—3762no—
443C23—3762no—
444C23—3762no—
445C23—1882no—
446C23#1309399611yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8#1309404912yescomplemented
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—3762no—
461D4—3762no—
462D4—3762no—
463D4—3762no—
464D4—1882no—
465D4—1882no—
466D4—1882no—
467D4—1882no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4#1309399611yescharacteristic
480C2×C4#1309399611yescharacteristic
481C2×C4#1309399611yescharacteristic
482C2×C4SD16×D4712yes—
483C2×C4#1309405112yes—
484C2×C4C2×D4×D4711yescharacteristic
485C2×C4C22×C22⋊D4711yescharacteristic
486C2×C4C2×#1062535211yescharacteristic
487C8—41no—
488C8—41no—
489C8—474no—
490C8—942no—
491C8—1882no—
492C8—1882no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—3762no—
499C22—3762no—
500C22—3762no—
501C22—7524no—
502C22#1568932112yes—
503C22C2×#1309399611yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4#1568932112yes—
513C4C2×SD16×D4711yescharacteristic
514C4C2×#1309404911yescharacteristic
515C4C2×#1309405111yescharacteristic
516C4#1568992111yescharacteristic
517C2—22no—
518C2—161no—
519C2—7522no—
520C2C2×#1568932111yescharacteristic
521C2#2025300911yescharacteristic
522C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3103C2
socleC22×C47C2×#121
prime radical#20253841C2
hypercenter#3103D47
hypocenterC47#19170
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×#15689323C2semidirect
3. Q8×#10625329C2semidirect
4. #20242870C2semidirect
5. #20242862C2semidirect
6. #20242746C2semidirect
7. #20242738C2semidirect
8. #20242141C2semidirect
9. #20242065C2semidirect
10. #20242055C2semidirect
11. C47×#3103C2semidirect
24. #15689254C22semidirect
25. Q8×C47⋊C8C22semidirect
32. #15688875C22semidirect
33. #15688873C22semidirect
34. #15688869C22semidirect
35. #15688867C22semidirect
41. C47×#475C22semidirect
42. C47×#475C22semidirect
43. C47×#460C22semidirect
44. C47×#460C22semidirect
173. #10625329SD16semidirect
191. Q8×C47AGL(1,ℤ/8)semidirect
230. C47⋊C8C2×SD16semidirect
257. #3103D47semidirect
283. #475C2×D47semidirect
284. #475C2×D47semidirect
289. #460C2×D47semidirect
290. #460C2×D47semidirect
310. C47#19170semidirect
453. Q8#13094049semidirect

Maximal subgroups

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

subgroupindex
#202538412
C2×#156893232
Q8×#106253292
#202428702
#202428622
#202427462
#202427382
#202421412
#202420652
#202420552
C47×#31032
#1917047