Finite Group Databasebeta

← #26202202

Subgroups — #26202202

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254326C211yesmaximal, characteristic, complemented
2C2×#15689318C211yesmaximal, characteristic
3#20252423C211yesmaximal, characteristic, complemented
4#20242951C212yesmaximal, complemented
5#20242855C212yesmaximal, complemented
6#20242827C212yesmaximal, complemented
7#20242731C212yesmaximal, complemented
8#20242174C211yesmaximal, characteristic, complemented
9#20242101C211yesmaximal, characteristic, complemented
10#20242088C211yesmaximal, characteristic, complemented
11C47×#3110C211yesmaximal, characteristic, complemented
12C22×#10625334—21no—
13C2×#13094021C2211yescharacteristic
14#15689497C2212yescomplemented
15#15689471C2212yescomplemented
16#15689323—22no—
17#15689318—21no—
18#15689318—21no—
19#15689318C2212yes—
20#15689318C2212yes—
21#15689273—21no—
22#15689273—21no—
23#15689201C2212yescomplemented
24#15689195—22no—
25D4×C47⋊C8C2212yescomplemented
26#15689180C2211yescharacteristic
27#15689176C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093944—21no—
30#15689087—21no—
31C2×C22⋊(C47⋊C8)—21no—
32#15688875C2212yescomplemented
33#15688873C2212yescomplemented
34#15688869C2212yescomplemented
35#15688869C2212yescomplemented
36#15688866C2212yescomplemented
37#15688866C2212yescomplemented
38C47×#533C2211yescharacteristic
39C2×C4×D4×C47C2211yescharacteristic
40C47×#484—21no—
41C47×#474C2211yescharacteristic, complemented
42C47×#474C2211yescharacteristic, complemented
43C47×#461C2211yescharacteristic, complemented
44C47×#461C2211yescharacteristic, complemented
45C47×#450—21no—
46C47×#440—21no—
47C47×#440—21no—
48C47×#432C2211yescharacteristic
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52C2×#10625336—22no—
53C2×#10625336—22no—
54C2×#10625334—22no—
55C2×#10625334—22no—
56C2×#10625334—44no—
57C2×#10625329—22no—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—22no—
60#13094021—21no—
61#13094021—21no—
62#13094021—21no—
63#13094021C2312yes—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)D412yes—
66C4⋊(C47⋊C8)C2312yes—
67C4⋊(C47⋊C8)C2312yes—
68#13093944—22no—
69#13093944—22no—
70#13093944—22no—
71#13093944—22no—
72C22⋊(C47⋊C8)—21no—
73C22⋊(C47⋊C8)—22no—
74C22⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)D411yescharacteristic
76C22⋊(C47⋊C8)D411yescharacteristic
77#13093935—21no—
78C4×C47⋊C8C2312yes—
79C22×D4×C47D411yescharacteristic
80C23×C4×C47—21no—
81C2×SD16×C47—21no—
82C2×SD16×C47—21no—
83C4⋊Q8×C47C2311yescharacteristic
84C4⋊Q8×C47C2311yescharacteristic
85#125×C47—21no—
86C22⋊Q8×C47—21no—
87C22⋊Q8×C47—21no—
88C4×Q8×C47—21no—
89C4×Q8×C47—21no—
90C4×D4×C47—21no—
91C4×D4×C47—21no—
92C4×D4×C47D411yescharacteristic
93C4×D4×C47D411yescharacteristic
94C4×D4×C47C2311yescharacteristic
95C4×D4×C47C2311yescharacteristic
96#117×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C22⋊C4×C47—21no—
99C2×C42×C47C2311yescharacteristic
100#107×C47—21no—
101#107×C47—21no—
102C4⋊C8×C47C2311yescharacteristic
103C4⋊C8×C47C2311yescharacteristic
104#103×C47—21no—
105#103×C47—21no—
106#102×C47—21no—
107#102×C47—21no—
108C22⋊C8×C47—21no—
109#97×C47—21no—
110C24×C47—21no—
111C23×D47—41no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×D4×C47—22no—
115C2×D4×C47—21no—
116C2×D4×C47—41no—
117C2×D4×C47C2×D411yescharacteristic
118C2×D4×C47C2×D411yescharacteristic
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—82no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47#4111yescharacteristic
128C2×D188—22no—
129C2×D188—22no—
130C2×D188—22no—
131C2×D188—42no—
132C2×D188—42no—
133C2×D188—41no—
134C2×D188—41no—
135C2×D188—41no—
136C2×D188—82no—
137C2×D188—82no—
138SD16×C47—42no—
139SD16×C47—42no—
140C2×C8×C47—21no—
141C2×C8×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—41no—
148C4⋊C4×C47—41no—
149C4⋊C4×C47C2×D411yescharacteristic
150C4⋊C4×C47C2×D411yescharacteristic
151C22⋊C4×C47—21no—
152C22⋊C4×C47—21no—
153C22⋊C4×C47—21no—
154C22⋊C4×C47—21no—
155C22⋊C4×C47—41no—
156C42×C47—41no—
157C42×C47C2×D411yescharacteristic
158C42×C47C2×D411yescharacteristic
159C42×C47#4111yescharacteristic
160C42×C47C2411yescharacteristic
161#10625336—82no—
162#10625336—82no—
163#10625334—44no—
164#10625334—44no—
165#10625329—82no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—42no—
169C2×C47⋊C8—82no—
170C2×C47⋊C8C2×D412yes—
171C2×C47⋊C8C2×D412yes—
172C23×C47—21no—
173C23×C47—21no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—81no—
177C23×C47#12111yescharacteristic
178C22×D47—42no—
179C22×D47—82no—
180C22×D47—82no—
181C22×D47—82no—
182Q8×C47—41no—
183Q8×C47—41no—
184Q8×C47—42no—
185D4×C47—21no—
186D4×C47—21no—
187D4×C47—21no—
188D4×C47—42no—
189D4×C47AGL(1,ℤ/8)12yes—
190C2×C4×C47—22no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—22no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—81no—
201C2×C4×C47—81no—
202C2×C4×C47—81no—
203C2×C4×C47#12111yescharacteristic
204C2×C4×C47#12111yescharacteristic
205C2×C4×C47#12111yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C22×D411yescharacteristic
208C2×C4×C47C2×#4111yescharacteristic
209D188—42no—
210D188—42no—
211D188—42no—
212D188—42no—
213D188—82no—
214D188—82no—
215D188—82no—
216D188—82no—
217C8×C47—41no—
218C8×C47—41no—
219C47⋊C8—22no—
220C47⋊C8—22no—
221C47⋊C8—42no—
222C47⋊C8—42no—
223#19456—471nomaximal, Sylow
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—22no—
227C22×C47—41no—
228C22×C47—41no—
229C22×C47—41no—
230C22×C47—41no—
231C22×C47—41no—
232C22×C47—84no—
233C22×C47#47112yes—
234C22×C47C2×#12111yescharacteristic
235C2×D47—82no—
236C2×D47—82no—
237C2×D47—82no—
238C2×D47—164no—
239C4×C47—21no—
240C4×C47—21no—
241C4×C47—21no—
242C4×C47—21no—
243C4×C47—41no—
244C4×C47—41no—
245C4×C47—81no—
246C4×C47—81no—
247C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
248C4×C47#57511yescharacteristic
249C4×C47#57612yes—
250#3208—471no—
251#3110D4711yescharacteristic, complemented
252C2×#459—471no—
253#2974—471no—
254#1701—472no—
255#1685—472no—
256#1681—472no—
257#1665—472no—
258#1573—471no—
259#1555—471no—
260#1552—471no—
261C2×C47—22no—
262C2×C47—42no—
263C2×C47—81no—
264C2×C47#304111yescharacteristic
265C2×C47#305511yescharacteristic
266C2×C47C2×#47111yescharacteristic
267D47—162no—
268C22×D8—941no—
269#533C2×D4711yescharacteristic
270C2×#127—471no—
271C2×C4×D4C2×D4711yescharacteristic
272#492—472no—
273#486—472no—
274#484—21no—
275#474C2×D4711yescharacteristic, complemented
276#474C2×D4711yescharacteristic, complemented
277#461C2×D4711yescharacteristic, complemented
278#461C2×D4711yescharacteristic, complemented
279#460—942no—
280#459—472no—
281#459—472no—
282#459—941no—
283#459—941no—
284#450—21no—
285#450—941no—
286#450—941no—
287#440—21no—
288#440—21no—
289#436—472no—
290#435—942no—
291C8×D4—472no—
292#432—471no—
293#432C2×D4711yescharacteristic
294#431—471no—
295C2×C4⋊C8—471no—
296C2×#102—941no—
297#407—941no—
298C2×C22⋊C8—941no—
299#335—472no—
300#334—472no—
301#332—472no—
302#332—472no—
303#331—472no—
304#331—472no—
305C47#1945611yescharacteristic, complemented, Sylow
306C22×D4—941no—
307C22×D4—941no—
308C22×D4—941no—
309C22×D4C22⋊D4711yescharacteristic
310C23×C4—21no—
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—942no—
314C2×SD16—942no—
315C2×D8—942no—
316C2×D8—942no—
317C2×D8—1884no—
318C2×M16—942no—
319C22×C8—942no—
320C22×C8—941no—
321C4⋊Q8C22×D4711yescharacteristic
322C4⋊Q8C22×D4711yescharacteristic
323#127—472no—
324#127—941no—
325#127—941no—
326#127—941no—
327#125—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4C22⋊D4711yescharacteristic
335C4×D4C22⋊D4711yescharacteristic
336C4×D4C22×D4711yescharacteristic
337C4×D4C22×D4711yescharacteristic
338#117—21no—
339C2×C4⋊C4C22⋊D4711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D4711yescharacteristic
342#107—21no—
343#107—21no—
344C4⋊C8—472no—
345C4⋊C8—472no—
346C4⋊C8—472no—
347C4⋊C8—941no—
348C4⋊C8C22×D4711yescharacteristic
349C4⋊C8C22×D4711yescharacteristic
350#103—21no—
351#103—21no—
352#102—21no—
353#102—21no—
354#102—942no—
355#102—942no—
356#102—942no—
357#102—942no—
358C22⋊C8—21no—
359C22⋊C8—471no—
360C22⋊C8—471no—
361C22⋊C8—942no—
362C22⋊C8—942no—
363C22⋊C8—941no—
364#97—21no—
365#97—941no—
366C4×C8—472no—
367C24—21no—
368C24—1881no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—942no—
375C2×D4—942no—
376C2×D4—942no—
377C2×D4—3762no—
378C2×D4—3762no—
379C2×D4—1882no—
380C2×D4—1882no—
381C2×D4—1881no—
382C2×D4—1881no—
383C2×D4—1881no—
384C2×D4C2×C22⋊D4711yescharacteristic
385C2×D4C2×C22⋊D4711yescharacteristic
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—82no—
392C22×C4D4×D4711yescharacteristic
393C22×C4#1062535211yescharacteristic
394C22×C4C2×C22⋊D4711yescharacteristic
395SD16—42no—
396SD16—42no—
397SD16—3762no—
398SD16—3762no—
399D8—1884no—
400D8—1884no—
401M16—3762no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—472no—
405C2×C8—472no—
406C2×C8—942no—
407C2×C8—942no—
408C2×C8—3762no—
409C2×C8—1882no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4C2×C22⋊D4711yescharacteristic
418C4⋊C4C2×C22⋊D4711yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—41no—
424C42—41no—
425C42D4×D4711yescharacteristic
426C42#1062535211yescharacteristic
427C42C2×C22⋊D4711yescharacteristic
428C42C23×D4711yescharacteristic
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—3762no—
435C23—3762no—
436C23—3762no—
437C23—1882no—
438C23#1309399611yescharacteristic
439Q8—41no—
440Q8—41no—
441Q8—42no—
442D4—21no—
443D4—21no—
444D4—21no—
445D4—42no—
446D4—3762no—
447D4—3762no—
448D4—3762no—
449D4—3762no—
450D4—1882no—
451D4—1882no—
452D4—1882no—
453D4—1882no—
454D4#1309404912yes—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#1309399611yescharacteristic
469C2×C4#1309399611yescharacteristic
470C2×C4#1309399611yescharacteristic
471C2×C4C2×D4×D4711yescharacteristic
472C2×C4C22×C22⋊D4711yescharacteristic
473C2×C4C2×#1062535211yescharacteristic
474C8—41no—
475C8—41no—
476C8—942no—
477C8—942no—
478C8—1882no—
479C8—1882no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—3762no—
490C22—3762no—
491C22—3762no—
492C22—7524no—
493C22#1568938812yes—
494C22C2×#1309399611yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—81no—
502C4—81no—
503C4C2×#1309404911yescharacteristic
504C4#1568992111yescharacteristic
505C4#1568992712yes—
506C2—22no—
507C2—42no—
508C2—81no—
509C2—7522no—
510C2#2025289311yescharacteristic
511C2#2025300911yescharacteristic
512C2C2×#1568938811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3110C2
socleC22×C47C2×#121
prime radicalC2×#15689318C2
hypercenter#3110D47
hypocenterC47#19456
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20254326C2semidirect
3. #20252423C2semidirect
4. #20242951C2semidirect
5. #20242855C2semidirect
6. #20242827C2semidirect
7. #20242731C2semidirect
8. #20242174C2semidirect
9. #20242101C2semidirect
10. #20242088C2semidirect
11. C47×#3110C2semidirect
14. #15689497C22semidirect
15. #15689471C22semidirect
23. #15689201C22semidirect
25. D4×C47⋊C8C22semidirect
32. #15688875C22semidirect
33. #15688873C22semidirect
34. #15688869C22semidirect
35. #15688869C22semidirect
36. #15688866C22semidirect
37. #15688866C22semidirect
41. C47×#474C22semidirect
42. C47×#474C22semidirect
43. C47×#461C22semidirect
44. C47×#461C22semidirect
251. #3110D47semidirect
275. #474C2×D47semidirect
276. #474C2×D47semidirect
277. #461C2×D47semidirect
278. #461C2×D47semidirect
305. C47#19456semidirect

Maximal subgroups

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

subgroupindex
#202543262
C2×#156893182
#202524232
#202429512
#202428552
#202428272
#202427312
#202421742
#202421012
#202420882
C47×#31102
#1945647