Finite Group Databasebeta

← #26199092

Subgroups — #26199092

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253764C211yesmaximal, characteristic, complemented
2C2×#15689318C211yesmaximal, characteristic
3D4×#10625329C211yesmaximal, characteristic, complemented
4#20242951C212yesmaximal, complemented
5#20242923C212yesmaximal, complemented
6#20242827C212yesmaximal, complemented
7#20242799C212yesmaximal, complemented
8#20242141C211yesmaximal, characteristic, complemented
9#20242078C211yesmaximal, characteristic, complemented
10#20242055C211yesmaximal, characteristic, complemented
11C47×#3101C211yesmaximal, characteristic, complemented
12C22×#10625334—21no—
13C22×#10625329—21no—
14C2×#13094021C2211yescharacteristic
15#15689489—22no—
16#15689471C2212yescomplemented
17#15689471C2212yescomplemented
18#15689318—21no—
19#15689318—21no—
20#15689318C2212yes—
21#15689318C2212yes—
22#15689201C2212yescomplemented
23#15689195—22no—
24D4×C47⋊C8C2212yescomplemented
25#15689147C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27#15689127—21no—
28#15689127—21no—
29C2×#13093944—21no—
30#15689087—21no—
31C4×#10625329C2211yescharacteristic
32#15688875C2212yescomplemented
33#15688873C2212yescomplemented
34#15688872C2212yescomplemented
35#15688869C2212yescomplemented
36#15688866C2212yescomplemented
37#15688866C2212yescomplemented
38C47×#533C2211yescharacteristic
39C2×C4×D4×C47C2211yescharacteristic
40C47×#483—21no—
41C47×#476C2211yescharacteristic, complemented
42C47×#476C2211yescharacteristic, complemented
43C47×#461C2211yescharacteristic, complemented
44C47×#461C2211yescharacteristic, complemented
45C47×#450—21no—
46C4×SD16×C47—21no—
47C4×SD16×C47—21no—
48C47×#431C2211yescharacteristic
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52C2×#10625334—22no—
53C2×#10625334—22no—
54C2×#10625334—44no—
55C2×#10625329—21no—
56C2×#10625329—22no—
57C2×#10625329—44no—
58C2×#10625329D411yescharacteristic
59C2×#10625329D411yescharacteristic
60C22×C47⋊C8—21no—
61C22×C47⋊C8—22no—
62#13094021—21no—
63#13094021—21no—
64#13094021—21no—
65#13094021C2312yes—
66C4⋊(C47⋊C8)—21no—
67C4⋊(C47⋊C8)—21no—
68C4⋊(C47⋊C8)C2312yes—
69C4⋊(C47⋊C8)C2312yes—
70#13093944—22no—
71#13093944—22no—
72#13093944—22no—
73#13093944—22no—
74#13093944—22no—
75#13093944—22no—
76C22⋊(C47⋊C8)—22no—
77C22⋊(C47⋊C8)—22no—
78#13093935D412yes—
79C4×C47⋊C8C2312yes—
80C22×D4×C47D411yescharacteristic
81C23×C4×C47—21no—
82C2×SD16×C47—21no—
83C2×SD16×C47—21no—
84C4⋊Q8×C47C2311yescharacteristic
85C4⋊Q8×C47C2311yescharacteristic
86#125×C47—21no—
87C22⋊Q8×C47—21no—
88C22⋊Q8×C47—21no—
89C4×Q8×C47—21no—
90C4×Q8×C47—21no—
91C4×D4×C47—21no—
92C4×D4×C47—21no—
93C4×D4×C47D411yescharacteristic
94C4×D4×C47D411yescharacteristic
95C4×D4×C47C2311yescharacteristic
96C4×D4×C47C2311yescharacteristic
97#117×C47—21no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C22⋊C4×C47—21no—
100C2×C42×C47C2311yescharacteristic
101#106×C47—21no—
102#106×C47—21no—
103C4⋊C8×C47C2311yescharacteristic
104C4⋊C8×C47C2311yescharacteristic
105#103×C47—21no—
106#103×C47—21no—
107#102×C47—21no—
108#102×C47—21no—
109C22⋊C8×C47—21no—
110C4×C8×C47—21no—
111C24×C47—21no—
112C23×D47—41no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×D4×C47—22no—
116C2×D4×C47—21no—
117C2×D4×C47—41no—
118C2×D4×C47C2×D411yescharacteristic
119C2×D4×C47C2×D411yescharacteristic
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47#4111yescharacteristic
130C2×D188—22no—
131C2×D188—22no—
132C2×D188—22no—
133C2×D188—42no—
134C2×D188—42no—
135C2×D188—42no—
136C2×D188—42no—
137C2×D188—41no—
138C2×D188—41no—
139C2×D188—41no—
140C2×D188—82no—
141SD16×C47—42no—
142SD16×C47—42no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—41no—
151C4⋊C4×C47—41no—
152C4⋊C4×C47C2×D411yescharacteristic
153C4⋊C4×C47C2×D411yescharacteristic
154C22⋊C4×C47—21no—
155C22⋊C4×C47—21no—
156C22⋊C4×C47—21no—
157C22⋊C4×C47—21no—
158C22⋊C4×C47—41no—
159C42×C47—41no—
160C42×C47C2×D411yescharacteristic
161C42×C47C2×D411yescharacteristic
162C42×C47#4111yescharacteristic
163C42×C47C2411yescharacteristic
164#10625334—44no—
165#10625334—44no—
166#10625329—22no—
167#10625329—22no—
168#10625329—44no—
169#10625329—42no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—44no—
173C2×C47⋊C8—42no—
174C2×C47⋊C8C2×D412yes—
175C2×C47⋊C8C2×D412yes—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—81no—
181C23×C47#12111yescharacteristic
182C22×D47—42no—
183C22×D47—82no—
184C22×D47—82no—
185C22×D47—82no—
186Q8×C47—41no—
187Q8×C47—41no—
188Q8×C47—81no—
189D4×C47—21no—
190D4×C47—21no—
191D4×C47—21no—
192D4×C47—42no—
193D4×C47AGL(1,ℤ/8)12yes—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—41no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47#12111yescharacteristic
209C2×C4×C47#12111yescharacteristic
210C2×C4×C47#12111yescharacteristic
211C2×C4×C47#13512yes—
212C2×C4×C47AGL(1,ℤ/8)12yes—
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216D188—42no—
217D188—42no—
218D188—42no—
219D188—42no—
220D188—82no—
221D188—82no—
222D188—82no—
223D188—82no—
224C8×C47—41no—
225C8×C47—41no—
226C47⋊C8—22no—
227C47⋊C8—22no—
228C47⋊C8—42no—
229C47⋊C8—42no—
230#19166—471nomaximal, Sylow
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—22no—
234C22×C47—41no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47—41no—
238C22×C47—41no—
239C22×C47—84no—
240C22×C47#46312yes—
241C22×C47C2×#12111yescharacteristic
242C2×D47—82no—
243C2×D47—82no—
244C2×D47—82no—
245C2×D47—164no—
246C4×C47—21no—
247C4×C47—21no—
248C4×C47—21no—
249C4×C47—41no—
250C4×C47—41no—
251C4×C47—81no—
252C4×C47—81no—
253C4×C47#46312yes—
254C4×C47C2×#13511yescharacteristic
255C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
257C4×C47#57511yescharacteristic
258#3138—471no—
259#3101D4711yescharacteristic, complemented
260C2×#459—471no—
261D4×M16—471no—
262#1701—472no—
263#1697—472no—
264#1681—472no—
265#1677—472no—
266#1565—471no—
267#1550—471no—
268#1545—471no—
269C2×C47—22no—
270C2×C47—42no—
271C2×C47—81no—
272C2×C47C2×#46311yescharacteristic
273C2×C47#305511yescharacteristic
274C2×C47#306811yescharacteristic
275D47—162no—
276C22×D8—941no—
277C22×M16—941no—
278#533C2×D4711yescharacteristic
279C2×#127—471no—
280C2×C4×D4C2×D4711yescharacteristic
281#490—942no—
282#486—472no—
283#486—472no—
284#483—21no—
285#476C2×D4711yescharacteristic, complemented
286#476C2×D4711yescharacteristic, complemented
287#461C2×D4711yescharacteristic, complemented
288#461C2×D4711yescharacteristic, complemented
289#459—472no—
290#459—472no—
291#459—941no—
292#459—941no—
293#450—21no—
294C4×SD16—21no—
295C4×SD16—21no—
296#436—472no—
297#435—942no—
298C8×D4—472no—
299#431C2×D4711yescharacteristic
300#423—471no—
301C2×C4⋊C8—471no—
302#418—941no—
303#418—941no—
304C2×#102—941no—
305#407—941no—
306C4×M16—471no—
307#335—472no—
308#334—472no—
309#333—472no—
310#332—472no—
311#331—472no—
312#331—472no—
313C47#1916611yescharacteristic, complemented, Sylow
314C22×D4—941no—
315C22×D4—941no—
316C22×D4—941no—
317C22×D4C22⋊D4711yescharacteristic
318C23×C4—21no—
319C2×SD16—21no—
320C2×SD16—21no—
321C2×D8—942no—
322C2×D8—942no—
323C2×D8—1884no—
324C2×M16—471no—
325C2×M16—471no—
326C2×M16—942no—
327C2×M16—941no—
328C2×M16—1884no—
329C22×C8—942no—
330C22×C8—941no—
331C4⋊Q8C22×D4711yescharacteristic
332C4⋊Q8C22×D4711yescharacteristic
333#127—472no—
334#127—941no—
335#127—941no—
336#127—941no—
337#125—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4C22⋊D4711yescharacteristic
345C4×D4C22⋊D4711yescharacteristic
346C4×D4C22×D4711yescharacteristic
347C4×D4C22×D4711yescharacteristic
348#117—21no—
349C2×C4⋊C4C22⋊D4711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D4711yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—472no—
355C4⋊C8—472no—
356C4⋊C8—941no—
357C4⋊C8—941no—
358C4⋊C8C22×D4711yescharacteristic
359C4⋊C8C22×D4711yescharacteristic
360#103—21no—
361#103—21no—
362#102—21no—
363#102—21no—
364#102—942no—
365#102—942no—
366#102—942no—
367#102—942no—
368#102—942no—
369#102—942no—
370C22⋊C8—21no—
371C22⋊C8—942no—
372C22⋊C8—942no—
373#97—472no—
374C4×C8—21no—
375C4×C8—472no—
376C24—21no—
377C24—1881no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—942no—
384C2×D4—942no—
385C2×D4—942no—
386C2×D4—3762no—
387C2×D4—1882no—
388C2×D4—1882no—
389C2×D4—1881no—
390C2×D4—1881no—
391C2×D4—1881no—
392C2×D4—1882no—
393C2×D4—1882no—
394C2×D4C2×C22⋊D4711yescharacteristic
395C2×D4C2×C22⋊D4711yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4D4×D4711yescharacteristic
404C22×C4#1062535211yescharacteristic
405C22×C4C2×C22⋊D4711yescharacteristic
406SD16—42no—
407SD16—42no—
408D8—1884no—
409D8—1884no—
410M16—942no—
411M16—942no—
412M16—1884no—
413M16—1882no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—472no—
417C2×C8—472no—
418C2×C8—942no—
419C2×C8—942no—
420C2×C8—1884no—
421C2×C8—1882no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C2×C22⋊D4711yescharacteristic
430C4⋊C4C2×C22⋊D4711yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C42—41no—
437C42D4×D4711yescharacteristic
438C42#1062535211yescharacteristic
439C42C2×C22⋊D4711yescharacteristic
440C42C23×D4711yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—3762no—
447C23—3762no—
448C23—3762no—
449C23—1882no—
450C23#1309399611yescharacteristic
451Q8—41no—
452Q8—41no—
453Q8—81no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—3762no—
459D4—3762no—
460D4—3762no—
461D4—3762no—
462D4—1882no—
463D4—1882no—
464D4—1882no—
465D4—1882no—
466D4#1309404912yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4#1309399611yescharacteristic
482C2×C4#1309399611yescharacteristic
483C2×C4#1309399611yescharacteristic
484C2×C4#1309404512yes—
485C2×C4#1309405112yes—
486C2×C4C2×D4×D4711yescharacteristic
487C2×C4C22×C22⋊D4711yescharacteristic
488C2×C4C2×#1062535211yescharacteristic
489C8—41no—
490C8—41no—
491C8—942no—
492C8—942no—
493C8—1882no—
494C8—1882no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—3762no—
505C22—3762no—
506C22—3762no—
507C22—7524no—
508C22#1568934012yes—
509C22C2×#1309399611yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4#1568934012yes—
518C4C2×#1309404511yescharacteristic
519C4C2×#1309404911yescharacteristic
520C4C2×#1309405111yescharacteristic
521C4#1568992111yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—7522no—
526C2C2×#1568934011yescharacteristic
527C2#2025300911yescharacteristic
528C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3101C2
socleC22×C47C2×#121
prime radicalC2×#15689318C2
hypercenter#3101D47
hypocenterC47#19166
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #20253764C2semidirect
3. D4×#10625329C2semidirect
4. #20242951C2semidirect
5. #20242923C2semidirect
6. #20242827C2semidirect
7. #20242799C2semidirect
8. #20242141C2semidirect
9. #20242078C2semidirect
10. #20242055C2semidirect
11. C47×#3101C2semidirect
16. #15689471C22semidirect
17. #15689471C22semidirect
22. #15689201C22semidirect
24. D4×C47⋊C8C22semidirect
32. #15688875C22semidirect
33. #15688873C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
36. #15688866C22semidirect
37. #15688866C22semidirect
41. C47×#476C22semidirect
42. C47×#476C22semidirect
43. C47×#461C22semidirect
44. C47×#461C22semidirect
259. #3101D47semidirect
285. #476C2×D47semidirect
286. #476C2×D47semidirect
287. #461C2×D47semidirect
288. #461C2×D47semidirect
313. C47#19166semidirect

Maximal subgroups

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

subgroupindex
#202537642
C2×#156893182
D4×#106253292
#202429512
#202429232
#202428272
#202427992
#202421412
#202420782
#202420552
C47×#31012
#1916647