Finite Group Databasebeta

← #26201690

Subgroups — #26201690

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254303C211yesmaximal, characteristic, complemented
2#20252868C211yesmaximal, characteristic, complemented
3#20252499C211yesmaximal, characteristic, complemented
4#20242834C212yesmaximal, complemented
5#20242827C212yesmaximal, complemented
6#20242746C212yesmaximal, complemented
7#20242741C212yesmaximal, complemented
8#20242162C211yesmaximal, characteristic, complemented
9#20242065C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic, complemented
11C47×#3035C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13C2×#13094021C2211yescharacteristic
14#15689499C2212yes—
15#15689471C2212yes—
16#15689323—21no—
17#15689323C2212yes—
18#15689318—21no—
19#15689318—21no—
20#15689318—21no—
21#15689318C2212yes—
22#15689256—22no—
23Q8×C47⋊C8C2212yescomplemented
24D4×C47⋊C8—22no—
25D4×C47⋊C8C2212yescomplemented
26#15689176—21no—
27#15689176C2211yescharacteristic
28#15689147C2211yescharacteristic
29C2×C4⋊(C47⋊C8)—21no—
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15689127—21no—
32#15688875C2212yescomplemented
33#15688875C2212yescomplemented
34#15688869C2212yescomplemented
35#15688867C2212yescomplemented
36#15688866C2212yes—
37#15688866C2212yes—
38C2×#135×C47—21no—
39C47×#532C2211yescharacteristic
40C4×#41×C47C2211yescharacteristic
41C47×#464—21no—
42C47×#463—21no—
43C47×#462C2211yescharacteristic, complemented
44C47×#461C2211yescharacteristic, complemented
45C47×#460C2211yescharacteristic, complemented
46C47×#459C2211yescharacteristic, complemented
47C2×C4⋊C8×C47C2211yescharacteristic
48C47×#416—21no—
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52#13094049—44no—
53C2×#10625336—22no—
54C2×#10625334—22no—
55C2×#10625334—22no—
56C2×#10625334—22no—
57C2×#10625329—21no—
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)—21no—
66C4⋊(C47⋊C8)—22no—
67C4⋊(C47⋊C8)D411yescharacteristic
68C4⋊(C47⋊C8)D411yescharacteristic
69C4⋊(C47⋊C8)C2312yes—
70C4⋊(C47⋊C8)C2312yes—
71#13093944—22no—
72#13093944—22no—
73#13093944—22no—
74#13093944—22no—
75C22⋊(C47⋊C8)—22no—
76C22⋊(C47⋊C8)D412yes—
77C4×C47⋊C8—22no—
78C4×C47⋊C8C2312yes—
79C2×#41×C47—21no—
80C2×#41×C47D411yescharacteristic
81#135×C47—44no—
82C2×Q16×C47—21no—
83C2×SD16×C47—21no—
84C2×SD16×C47—21no—
85C2×D8×C47—21no—
86C22×C8×C47—21no—
87C4⋊Q8×C47C2311yescharacteristic
88#127×C47C2311yescharacteristic
89#124×C47—21no—
90#121×C47—21no—
91#121×C47—21no—
92C4×Q8×C47D411yescharacteristic
93C4×Q8×C47C2311yescharacteristic
94C4×D4×C47—21no—
95C4×D4×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47—21no—
98C4×D4×C47D411yescharacteristic
99C4×D4×C47C2311yescharacteristic
100#117×C47—21no—
101#117×C47D411yescharacteristic
102C2×C42×C47—21no—
103C2×C42×C47C2311yescharacteristic
104C4⋊C8×C47—21no—
105C4⋊C8×C47C2311yescharacteristic
106C4⋊C8×C47C2311yescharacteristic
107#103×C47—21no—
108#103×C47—21no—
109#102×C47—21no—
110#102×C47—21no—
111C23×D47—41no—
112#41×C47—22no—
113#41×C47—42no—
114#41×C47—42no—
115#41×C47D814yescomplemented
116C2×Q8×C47—21no—
117C2×Q8×C47C2×D411yescharacteristic
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—41no—
123C2×D4×C47C2×D411yescharacteristic
124C22×C4×C47—21no—
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—41no—
137C2×D188—41no—
138C2×D188—41no—
139C2×D188—81no—
140C2×D188—81no—
141C2×D188—82no—
142Q16×C47—42no—
143SD16×C47—42no—
144SD16×C47—42no—
145D8×C47—42no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—42no—
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—21no—
156C22⋊C4×C47—21no—
157C22⋊C4×C47—21no—
158C22⋊C4×C47—41no—
159C22⋊C4×C47—41no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47C2×D411yescharacteristic
164C42×C47C2×D411yescharacteristic
165C42×C47#4111yescharacteristic
166C42×C47C2411yescharacteristic
167#10625336—44no—
168#10625334—44no—
169#10625334—82no—
170#10625334—82no—
171#10625329—42no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—82no—
175C2×C47⋊C8—81no—
176C2×C47⋊C8C2×D412yes—
177C2×C47⋊C8C2×D412yes—
178C23×C47—21no—
179C23×C47—41no—
180C23×C47#12111yescharacteristic
181C22×D47—42no—
182C22×D47—82no—
183C22×D47—82no—
184C22×D47—82no—
185Q8×C47—21no—
186Q8×C47—41no—
187Q8×C47—41no—
188Q8×C47C2×D812yes—
189D4×C47—24no—
190D4×C47—21no—
191D4×C47—41no—
192D4×C47—41no—
193D4×C47—41no—
194D4×C47—41no—
195D4×C47—81no—
196D4×C47—82no—
197D4×C47C2×D812yes—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—24no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—81no—
209C2×C4×C47—82no—
210C2×C4×C47#12111yescharacteristic
211C2×C4×C47#12111yescharacteristic
212C2×C4×C47#12111yescharacteristic
213C2×C4×C47C2×D812yes—
214C2×C4×C47#13512yes—
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218D188—42no—
219D188—42no—
220D188—42no—
221D188—42no—
222D188—82no—
223D188—82no—
224D188—82no—
225D188—82no—
226C8×C47—41no—
227C8×C47—41no—
228C47⋊C8—22no—
229C47⋊C8—22no—
230C47⋊C8—42no—
231C47⋊C8—42no—
232#19410—471nomaximal, Sylow
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—22no—
236C22×C47—41no—
237C22×C47—41no—
238C22×C47—81no—
239C22×C47—81no—
240C22×C47—81no—
241C22×C47#46612yes—
242C22×C47C2×#12111yescharacteristic
243C2×D47—82no—
244C2×D47—82no—
245C2×D47—82no—
246C2×D47—164no—
247C4×C47—21no—
248C4×C47—22no—
249C4×C47—21no—
250C4×C47—21no—
251C4×C47—41no—
252C4×C47—41no—
253C4×C47—81no—
254C4×C47#46612yes—
255C4×C47C22×D811yescharacteristic
256C4×C47C2×#13511yescharacteristic
257C4×C47#57611yescharacteristic
258C4×C47#57711yescharacteristic
259#3205—471no—
260#3038—471no—
261#3035D4711yescharacteristic, complemented
262#2988—471no—
263#1682—472no—
264#1681—472no—
265#1668—472no—
266#1667—472no—
267#1571—471no—
268#1547—471no—
269C2×#331—471no—
270C2×C47—22no—
271C2×C47—42no—
272C2×C47—161no—
273C2×C47#304511yescharacteristic
274C2×C47C2×#46611yescharacteristic
275C2×C47#306811yescharacteristic
276D47—162no—
277C2×AGL(1,ℤ/8)—941no—
278C2×#135—21no—
279#532C2×D4711yescharacteristic
280C2×#127—471no—
281C4×#41C2×D4711yescharacteristic
282#493—472no—
283#486—472no—
284#464—21no—
285#463—21no—
286#462C2×D4711yescharacteristic, complemented
287#461C2×D4711yescharacteristic, complemented
288#460—472no—
289#460—941no—
290#460C2×D4711yescharacteristic, complemented
291#459—472no—
292#459—941no—
293#459—941no—
294#459—941no—
295#459C2×D4711yescharacteristic, complemented
296#447—942no—
297C8×Q8—472no—
298C8×D4—472no—
299C8×D4—942no—
300#431—471no—
301#431—941no—
302#423—471no—
303C2×C4⋊C8—471no—
304C2×C4⋊C8—941no—
305C2×C4⋊C8C2×D4711yescharacteristic
306#418—941no—
307#416—21no—
308#335—472no—
309#335—472no—
310#332—472no—
311#331—472no—
312#331—472no—
313#331—472no—
314C47#1941011yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41C22⋊D4711yescharacteristic
317C22×D4—941no—
318C22×D4—941no—
319C22×D4—941no—
320AGL(1,ℤ/8)—1884no—
321#135—44no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—942no—
326C2×D8—21no—
327C2×D8—942no—
328C2×D8—942no—
329C2×D8—942no—
330C2×M16—941no—
331C22×C8—21no—
332C22×C8—942no—
333C22×C8—941no—
334C4⋊Q8C22×D4711yescharacteristic
335#127—472no—
336#127—941no—
337#127—941no—
338#127—941no—
339#127C22×D4711yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8C22⋊D4711yescharacteristic
344C4×Q8C22×D4711yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4C22⋊D4711yescharacteristic
350C4×D4C22×D4711yescharacteristic
351#117—21no—
352#117C22⋊D4711yescharacteristic
353C2×C42—21no—
354C2×C42C22×D4711yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—471no—
357C4⋊C8—471no—
358C4⋊C8—472no—
359C4⋊C8—472no—
360C4⋊C8—942no—
361C4⋊C8—941no—
362C4⋊C8—941no—
363C4⋊C8C22×D4711yescharacteristic
364C4⋊C8C22×D4711yescharacteristic
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369#102—942no—
370#102—942no—
371#102—942no—
372#102—942no—
373C22⋊C8—472no—
374C22⋊C8—942no—
375C4×C8—472no—
376C4×C8—942no—
377C24—1881no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41#1062533414yescomplemented
382C2×Q8—21no—
383C2×Q8C2×C22⋊D4711yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—942no—
390C2×D4—942no—
391C2×D4—942no—
392C2×D4—3761no—
393C2×D4—3761no—
394C2×D4—3762no—
395C2×D4—1881no—
396C2×D4—1881no—
397C2×D4—1881no—
398C2×D4—1882no—
399C2×D4—1882no—
400C2×D4C2×C22⋊D4711yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4D4×D4711yescharacteristic
406C22×C4#1062535211yescharacteristic
407C22×C4C2×C22⋊D4711yescharacteristic
408Q16—42no—
409SD16—42no—
410SD16—42no—
411SD16—1884no—
412D8—42no—
413D8—3762no—
414D8—3762no—
415D8—1884no—
416M16—1882no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—472no—
421C2×C8—472no—
422C2×C8—942no—
423C2×C8—942no—
424C2×C8—3762no—
425C2×C8—3761no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4C2×C22⋊D4711yescharacteristic
431C4⋊C4C2×C22⋊D4711yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D4711yescharacteristic
441C42#1062535211yescharacteristic
442C42C2×C22⋊D4711yescharacteristic
443C42C23×D4711yescharacteristic
444C23—21no—
445C23—41no—
446C23—3762no—
447C23—3762no—
448C23—3762no—
449C23—1882no—
450C23#1309399611yescharacteristic
451Q8—21no—
452Q8—41no—
453Q8—41no—
454Q8C2×#1062533412yes—
455D4—24no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—81no—
462D4—82no—
463D4—3762no—
464D4—3762no—
465D4—3762no—
466D4—3762no—
467D4—1882no—
468D4—1882no—
469D4—1882no—
470D4—1882no—
471D4C2×#1062533412yes—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#1309399611yescharacteristic
485C2×C4#1309399611yescharacteristic
486C2×C4#1309399611yescharacteristic
487C2×C4C2×#1062533412yes—
488C2×C4#1309404412yes—
489C2×C4C2×D4×D4711yescharacteristic
490C2×C4C22×C22⋊D4711yescharacteristic
491C2×C4C2×#1062535211yescharacteristic
492C8—41no—
493C8—41no—
494C8—942no—
495C8—942no—
496C8—1882no—
497C8—1882no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—3762no—
507C22—3762no—
508C22—3762no—
509C22—7524no—
510C22#1568935712yes—
511C22C2×#1309399611yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—81no—
519C4#1568935712yes—
520C4C22×#1062533411yescharacteristic
521C4C2×#1309404411yescharacteristic
522C4#1568992511yescharacteristic
523C4#1568993011yescharacteristic
524C2—22no—
525C2—42no—
526C2—161no—
527C2—7522no—
528C2#2025292511yescharacteristic
529C2C2×#1568935711yescharacteristic
530C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3035C2
socleC22×C47C2×#121
hypercenter#3035D47
hypocenterC47#19410
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20254303C2semidirect
2. #20252868C2semidirect
3. #20252499C2semidirect
4. #20242834C2semidirect
5. #20242827C2semidirect
6. #20242746C2semidirect
7. #20242741C2semidirect
8. #20242162C2semidirect
9. #20242065C2semidirect
10. C2×#15688866C2semidirect
11. C47×#3035C2semidirect
23. Q8×C47⋊C8C22semidirect
25. D4×C47⋊C8C22semidirect
32. #15688875C22semidirect
33. #15688875C22semidirect
34. #15688869C22semidirect
35. #15688867C22semidirect
43. C47×#462C22semidirect
44. C47×#461C22semidirect
45. C47×#460C22semidirect
46. C47×#459C22semidirect
115. #41×C47D8semidirect
261. #3035D47semidirect
286. #462C2×D47semidirect
287. #461C2×D47semidirect
290. #460C2×D47semidirect
295. #459C2×D47semidirect
314. C47#19410semidirect
381. #41#10625334semidirect

Maximal subgroups

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

subgroupindex
#202543032
#202528682
#202524992
#202428342
#202428272
#202427462
#202427412
#202421622
#202420652
C2×#156888662
C47×#30352
#1941047