Finite Group Databasebeta

← #26220187

Subgroups — #26220187

Order 12032.

1678 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic
2C2×#15689163C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic, complemented
5C2×#15689079C211yesmaximal, characteristic, complemented
6#20245968C211yesmaximal, characteristic, complemented
7#20244272C212yesmaximal, complemented
8#20244263C212yesmaximal, complemented
9#20244253C212yesmaximal, complemented
10#20244245C212yesmaximal, complemented
11C47×#2311C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C2×#13094049—24no—
14C2×#13094049—24no—
15C2×#13094049C418yescomplemented
16C22×#10625336C2212yes—
17C22×#10625334C2212yes—
18C22×#10625329C2211yescharacteristic
19C23×C47⋊C8C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21#15689163—24no—
22C2×#13093957C2212yescomplemented
23C2×#13093955C2212yescomplemented
24#15689119—24no—
25C2×#13093948C2211yescharacteristic, complemented
26C2×#13093948C2211yescharacteristic, complemented
27C2×#13093945C2211yescharacteristic, complemented
28C2×#13093945C2211yescharacteristic, complemented
29C2×#13093944—24no—
30C2×#13093944C2212yes—
31C2×#13093944C2212yes—
32#15689079—24no—
33C2×#13093935C2212yescomplemented
34C2×C4×C47⋊C8C2212yescomplemented
35#15689025C2212yes—
36#15689025C2212yes—
37#15689024—22no—
38#15688978—22no—
39C22×#41×C47C2211yescharacteristic
40C2×#117×C47C2211yescharacteristic
41C22×C4⋊C4×C47C2211yescharacteristic
42C47×#391C2212yescomplemented
43C47×#390C2212yescomplemented
44C47×#386—22no—
45C47×#380—22no—
46C24×D47—21no—
47C22×D188—22no—
48C22×D188—22no—
49C22×D188—22no—
50C22×D188—22no—
51C22×D188—21no—
52C22×D188—22no—
53C22×D188C2×C414yes—
54C22×D188C2312yes—
55#13094049—232no—
56C2×#10625336—24no—
57C2×#10625336—24no—
58C2×#10625336C2×C418yescomplemented
59C2×#10625334—24no—
60C2×#10625334—24no—
61C2×#10625334C2×C418yescomplemented
62C2×#10625329C2×C414yes—
63C2×#10625329D414yes—
64C2×#10625329D414yes—
65C22×C47⋊C8—22no—
66C22×C47⋊C8—22no—
67C22×C47⋊C8D414yes—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2312yes—
71#13093983—22no—
72#13093983—22no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C22⋊Dic47—21no—
76C2×(C2×D47)⋊C4—24no—
77C2×(C2×D47)⋊C4—24no—
78C2×C4×Dic47—21no—
79#13093957—24no—
80#13093955—24no—
81#13093948—22no—
82#13093948—22no—
83#13093945—22no—
84#13093945—22no—
85#13093944—24no—
86#13093944—24no—
87#13093935—24no—
88C4×C47⋊C8—24no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47C2×C414yes—
95C22×Q8×C47C2311yescharacteristic
96C22×D4×C47—21no—
97C22×D4×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101#117×C47—22no—
102#117×C47—22no—
103C2×C4⋊C4×C47—22no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47D412yes—
107C2×C4⋊C4×C47C2312yes—
108C2×C22⋊C4×C47—24no—
109C2×C22⋊C4×C47D412yes—
110C2×C42×C47C2312yes—
111#95×C47—24no—
112C24×C47—21no—
113C24×C47C2×D411yescharacteristic
114C23×D47—22no—
115C23×D47—24no—
116C23×D47—42no—
117C23×D47—42no—
118C23×D47—82no—
119C23×D47—82no—
120#41×C47—28no—
121#41×C47—28no—
122#41×C47—44no—
123#41×C47—44no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47C22×C412yes—
130C2×D4×C47—24no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—42no—
137C2×D4×C47—42no—
138C2×D4×C47C22×C412yes—
139C22×C4×C47—21no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—42no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—82no—
146C22×C4×C47C22×C412yes—
147C22×C4×C47C2×D414yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C2×D188—28no—
158C2×D188—28no—
159C2×D188—22no—
160C2×D188—22no—
161C2×D188—22no—
162C2×D188—22no—
163C2×D188—24no—
164C2×D188—22no—
165C2×D188—44no—
166C2×D188—42no—
167C2×D188—44no—
168C2×D188—42no—
169C2×D188C22×C414yes—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—42no—
174C22⋊C4×C47—24no—
175C22⋊C4×C47—84no—
176C42×C47—22no—
177C42×C47—22no—
178C22⋊Dic47—42no—
179#10625336—216no—
180#10625334—216no—
181(C2×D47)⋊C4—84no—
182(C2×D47)⋊C4—84no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4×Dic47—42no—
188#10625329C22⋊C4116yescomplemented
189C2×C47⋊C8—28no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8C22×C414yes—
195C2×C47⋊C8C2×D412yes—
196C2×C47⋊C8C2×D412yes—
197C2×C47⋊C8C2×D414yes—
198C2×C47⋊C8C2×D414yes—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47C4×D412yes—
207C23×C47#12112yes—
208C23×C47#12712yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—24no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—44no—
214C22×D47—44no—
215C22×D47—48no—
216C22×D47—88no—
217C22×D47—84no—
218C22×D47—84no—
219C22×D47—88no—
220Q8×C47—22no—
221Q8×C47—22no—
222Q8×C47—41no—
223Q8×C47—41no—
224D4×C47—22no—
225D4×C47—22no—
226D4×C47—48no—
227D4×C47—41no—
228D4×C47—41no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—48no—
236C2×C4×C47—42no—
237C2×C4×C47—44no—
238C2×C4×C47—84no—
239C2×C4×C47C2×C22⋊C414yes—
240C2×C4×C47C4×D414yes—
241C2×C4×C47C4×D412yes—
242C2×C4×C47#12112yes—
243C2×C4×C47#12114yes—
244C2×C4×C47#12414yes—
245C2×C4×C47#12712yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256D188—24no—
257D188—24no—
258D188—42no—
259D188—42no—
260D188—48no—
261C47⋊C8—22no—
262C47⋊C8—22no—
263C47⋊C8C2×C22⋊C418yes—
264#21235—471nomaximal, Sylow
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—24no—
268C22×C47—24no—
269C22×C47—44no—
270C22×C47—42no—
271C22×C47—81no—
272C22×C47—81no—
273C22×C47—82no—
274C22×C47#39014yes—
275C22×C47C2×C4×D411yescharacteristic
276C22×C47C2×#12111yescharacteristic
277C22×C47C2×#12711yescharacteristic
278C22×C47#57612yes—
279C22×C47#57712yes—
280C2×D47—48no—
281C2×D47—44no—
282C2×D47—816no—
283C2×D47—82no—
284C2×D47—82no—
285C2×D47—84no—
286C4×C47—42no—
287C4×C47—44no—
288C4×C47—82no—
289C4×C47#39014yes—
290C4×C47C22×C22⋊C411yescharacteristic
291C4×C47C2×C4×D411yescharacteristic
292C4×C47C2×#12111yescharacteristic
293C4×C47C2×#12411yescharacteristic
294Dic47—81no—
295Dic47—81no—
296C22×AGL(1,ℤ/8)—471no—
297C2×#428—471no—
298C2×#416—471no—
299C22×#102—471no—
300C2×#405—471no—
301#2311D4711yescharacteristic, complemented
302#2309—471no—
303#1973—472no—
304#1972—472no—
305#1970—472no—
306#1968—472no—
307C2×C47—24no—
308C2×C47—82no—
309C2×C47C2×#39011yescharacteristic
310C2×C47#294612yes—
311C2×C47#306212yes—
312C2×C47#315411yescharacteristic
313C2×C47#315511yescharacteristic
314D47—84no—
315C22×#41C2×D4711yescharacteristic
316C23×D4—471no—
317C2×AGL(1,ℤ/8)—478no—
318C2×AGL(1,ℤ/8)—944no—
319C2×AGL(1,ℤ/8)—944no—
320C22×SD16—472no—
321C22×D8—472no—
322C22×M16—471no—
323C23×C8—471no—
324C2×#117—471no—
325C2×#117C2×D4711yescharacteristic
326C22×C4⋊C4C2×D4711yescharacteristic
327#428—944no—
328C2×#107—472no—
329C2×#106—472no—
330#416—944no—
331C2×#103—471no—
332C2×#103—471no—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102—472no—
336C2×#102—472no—
337C2×#102—944no—
338#405—944no—
339C2×#97—472no—
340C2×C4×C8—472no—
341#391C2×D4712yescomplemented
342#390—472no—
343#390—472no—
344#390—942no—
345#390C2×D4712yescomplemented
346#386—22no—
347#380—22no—
348#379—942no—
349C47#2123511yescharacteristic, complemented, Sylow
350C25—941no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41C4×D4714yes—
357C22×Q8C22×D4711yescharacteristic
358C22×D4—21no—
359C22×D4—474no—
360C22×D4—472no—
361C22×D4—942no—
362C22×D4—942no—
363C22×D4—942no—
364C22×D4—942no—
365C22×D4—942no—
366C22×D4—941no—
367C22×D4C22×D4711yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4C22×D4711yescharacteristic
371AGL(1,ℤ/8)—9432no—
372C2×SD16—478no—
373C2×SD16—944no—
374C2×SD16—944no—
375C2×D8—478no—
376C2×D8—944no—
377C2×D8—944no—
378C2×M16—474no—
379C2×M16—474no—
380C2×M16—474no—
381C22×C8—472no—
382C22×C8—474no—
383C22×C8—471no—
384C22×C8—471no—
385C22×C8—942no—
386C22×C8—942no—
387#117—22no—
388#117—22no—
389#117—942no—
390#117—942no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4C22⋊D4712yes—
397C2×C4⋊C4C22×D4712yes—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4C22⋊D4712yes—
403C2×C42—941no—
404C2×C42C22×D4712yes—
405#107—944no—
406#106—944no—
407#103—942no—
408#103—942no—
409#102—944no—
410#102—944no—
411#102—942no—
412#102—942no—
413#97—944no—
414C4×C8—944no—
415#95—24no—
416C24—21no—
417C24—944no—
418C24—942no—
419C24—3762no—
420C24—3762no—
421C24—1882no—
422C24—1882no—
423C24C2×C22⋊D4711yescharacteristic
424#41—28no—
425#41—28no—
426#41—44no—
427#41—44no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D4712yes—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—474no—
443C2×D4—948no—
444C2×D4—948no—
445C2×D4—944no—
446C2×D4—942no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—942no—
451C2×D4—1882no—
452C2×D4—1884no—
453C2×D4—1884no—
454C2×D4—1882no—
455C2×D4C2×C4×D4712yes—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4C2×C4×D4712yes—
466C22×C4#1062535012yes—
467C22×C4D4×D4712yes—
468C22×C4D4×D4712yes—
469C22×C4D4×D4712yes—
470C22×C4#1062535212yes—
471C22×C4C2×C22⋊D4714yes—
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C23×D4711yescharacteristic
474SD16—9416no—
475D8—9416no—
476M16—4716no—
477C2×C8—474no—
478C2×C8—474no—
479C2×C8—474no—
480C2×C8—472no—
481C2×C8—472no—
482C2×C8—948no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C22⋊C4—24no—
496C22⋊C4—84no—
497C22⋊C4—3764no—
498C22⋊C4—3764no—
499C22⋊C4—1882no—
500C42—22no—
501C42—22no—
502C42—1882no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—944no—
511C23—3768no—
512C23—3764no—
513C23—3764no—
514C23—3768no—
515C23—1888no—
516C23—1882no—
517C23—1882no—
518C23—1884no—
519C23—1884no—
520C23C4×C22⋊D4712yes—
521C23#1309399612yes—
522C23#1309402012yes—
523C23C22×C22⋊D4711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528D4—22no—
529D4—22no—
530D4—48no—
531D4—41no—
532D4—41no—
533D4—944no—
534D4—944no—
535D4—1888no—
536D4—1882no—
537D4—1882no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—84no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4C22⋊C4×D4714yes—
553C2×C4(C22⋊D47)⋊C414yes—
554C2×C4C4×C22⋊D4712yes—
555C2×C4#1309399612yes—
556C2×C4#1309399714yes—
557C2×C4#1309401014yes—
558C2×C4#1309402012yes—
559C2×C4C22×C4×D4711yescharacteristic
560C2×C4C2×D4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C2×D4×D4711yescharacteristic
563C2×C4C2×#1062535211yescharacteristic
564C2×C4C2×#1062535011yescharacteristic
565C8—478no—
566C8—942no—
567C8—942no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22—37616no—
578C22—3764no—
579C22—3762no—
580C22—3762no—
581C22—1888no—
582C22—1884no—
583C22#1568902314yes—
584C22C2×C4×C22⋊D4711yescharacteristic
585C22C2×#1309399611yescharacteristic
586C22C2×#1309402011yescharacteristic
587C22#1568992612yes—
588C22#1568993112yes—
589C4—42no—
590C4—44no—
591C4—82no—
592C4—3761no—
593C4—3761no—
594C4#1568902314yes—
595C4C2×C22⋊C4×D4711yescharacteristic
596C4C2×(C22⋊D47)⋊C411yescharacteristic
597C4C2×#1309399711yescharacteristic
598C4C2×#1309401011yescharacteristic
599C2—24no—
600C2—82no—
601C2—3764no—
602C2C2×#1568902311yescharacteristic
603C2#2025221612yes—
604C2#2025307012yes—
605C2#2025386611yescharacteristic
606C2#2025387111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC22×#13094049C2
cosocleC22×C4×C47C24
hypercenter#2311D47
hypocenterC47#21235
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#15689163C2semidirect
3. C2×#15689119C2semidirect
4. C22×#13093944C2semidirect
5. C2×#15689079C2semidirect
6. #20245968C2semidirect
7. #20244272C2semidirect
8. #20244263C2semidirect
9. #20244253C2semidirect
10. #20244245C2semidirect
11. C47×#2311C2semidirect
15. C2×#13094049C4semidirect
22. C2×#13093957C22semidirect
23. C2×#13093955C22semidirect
25. C2×#13093948C22semidirect
26. C2×#13093948C22semidirect
27. C2×#13093945C22semidirect
28. C2×#13093945C22semidirect
33. C2×#13093935C22semidirect
34. C2×C4×C47⋊C8C22semidirect
42. C47×#391C22semidirect
43. C47×#390C22semidirect
58. C2×#10625336C2×C4semidirect
61. C2×#10625334C2×C4semidirect
188. #10625329C22⋊C4semidirect
301. #2311D47semidirect
341. #391C2×D47semidirect
345. #390C2×D47semidirect
349. C47#21235semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156891632
C2×#156891192
C22×#130939442
C2×#156890792
#202459682
#202442722
#202442632
#202442532
#202442452
C47×#23112
#2123547