Finite Group Databasebeta

← #26218227

Subgroups — #26218227

Order 12032.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689921C211yesmaximal, characteristic
2C2×#15689125C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689105C211yesmaximal, characteristic, complemented
5#20243903C214yesmaximal, complemented
6#20243896C214yesmaximal, complemented
7C2×#15688891C211yesmaximal, characteristic, complemented
8#20243115C211yesmaximal, characteristic, complemented
9C2×C47×#568C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11#15689921—24no—
12#15689921C418yescomplemented
13C2×#13094055—21no—
14C2×#13094054C2212yes—
15C2×#13094050—21no—
16C2×#13094049C2212yes—
17C2×#13094044—22no—
18C22×#10625329C2211yescharacteristic
19C22×#10625329C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21#15689125—22no—
22#15689125—22no—
23#15689124—22no—
24#15689124C2214yes—
25C2×#13093949C2212yescomplemented
26C2×#13093948—21no—
27C2×#13093946C2212yescomplemented
28C2×#13093945—21no—
29#15689105—24no—
30C2×#13093942C2212yescomplemented
31C2×#13093940C2212yescomplemented
32#15688891C2214yescomplemented
33#15688888C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C47×#568—24no—
36C47×#568—24no—
37C2×#131×C47—22no—
38C2×#131×C47C2214yes—
39C22×M16×C47—22no—
40C22×M16×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C2×#10625350—22no—
43C2×#10625350—21no—
44C2×#10625350—21no—
45C2×#10625350—21no—
46C2×#10625350—21no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350C2×C414yes—
50C22×Dic94C2311yescharacteristic
51C22×D188C2311yescharacteristic
52C22×C22⋊D47—21no—
53C23×Dic47—21no—
54C22×C4×D47—21no—
55#13094055—28no—
56#13094054—24no—
57#13094054C2×C418yescomplemented
58#13094050—28no—
59#13094049—24no—
60#13094049C2×C418yescomplemented
61#13094044—216no—
62C2×#10625337—22no—
63C2×#10625335—22no—
64C2×#10625336—22no—
65C2×#10625334—22no—
66C2×#10625329—21no—
67C2×#10625329—22no—
68C2×#10625329—21no—
69C2×#10625329—22no—
70C2×#10625329—22no—
71C2×#10625329C2×C414yes—
72C2×#10625329C2312yes—
73C2×#10625329C2312yes—
74C22×C47⋊C8—21no—
75C22×C47⋊C8—21no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47C2312yes—
82#13093949—24no—
83#13093948—24no—
84#13093946—24no—
85#13093945—24no—
86#13093942—24no—
87#13093940—24no—
88C2×#41×C47—21no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—21no—
92C2×#41×C47C2×C414yes—
93C2×#41×C47D414yes—
94C2×#41×C47D412yes—
95C2×#41×C47C2312yes—
96C22×Q8×C47D411yescharacteristic
97C22×D4×C47—21no—
98C22×D4×C47D411yescharacteristic
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101#131×C47—216no—
102#131×C47—28no—
103#131×C47—28no—
104C2×M16×C47—216no—
105C2×M16×C47—21no—
106C2×M16×C47—22no—
107C2×M16×C47—22no—
108C2×M16×C47—22no—
109C2×M16×C47—24no—
110C2×M16×C47D414yes—
111C2×M16×C47D412yes—
112C2×M16×C47D412yes—
113C2×M16×C47C2312yes—
114C22×C8×C47—21no—
115C22×C8×C47—24no—
116C22×C8×C47C2312yes—
117C24×C47—21no—
118C24×C47#4111yescharacteristic
119C23×D47—21no—
120#41×C47—24no—
121#41×C47—24no—
122#41×C47—24no—
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—22no—
126#41×C47—22no—
127#41×C47C22⋊C418yes—
128#41×C47C22×C414yes—
129#41×C47C2×D414yes—
130C2×Q8×C47—21no—
131C2×Q8×C47—22no—
132C2×Q8×C47C22⋊C414yes—
133C2×Q8×C47C2×D412yes—
134C2×D4×C47—24no—
135C2×D4×C47—22no—
136C2×D4×C47—24no—
137C2×D4×C47—22no—
138C2×D4×C47—22no—
139C2×D4×C47—21no—
140C2×D4×C47C22⋊C414yes—
141C2×D4×C47C2×D412yes—
142C22×C4×C47—21no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47#4111yescharacteristic
155C22×C4×C47#4112yes—
156C22×C4×C47C2411yescharacteristic
157C2×C22⋊D47—24no—
158C2×C22⋊D47—41no—
159C2×C22⋊D47—41no—
160C2×C22⋊D47—42no—
161C22×Dic47—21no—
162C22×Dic47—22no—
163C22×Dic47—41no—
164C22×Dic47—41no—
165C22×Dic47—82no—
166#10625350—24no—
167#10625350—24no—
168#10625350—24no—
169#10625350—24no—
170#10625350—42no—
171#10625350—42no—
172#10625350—42no—
173#10625350—42no—
174C2×D188—21no—
175C2×D188—21no—
176C2×D188—21no—
177C2×D188—21no—
178C2×D188—21no—
179C2×D188C22×C412yes—
180C2×C4×D47—24no—
181C2×C4×D47—41no—
182C2×C4×D47—41no—
183C2×C4×D47—42no—
184C2×Dic94—21no—
185C2×Dic94—21no—
186C2×Dic94—21no—
187C2×Dic94—21no—
188C2×Dic94—21no—
189C2×Dic94C22×C412yes—
190M16×C47—216no—
191M16×C47—24no—
192M16×C47—24no—
193M16×C47—24no—
194M16×C47—22no—
195M16×C47C2×D414yes—
196C2×C8×C47—216no—
197C2×C8×C47—24no—
198C2×C8×C47—22no—
199C2×C8×C47—22no—
200C2×C8×C47—22no—
201C2×C8×C47C2×D414yes—
202#10625337—28no—
203#10625336—28no—
204#10625335—28no—
205#10625334—28no—
206C4⋊Dic47—22no—
207C4⋊Dic47—22no—
208C4⋊Dic47—41no—
209C4⋊Dic47—41no—
210#10625329—24no—
211#10625329—22no—
212#10625329—22no—
213#10625329—22no—
214#10625329—42no—
215#10625329C22×C414yes—
216C2×C47⋊C8—24no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—42no—
220C23×C47—22no—
221C23×C47—24no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—81no—
225C23×C47—81no—
226C23×C47C4×D412yes—
227C23×C47#12111yescharacteristic
228C23×C47#12111yescharacteristic
229C23×C47#12312yes—
230C23×C47C2×#4111yescharacteristic
231C22×D47—22no—
232C22×D47—41no—
233C22×D47—41no—
234C22×D47—81no—
235C22×D47—81no—
236Q8×C47—22no—
237Q8×C47—24no—
238Q8×C47C2×C22⋊C414yes—
239D4×C47—24no—
240D4×C47—24no—
241D4×C47—28no—
242D4×C47—22no—
243D4×C47—24no—
244D4×C47C2×C22⋊C414yes—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—28no—
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C2×C22⋊C412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47C4×D412yes—
254C2×C4×C47C4×D412yes—
255C2×C4×C47#12012yes—
256C2×C4×C47#12012yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12111yescharacteristic
260C2×C4×C47#12111yescharacteristic
261C2×C4×C47#12312yes—
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C22⋊D47—44no—
269C22⋊D47—44no—
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—42no—
274C2×Dic47—82no—
275C2×Dic47—82no—
276C2×Dic47—82no—
277D188—22no—
278D188—22no—
279D188—41no—
280D188—41no—
281C4×D47—44no—
282C4×D47—44no—
283Dic94—22no—
284Dic94—22no—
285Dic94—41no—
286Dic94—41no—
287C8×C47—24no—
288C8×C47—24no—
289C47⋊C8—24no—
290C47⋊C8—42no—
291#21004—471nomaximal, Sylow
292C22×C47—21no—
293C22×C47—21no—
294C22×C47—21no—
295C22×C47—21no—
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—24no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47—42no—
302C22×C47—42no—
303C22×C47—84no—
304C22×C47#38612yes—
305C22×C47#38612yes—
306C22×C47C2×C4×D411yescharacteristic
307C22×C47C2×#12111yescharacteristic
308C22×C47C2×#12311yescharacteristic
309C2×D47—42no—
310C2×D47—82no—
311C2×D47—81no—
312C2×D47—81no—
313C2×D47—82no—
314C4×C47—24no—
315C4×C47#38612yes—
316C4×C47#38612yes—
317C4×C47C22×C22⋊C411yescharacteristic
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C2×#12011yescharacteristic
320C4×C47C2×#12111yescharacteristic
321Dic47—42no—
322Dic47—82no—
323C2×#575—471no—
324C2×#568D4711yescharacteristic, complemented
325C2×#417—471no—
326C2×#417—471no—
327C2×#413—471no—
328#1908—474no—
329#1907—474no—
330C2×#343—471no—
331#1738—471no—
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—22no—
335C2×C47—22no—
336C2×C47—44no—
337C2×C47C2×#38611yescharacteristic
338C2×C47#306811yescharacteristic
339C2×C47#306911yescharacteristic
340D47—82no—
341C22×#41—471no—
342C22×#41C2×D4711yescharacteristic
343#575—478no—
344#575—944no—
345C2×#137—472no—
346C2×#137—941no—
347C2×AGL(1,ℤ/8)—472no—
348C2×AGL(1,ℤ/8)—941no—
349C2×#135—942no—
350#568—24no—
351#568—24no—
352C2×#131—22no—
353C2×#131C2×D4714yes—
354C22×M16—22no—
355C22×M16—471no—
356C22×M16—471no—
357C22×M16C2×D4711yescharacteristic
358C22×M16C2×D4711yescharacteristic
359C22×C4⋊C4—471no—
360#417—474no—
361#417—942no—
362#417—942no—
363#417—942no—
364C2×#103—472no—
365C2×#103—941no—
366C2×#102—472no—
367C2×#102—941no—
368#413—944no—
369C2×#101—472no—
370C2×#100—472no—
371#343—474no—
372#341—474no—
373C47#2100411yescharacteristic, complemented, Sylow
374C2×#41—22no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—22no—
378C2×#41—474no—
379C2×#41—942no—
380C2×#41—941no—
381C2×#41—941no—
382C2×#41—942no—
383C2×#41—942no—
384C2×#41—941no—
385C2×#41—941no—
386C2×#41C4×D4714yes—
387C2×#41D18812yes—
388C2×#41C22⋊D4714yes—
389C2×#41C22×D4712yes—
390C22×Q8—471no—
391C22×Q8D18811yescharacteristic
392C22×D4—21no—
393C22×D4—471no—
394C22×D4—941no—
395C22×D4D18811yescharacteristic
396C23×C4—21no—
397C23×C4—941no—
398C23×C4—941no—
399C23×C4C22×D4711yescharacteristic
400#137—478no—
401#137—944no—
402#137—948no—
403AGL(1,ℤ/8)—478no—
404AGL(1,ℤ/8)—944no—
405AGL(1,ℤ/8)—948no—
406#135—9416no—
407C2×Q16—942no—
408C2×SD16—942no—
409C2×SD16—942no—
410C2×D8—942no—
411#131—216no—
412#131—28no—
413#131—28no—
414C2×M16—216no—
415C2×M16—21no—
416C2×M16—22no—
417C2×M16—22no—
418C2×M16—22no—
419C2×M16—24no—
420C2×M16—474no—
421C2×M16—472no—
422C2×M16—472no—
423C2×M16—941no—
424C2×M16—942no—
425C2×M16—941no—
426C2×M16—942no—
427C2×M16—942no—
428C2×M16C22⋊D4714yes—
429C2×M16C22⋊D4712yes—
430C2×M16C22⋊D4712yes—
431C2×M16C22×D4712yes—
432C22×C8—21no—
433C22×C8—24no—
434C22×C8—941no—
435C22×C8—941no—
436C22×C8C22×D4712yes—
437C2×C4⋊C4—472no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4—941no—
443#103—944no—
444#103—944no—
445#102—944no—
446#102—944no—
447#101—944no—
448#100—944no—
449C24—21no—
450C24—941no—
451C24#1062535011yescharacteristic
452#41—24no—
453#41—24no—
454#41—24no—
455#41—24no—
456#41—24no—
457#41—22no—
458#41—22no—
459#41—944no—
460#41—944no—
461#41—944no—
462#41—944no—
463#41—1882no—
464#41—1882no—
465#41—1882no—
466#41—1882no—
467#41(C2×D47)⋊C418yes—
468#41C2×C4×D4714yes—
469#41C2×C22⋊D4714yes—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—472no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8(C2×D47)⋊C414yes—
479C2×Q8C2×D18812yes—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—24no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—472no—
487C2×D4—944no—
488C2×D4—941no—
489C2×D4—941no—
490C2×D4—941no—
491C2×D4—941no—
492C2×D4—941no—
493C2×D4—1881no—
494C2×D4—1881no—
495C2×D4—1882no—
496C2×D4(C2×D47)⋊C414yes—
497C2×D4C2×D18812yes—
498C22×C4—21no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—941no—
504C22×C4—944no—
505C22×C4—942no—
506C22×C4—3762no—
507C22×C4—1881no—
508C22×C4—1881no—
509C22×C4—1881no—
510C22×C4—1881no—
511C22×C4—1882no—
512C22×C4C2×C4×D4712yes—
513C22×C4C2×D18811yescharacteristic
514C22×C4C2×D18811yescharacteristic
515C22×C4#1062535011yescharacteristic
516C22×C4#1062535012yes—
517C22×C4C2×C22⋊D4712yes—
518C22×C4C2×C22⋊D4711yescharacteristic
519C22×C4C2×C22⋊D4711yescharacteristic
520C22×C4C2×C22⋊D4712yes—
521C22×C4C23×D4711yescharacteristic
522Q16—948no—
523SD16—948no—
524SD16—948no—
525D8—948no—
526M16—216no—
527M16—24no—
528M16—24no—
529M16—24no—
530M16—22no—
531M16—474no—
532M16—944no—
533M16—942no—
534M16—942no—
535M16—942no—
536M16—1882no—
537M16C2×C22⋊D4714yes—
538C2×C8—216no—
539C2×C8—24no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—944no—
544C2×C8—942no—
545C2×C8—942no—
546C2×C8—1882no—
547C2×C8C2×C22⋊D4714yes—
548C4⋊C4—942no—
549C4⋊C4—942no—
550C4⋊C4—1881no—
551C4⋊C4—1881no—
552C23—22no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—81no—
558C23—942no—
559C23—3761no—
560C23—3761no—
561C23—1881no—
562C23—1881no—
563C23C4×C22⋊D4712yes—
564C23#1309399811yescharacteristic
565C23#1309399811yescharacteristic
566C23#1309400712yes—
567C23C2×#1062535011yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—942no—
571Q8—942no—
572Q8—1881no—
573Q8—1881no—
574Q8C2×(C2×D47)⋊C414yes—
575D4—28no—
576D4—24no—
577D4—24no—
578D4—24no—
579D4—22no—
580D4—942no—
581D4—942no—
582D4—1884no—
583D4—1884no—
584D4—1881no—
585D4—1881no—
586D4C2×(C2×D47)⋊C414yes—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—942no—
593C2×C4—3762no—
594C2×C4—3762no—
595C2×C4—3762no—
596C2×C4—1882no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1884no—
600C2×C4—1884no—
601C2×C4C2×(C2×D47)⋊C412yes—
602C2×C4C2×(C2×D47)⋊C412yes—
603C2×C4C4×C22⋊D4712yes—
604C2×C4C4×C22⋊D4712yes—
605C2×C4C4×C22⋊D4712yes—
606C2×C4#1309399312yes—
607C2×C4#1309399312yes—
608C2×C4#1309399811yescharacteristic
609C2×C4#1309399811yescharacteristic
610C2×C4#1309399812yes—
611C2×C4#1309399812yes—
612C2×C4#1309400712yes—
613C2×C4C22×C4×D4711yescharacteristic
614C2×C4C22×C22⋊D4711yescharacteristic
615C2×C4C22×C22⋊D4711yescharacteristic
616C2×C4C22×C22⋊D4711yescharacteristic
617C2×C4C22×D18811yescharacteristic
618C2×C4C2×#1062535011yescharacteristic
619C8—24no—
620C8—24no—
621C8—944no—
622C8—1882no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—84no—
635C22—3762no—
636C22—3762no—
637C22—3761no—
638C22—3761no—
639C22—1882no—
640C22#1568900212yes—
641C22#1568900212yes—
642C22C2×C4×C22⋊D4711yescharacteristic
643C22C2×#1309399811yescharacteristic
644C22C2×#1309400711yescharacteristic
645C4—24no—
646C4—3762no—
647C4—1882no—
648C4#1568900212yes—
649C4#1568900212yes—
650C4C22×(C2×D47)⋊C411yescharacteristic
651C4C2×C4×C22⋊D4711yescharacteristic
652C4C2×#1309399311yescharacteristic
653C4C2×#1309399811yescharacteristic
654C2—22no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—44no—
659C2—3762no—
660C2C2×#1568900211yescharacteristic
661C2#2025311611yescharacteristic
662C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#568C2
socleC22×C47C2×#121
prime radicalC2×#15689921C2
cosocleC22×C4×C47C24
hypercenterC2×#568D47
hypocenterC47#21004
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
2. C2×#15689125C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689105C2semidirect
5. #20243903C2semidirect
6. #20243896C2semidirect
7. C2×#15688891C2semidirect
8. #20243115C2semidirect
9. C2×C47×#568C2semidirect
12. #15689921C4semidirect
25. C2×#13093949C22semidirect
27. C2×#13093946C22semidirect
30. C2×#13093942C22semidirect
31. C2×#13093940C22semidirect
32. #15688891C22semidirect
33. #15688888C22semidirect
57. #13094054C2×C4semidirect
60. #13094049C2×C4semidirect
324. C2×#568D47semidirect
373. C47#21004semidirect

Maximal subgroups

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

subgroupindex
C2×#156899212
C2×#156891252
C2×#156891242
C2×#156891052
#202439032
#202438962
C2×#156888912
#202431152
C2×C47×#5682
#2100447