Finite Group Databasebeta

← #26123433

Subgroups — #26123433

Order 12032.

974 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689141C211yesmaximal, characteristic
2#20251905C211yesmaximal, characteristic
3#20245878C211yesmaximal, characteristic
4#20243097C211yesmaximal, characteristic, complemented
5C2×#15688879C211yesmaximal, characteristic, complemented
6#20243082C211yesmaximal, characteristic, complemented
7C47×#2296C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689141—22no—
13#15689141C414yescomplemented
14C2×#13093952—22no—
15#15689131—21no—
16#15689127—21no—
17#15689122—22no—
18#15689117—22no—
19#15689029—21no—
20#15689025—21no—
21#15689002—22no—
22C4×(C2×D47)⋊C4—22no—
23#15688882—22no—
24#15688879—22no—
25#15688878—22no—
26C2×#117×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C22×C42×C47C2211yescharacteristic
29C47×#389—22no—
30C47×#385—22no—
31C4×C4⋊C4×C47—22no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350C2×C412yes—
43C2×#10625350Q812yes—
44C22×Dic94D411yescharacteristic
45C22×D188D411yescharacteristic
46C22×C22⋊D47—21no—
47C22×C4×D47—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329C2×C412yes—
60C22×C47⋊C8—21no—
61C22×C47⋊C8D412yes—
62#13093983—21no—
63#13093983—21no—
64#13093983—21no—
65#13093983—21no—
66C2×C4⋊Dic47—21no—
67C2×C22⋊Dic47—21no—
68C2×(C2×D47)⋊C4—24no—
69C2×C4×Dic47D412yes—
70#13093952—28no—
71#13093947—24no—
72#13093944—24no—
73C47⋊#95—24no—
74C23×C4×C47D412yes—
75C23×C4×C47C2311yescharacteristic
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47C2×C412yes—
83#117×C47D412yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47D412yes—
87C2×C22⋊C4×C47—21no—
88C2×C22⋊C4×C47—21no—
89C2×C42×C47—21no—
90C2×C42×C47—21no—
91C2×C42×C47—21no—
92C2×C42×C47—21no—
93C2×C42×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47D412yes—
97#95×C47—24no—
98C24×C47C2×D411yescharacteristic
99C23×D47—21no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—44no—
106C22×C4×C47C22×C411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×Q811yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C22×C4×C47#4111yescharacteristic
116C22×C4×C47#4111yescharacteristic
117C22×C4×C47#4111yescharacteristic
118C22×C4×C47#4111yescharacteristic
119C22×C4×C47#4111yescharacteristic
120C2×C22⋊D47—22no—
121C2×C22⋊D47—22no—
122C2×C22⋊D47—41no—
123C2×C22⋊D47—41no—
124C2×C22⋊D47—41no—
125C2×C22⋊D47—41no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—22no—
134#10625350—22no—
135#10625350—22no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141#10625350—41no—
142#10625350—41no—
143#10625350—41no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188C4⋊C412yescomplemented
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—41no—
153C2×C4×D47—41no—
154C2×C4×D47—41no—
155C2×C4×D47—41no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94C4⋊C412yescomplemented
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—48no—
167C22⋊C4×C47—22no—
168C22⋊C4×C47—22no—
169C22⋊C4×C47—41no—
170C22⋊C4×C47—41no—
171C42×C47—22no—
172C42×C47—41no—
173C42×C47—41no—
174C42×C47—41no—
175C42×C47—41no—
176C42×C47—41no—
177C42×C47—41no—
178C42×C47C22⋊C414yescomplemented
179C22⋊Dic47—41no—
180C22⋊Dic47—41no—
181(C2×D47)⋊C4—48no—
182C4⋊Dic47—41no—
183C4⋊Dic47—41no—
184C4×Dic47—22no—
185C4×Dic47—22no—
186#10625329—22no—
187#10625329—22no—
188#10625329—41no—
189#10625329—41no—
190#10625329—41no—
191#10625329—41no—
192#10625329—41no—
193#10625329—41no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—41no—
199C2×C47⋊C8—41no—
200C23×C47—42no—
201C23×C47C4×D411yescharacteristic
202C23×C47#12011yescharacteristic
203C23×C47#12011yescharacteristic
204C23×C47#12111yescharacteristic
205C23×C47#12111yescharacteristic
206C23×C47C22⋊Q811yescharacteristic
207C23×C47#12411yescharacteristic
208C22×D47—22no—
209C22×D47—44no—
210C22×D47—41no—
211C22×D47—41no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—44no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—44no—
222C2×C4×C47—42no—
223C2×C4×C47C2×C22⋊C411yescharacteristic
224C2×C4×C47C2×C4⋊C411yescharacteristic
225C2×C4×C47#11711yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47#12011yescharacteristic
230C2×C4×C47#12111yescharacteristic
231C2×C4×C47#12111yescharacteristic
232C2×C4×C47#12111yescharacteristic
233C2×C4×C47#12111yescharacteristic
234C2×C4×C47#12111yescharacteristic
235C2×C4×C47#12111yescharacteristic
236C2×C4×C47C22⋊Q811yescharacteristic
237C2×C4×C47C22⋊Q811yescharacteristic
238C2×C4×C47C22⋊Q811yescharacteristic
239C2×C4×C47C22⋊Q811yescharacteristic
240C2×C4×C47C22⋊Q811yescharacteristic
241C2×C4×C47#12311yescharacteristic
242C2×C4×C47#12311yescharacteristic
243C2×C4×C47#12311yescharacteristic
244C2×C4×C47#12311yescharacteristic
245C2×C4×C47#12311yescharacteristic
246C2×C4×C47#12411yescharacteristic
247C2×C4×C47#12411yescharacteristic
248C2×C4×C47#12611yescharacteristic
249C2×C4×C47#12611yescharacteristic
250C2×C4×C47#12711yescharacteristic
251C22⋊D47—42no—
252C22⋊D47—42no—
253C22⋊D47—42no—
254C22⋊D47—42no—
255C2×Dic47—22no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—44no—
259C2×Dic47—42no—
260D188—22no—
261D188—22no—
262D188—41no—
263D188—41no—
264C4×D47—42no—
265C4×D47—42no—
266C4×D47—42no—
267C4×D47—42no—
268Dic94—22no—
269Dic94—22no—
270Dic94—41no—
271Dic94—41no—
272C47⋊C8—42no—
273C47⋊C8—42no—
274#11083—471nomaximal, Sylow
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47#38511yescharacteristic
283C22×C47#38611yescharacteristic
284C22×C47#38811yescharacteristic
285C22×C47#39211yescharacteristic
286C22×C47#39311yescharacteristic
287C22×C47#39411yescharacteristic
288C22×C47#39611yescharacteristic
289C2×D47—44no—
290C2×D47—42no—
291C2×D47—82no—
292C2×D47—82no—
293C4×C47—44no—
294C4×C47—44no—
295C4×C47—82no—
296C4×C47#38011yescharacteristic
297C4×C47#38511yescharacteristic
298C4×C47#38811yescharacteristic
299C4×C47#39011yescharacteristic
300C4×C47#39411yescharacteristic
301C4×C47#39611yescharacteristic
302C4×C47#39711yescharacteristic
303C4×C47#39911yescharacteristic
304Dic47—42no—
305Dic47—82no—
306C2×#421—471no—
307#2896—471no—
308#2297—471no—
309#2296D4711yescharacteristic, complemented
310#1733—471no—
311C2×H(3,ℤ/4)—471no—
312#1728—471no—
313C2×C47—21no—
314C2×C47—21no—
315C2×C47—21no—
316C2×C47—21no—
317C2×C47—21no—
318C2×C47—21no—
319C2×C47#143911yescharacteristic
320C2×C47#201012yes—
321D47—82no—
322C22×#41—471no—
323C22×M16—471no—
324C22×M16—471no—
325C2×#117—471no—
326C2×#117C2×D4711yescharacteristic
327C2×#117C2×D4711yescharacteristic
328C22×C42C2×D4711yescharacteristic
329#421—474no—
330#421—942no—
331C2×#104—942no—
332#419—941no—
333#418—941no—
334#417—942no—
335#416—942no—
336#391—941no—
337#390—941no—
338#389—22no—
339#386—942no—
340#385—22no—
341C4×C4⋊C4—22no—
342C4×C22⋊C4—942no—
343#339—942no—
344H(3,ℤ/4)—942no—
345#336—942no—
346C47#1108311yescharacteristic, complemented, Sylow
347C2×#41—472no—
348C2×#41—472no—
349C2×#41—941no—
350C2×#41—941no—
351C2×#41—941no—
352C2×#41—941no—
353C2×#41—941no—
354C2×#41—941no—
355C2×#41—941no—
356C2×#41—941no—
357C2×#41—941no—
358C2×#41—941no—
359C22×Q8—471no—
360C22×D4—471no—
361C22×D4—941no—
362C23×C4—941no—
363C23×C4C22⋊D4712yes—
364C23×C4C22×D4711yescharacteristic
365C2×M16—472no—
366C2×M16—941no—
367C2×M16—941no—
368C2×M16—941no—
369C2×M16—941no—
370C2×M16—941no—
371C2×M16—941no—
372C2×M16—941no—
373C2×M16—941no—
374C2×M16—941no—
375C2×M16—941no—
376C2×M16—941no—
377C22×C8—472no—
378C22×C8—941no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—941no—
386#117—941no—
387#117—941no—
388#117—941no—
389#117C4×D4712yes—
390#117D18812yes—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22⋊D4712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—941no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—472no—
407C2×C42C22⋊D4712yes—
408#104—948no—
409#103—944no—
410#102—944no—
411#95—24no—
412#95—944no—
413C24—941no—
414C24C2×C22⋊D4711yescharacteristic
415#41—942no—
416#41—942no—
417#41—942no—
418#41—942no—
419#41—942no—
420#41—942no—
421#41—942no—
422#41—942no—
423#41—1881no—
424#41—1881no—
425#41—1881no—
426#41—1881no—
427#41—1881no—
428#41—1881no—
429#41—1881no—
430#41—1881no—
431C2×Q8—472no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×Q8—941no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×D4—472no—
438C2×D4—942no—
439C2×D4—942no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—941no—
445C2×D4—1881no—
446C2×D4—1881no—
447C2×D4—1881no—
448C2×D4—1881no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—941no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—1881no—
460C22×C4—1881no—
461C22×C4—1881no—
462C22×C4—1881no—
463C22×C4C2×C4×D4711yescharacteristic
464C22×C4C2×D18811yescharacteristic
465C22×C4#1062535011yescharacteristic
466C22×C4#1062535011yescharacteristic
467C22×C4D4×D4711yescharacteristic
468C22×C4D4×D4711yescharacteristic
469C22×C4D4×D4711yescharacteristic
470C22×C4#1062535211yescharacteristic
471C22×C4Q8×D4711yescharacteristic
472C22×C4#1062535411yescharacteristic
473C22×C4#1062535411yescharacteristic
474C22×C4#1062535411yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477M16—942no—
478M16—942no—
479M16—1881no—
480M16—1881no—
481M16—1881no—
482M16—1881no—
483M16—1881no—
484M16—1881no—
485C2×C8—942no—
486C2×C8—942no—
487C2×C8—942no—
488C2×C8—942no—
489C2×C8—1881no—
490C2×C8—1881no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—1881no—
497C4⋊C4—1881no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—1888no—
503C22⋊C4—1881no—
504C22⋊C4—1881no—
505C42—22no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—942no—
513C42—942no—
514C42(C2×D47)⋊C414yescomplemented
515C23—42no—
516C23—942no—
517C23—1884no—
518C23—1881no—
519C23—1881no—
520C23C4×C22⋊D4711yescharacteristic
521C23#1309399211yescharacteristic
522C23#1309399311yescharacteristic
523C23#1309399611yescharacteristic
524C23#1309399811yescharacteristic
525C23#1309400011yescharacteristic
526C23#1309401311yescharacteristic
527Q8—942no—
528Q8—942no—
529Q8—1881no—
530Q8—1881no—
531D4—942no—
532D4—942no—
533D4—1882no—
534D4—1882no—
535D4—1882no—
536D4—1882no—
537D4—1881no—
538D4—1881no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—942no—
551C2×C4—1884no—
552C2×C4—1882no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4C2×(C2×D47)⋊C411yescharacteristic
560C2×C4C4⋊C4×D4711yescharacteristic
561C2×C4#1309398211yescharacteristic
562C2×C4C4×C22⋊D4711yescharacteristic
563C2×C4#1309398611yescharacteristic
564C2×C4#1309398611yescharacteristic
565C2×C4#1309399211yescharacteristic
566C2×C4#1309399511yescharacteristic
567C2×C4#1309399611yescharacteristic
568C2×C4#1309399811yescharacteristic
569C2×C4#1309399911yescharacteristic
570C2×C4#1309399911yescharacteristic
571C2×C4#1309399911yescharacteristic
572C2×C4#1309400011yescharacteristic
573C2×C4#1309400011yescharacteristic
574C2×C4#1309400211yescharacteristic
575C2×C4#1309400211yescharacteristic
576C2×C4#1309400411yescharacteristic
577C2×C4#1309400711yescharacteristic
578C2×C4#1309400911yescharacteristic
579C2×C4#1309400911yescharacteristic
580C2×C4#1309400911yescharacteristic
581C2×C4#1309400911yescharacteristic
582C2×C4#1309401311yescharacteristic
583C2×C4#1309401311yescharacteristic
584C2×C4#1309401911yescharacteristic
585C2×C4#1309401911yescharacteristic
586C2×C4#1309402011yescharacteristic
587C8—1882no—
588C8—1882no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—3762no—
597C22—3762no—
598C22—1884no—
599C22—1882no—
600C22#1568899411yescharacteristic
601C22#1568900211yescharacteristic
602C22#1568901311yescharacteristic
603C22#1568903111yescharacteristic
604C22#1568903411yescharacteristic
605C22#1568904011yescharacteristic
606C22#1568904511yescharacteristic
607C4—44no—
608C4—44no—
609C4—82no—
610C4—3762no—
611C4—1882no—
612C4#1568898011yescharacteristic
613C4#1568899411yescharacteristic
614C4#1568901311yescharacteristic
615C4#1568902411yescharacteristic
616C4#1568904011yescharacteristic
617C4#1568904511yescharacteristic
618C4#1568904911yescharacteristic
619C4#1568905811yescharacteristic
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—3762no—
627C2#2024163011yescharacteristic
628C2#2024445312yes—
subgroupG/H
centerC2×C4#13094020
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2296C2
socleC22×C47#392
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#2296D47
hypocenterC47#11083
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
4. #20243097C2semidirect
5. C2×#15688879C2semidirect
6. #20243082C2semidirect
7. C47×#2296C2semidirect
13. #15689141C4semidirect
149. C2×D188C4⋊C4semidirect
161. C2×Dic94C4⋊C4semidirect
178. C42×C47C22⋊C4semidirect
309. #2296D47semidirect
346. C47#11083semidirect
514. C42(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156891412
#202519052
#202458782
#202430972
C2×#156888792
#202430822
C47×#22962
#1108347