Finite Group Databasebeta

← #26123569

Subgroups — #26123569

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093959C211yesmaximal, characteristic, complemented
2C2×#15689125C211yesmaximal, characteristic
3C2×#15689122C211yesmaximal, characteristic
4C2×#15689091C211yesmaximal, characteristic
5C2×#15688891C211yesmaximal, characteristic, complemented
6C2×#15688885C211yesmaximal, characteristic, complemented
7C2×C47×#343C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C2×#13093959—22no—
13C2×#13093959—22no—
14C2×#13093959—22no—
15#15689125—22no—
16#15689125—22no—
17#15689122—22no—
18#15689122—22no—
19C2×#13093949—21no—
20C2×#13093947—21no—
21C2×#13093946—21no—
22C2×#13093944—21no—
23#15689091—22no—
24#15689091C414yescomplemented
25C2×(C2×D47)⋊C8—22no—
26#15688891—22no—
27#15688885—22no—
28C22×M16×C47C2211yescharacteristic
29C22×M16×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C47×#343—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—
47C23×Dic47—21no—
48C22×C4×D47—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329D412yes—
55C22×C47⋊C8—21no—
56C22×C47⋊C8—21no—
57C22×C47⋊C8—21no—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—21no—
60C22×C47⋊C8C2×C412yes—
61C22×C47⋊C8D412yes—
62C22×C47⋊C8D412yes—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47D412yes—
69#13093959—28no—
70#13093949—24no—
71#13093947—24no—
72#13093946—24no—
73#13093944—24no—
74(C2×D47)⋊C8—28no—
75C23×C4×C47—21no—
76C23×C4×C47C2311yescharacteristic
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47—21no—
81C2×M16×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47—21no—
84C2×M16×C47—21no—
85C2×M16×C47—21no—
86C2×M16×C47—21no—
87C2×M16×C47C2×C412yes—
88C2×M16×C47D412yes—
89C22×C8×C47—21no—
90C22×C8×C47—21no—
91C2×C4⋊C4×C47—21no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47D412yes—
97C24×C47#4111yescharacteristic
98C23×D47—21no—
99C22×C4×C47—22no—
100C22×C4×C47—41no—
101C22×C4×C47—41no—
102C22×C4×C47—81no—
103C22×C4×C47—81no—
104C22×C4×C47C22×C411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×Q811yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C22×C4×C47#4111yescharacteristic
116C22×C4×C47#4111yescharacteristic
117C22×C4×C47#4111yescharacteristic
118C2×C22⋊D47—22no—
119C2×C22⋊D47—22no—
120C2×C22⋊D47—41no—
121C2×C22⋊D47—41no—
122C2×C22⋊D47—41no—
123C2×C22⋊D47—41no—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C22×Dic47—41no—
127C22×Dic47—41no—
128C22×Dic47—81no—
129C22×Dic47—81no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—22no—
134#10625350—22no—
135#10625350—22no—
136#10625350—22no—
137#10625350—22no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141#10625350—41no—
142#10625350—41no—
143#10625350—41no—
144#10625350—41no—
145#10625350—41no—
146C2×D188—21no—
147C2×D188—21no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188C4⋊C412yes—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—41no—
155C2×C4×D47—41no—
156C2×C4×D47—41no—
157C2×C4×D47—41no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94C4⋊C412yes—
164M16×C47—22no—
165M16×C47—22no—
166M16×C47—22no—
167M16×C47—22no—
168M16×C47—41no—
169M16×C47—41no—
170M16×C47—41no—
171M16×C47—41no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—41no—
175C2×C8×C47—41no—
176C2×C8×C47—41no—
177C2×C8×C47—41no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—41no—
181C4⋊C4×C47—41no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—41no—
185C4⋊Dic47—41no—
186#10625329—22no—
187#10625329—22no—
188#10625329—41no—
189#10625329—41no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—41no—
198C2×C47⋊C8—41no—
199C2×C47⋊C8C22⋊C414yes—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47#11711yescharacteristic
205C23×C47#12111yescharacteristic
206C23×C47C22⋊Q811yescharacteristic
207C23×C47#12311yescharacteristic
208C23×C47#12311yescharacteristic
209C23×C47#12411yescharacteristic
210C23×C47#12411yescharacteristic
211C22×D47—22no—
212C22×D47—41no—
213C22×D47—41no—
214C22×D47—41no—
215C22×D47—41no—
216C22×D47—41no—
217C22×D47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—82no—
220C2×C4×C47—81no—
221C2×C4×C47—81no—
222C2×C4×C47—82no—
223C2×C4×C47C2×C22⋊C411yescharacteristic
224C2×C4×C47C2×C4⋊C411yescharacteristic
225C2×C4×C47C4×D411yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47#12011yescharacteristic
230C2×C4×C47#12011yescharacteristic
231C2×C4×C47#12011yescharacteristic
232C2×C4×C47#12111yescharacteristic
233C2×C4×C47#12111yescharacteristic
234C2×C4×C47#12111yescharacteristic
235C2×C4×C47#12111yescharacteristic
236C2×C4×C47#12111yescharacteristic
237C2×C4×C47#12111yescharacteristic
238C2×C4×C47#12111yescharacteristic
239C2×C4×C47C22⋊Q811yescharacteristic
240C2×C4×C47C22⋊Q811yescharacteristic
241C2×C4×C47C22⋊Q811yescharacteristic
242C2×C4×C47C22⋊Q811yescharacteristic
243C2×C4×C47C22⋊Q811yescharacteristic
244C2×C4×C47#12311yescharacteristic
245C2×C4×C47#12311yescharacteristic
246C2×C4×C47#12311yescharacteristic
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—42no—
259C2×Dic47—82no—
260C2×Dic47—81no—
261C2×Dic47—81no—
262C2×Dic47—82no—
263D188—22no—
264D188—22no—
265D188—41no—
266D188—41no—
267C4×D47—42no—
268C4×D47—42no—
269C4×D47—42no—
270C4×D47—42no—
271Dic94—22no—
272Dic94—22no—
273Dic94—41no—
274Dic94—41no—
275C8×C47—42no—
276C8×C47—42no—
277C47⋊C8—24no—
278C47⋊C8—42no—
279#11105—471nomaximal, Sylow
280C22×C47—21no—
281C22×C47—21no—
282C22×C47—21no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—21no—
287C22×C47—21no—
288C22×C47—21no—
289C22×C47—21no—
290C22×C47—21no—
291C22×C47—21no—
292C22×C47#38011yescharacteristic
293C22×C47#38811yescharacteristic
294C22×C47#38811yescharacteristic
295C22×C47#39311yescharacteristic
296C22×C47#39411yescharacteristic
297C22×C47#39411yescharacteristic
298C22×C47#39911yescharacteristic
299C22×C47#47111yescharacteristic
300C22×C47#47111yescharacteristic
301C22×C47#47211yescharacteristic
302C22×C47#47311yescharacteristic
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—81no—
307C2×D47—81no—
308C2×D47—81no—
309C2×D47—81no—
310C4×C47—82no—
311C4×C47#38511yescharacteristic
312C4×C47#38511yescharacteristic
313C4×C47#38611yescharacteristic
314C4×C47#39011yescharacteristic
315C4×C47#39211yescharacteristic
316C4×C47#39611yescharacteristic
317C4×C47#39611yescharacteristic
318C4×C47#39711yescharacteristic
319Dic47—42no—
320Dic47—82no—
321C22×#108—471no—
322C2×#417—471no—
323C2×#417—471no—
324C2×#408—471no—
325C2×#343—471no—
326C2×#343D4711yescharacteristic, complemented
327C2×#340—471no—
328C2×C47—21no—
329C2×C47—21no—
330C2×C47—21no—
331C2×C47—21no—
332C2×C47#143911yescharacteristic
333C2×C47#181511yescharacteristic
334C2×C47#181611yescharacteristic
335C2×C47#205111yescharacteristic
336C2×C47#205111yescharacteristic
337C2×C47#205211yescharacteristic
338C2×C47#205211yescharacteristic
339D47—82no—
340C22×#41—471no—
341C22×M16—471no—
342C22×M16C2×D4711yescharacteristic
343C22×M16C2×D4711yescharacteristic
344C23×C8—471no—
345C22×C4⋊C4—471no—
346C22×C4⋊C4C2×D4711yescharacteristic
347C2×#108—942no—
348C2×#108—942no—
349C2×#108—942no—
350#417—942no—
351#417—942no—
352#417—942no—
353#417—942no—
354C2×#103—941no—
355C2×#103—941no—
356C2×#102—941no—
357C2×#102—941no—
358#408—474no—
359#408—942no—
360C2×C22⋊C8—942no—
361#343—22no—
362#343—942no—
363#340—942no—
364C47#1110511yescharacteristic, complemented, Sylow
365C2×#41—472no—
366C2×#41—472no—
367C2×#41—941no—
368C2×#41—941no—
369C2×#41—941no—
370C2×#41—941no—
371C2×#41—941no—
372C2×#41—941no—
373C2×#41—941no—
374C2×#41—941no—
375C2×#41—941no—
376C2×#41—941no—
377C22×Q8—471no—
378C22×D4—471no—
379C22×D4—941no—
380C23×C4—21no—
381C23×C4—941no—
382C23×C4—941no—
383C23×C4C22×D4711yescharacteristic
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—472no—
395C2×M16—941no—
396C2×M16—941no—
397C2×M16—941no—
398C2×M16—941no—
399C2×M16—941no—
400C2×M16C4×D4712yes—
401C2×M16D18812yes—
402C22×C8—21no—
403C22×C8—21no—
404C22×C8—472no—
405C22×C8—472no—
406C22×C8—472no—
407C22×C8—941no—
408C22×C8—941no—
409C22×C8—941no—
410C22×C8—941no—
411C22×C8—941no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—472no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4C22⋊D4712yes—
424#108—948no—
425#103—944no—
426#103—944no—
427#102—944no—
428#102—944no—
429C22⋊C8—948no—
430C24—941no—
431C24#1062535011yescharacteristic
432#41—942no—
433#41—942no—
434#41—942no—
435#41—942no—
436#41—942no—
437#41—942no—
438#41—942no—
439#41—942no—
440#41—1881no—
441#41—1881no—
442#41—1881no—
443#41—1881no—
444#41—1881no—
445#41—1881no—
446#41—1881no—
447#41—1881no—
448C2×Q8—472no—
449C2×Q8—941no—
450C2×Q8—941no—
451C2×Q8—941no—
452C2×Q8—941no—
453C2×Q8—941no—
454C2×D4—472no—
455C2×D4—942no—
456C2×D4—942no—
457C2×D4—941no—
458C2×D4—941no—
459C2×D4—941no—
460C2×D4—941no—
461C2×D4—941no—
462C2×D4—1881no—
463C2×D4—1881no—
464C2×D4—1881no—
465C2×D4—1881no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—941no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—3761no—
476C22×C4—3761no—
477C22×C4—1881no—
478C22×C4—1881no—
479C22×C4—1881no—
480C22×C4—1881no—
481C22×C4—1881no—
482C22×C4—1881no—
483C22×C4C2×C4×D4711yescharacteristic
484C22×C4C2×D18811yescharacteristic
485C22×C4#1062535011yescharacteristic
486C22×C4#1062535011yescharacteristic
487C22×C4#1062535011yescharacteristic
488C22×C4D4×D4711yescharacteristic
489C22×C4D4×D4711yescharacteristic
490C22×C4D4×D4711yescharacteristic
491C22×C4D4×D4711yescharacteristic
492C22×C4D4×D4711yescharacteristic
493C22×C4#1062535211yescharacteristic
494C22×C4Q8×D4711yescharacteristic
495C22×C4#1062535411yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—942no—
506M16—942no—
507M16—1881no—
508M16—1881no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—474no—
516C2×C8—942no—
517C2×C8—942no—
518C2×C8—942no—
519C2×C8—942no—
520C2×C8—942no—
521C2×C8—942no—
522C2×C8—942no—
523C2×C8—1881no—
524C2×C8—1881no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—942no—
530C4⋊C4—942no—
531C4⋊C4—1881no—
532C4⋊C4—1881no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—942no—
538C23—1881no—
539C23—1881no—
540C23—1881no—
541C23—1881no—
542C23—1881no—
543C23—1881no—
544C23#1309398111yescharacteristic
545C23#1309399711yescharacteristic
546C23#1309400211yescharacteristic
547C23#1309400711yescharacteristic
548C23#1309400911yescharacteristic
549C23#1309401011yescharacteristic
550C23#1309401211yescharacteristic
551Q8—942no—
552Q8—942no—
553Q8—1881no—
554Q8—1881no—
555D4—942no—
556D4—942no—
557D4—1882no—
558D4—1882no—
559D4—1882no—
560D4—1882no—
561D4—1881no—
562D4—1881no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—942no—
569C2×C4—3762no—
570C2×C4—3761no—
571C2×C4—3761no—
572C2×C4—3762no—
573C2×C4—1882no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4C22⋊C4×D4711yescharacteristic
581C2×C4C4⋊C4×D4711yescharacteristic
582C2×C4(C22⋊D47)⋊C411yescharacteristic
583C2×C4C4×C22⋊D4711yescharacteristic
584C2×C4#1309398611yescharacteristic
585C2×C4C4×D18811yescharacteristic
586C2×C4#1309399211yescharacteristic
587C2×C4#1309399411yescharacteristic
588C2×C4#1309399411yescharacteristic
589C2×C4#1309399611yescharacteristic
590C2×C4#1309399711yescharacteristic
591C2×C4#1309399711yescharacteristic
592C2×C4#1309399711yescharacteristic
593C2×C4#1309399711yescharacteristic
594C2×C4#1309399811yescharacteristic
595C2×C4#1309399911yescharacteristic
596C2×C4#1309400011yescharacteristic
597C2×C4#1309400211yescharacteristic
598C2×C4#1309400211yescharacteristic
599C2×C4#1309400211yescharacteristic
600C2×C4#1309400411yescharacteristic
601C2×C4#1309400611yescharacteristic
602C2×C4#1309400911yescharacteristic
603C2×C4#1309400911yescharacteristic
604C2×C4#1309401011yescharacteristic
605C2×C4#1309401811yescharacteristic
606C2×C4#1309401911yescharacteristic
607C2×C4#1309402011yescharacteristic
608C8—42no—
609C8—42no—
610C8—944no—
611C8—1882no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—3761no—
625C22—3761no—
626C22—3761no—
627C22—3761no—
628C22—1882no—
629C22—1882no—
630C22—1882no—
631C22#1568898211yescharacteristic
632C22#1568901211yescharacteristic
633C22#1568901411yescharacteristic
634C22#1568903611yescharacteristic
635C22#1568903811yescharacteristic
636C22#1568904111yescharacteristic
637C22#1568905811yescharacteristic
638C22#1568938511yescharacteristic
639C22#1568938911yescharacteristic
640C22#1568939211yescharacteristic
641C22#1568939711yescharacteristic
642C4—82no—
643C4—3762no—
644C4—1882no—
645C4#1568899411yescharacteristic
646C4#1568899711yescharacteristic
647C4#1568900111yescharacteristic
648C4#1568902311yescharacteristic
649C4#1568903211yescharacteristic
650C4#1568904511yescharacteristic
651C4#1568904711yescharacteristic
652C4#1568905211yescharacteristic
653C2—21no—
654C2—21no—
655C2—21no—
656C2—21no—
657C2—3762no—
658C2#2024163211yescharacteristic
659C2#2024344611yescharacteristic
660C2#2024345111yescharacteristic
661C2#2024468011yescharacteristic
662C2#2024468311yescharacteristic
663C2#2024468711yescharacteristic
664C2#2024469011yescharacteristic
subgroupG/H
centerC23#13093997
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#343C2
socleC23×C47#121
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×#343D47
hypocenterC47#11105
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
1. C22×#13093959C2semidirect
5. C2×#15688891C2semidirect
6. C2×#15688885C2semidirect
7. C2×C47×#343C2semidirect
24. #15689091C4semidirect
326. C2×#343D47semidirect
364. C47#11105semidirect

Maximal subgroups

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

subgroupindex
C22×#130939592
C2×#156891252
C2×#156891222
C2×#156890912
C2×#156888912
C2×#156888852
C2×C47×#3432
#1110547