Finite Group Databasebeta

← #26123662

Subgroups — #26123662

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689171C211yesmaximal, characteristic, complemented
2C2×#15689141C211yesmaximal, characteristic
3C2×#15689125C211yesmaximal, characteristic
4C2×#15689105C211yesmaximal, characteristic
5C2×#15688891C211yesmaximal, characteristic, complemented
6#20243109C211yesmaximal, characteristic, complemented
7C2×C47×#344C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12#15689171—22no—
13#15689171—22no—
14C2×#13093959—22no—
15#15689141—22no—
16#15689141C414yescomplemented
17C2×#13093952—22no—
18#15689125—22no—
19#15689125—22no—
20C2×#13093949—21no—
21C2×#13093946—21no—
22#15689105—22no—
23#15689105—22no—
24C2×#13093942—21no—
25C2×#13093940—21no—
26#15688891—22no—
27#15688885—22no—
28C22×M16×C47C2211yescharacteristic
29C22×M16×C47C2211yescharacteristic
30C2×#117×C47C2211yescharacteristic
31C47×#344—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×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329C2×C412yes—
58C2×#10625329D412yes—
59C2×#10625329D412yes—
60C2×#10625329D412yes—
61C22×C47⋊C8—21no—
62C22×C47⋊C8—21no—
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#13093952—28no—
71#13093949—24no—
72#13093946—24no—
73#13093942—24no—
74#13093940—24no—
75C23×C4×C47C2311yescharacteristic
76C2×M16×C47—21no—
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×C47D412yes—
87C2×M16×C47D412yes—
88C22×C8×C47—21no—
89C22×C8×C47—21no—
90#117×C47—21no—
91#117×C47—21no—
92#117×C47—21no—
93#117×C47C2×C412yes—
94C2×C4⋊C4×C47—21no—
95C2×C22⋊C4×C47—21no—
96C2×C42×C47—21no—
97C24×C47#4111yescharacteristic
98C23×D47—21no—
99C22×C4×C47—41no—
100C22×C4×C47C22×C411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×Q811yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C2×C22⋊D47—22no—
115C2×C22⋊D47—22no—
116C2×C22⋊D47—41no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C22×Dic47—21no—
121C22×Dic47—22no—
122C22×Dic47—41no—
123C22×Dic47—41no—
124C22×Dic47—82no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188C4⋊C412yes—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×C4×D47—41no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94C4⋊C412yes—
159M16×C47—22no—
160M16×C47—22no—
161M16×C47—22no—
162M16×C47—22no—
163M16×C47—41no—
164M16×C47—41no—
165M16×C47—41no—
166M16×C47—41no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—41no—
170C2×C8×C47—41no—
171C2×C8×C47—41no—
172C2×C8×C47—41no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—41no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—41no—
177C42×C47—22no—
178C42×C47—41no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47—41no—
182C4⋊Dic47—41no—
183#10625329—22no—
184#10625329—22no—
185#10625329—22no—
186#10625329—22no—
187#10625329—22no—
188#10625329—41no—
189#10625329—41no—
190#10625329C22⋊C414yes—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—41no—
196C2×C47⋊C8—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47C4×D411yescharacteristic
200C23×C47#12111yescharacteristic
201C23×C47C22⋊Q811yescharacteristic
202C23×C47#12311yescharacteristic
203C23×C47#12411yescharacteristic
204C23×C47#12411yescharacteristic
205C23×C47#12611yescharacteristic
206C22×D47—22no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—81no—
210C22×D47—81no—
211C2×C4×C47—42no—
212C2×C4×C47—81no—
213C2×C4×C47—81no—
214C2×C4×C47C2×C22⋊C411yescharacteristic
215C2×C4×C47C2×C4⋊C411yescharacteristic
216C2×C4×C47#11711yescharacteristic
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47#12011yescharacteristic
221C2×C4×C47#12011yescharacteristic
222C2×C4×C47#12011yescharacteristic
223C2×C4×C47#12111yescharacteristic
224C2×C4×C47#12111yescharacteristic
225C2×C4×C47#12111yescharacteristic
226C2×C4×C47#12111yescharacteristic
227C2×C4×C47#12111yescharacteristic
228C2×C4×C47#12111yescharacteristic
229C2×C4×C47#12111yescharacteristic
230C2×C4×C47C22⋊Q811yescharacteristic
231C2×C4×C47C22⋊Q811yescharacteristic
232C2×C4×C47C22⋊Q811yescharacteristic
233C2×C4×C47C22⋊Q811yescharacteristic
234C2×C4×C47C22⋊Q811yescharacteristic
235C2×C4×C47#12311yescharacteristic
236C2×C4×C47#12311yescharacteristic
237C2×C4×C47#12311yescharacteristic
238C2×C4×C47#12311yescharacteristic
239C2×C4×C47#12411yescharacteristic
240C2×C4×C47#12611yescharacteristic
241C2×C4×C47#12711yescharacteristic
242C22⋊D47—42no—
243C22⋊D47—42no—
244C22⋊D47—42no—
245C22⋊D47—42no—
246C2×Dic47—22no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—42no—
250C2×Dic47—82no—
251C2×Dic47—82no—
252C2×Dic47—82no—
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—42no—
258C4×D47—42no—
259C4×D47—42no—
260C4×D47—42no—
261Dic94—22no—
262Dic94—22no—
263Dic94—41no—
264Dic94—41no—
265C8×C47—42no—
266C8×C47—42no—
267C47⋊C8—24no—
268C47⋊C8—42no—
269#11120—471nomaximal, Sylow
270C22×C47—21no—
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47#38511yescharacteristic
281C22×C47#38811yescharacteristic
282C22×C47#39011yescharacteristic
283C22×C47#39311yescharacteristic
284C22×C47#39411yescharacteristic
285C22×C47#39611yescharacteristic
286C22×C47#39711yescharacteristic
287C2×D47—42no—
288C2×D47—82no—
289C2×D47—81no—
290C2×D47—81no—
291C2×D47—82no—
292C4×C47—82no—
293C4×C47#38011yescharacteristic
294C4×C47#38511yescharacteristic
295C4×C47#38611yescharacteristic
296C4×C47#38811yescharacteristic
297C4×C47#39211yescharacteristic
298C4×C47#39411yescharacteristic
299C4×C47#39611yescharacteristic
300C4×C47#39911yescharacteristic
301Dic47—42no—
302Dic47—82no—
303C2×#430—471no—
304C2×#421—471no—
305C2×#417—471no—
306C2×#413—471no—
307C2×#344D4711yescharacteristic, complemented
308C2×#343—471no—
309#1736—471no—
310C2×C47—21no—
311C2×C47—21no—
312C2×C47—21no—
313C2×C47—21no—
314C2×C47—21no—
315C2×C47—21no—
316C2×C47#143911yescharacteristic
317C2×C47#204811yescharacteristic
318C2×C47#204911yescharacteristic
319D47—82no—
320C22×#41—471no—
321C22×M16—471no—
322C22×M16—471no—
323C22×M16C2×D4711yescharacteristic
324C22×M16C2×D4711yescharacteristic
325C2×#117C2×D4711yescharacteristic
326C22×C4⋊C4—471no—
327#430—942no—
328#430—942no—
329C2×#108—942no—
330#421—474no—
331#421—942no—
332C2×#104—942no—
333#417—942no—
334#417—942no—
335C2×#103—941no—
336C2×#102—941no—
337#413—942no—
338#413—942no—
339C2×#101—941no—
340C2×#100—941no—
341#344—22no—
342#343—942no—
343#340—942no—
344C47#1112011yescharacteristic, complemented, Sylow
345C2×#41—472no—
346C2×#41—472no—
347C2×#41—941no—
348C2×#41—941no—
349C2×#41—941no—
350C2×#41—941no—
351C2×#41—941no—
352C2×#41—941no—
353C2×#41—941no—
354C2×#41—941no—
355C2×#41—941no—
356C2×#41—941no—
357C22×Q8—471no—
358C22×D4—471no—
359C22×D4—941no—
360C23×C4—941no—
361C23×C4—941no—
362C23×C4C22×D4711yescharacteristic
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—472no—
374C2×M16—472no—
375C2×M16—472no—
376C2×M16—472no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C2×M16—941no—
381C2×M16—941no—
382C2×M16—941no—
383C2×M16—941no—
384C2×M16—941no—
385C2×M16D18812yes—
386C2×M16C22⋊D4712yes—
387C22×C8—21no—
388C22×C8—21no—
389C22×C8—941no—
390C22×C8—941no—
391#117—21no—
392#117—21no—
393#117—21no—
394#117C4×D4712yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—472no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C22⋊C4—21no—
403C2×C42—21no—
404#108—948no—
405#104—948no—
406#103—944no—
407#102—944no—
408#101—944no—
409#100—944no—
410C24—941no—
411C24#1062535011yescharacteristic
412#41—942no—
413#41—942no—
414#41—942no—
415#41—942no—
416#41—942no—
417#41—942no—
418#41—942no—
419#41—942no—
420#41—1881no—
421#41—1881no—
422#41—1881no—
423#41—1881no—
424#41—1881no—
425#41—1881no—
426#41—1881no—
427#41—1881no—
428C2×Q8—472no—
429C2×Q8—941no—
430C2×Q8—941no—
431C2×Q8—941no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×D4—472no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—941no—
438C2×D4—941no—
439C2×D4—941no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—1881no—
443C2×D4—1881no—
444C2×D4—1881no—
445C2×D4—1881no—
446C22×C4—41no—
447C22×C4—941no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—3762no—
452C22×C4—1881no—
453C22×C4—1881no—
454C22×C4—1881no—
455C22×C4—1881no—
456C22×C4—1881no—
457C22×C4—1881no—
458C22×C4C2×C4×D4711yescharacteristic
459C22×C4C2×D18811yescharacteristic
460C22×C4#1062535011yescharacteristic
461C22×C4#1062535011yescharacteristic
462C22×C4#1062535011yescharacteristic
463C22×C4D4×D4711yescharacteristic
464C22×C4D4×D4711yescharacteristic
465C22×C4D4×D4711yescharacteristic
466C22×C4D4×D4711yescharacteristic
467C22×C4D4×D4711yescharacteristic
468C22×C4#1062535211yescharacteristic
469C22×C4Q8×D4711yescharacteristic
470C22×C4#1062535411yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—41no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16—474no—
481M16—942no—
482M16—942no—
483M16—942no—
484M16—942no—
485M16—942no—
486M16—1881no—
487M16—1881no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—942no—
495C2×C8—942no—
496C2×C8—942no—
497C2×C8—942no—
498C2×C8—1881no—
499C2×C8—1881no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C4⋊C4—1881no—
505C4⋊C4—1881no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C42—22no—
509C42—41no—
510C23—41no—
511C23—41no—
512C23—942no—
513C23—3761no—
514C23—3761no—
515C23—1881no—
516C23—1881no—
517C23C4×C22⋊D4711yescharacteristic
518C23#1309399711yescharacteristic
519C23#1309400211yescharacteristic
520C23#1309400911yescharacteristic
521C23#1309401011yescharacteristic
522C23#1309401211yescharacteristic
523C23#1309401811yescharacteristic
524Q8—942no—
525Q8—942no—
526Q8—1881no—
527Q8—1881no—
528D4—942no—
529D4—942no—
530D4—1882no—
531D4—1882no—
532D4—1882no—
533D4—1882no—
534D4—1881no—
535D4—1881no—
536C2×C4—42no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—942no—
540C2×C4—3762no—
541C2×C4—3762no—
542C2×C4—3762no—
543C2×C4—1882no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4—1882no—
550C2×C4C22⋊C4×D4711yescharacteristic
551C2×C4C4⋊C4×D4711yescharacteristic
552C2×C4#1309398111yescharacteristic
553C2×C4(C22⋊D47)⋊C411yescharacteristic
554C2×C4#1309398611yescharacteristic
555C2×C4C4×D18811yescharacteristic
556C2×C4#1309399211yescharacteristic
557C2×C4#1309399411yescharacteristic
558C2×C4#1309399411yescharacteristic
559C2×C4#1309399611yescharacteristic
560C2×C4#1309399711yescharacteristic
561C2×C4#1309399711yescharacteristic
562C2×C4#1309399711yescharacteristic
563C2×C4#1309399711yescharacteristic
564C2×C4#1309399811yescharacteristic
565C2×C4#1309399911yescharacteristic
566C2×C4#1309400011yescharacteristic
567C2×C4#1309400211yescharacteristic
568C2×C4#1309400211yescharacteristic
569C2×C4#1309400211yescharacteristic
570C2×C4#1309400411yescharacteristic
571C2×C4#1309400611yescharacteristic
572C2×C4#1309400711yescharacteristic
573C2×C4#1309400911yescharacteristic
574C2×C4#1309400911yescharacteristic
575C2×C4#1309401011yescharacteristic
576C2×C4#1309401911yescharacteristic
577C2×C4#1309402011yescharacteristic
578C8—42no—
579C8—42no—
580C8—944no—
581C8—1882no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—3762no—
593C22—3762no—
594C22—3761no—
595C22—3761no—
596C22—1882no—
597C22#1568899411yescharacteristic
598C22#1568901411yescharacteristic
599C22#1568902311yescharacteristic
600C22#1568903611yescharacteristic
601C22#1568904111yescharacteristic
602C22#1568904711yescharacteristic
603C22#1568905211yescharacteristic
604C4—82no—
605C4—3762no—
606C4—1882no—
607C4#1568898211yescharacteristic
608C4#1568899711yescharacteristic
609C4#1568900111yescharacteristic
610C4#1568901211yescharacteristic
611C4#1568903211yescharacteristic
612C4#1568903811yescharacteristic
613C4#1568904511yescharacteristic
614C4#1568905811yescharacteristic
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—3762no—
622C2#2024163211yescharacteristic
623C2#2024466411yescharacteristic
624C2#2024467111yescharacteristic
subgroupG/H
centerC22#15689041
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#344C2
socleC22×C47#394
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×#344D47
hypocenterC47#11120
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
1. C2×#15689171C2semidirect
5. C2×#15688891C2semidirect
6. #20243109C2semidirect
7. C2×C47×#344C2semidirect
16. #15689141C4semidirect
307. C2×#344D47semidirect
344. C47#11120semidirect

Maximal subgroups

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

subgroupindex
C2×#156891712
C2×#156891412
C2×#156891252
C2×#156891052
C2×#156888912
#202431092
C2×C47×#3442
#1112047