Finite Group Databasebeta

← #26122675

Subgroups — #26122675

Order 12032.

990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689141C211yesmaximal, characteristic
2C2×#15689139C211yesmaximal, characteristic
3#20245976C211yesmaximal, characteristic
4#20245744C211yesmaximal, characteristic, complemented
5C2×#15688891C211yesmaximal, characteristic, complemented
6C2×#15688879C211yesmaximal, characteristic, complemented
7C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12#15689141—22no—
13#15689141C414yescomplemented
14#15689139—22no—
15#15689139C414yescomplemented
16C2×#13093952—22no—
17C2×#13093950—22no—
18#15689029C412yescomplemented
19#15689025C412yescomplemented
20#15689000—22no—
21#15688980—22no—
22C4⋊C4×Dic47—22no—
23C4×C4⋊Dic47—22no—
24C22⋊C4×Dic47—22no—
25#15688891—22no—
26#15688879—22no—
27C22×M16×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C47×H(3,ℤ/4)—22no—
31C2×#10625350—21no—
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×#10625350C2×C412yes—
42C2×#10625350C2×C412yes—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47—21no—
46C23×Dic47D412yes—
47C22×C4×D47—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329C2×C412yes—
54C22×C47⋊C8—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983C2×C412yes—
58#13093983D412yes—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47C2×C412yes—
65C2×C4⋊Dic47D412yes—
66C2×C22⋊Dic47—21no—
67C2×(C2×D47)⋊C4—24no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47D412yes—
70#13093952—28no—
71#13093950—28no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C23×C4×C47C2311yescharacteristic
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47C2×C412yes—
81C22×C8×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47—21no—
85#117×C47—21no—
86#117×C47—21no—
87#117×C47—21no—
88#117×C47C2×C412yes—
89#117×C47D412yes—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—21no—
92C2×C22⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C42×C47C2×C412yes—
95C2×C42×C47D412yes—
96C24×C47C2×D411yescharacteristic
97C23×D47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47C22×C411yescharacteristic
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C22×C411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47#4111yescharacteristic
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—22no—
123C22×Dic47—22no—
124C22×Dic47—82no—
125C22×Dic47C22⋊C414yes—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—22no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141#10625350—41no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188C4212yescomplemented
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×C4×D47—41no—
153C2×C4×D47—41no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94C4212yescomplemented
160M16×C47—22no—
161M16×C47—22no—
162M16×C47—41no—
163M16×C47—41no—
164C2×C8×C47—22no—
165C2×C8×C47—41no—
166C2×C8×C47—41no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47—41no—
171C22⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—41no—
174C22⋊C4×C47—41no—
175C42×C47—22no—
176C42×C47—22no—
177C42×C47—22no—
178C42×C47C22⋊C414yescomplemented
179C22⋊Dic47—22no—
180C22⋊Dic47—22no—
181(C2×D47)⋊C4—84no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187C4⋊Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—84no—
190C4×Dic47C22⋊C414yescomplemented
191#10625329—22no—
192#10625329—22no—
193#10625329—41no—
194#10625329—41no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—41no—
197C2×C47⋊C8—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47C2×C22⋊C411yescharacteristic
201C23×C47C4×D411yescharacteristic
202C23×C47C4×D411yescharacteristic
203C23×C47#12011yescharacteristic
204C23×C47#12111yescharacteristic
205C23×C47#12111yescharacteristic
206C23×C47#12311yescharacteristic
207C22×D47—22no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—41no—
211C22×D47—41no—
212C2×C4×C47—24no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47C2×C4211yescharacteristic
219C2×C4×C47C2×C22⋊C411yescharacteristic
220C2×C4×C47C2×C22⋊C411yescharacteristic
221C2×C4×C47#11711yescharacteristic
222C2×C4×C47#11711yescharacteristic
223C2×C4×C47#11711yescharacteristic
224C2×C4×C47C4×D411yescharacteristic
225C2×C4×C47C4×D411yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47C4×D411yescharacteristic
230C2×C4×C47C4×D411yescharacteristic
231C2×C4×C47C4×D411yescharacteristic
232C2×C4×C47C4×D411yescharacteristic
233C2×C4×C47C4×D411yescharacteristic
234C2×C4×C47#12111yescharacteristic
235C2×C4×C47#12111yescharacteristic
236C2×C4×C47#12111yescharacteristic
237C2×C4×C47#12111yescharacteristic
238C2×C4×C47#12311yescharacteristic
239C2×C4×C47#12311yescharacteristic
240C2×C4×C47#12411yescharacteristic
241C2×C4×C47#12411yescharacteristic
242C2×C4×C47#12411yescharacteristic
243C2×C4×C47#12611yescharacteristic
244C2×C4×C47#12611yescharacteristic
245C2×C4×C47#12711yescharacteristic
246C22⋊D47—42no—
247C22⋊D47—42no—
248C22⋊D47—42no—
249C22⋊D47—42no—
250C2×Dic47—22no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—42no—
254C2×Dic47—44no—
255C2×Dic47—42no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—44no—
259C2×Dic47—82no—
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—
272C8×C47—42no—
273C47⋊C8—42no—
274#10949—471nomaximal, Sylow
275C22×C47—21no—
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—21no—
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47C4×C22⋊C411yescharacteristic
286C22×C47#38611yescharacteristic
287C22×C47#38611yescharacteristic
288C22×C47#38611yescharacteristic
289C22×C47#38811yescharacteristic
290C22×C47#38811yescharacteristic
291C22×C47#39011yescharacteristic
292C2×D47—44no—
293C2×D47—42no—
294C2×D47—82no—
295C2×D47—81no—
296C2×D47—81no—
297C4×C47—44no—
298C4×C47—82no—
299C4×C47C4×C22⋊C411yescharacteristic
300C4×C47C4×C22⋊C411yescharacteristic
301C4×C47#38811yescharacteristic
302C4×C47#38811yescharacteristic
303C4×C47#38811yescharacteristic
304C4×C47#38811yescharacteristic
305C4×C47#39011yescharacteristic
306C4×C47#39011yescharacteristic
307Dic47—44no—
308Dic47—44no—
309Dic47—42no—
310C2×#421—471no—
311C2×#421—471no—
312#2311—471no—
313#2274—471no—
314C2×#343—471no—
315C2×H(3,ℤ/4)—471no—
316C2×H(3,ℤ/4)D4711yescharacteristic, complemented
317C2×C47—21no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—21no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47#143711yescharacteristic
324C2×C47#191111yescharacteristic
325C2×C47#191211yescharacteristic
326D47—82no—
327C22×#41—471no—
328C22×M16—471no—
329C22×M16C2×D4711yescharacteristic
330C2×#117—471no—
331C2×#117C2×D4711yescharacteristic
332C2×#117C2×D4711yescharacteristic
333C22×C4⋊C4—471no—
334#421—474no—
335#421—474no—
336#421—942no—
337#421—942no—
338C2×#104—942no—
339C2×#104—942no—
340#391—472no—
341#390—472no—
342#386—942no—
343#380—942no—
344C4×C4⋊C4—942no—
345C4×C4⋊C4—942no—
346C4×C22⋊C4—942no—
347#343—942no—
348H(3,ℤ/4)—22no—
349H(3,ℤ/4)—942no—
350C47#1094911yescharacteristic, complemented, Sylow
351C2×#41—472no—
352C2×#41—472no—
353C2×#41—941no—
354C2×#41—941no—
355C2×#41—941no—
356C2×#41—941no—
357C2×#41—941no—
358C2×#41—941no—
359C2×#41—941no—
360C2×#41—941no—
361C2×#41—941no—
362C2×#41—941no—
363C22×Q8—471no—
364C22×D4—471no—
365C22×D4—941no—
366C23×C4—472no—
367C23×C4—941no—
368C23×C4C22×D4711yescharacteristic
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—472no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C2×M16C4×D4712yes—
381C22×C8—21no—
382C22×C8—941no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—472no—
390#117—472no—
391#117—941no—
392#117—941no—
393#117C4×D4712yes—
394#117C22⋊D4712yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—472no—
398C2×C4⋊C4—472no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—944no—
408C2×C42—472no—
409C2×C42—944no—
410C2×C42C4×D4712yes—
411C2×C42D18812yes—
412#104—948no—
413#104—948no—
414#95—944no—
415#95—944no—
416C24—941no—
417C24D4×D4711yescharacteristic
418#41—942no—
419#41—942no—
420#41—942no—
421#41—942no—
422#41—942no—
423#41—942no—
424#41—942no—
425#41—942no—
426#41—1881no—
427#41—1881no—
428#41—1881no—
429#41—1881no—
430#41—1881no—
431#41—1881no—
432#41—1881no—
433#41—1881no—
434C2×Q8—472no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×D4—472no—
441C2×D4—942no—
442C2×D4—942no—
443C2×D4—941no—
444C2×D4—941no—
445C2×D4—941no—
446C2×D4—941no—
447C2×D4—941no—
448C2×D4—1881no—
449C2×D4—1881no—
450C2×D4—1881no—
451C2×D4—1881no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—474no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—3762no—
462C22×C4—1881no—
463C22×C4—1881no—
464C22×C4—1881no—
465C22×C4—1881no—
466C22×C4C2×C4×D4711yescharacteristic
467C22×C4C2×C4×D4711yescharacteristic
468C22×C4C2×C4×D4711yescharacteristic
469C22×C4C2×D18811yescharacteristic
470C22×C4#1062535011yescharacteristic
471C22×C4#1062535011yescharacteristic
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4711yescharacteristic
474C22×C4D4×D4711yescharacteristic
475C22×C4#1062535211yescharacteristic
476C22×C4#1062535211yescharacteristic
477C22×C4#1062535411yescharacteristic
478C22×C4#1062535411yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480M16—22no—
481M16—22no—
482M16—41no—
483M16—41no—
484M16—942no—
485M16—942no—
486M16—1881no—
487M16—1881no—
488C2×C8—22no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—942no—
492C2×C8—1881no—
493C2×C8—1881no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—942no—
501C4⋊C4—942no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—942no—
509C22⋊C4—942no—
510C22⋊C4—3764no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—474no—
515C42—942no—
516C42—3764no—
517C42(C2×D47)⋊C414yescomplemented
518C23—41no—
519C23—41no—
520C23—942no—
521C23—1882no—
522C23—1882no—
523C23—1881no—
524C23—1881no—
525C23C22⋊C4×D4711yescharacteristic
526C23(C22⋊D47)⋊C411yescharacteristic
527C23#1309398611yescharacteristic
528C23#1309399411yescharacteristic
529C23#1309399611yescharacteristic
530C23#1309399711yescharacteristic
531C23#1309400911yescharacteristic
532Q8—942no—
533Q8—942no—
534Q8—1881no—
535Q8—1881no—
536D4—942no—
537D4—942no—
538D4—1882no—
539D4—1882no—
540D4—1882no—
541D4—1882no—
542D4—1881no—
543D4—1881no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—942no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—3762no—
554C2×C4—1882no—
555C2×C4—1884no—
556C2×C4—1882no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1882no—
560C2×C4—1882no—
561C2×C4—1882no—
562C2×C4—1882no—
563C2×C4—1884no—
564C2×C4C42×D4711yescharacteristic
565C2×C4C22⋊C4×D4711yescharacteristic
566C2×C4C2×(C2×D47)⋊C411yescharacteristic
567C2×C4#1309398111yescharacteristic
568C2×C4#1309398211yescharacteristic
569C2×C4#1309398211yescharacteristic
570C2×C4(C22⋊D47)⋊C411yescharacteristic
571C2×C4(C22⋊D47)⋊C411yescharacteristic
572C2×C4(C22⋊D47)⋊C411yescharacteristic
573C2×C4C4×C22⋊D4711yescharacteristic
574C2×C4C4×C22⋊D4711yescharacteristic
575C2×C4#1309398611yescharacteristic
576C2×C4#1309398611yescharacteristic
577C2×C4#1309398611yescharacteristic
578C2×C4C4×D18811yescharacteristic
579C2×C4C4×D18811yescharacteristic
580C2×C4#1309399611yescharacteristic
581C2×C4#1309399711yescharacteristic
582C2×C4#1309399911yescharacteristic
583C2×C4#1309399911yescharacteristic
584C2×C4#1309400811yescharacteristic
585C2×C4#1309400911yescharacteristic
586C2×C4#1309401011yescharacteristic
587C2×C4#1309401011yescharacteristic
588C2×C4#1309401311yescharacteristic
589C2×C4#1309401911yescharacteristic
590C2×C4#1309401911yescharacteristic
591C2×C4#1309402011yescharacteristic
592C8—42no—
593C8—1882no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—3762no—
605C22—3761no—
606C22—3761no—
607C22—1884no—
608C22—1882no—
609C22#1568897011yescharacteristic
610C22#1568900011yescharacteristic
611C22#1568900111yescharacteristic
612C22#1568900111yescharacteristic
613C22#1568901211yescharacteristic
614C22#1568901311yescharacteristic
615C22#1568902311yescharacteristic
616C4—44no—
617C4—82no—
618C4—1884no—
619C4—1884no—
620C4—1882no—
621C4#1568897011yescharacteristic
622C4C4×(C2×D47)⋊C411yescharacteristic
623C4#1568901211yescharacteristic
624C4#1568901311yescharacteristic
625C4#1568901511yescharacteristic
626C4#1568901511yescharacteristic
627C4#1568902311yescharacteristic
628C4#1568902411yescharacteristic
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—3762no—
636C2#2024162411yescharacteristic
637C2#2024391911yescharacteristic
638C2#2024392411yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×H(3,ℤ/4)C2
socleC22×C47#386
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenterC2×H(3,ℤ/4)D47
hypocenterC47#10949
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #20245744C2semidirect
5. C2×#15688891C2semidirect
6. C2×#15688879C2semidirect
7. C2×C47×H(3,ℤ/4)C2semidirect
13. #15689141C4semidirect
15. #15689139C4semidirect
18. #15689029C4semidirect
19. #15689025C4semidirect
147. C2×D188C42semidirect
159. C2×Dic94C42semidirect
178. C42×C47C22⋊C4semidirect
190. C4×Dic47C22⋊C4semidirect
316. C2×H(3,ℤ/4)D47semidirect
350. C47#10949semidirect
517. 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
C2×#156891392
#202459762
#202457442
C2×#156888912
C2×#156888792
C2×C47×H(3,ℤ/4)2
#1094947