Finite Group Databasebeta

← #26217505

Subgroups — #26217505

Order 12032.

2258 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×#15689635C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3C2×#15689125C211yesmaximal, characteristic, complemented
4C2×#15689122C211yesmaximal, characteristic, complemented
5C2×#15689086C211yesmaximal, characteristic, complemented
6#20243704C214yesmaximal, complemented
7#20243694C214yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C23×C4×D47—21no—
12C23×C4×D47C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689635—24no—
15#15689635C418yescomplemented
16C2×#13094008—21no—
17C2×#13094004C2212yes—
18C2×#13093999C2212yes—
19C2×#13093994—21no—
20C2×C4×D188—22no—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×Dic47⋊C4—21no—
23C2×C4⋊C4×D47—22no—
24C2×C4⋊C4×D47—24no—
25C2×C4⋊C4×D47—24no—
26C2×C4⋊C4×D47C2214yes—
27#15689125—24no—
28#15689122—24no—
29C2×#13093949C2212yescomplemented
30C2×#13093947C2212yescomplemented
31C2×#13093946C2212yescomplemented
32C2×#13093944C2212yescomplemented
33#15689086—24no—
34C2×(C2×D47)⋊C8C2214yes—
35#15688885C2214yescomplemented
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C22×C4⋊C4×C47C2211yescharacteristic
39C47×#340C2214yescomplemented
40C24×D47D411yescharacteristic
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350C2×C414yescomplemented
47C22×Dic94C2311yescharacteristic
48C22×D188—22no—
49C22×D188C2311yescharacteristic
50C22×C22⋊D47—21no—
51C23×Dic47—21no—
52C23×Dic47D411yescharacteristic
53C22×C4×D47—22no—
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47—22no—
57C22×C4×D47—28no—
58C22×C4×D47—24no—
59C22×C4×D47—21no—
60C22×C4×D47—48no—
61C22×C4×D47C2×C414yes—
62C22×C4×D47D412yes—
63C22×C4×D47D414yes—
64C22×C4×D47C2312yes—
65C2×#10625329—22no—
66C2×#10625329—22no—
67C2×#10625329—22no—
68C22×C47⋊C8C2312yes—
69#13094008—28no—
70#13094004—24no—
71#13094004C2×C418yescomplemented
72#13093999—24no—
73#13093999C2×C418yescomplemented
74#13093994—28no—
75C4×D188—216no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—22no—
78C2×C4⋊Dic47—22no—
79C2×C4⋊Dic47D414yescomplemented
80C2×C4⋊Dic47C2312yes—
81C2×Dic47⋊C4—28no—
82C2×Dic47⋊C4—24no—
83C4⋊C4×D47—216no—
84C4⋊C4×D47—28no—
85C4⋊C4×D47—28no—
86C2×(C2×D47)⋊C4—22no—
87C2×(C2×D47)⋊C4—22no—
88#13093949—24no—
89#13093947—24no—
90#13093946—24no—
91#13093944—24no—
92(C2×D47)⋊C8—28no—
93C23×C4×C47—21no—
94C23×C4×C47C2311yescharacteristic
95C2×M16×C47—22no—
96C2×M16×C47—22no—
97C2×M16×C47—22no—
98C22×C8×C47C2312yes—
99#117×C47—22no—
100#117×C47C2×C414yescomplemented
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—22no—
103C2×C4⋊C4×C47D414yescomplemented
104C2×C4⋊C4×C47C2312yes—
105C2×C4⋊C4×C47C2312yes—
106C2×C22⋊C4×C47—21no—
107C2×C42×C47—21no—
108C24×C47C2×D411yescharacteristic
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—24no—
112C23×D47—24no—
113C23×D47—21no—
114C23×D47C22⋊C414yes—
115C23×D47C2×D412yes—
116C22×C4×C47—24no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—44no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47#4112yes—
126C22×C4×C47#4112yes—
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—42no—
130C2×C22⋊D47—42no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—24no—
134C22×Dic47—21no—
135C22×Dic47—22no—
136C22×Dic47—44no—
137C22×Dic47C22⋊C414yes—
138C22×Dic47C2×D412yes—
139#10625350—28no—
140#10625350—28no—
141#10625350—44no—
142#10625350—44no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188—28no—
148C2×D188—21no—
149C2×D188—44no—
150C2×D188—44no—
151C2×D188C22×C412yescomplemented
152C2×C4×D47—28no—
153C2×C4×D47—28no—
154C2×C4×D47—24no—
155C2×C4×D47—28no—
156C2×C4×D47—28no—
157C2×C4×D47—24no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—216no—
163C2×C4×D47—24no—
164C2×C4×D47—432no—
165C2×C4×D47—48no—
166C2×C4×D47—48no—
167C2×C4×D47—42no—
168C2×C4×D47—42no—
169C2×C4×D47C22⋊C418yes—
170C2×C4×D47C22×C414yes—
171C2×C4×D47C2×D414yes—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94—21no—
177C2×Dic94C22×C412yescomplemented
178M16×C47—44no—
179C2×C8×C47—22no—
180C2×C8×C47—22no—
181C2×C8×C47—22no—
182C4⋊C4×C47—24no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47—22no—
185C4⋊C4×C47C22×C414yescomplemented
186C4⋊C4×C47C2×D414yes—
187C22⋊C4×C47—24no—
188C42×C47—24no—
189(C2×D47)⋊C4—28no—
190(C2×D47)⋊C4—28no—
191C4⋊Dic47—24no—
192C4⋊Dic47—22no—
193C4⋊Dic47—28no—
194C4⋊Dic47C2×D414yes—
195Dic47⋊C4—216no—
196#10625329—44no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47C2×C22⋊C412yes—
203C23×C47#12012yes—
204C23×C47#12312yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—28no—
207C22×D47—216no—
208C22×D47—28no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—28no—
214C22×D47—216no—
215C22×D47—22no—
216C22×D47—41no—
217C22×D47—41no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47C2×C22⋊C414yes—
221C22×D47AGL(1,ℤ/8)14yes—
222C22×D47#13714yes—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—48no—
226C2×C4×C47—44no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47C2×C22⋊C412yes—
230C2×C4×C47C4×D414yes—
231C2×C4×C47C4×D414yes—
232C2×C4×C47#12012yes—
233C2×C4×C47#12114yes—
234C2×C4×C47#12114yes—
235C2×C4×C47#12312yes—
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C22×D411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C22⋊D47—48no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—28no—
246C2×Dic47—22no—
247C2×Dic47—24no—
248C2×Dic47—48no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—44no—
252C2×Dic47C2×C22⋊C414yes—
253D188—22no—
254D188—22no—
255D188—416no—
256D188—41no—
257D188—41no—
258C4×D47—24no—
259C4×D47—24no—
260C4×D47—24no—
261C4×D47—24no—
262C4×D47—216no—
263C4×D47—432no—
264C4×D47—48no—
265Dic94—22no—
266Dic94—22no—
267Dic94—41no—
268Dic94—41no—
269C8×C47—42no—
270C47⋊C8—42no—
271#20909—471nomaximal, Sylow
272C22×C47—24no—
273C22×C47—24no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47#38614yes—
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×#12011yescharacteristic
279C22×C47C2×#12311yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)12yes—
281C22×C47C2×#13712yes—
282C2×D47—232no—
283C2×D47—24no—
284C2×D47—24no—
285C2×D47—24no—
286C2×D47—24no—
287C2×D47—24no—
288C2×D47—24no—
289C2×D47—44no—
290C2×D47—42no—
291C2×D47—82no—
292C2×D47—81no—
293C2×D47—81no—
294C2×D47#41718yescomplemented
295C4×C47—44no—
296C4×C47—44no—
297C4×C47#38614yes—
298C4×C47C2×C4×D411yescharacteristic
299C4×C47C2×C4×D411yescharacteristic
300C4×C47C2×#12111yescharacteristic
301C4×C47C2×#12111yescharacteristic
302Dic47—24no—
303Dic47—44no—
304Dic47—42no—
305C2×#525—471no—
306C23×C4⋊C4—471no—
307C2×#417—471no—
308C2×#417—471no—
309C2×#407—471no—
310#1874—474no—
311#1872—474no—
312#1736—471no—
313#1736D4711yescharacteristic, complemented
314C2×C47—24no—
315C2×C47C2×#38611yescharacteristic
316C2×C47C2×#41712yes—
317C2×C47#300311yescharacteristic
318C2×C47#300411yescharacteristic
319C2×C47#309212yes—
320D47—28no—
321D47—82no—
322C22×#41—471no—
323C24×C4—471no—
324C24×C4—941no—
325C22×M16—471no—
326C22×M16C2×D4711yescharacteristic
327#525—478no—
328#525—944no—
329C2×#123—941no—
330C2×C22⋊Q8—472no—
331C2×#121—472no—
332C2×#120—941no—
333C2×C4×D4—942no—
334C2×#117C2×D4711yescharacteristic
335C22×C4⋊C4—471no—
336C22×C4⋊C4—474no—
337C22×C4⋊C4—944no—
338C22×C4⋊C4—942no—
339C22×C4⋊C4—941no—
340C22×C4⋊C4—944no—
341C22×C4⋊C4C2×D4711yescharacteristic
342#417—944no—
343#417—944no—
344C2×#103—472no—
345C2×#103—472no—
346C2×#102—472no—
347C2×#102—472no—
348#407—944no—
349C2×C22⋊C8—474no—
350#340—474no—
351#340C2×D4714yescomplemented
352C47#2090911yescharacteristic, complemented, Sylow
353C25—471no—
354C2×#41—474no—
355C2×#41—942no—
356C2×#41—942no—
357C2×#41—942no—
358C2×#41—942no—
359C2×#41—942no—
360C22×Q8—471no—
361C22×D4—471no—
362C22×D4—942no—
363C22×D4—941no—
364C23×C4—21no—
365C23×C4—471no—
366C23×C4—474no—
367C23×C4—472no—
368C23×C4—474no—
369C23×C4—472no—
370C23×C4—944no—
371C23×C4—948no—
372C23×C4—942no—
373C23×C4—942no—
374C23×C4—942no—
375C23×C4—942no—
376C23×C4—941no—
377C23×C4—941no—
378C23×C4—1888no—
379C23×C4C22×D4711yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—942no—
384C2×M16—942no—
385C2×M16—942no—
386C22×C8—472no—
387C22×C8C22×D4712yes—
388#123—948no—
389C22⋊Q8—478no—
390C22⋊Q8—944no—
391#121—478no—
392#121—944no—
393#120—948no—
394C4×D4—9416no—
395#117—22no—
396#117C4×D4714yescomplemented
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—474no—
400C2×C4⋊C4—472no—
401C2×C4⋊C4—948no—
402C2×C4⋊C4—9416no—
403C2×C4⋊C4—944no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—948no—
406C2×C4⋊C4—942no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—948no—
409C2×C4⋊C4D18814yescomplemented
410C2×C4⋊C4C22×D4712yes—
411C2×C4⋊C4C22×D4712yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—942no—
414C2×C22⋊C4—942no—
415C2×C42—21no—
416#103—944no—
417#103—944no—
418#102—944no—
419#102—944no—
420C22⋊C8—948no—
421C24—472no—
422C24—474no—
423C24—944no—
424C24—944no—
425C24—942no—
426C24—942no—
427C24—941no—
428C24D4×D4711yescharacteristic
429#41—948no—
430#41—948no—
431#41—1884no—
432#41—1884no—
433C2×Q8—472no—
434C2×Q8—941no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×D4—472no—
440C2×D4—948no—
441C2×D4—944no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—941no—
445C2×D4—941no—
446C2×D4—941no—
447C2×D4—1884no—
448C2×D4—1884no—
449C2×D4—1882no—
450C2×D4—1882no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—474no—
456C22×C4—472no—
457C22×C4—478no—
458C22×C4—474no—
459C22×C4—474no—
460C22×C4—9416no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—948no—
464C22×C4—948no—
465C22×C4—944no—
466C22×C4—948no—
467C22×C4—948no—
468C22×C4—944no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—944no—
474C22×C4—941no—
475C22×C4—942no—
476C22×C4—944no—
477C22×C4—1888no—
478C22×C4—18832no—
479C22×C4—1888no—
480C22×C4—1884no—
481C22×C4—1882no—
482C22×C4—1882no—
483C22×C4C2×C4×D4712yes—
484C22×C4C2×D18812yes—
485C22×C4#1062535012yes—
486C22×C4D4×D4711yescharacteristic
487C22×C4D4×D4712yes—
488C22×C4D4×D4712yes—
489C22×C4#1062535412yes—
490C22×C4C23×D4711yescharacteristic
491M16—44no—
492M16—1884no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—942no—
497C2×C8—942no—
498C2×C8—942no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—474no—
503C4⋊C4—948no—
504C4⋊C4—9416no—
505C4⋊C4—944no—
506C4⋊C4—942no—
507C4⋊C4C2×C4×D4714yescomplemented
508C4⋊C4C2×D18814yes—
509C22⋊C4—24no—
510C22⋊C4—948no—
511C22⋊C4—948no—
512C42—24no—
513C23—22no—
514C23—22no—
515C23—474no—
516C23—474no—
517C23—474no—
518C23—9416no—
519C23—948no—
520C23—9416no—
521C23—948no—
522C23—948no—
523C23—942no—
524C23—942no—
525C23—942no—
526C23—942no—
527C23—942no—
528C23—1881no—
529C23—1881no—
530C23—1882no—
531C23—1882no—
532C23C22⋊C4×D4712yes—
533C23#1309399412yes—
534C23#1309400912yes—
535C23C2×D4×D4711yescharacteristic
536Q8—942no—
537Q8—942no—
538Q8—1881no—
539Q8—1881no—
540D4—942no—
541D4—942no—
542D4—18816no—
543D4—1888no—
544D4—1881no—
545D4—1881no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—474no—
553C2×C4—948no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—9416no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—942no—
562C2×C4—944no—
563C2×C4—18832no—
564C2×C4—1888no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1884no—
568C2×C4—1888no—
569C2×C4C22⋊C4×D4712yes—
570C2×C4#1309398614yes—
571C2×C4C4×D18814yes—
572C2×C4#1309399412yes—
573C2×C4#1309399714yes—
574C2×C4#1309399914yes—
575C2×C4#1309400912yes—
576C2×C4C22×C4×D4711yescharacteristic
577C2×C4C2×D4×D4711yescharacteristic
578C2×C4C2×D4×D4711yescharacteristic
579C2×C4C22×D18811yescharacteristic
580C2×C4C2×#1062535011yescharacteristic
581C2×C4C2×#1062535411yescharacteristic
582C8—42no—
583C8—1882no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—478no—
589C22—9432no—
590C22—944no—
591C22—944no—
592C22—944no—
593C22—944no—
594C22—944no—
595C22—944no—
596C22—3762no—
597C22—3761no—
598C22—3761no—
599C22—1882no—
600C22—1884no—
601C22#1568900114yes—
602C22C2×C22⋊C4×D4711yescharacteristic
603C22C2×#1309399411yescharacteristic
604C22C2×#1309400911yescharacteristic
605C22AGL(1,ℤ/8)×D4712yes—
606C22#137×D4712yes—
607C4—44no—
608C4—44no—
609C4—944no—
610C4—1882no—
611C4—1884no—
612C4#1568900114yes—
613C4C2×#1309398611yescharacteristic
614C4C2×C4×D18811yescharacteristic
615C4C2×#1309399711yescharacteristic
616C4C2×#1309399911yescharacteristic
617C2—24no—
618C2—948no—
619C2—3762no—
620C2C2×#1568900111yescharacteristic
621C2#417×D4712yes—
622C2#2025261711yescharacteristic
623C2#2025262411yescharacteristic
624C2#2025331412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC2×#15689635C2
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20909
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D47C2semidirect
3. C2×#15689125C2semidirect
4. C2×#15689122C2semidirect
5. C2×#15689086C2semidirect
6. #20243704C2semidirect
7. #20243694C2semidirect
8. #20243109C2semidirect
9. C47×#1736C2semidirect
15. #15689635C4semidirect
29. C2×#13093949C22semidirect
30. C2×#13093947C22semidirect
31. C2×#13093946C22semidirect
32. C2×#13093944C22semidirect
35. #15688885C22semidirect
39. C47×#340C22semidirect
46. C2×#10625350C2×C4semidirect
71. #13094004C2×C4semidirect
73. #13093999C2×C4semidirect
79. C2×C4⋊Dic47D4semidirect
100. #117×C47C2×C4semidirect
103. C2×C4⋊C4×C47D4semidirect
151. C2×D188C22×C4semidirect
177. C2×Dic94C22×C4semidirect
185. C4⋊C4×C47C22×C4semidirect
294. C2×D47#417semidirect
313. #1736D47semidirect
351. #340C2×D47semidirect
352. C47#20909semidirect
396. #117C4×D47semidirect
409. C2×C4⋊C4D188semidirect
507. C4⋊C4C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896352
C22×C4⋊C4×D472
C2×#156891252
C2×#156891222
C2×#156890862
#202437042
#202436942
#202431092
C47×#17362
#2090947