Finite Group Databasebeta

← #26217166

Subgroups — #26217166

Order 12032.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256589C211yesmaximal, characteristic
2C2×#15689163C211yesmaximal, characteristic, complemented
3C2×#15689141C211yesmaximal, characteristic
4C2×#15689122C211yesmaximal, characteristic
5#20243620C214yesmaximal, complemented
6#20243613C214yesmaximal, complemented
7#20243127C211yesmaximal, characteristic, complemented
8#20243098C211yesmaximal, characteristic, complemented
9C47×#2317C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13#15689604C2212yes—
14#15689604C2212yes—
15#15689603—21no—
16#15689603—21no—
17#15689600—21no—
18#15689597—21no—
19#15689596—22no—
20C4×#10625350—22no—
21#15689591—22no—
22C2×#13093983C2211yescharacteristic
23#117×D47—22no—
24#15689163—22no—
25#15689163C2214yescomplemented
26C2×#13093957—21no—
27C2×#13093955—21no—
28#15689141C418yescomplemented
29C2×#13093952C2214yes—
30#15689122—22no—
31#15689122C2214yes—
32C2×#13093947—21no—
33C2×#13093944—21no—
34#15688892C2212yescomplemented
35#15688892C2212yescomplemented
36#15688883C2214yescomplemented
37C2×#117×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2211yescharacteristic
40C47×#389—22no—
41C47×#385—22no—
42C47×#384C2212yescomplemented
43C47×#382C2212yescomplemented
44C2×#10625350—21no—
45C2×#10625350—21no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350C2×C414yes—
49C2×#10625350D412yes—
50C2×#10625350Q814yes—
51C2×#10625350C2312yes—
52C22×Dic94D411yescharacteristic
53C22×D188D411yescharacteristic
54C22×C22⋊D47—21no—
55C22×C4×D47—21no—
56C2×#10625329—21no—
57C2×#10625329—22no—
58C2×#10625329—21no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329C2×C414yescomplemented
62C2×#10625329C2312yes—
63C2×#10625329C2312yes—
64C22×C47⋊C8—21no—
65C22×C47⋊C8—21no—
66C4×Dic94—22no—
67C4×Dic94—22no—
68#13093989—22no—
69#13093989—22no—
70C4×D188—22no—
71C4×D188—22no—
72#13093986—22no—
73#13093986—22no—
74C4×C22⋊D47—24no—
75C4×C22⋊D47—24no—
76(C22⋊D47)⋊C4—24no—
77(C22⋊D47)⋊C4—24no—
78#13093983—21no—
79#13093983—21no—
80#13093983D412yes—
81#13093983C2312yes—
82#13093982—22no—
83#13093982—22no—
84#13093981—24no—
85#13093980—24no—
86C2×C4⋊Dic47D411yescharacteristic
87C2×C4⋊Dic47D411yescharacteristic
88C4⋊C4×D47—22no—
89C4⋊C4×D47—22no—
90C2×C22⋊Dic47—21no—
91C22⋊C4×D47—24no—
92C2×C4×Dic47—21no—
93C42×D47—24no—
94#13093957—24no—
95#13093955—24no—
96#13093952C2×C4116yescomplemented
97#13093947—24no—
98#13093944—24no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101#117×C47—21no—
102#117×C47—21no—
103#117×C47—21no—
104#117×C47—21no—
105#117×C47—22no—
106#117×C47C2×C414yescomplemented
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—24no—
114C2×C4⋊C4×C47C2312yes—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47D412yes—
119C2×C42×C47D412yes—
120C2×C42×C47C2312yes—
121#95×C47—24no—
122C24×C47C2×D411yescharacteristic
123C23×D47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—41no—
127C22×C4×C47—41no—
128C22×C4×C47—82no—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×Q812yes—
138C22×C4×C47#4112yes—
139C22×C4×C47#4112yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—24no—
142C2×C22⋊D47—22no—
143C2×C22⋊D47—24no—
144C2×C22⋊D47—22no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148#10625350—24no—
149#10625350—24no—
150#10625350—24no—
151#10625350—24no—
152#10625350—24no—
153#10625350—22no—
154#10625350—22no—
155#10625350C4⋊C418yescomplemented
156#10625350C22×C414yes—
157#10625350C2×Q814yes—
158C2×D188—21no—
159C2×D188—22no—
160C2×D188C4⋊C414yescomplemented
161C2×D188C2×D412yes—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—24no—
165C2×C4×D47—22no—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×Dic94—21no—
169C2×Dic94—22no—
170C2×Dic94C4⋊C414yescomplemented
171C2×Dic94C2×D412yes—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—41no—
180C4⋊C4×C47—41no—
181C4⋊C4×C47—84no—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—22no—
185C42×C47—22no—
186C42×C47—22no—
187C42×C47—22no—
188C42×C47C22×C414yes—
189C22⋊Dic47—22no—
190C22⋊Dic47—22no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C2×D412yes—
196C4⋊Dic47C2×D412yes—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199C4×Dic47—24no—
200C4×Dic47—24no—
201C4×Dic47—22no—
202C4×Dic47—22no—
203#10625329—24no—
204#10625329—22no—
205#10625329—22no—
206#10625329—22no—
207#10625329—42no—
208#10625329C22×C414yes—
209C2×C47⋊C8—24no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—42no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C4×D412yes—
216C23×C47#12011yescharacteristic
217C23×C47#12011yescharacteristic
218C23×C47C22⋊Q812yes—
219C23×C47C22×D411yescharacteristic
220C22×D47—22no—
221C22×D47—24no—
222C22×D47—81no—
223C22×D47—81no—
224C2×C4×C47—24no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—44no—
232C2×C4×C47—42no—
233C2×C4×C47—82no—
234C2×C4×C47—82no—
235C2×C4×C47C2×C4⋊C412yes—
236C2×C4×C47C2×C4⋊C412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47C4×D412yes—
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47#12011yescharacteristic
242C2×C4×C47C22⋊Q812yes—
243C2×C4×C47C22⋊Q812yes—
244C2×C4×C47C22⋊Q812yes—
245C2×C4×C47#12312yes—
246C2×C4×C47#12312yes—
247C2×C4×C47C23×C411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C22×Q811yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C22⋊D47—24no—
254C22⋊D47—24no—
255C22⋊D47—28no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—81no—
261C2×Dic47—81no—
262D188—22no—
263D188—24no—
264D188C2×C4⋊C414yes—
265C4×D47—24no—
266C4×D47—24no—
267C4×D47—28no—
268C4×D47—84no—
269C4×D47—84no—
270Dic94—22no—
271Dic94—24no—
272Dic94C2×C4⋊C414yes—
273C47⋊C8—24no—
274C47⋊C8—42no—
275#20865—471nomaximal, Sylow
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—21no—
279C22×C47—21no—
280C22×C47—21no—
281C22×C47—21no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47#38512yes—
285C22×C47#38512yes—
286C22×C47C2×C4×D411yescharacteristic
287C22×C47C2×#12011yescharacteristic
288C22×C47C2×C22⋊Q811yescharacteristic
289C2×D47—24no—
290C2×D47—42no—
291C2×D47—42no—
292C2×D47—84no—
293C4×C47—44no—
294C4×C47—42no—
295C4×C47—42no—
296C4×C47—82no—
297C4×C47#38512yes—
298C4×C47#38512yes—
299C4×C47C22×C4⋊C411yescharacteristic
300C4×C47C2×C4×D411yescharacteristic
301C4×C47C2×C22⋊Q811yescharacteristic
302C4×C47C2×#12311yescharacteristic
303Dic47—24no—
304Dic47—42no—
305Dic47—42no—
306#3428—471no—
307C2×#428—471no—
308C2×#421—471no—
309C2×#417—471no—
310#2317D4711yescharacteristic, complemented
311#1860—474no—
312#1859—474no—
313#1742—471no—
314#1733—471no—
315C2×C47—21no—
316C2×C47—22no—
317C2×C47—21no—
318C2×C47—22no—
319C2×C47C2×#38511yescharacteristic
320C2×C47#302111yescharacteristic
321C2×C47#302211yescharacteristic
322D47—44no—
323C22×#41—471no—
324C22×M16—471no—
325C22×M16—471no—
326#520—472no—
327#520—472no—
328#520—941no—
329#520—941no—
330#519—941no—
331#518—941no—
332#518—942no—
333C4×#41—942no—
334C4×#41—942no—
335C2×#117—471no—
336C2×#117—942no—
337C2×#117C2×D4711yescharacteristic
338C2×#117C2×D4711yescharacteristic
339C22×C4⋊C4C2×D4711yescharacteristic
340#428—474no—
341#428—942no—
342C2×#107—941no—
343C2×#106—941no—
344#421—478no—
345C2×#104—474no—
346#417—474no—
347#417—942no—
348C2×#103—941no—
349C2×#102—941no—
350#389—22no—
351#385—22no—
352#384C2×D4712yescomplemented
353#382C2×D4712yescomplemented
354#343—472no—
355#343—472no—
356#339—474no—
357C47#2086511yescharacteristic, complemented, Sylow
358C2×#41—474no—
359C2×#41—472no—
360C2×#41—474no—
361C2×#41—472no—
362C2×#41—942no—
363C2×#41—941no—
364C2×#41—942no—
365C2×#41—941no—
366C22×Q8—471no—
367C22×D4—471no—
368C22×D4—941no—
369C23×C4—21no—
370C23×C4—941no—
371C23×C4C22×D4711yescharacteristic
372C2×M16—474no—
373C2×M16—472no—
374C2×M16—472no—
375C2×M16—941no—
376C2×M16—942no—
377C2×M16—941no—
378C2×M16—942no—
379C2×M16—942no—
380C22×C8—941no—
381C22×C8—941no—
382C4×Q8—942no—
383C4×Q8—942no—
384C4×Q8—942no—
385C4×Q8—942no—
386C4×D4—942no—
387C4×D4—942no—
388C4×D4—944no—
389C4×D4—944no—
390C4×D4—942no—
391C4×D4—942no—
392C4×D4—944no—
393C4×D4—944no—
394#117—22no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—472no—
400#117—472no—
401#117—944no—
402#117—944no—
403#117—942no—
404#117—942no—
405#117—941no—
406#117—941no—
407#117C4×D4714yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—24no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4C22×D4712yes—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4C22⋊D4712yes—
426C2×C42—944no—
427C2×C42—941no—
428C2×C42C22⋊D4712yes—
429C2×C42C22×D4712yes—
430#107—944no—
431#106—944no—
432#104—4716no—
433#103—944no—
434#102—944no—
435#95—24no—
436C24—941no—
437C24C2×C22⋊D4711yescharacteristic
438#41—478no—
439#41—474no—
440#41—474no—
441#41—944no—
442#41—944no—
443#41—944no—
444#41—944no—
445#41—944no—
446#41—942no—
447#41—942no—
448C2×Q8—474no—
449C2×Q8—472no—
450C2×Q8—941no—
451C2×Q8—942no—
452C2×D4—474no—
453C2×D4—472no—
454C2×D4—944no—
455C2×D4—942no—
456C2×D4—944no—
457C2×D4—942no—
458C2×D4—941no—
459C2×D4—942no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—82no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—941no—
468C22×C4—944no—
469C22×C4—942no—
470C22×C4—944no—
471C22×C4—942no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4C2×C4×D4712yes—
475C22×C4#1062535012yes—
476C22×C4D4×D4711yescharacteristic
477C22×C4D4×D4711yescharacteristic
478C22×C4D4×D4711yescharacteristic
479C22×C4D4×D4711yescharacteristic
480C22×C4Q8×D4712yes—
481C22×C4#1062535412yes—
482C22×C4C2×C22⋊D4712yes—
483C22×C4C2×C22⋊D4712yes—
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486M16—474no—
487M16—944no—
488M16—942no—
489M16—942no—
490M16—942no—
491M16—1882no—
492C2×C8—944no—
493C2×C8—942no—
494C2×C8—942no—
495C2×C8—1882no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—84no—
506C4⋊C4—472no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—944no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—944no—
516C22⋊C4—944no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—944no—
523C42—944no—
524C42—942no—
525C42—942no—
526C42C2×C4×D4714yes—
527C23—41no—
528C23—41no—
529C23—944no—
530C23—942no—
531C23—3761no—
532C23—3761no—
533C23C4×C22⋊D4712yes—
534C23#1309399211yescharacteristic
535C23#1309399211yescharacteristic
536C23#1309400012yes—
537C23C22×C22⋊D4711yescharacteristic
538Q8—474no—
539Q8—944no—
540Q8—942no—
541D4—474no—
542D4—948no—
543D4—944no—
544D4—944no—
545D4—944no—
546D4—942no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—944no—
559C2×C4—942no—
560C2×C4—948no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—3764no—
566C2×C4—3764no—
567C2×C4—3761no—
568C2×C4—3761no—
569C2×C4C4⋊C4×D4712yes—
570C2×C4C4⋊C4×D4712yes—
571C2×C4C4×C22⋊D4712yes—
572C2×C4#1309398612yes—
573C2×C4#1309398612yes—
574C2×C4#1309399211yescharacteristic
575C2×C4#1309399211yescharacteristic
576C2×C4#1309400012yes—
577C2×C4#1309400212yes—
578C2×C4#1309400212yes—
579C2×C4#1309400912yes—
580C2×C4#1309400912yes—
581C2×C4C22×C4×D4711yescharacteristic
582C2×C4C2×D4×D4711yescharacteristic
583C2×C4C2×D4×D4711yescharacteristic
584C2×C4C2×Q8×D4711yescharacteristic
585C2×C4C2×#1062535011yescharacteristic
586C2×C4C2×#1062535411yescharacteristic
587C8—944no—
588C8—1882no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—944no—
598C22—3764no—
599C22—1882no—
600C22—1882no—
601C22#1568899412yes—
602C22#1568899412yes—
603C22C2×C4×C22⋊D4711yescharacteristic
604C22C2×#1309399211yescharacteristic
605C22C2×#1309400011yescharacteristic
606C4—44no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—944no—
611C4—1882no—
612C4—1882no—
613C4#1568899412yes—
614C4#1568899412yes—
615C4C2×C4⋊C4×D4711yescharacteristic
616C4C2×#1309398611yescharacteristic
617C4C2×#1309400211yescharacteristic
618C4C2×#1309400911yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—1884no—
624C2C2×#1568899411yescharacteristic
625C2#2025275911yescharacteristic
626C2#2025276611yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2317C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#2317D47
hypocenterC47#20865
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#15689163C2semidirect
5. #20243620C2semidirect
6. #20243613C2semidirect
7. #20243127C2semidirect
8. #20243098C2semidirect
9. C47×#2317C2semidirect
25. #15689163C22semidirect
28. #15689141C4semidirect
34. #15688892C22semidirect
35. #15688892C22semidirect
36. #15688883C22semidirect
42. C47×#384C22semidirect
43. C47×#382C22semidirect
61. C2×#10625329C2×C4semidirect
96. #13093952C2×C4semidirect
106. #117×C47C2×C4semidirect
155. #10625350C4⋊C4semidirect
160. C2×D188C4⋊C4semidirect
170. C2×Dic94C4⋊C4semidirect
310. #2317D47semidirect
352. #384C2×D47semidirect
353. #382C2×D47semidirect
357. C47#20865semidirect
407. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565892
C2×#156891632
C2×#156891412
C2×#156891222
#202436202
#202436132
#202431272
#202430982
C47×#23172
#2086547