Finite Group Databasebeta

← #26217167

Subgroups — #26217167

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
1C2×#15689166C211yesmaximal, characteristic, complemented
2C2×#15689140C211yesmaximal, characteristic
3C2×#15689124C211yesmaximal, characteristic
4#20246008C211yesmaximal, characteristic, complemented
5#20243621C214yesmaximal, complemented
6#20243614C214yesmaximal, complemented
7#20243127C211yesmaximal, characteristic, complemented
8#20243097C211yesmaximal, characteristic, complemented
9C47×#3428C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689166—22no—
15#15689166C2214yescomplemented
16C2×#13093957—21no—
17C2×#13093955—21no—
18#15689140C418yescomplemented
19C2×#13093951C2214yes—
20#15689124—22no—
21#15689124C2214yes—
22C2×#13093948—21no—
23C2×#13093945—21no—
24#15689021—22no—
25#15688995—22no—
26#15688992C2212yescomplemented
27#15688988C2212yescomplemented
28#15688892C2212yescomplemented
29#15688892C2212yescomplemented
30#15688882C2214yescomplemented
31C22×#41×C47C2211yescharacteristic
32C47×#520—21no—
33C47×#520—21no—
34C47×#520C2212yes—
35C47×#520C2212yes—
36C47×#519—21no—
37C47×#518—21no—
38C47×#518—22no—
39C4×#41×C47—22no—
40C4×#41×C47—22no—
41C2×#117×C47—22no—
42C2×#117×C47C2211yescharacteristic
43C2×#117×C47C2211yescharacteristic
44C23×Dic47—21no—
45C2×#10625329—21no—
46C2×#10625329—22no—
47C2×#10625329—21no—
48C2×#10625329—22no—
49C2×#10625329—22no—
50C2×#10625329C2×C414yescomplemented
51C2×#10625329C2312yes—
52C2×#10625329C2312yes—
53C22×C47⋊C8—21no—
54C22×C47⋊C8—21no—
55#13093983—22no—
56#13093983C2×C414yescomplemented
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47C2312yes—
64C2×Dic47⋊C4—24no—
65C2×C22⋊Dic47—21no—
66C2×C4×Dic47C2312yes—
67#13093957—24no—
68#13093955—24no—
69#13093951C2×C4116yescomplemented
70#13093948—24no—
71#13093945—24no—
72C47⋊#95—24no—
73C2×#41×C47—21no—
74C2×#41×C47—21no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47C2×C414yes—
78C2×#41×C47D412yes—
79C2×#41×C47Q814yes—
80C2×#41×C47C2312yes—
81C22×Q8×C47D411yescharacteristic
82C22×D4×C47—21no—
83C22×D4×C47D411yescharacteristic
84C23×C4×C47—21no—
85C23×C4×C47C2311yescharacteristic
86C4×Q8×C47—22no—
87C4×Q8×C47—22no—
88C4×Q8×C47—22no—
89C4×Q8×C47—22no—
90C4×D4×C47—24no—
91C4×D4×C47—24no—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95C4×D4×C47—22no—
96C4×D4×C47—24no—
97C4×D4×C47—24no—
98#117×C47—24no—
99#117×C47—24no—
100#117×C47—21no—
101#117×C47—21no—
102#117×C47—21no—
103#117×C47—21no—
104#117×C47—21no—
105#117×C47—21no—
106#117×C47—22no—
107#117×C47—22no—
108#117×C47D412yes—
109#117×C47C2312yes—
110C2×C4⋊C4×C47—22no—
111C2×C4⋊C4×C47—22no—
112C2×C4⋊C4×C47D411yescharacteristic
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C22⋊C4×C47—24no—
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47D412yes—
119C2×C42×C47—24no—
120C2×C42×C47—21no—
121C2×C42×C47D412yes—
122C24×C47—21no—
123C24×C47C2×D411yescharacteristic
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—24no—
129#41×C47—22no—
130#41×C47—22no—
131#41×C47C4⋊C418yescomplemented
132#41×C47C22×C414yes—
133#41×C47C2×Q814yes—
134C2×Q8×C47—21no—
135C2×Q8×C47—22no—
136C2×Q8×C47C4⋊C414yescomplemented
137C2×Q8×C47C2×D412yes—
138C2×D4×C47—24no—
139C2×D4×C47—22no—
140C2×D4×C47—24no—
141C2×D4×C47—22no—
142C2×D4×C47—21no—
143C2×D4×C47—22no—
144C2×D4×C47C4⋊C414yescomplemented
145C2×D4×C47C2×D412yes—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—24no—
150C22×C4×C47—22no—
151C22×C4×C47—24no—
152C22×C4×C47—24no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47C22×C412yes—
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×Q812yes—
164C22×C4×C47#4112yes—
165C22×C4×C47#4112yes—
166C22×C4×C47C2411yescharacteristic
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—41no—
170C22×Dic47—41no—
171C22×Dic47—82no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47C2×D412yes—
181C4⋊C4×C47C2×D412yes—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—24no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47—22no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—22no—
188C42×C47—24no—
189C42×C47—24no—
190C42×C47—22no—
191C42×C47—22no—
192C42×C47—22no—
193C42×C47—22no—
194C22⋊Dic47—24no—
195C4⋊Dic47—24no—
196C4⋊Dic47—22no—
197C4⋊Dic47—22no—
198C4⋊Dic47—41no—
199C4⋊Dic47—41no—
200Dic47⋊C4—84no—
201C4×Dic47—22no—
202C4×Dic47C22×C414yes—
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—22no—
214C23×C47—24no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—81no—
218C23×C47—81no—
219C23×C47C4×D412yes—
220C23×C47#12011yescharacteristic
221C23×C47#12011yescharacteristic
222C23×C47C22⋊Q812yes—
223C23×C47C22×D411yescharacteristic
224Q8×C47—22no—
225Q8×C47—24no—
226Q8×C47C2×C4⋊C414yes—
227D4×C47—24no—
228D4×C47—24no—
229D4×C47—28no—
230D4×C47—22no—
231D4×C47—24no—
232D4×C47C2×C4⋊C414yes—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—28no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—81no—
245C2×C4×C47—81no—
246C2×C4×C47—84no—
247C2×C4×C47—84no—
248C2×C4×C47C2×C4⋊C412yes—
249C2×C4×C47C2×C4⋊C412yes—
250C2×C4×C47C4×D412yes—
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47#12011yescharacteristic
254C2×C4×C47#12011yescharacteristic
255C2×C4×C47C22⋊Q812yes—
256C2×C4×C47C22⋊Q812yes—
257C2×C4×C47C22⋊Q812yes—
258C2×C4×C47#12312yes—
259C2×C4×C47#12312yes—
260C2×C4×C47C23×C411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×Q811yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×Dic47—24no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—44no—
270C2×Dic47—42no—
271C2×Dic47—82no—
272C2×Dic47—82no—
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—24no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—84no—
288C22×C47#38512yes—
289C22×C47#38512yes—
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×#12011yescharacteristic
292C22×C47C2×C22⋊Q811yescharacteristic
293C4×C47—24no—
294C4×C47—42no—
295C4×C47—42no—
296C4×C47—42no—
297C4×C47—42no—
298C4×C47#38512yes—
299C4×C47#38512yes—
300C4×C47C22×C4⋊C411yescharacteristic
301C4×C47C2×C4×D411yescharacteristic
302C4×C47C2×C22⋊Q811yescharacteristic
303C4×C47C2×#12311yescharacteristic
304Dic47—44no—
305Dic47—82no—
306#3428D4711yescharacteristic
307C2×#428—471no—
308C2×#421—471no—
309C2×#417—471no—
310#2317—471no—
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×C47—44no—
320C2×C47C2×#38511yescharacteristic
321C2×C47#302111yescharacteristic
322C2×C47#302211yescharacteristic
323C22×#41C2×D4711yescharacteristic
324C22×M16—471no—
325C22×M16—471no—
326#520—21no—
327#520—21no—
328#520C2×D4712yes—
329#520C2×D4712yes—
330#519—21no—
331#518—21no—
332#518—22no—
333C4×#41—22no—
334C4×#41—22no—
335C2×#117—22no—
336C2×#117—471no—
337C2×#117C2×D4711yescharacteristic
338C2×#117C2×D4711yescharacteristic
339C22×C4⋊C4—471no—
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—942no—
351#385—942no—
352#384—472no—
353#382—472no—
354#343—472no—
355#343—472no—
356#339—474no—
357C47#2086511yescharacteristic, complemented, Sylow
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41Dic9414yes—
363C2×#41C4×D4714yes—
364C2×#41C22⋊D4712yes—
365C2×#41C22×D4712yes—
366C22×Q8C22⋊D4711yescharacteristic
367C22×D4—21no—
368C22×D4C22⋊D4711yescharacteristic
369C23×C4—21no—
370C23×C4—941no—
371C23×C4C22×D4711yescharacteristic
372C2×M16—474no—
373C2×M16—472no—
374C2×M16—472no—
375C2×M16—942no—
376C2×M16—941no—
377C2×M16—941no—
378C2×M16—942no—
379C2×M16—942no—
380C22×C8—941no—
381C22×C8—941no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×D4—24no—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—24no—
393C4×D4—24no—
394#117—24no—
395#117—24no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117—474no—
405#117—942no—
406#117C22⋊D4712yes—
407#117C22×D4712yes—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—472no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—944no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4C22⋊D4711yescharacteristic
419C2×C4⋊C4C22⋊D4711yescharacteristic
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C4⋊C4C22×D4711yescharacteristic
422C2×C22⋊C4—24no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4C22⋊D4712yes—
426C2×C42—24no—
427C2×C42—21no—
428C2×C42—472no—
429C2×C42C22⋊D4712yes—
430#107—944no—
431#106—944no—
432#104—4716no—
433#103—944no—
434#102—944no—
435#95—944no—
436C24—21no—
437C24C2×C22⋊D4711yescharacteristic
438#41—24no—
439#41—24no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—22no—
444#41—22no—
445#41Dic47⋊C418yescomplemented
446#41C2×Dic9414yes—
447#41C2×C4×D4714yes—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8Dic47⋊C414yescomplemented
451C2×Q8C2×C22⋊D4712yes—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4Dic47⋊C414yescomplemented
459C2×D4C2×C22⋊D4712yes—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—3762no—
472C22×C4—1881no—
473C22×C4—1881no—
474C22×C4C2×Dic9412yes—
475C22×C4C2×C4×D4712yes—
476C22×C4#1062535012yes—
477C22×C4#1062535012yes—
478C22×C4C2×C22⋊D4712yes—
479C22×C4C2×C22⋊D4712yes—
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
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—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—944no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—3764no—
508C4⋊C4—1881no—
509C4⋊C4—1881no—
510C4⋊C4C2×C22⋊D4712yes—
511C4⋊C4C2×C22⋊D4712yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—944no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—474no—
526C42—942no—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23C4×C22⋊D4712yes—
534C23#1309399311yescharacteristic
535C23#1309399311yescharacteristic
536C23#1309400312yes—
537C23C22×C22⋊D4711yescharacteristic
538Q8—24no—
539Q8—22no—
540Q8C2×Dic47⋊C414yes—
541D4—28no—
542D4—24no—
543D4—24no—
544D4—24no—
545D4—22no—
546D4C2×Dic47⋊C414yes—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—944no—
563C2×C4—3762no—
564C2×C4—3762no—
565C2×C4—1884no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4C2×Dic47⋊C412yes—
570C2×C4C2×Dic47⋊C412yes—
571C2×C4C4×C22⋊D4712yes—
572C2×C4C4×C22⋊D4712yes—
573C2×C4C4×C22⋊D4712yes—
574C2×C4#1309399311yescharacteristic
575C2×C4#1309399311yescharacteristic
576C2×C4#1309400312yes—
577C2×C4#1309400312yes—
578C2×C4#1309400312yes—
579C2×C4#1309400712yes—
580C2×C4#1309400712yes—
581C2×C4C22×C4×D4711yescharacteristic
582C2×C4C22×C22⋊D4711yescharacteristic
583C2×C4C22×C22⋊D4711yescharacteristic
584C2×C4C22×Dic9411yescharacteristic
585C2×C4C2×#1062535011yescharacteristic
586C2×C4C2×#1062535011yescharacteristic
587C8—944no—
588C8—1882no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1568899612yes—
602C22#1568899612yes—
603C22C2×C4×C22⋊D4711yescharacteristic
604C22C2×#1309399311yescharacteristic
605C22C2×#1309400311yescharacteristic
606C4—24no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—3762no—
612C4—1884no—
613C4#1568899612yes—
614C4#1568899612yes—
615C4C22×Dic47⋊C411yescharacteristic
616C4C2×C4×C22⋊D4711yescharacteristic
617C4C2×#1309400311yescharacteristic
618C4C2×#1309400711yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—44no—
624C2C2×#1568899611yescharacteristic
625C2#2025276011yescharacteristic
626C2#2025276711yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#3428C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenter#3428D47
hypocenterC47#20865
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
4. #20246008C2semidirect
5. #20243621C2semidirect
6. #20243614C2semidirect
7. #20243127C2semidirect
8. #20243097C2semidirect
15. #15689166C22semidirect
18. #15689140C4semidirect
26. #15688992C22semidirect
27. #15688988C22semidirect
28. #15688892C22semidirect
29. #15688892C22semidirect
30. #15688882C22semidirect
50. C2×#10625329C2×C4semidirect
56. #13093983C2×C4semidirect
69. #13093951C2×C4semidirect
131. #41×C47C4⋊C4semidirect
136. C2×Q8×C47C4⋊C4semidirect
144. C2×D4×C47C4⋊C4semidirect
357. C47#20865semidirect
445. #41Dic47⋊C4semidirect
450. C2×Q8Dic47⋊C4semidirect
458. C2×D4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156891662
C2×#156891402
C2×#156891242
#202460082
#202436212
#202436142
#202431272
#202430972
C47×#34282
#2086547