Finite Group Databasebeta

← #26218082

Subgroups — #26218082

Order 12032.

2030 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689687C211yesmaximal, characteristic
2C2×#15689597C211yesmaximal, characteristic, complemented
3C2×#15689097C211yesmaximal, characteristic, complemented
4#20245968C211yesmaximal, characteristic, complemented
5#20243838C218yesmaximal, complemented
6#20243120C211yesmaximal, characteristic, complemented
7#20243120C211yesmaximal, characteristic, complemented
8C2×C47×#410C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10#15689687—22no—
11#15689687—22no—
12#15689687—22no—
13#15689687—22no—
14C2×#13094020C412yescomplemented
15C2×#13094013C412yescomplemented
16C2×#13093998C414yescomplemented
17C2×#13093992C2214yes—
18#15689597—24no—
19#15689597—24no—
20C2×C4×D188—22no—
21C2×#13093986—22no—
22C2×#13093983C2211yescharacteristic
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×C22⋊C4×D47C2214yes—
26#15689097—24no—
27C2×#13093939C2214yes—
28#15689025—22no—
29#15689024—22no—
30#15688978C2214yes—
31#15688889C2214yes—
32#15688889C2214yes—
33C22×#41×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#410—24no—
37C2×#99×C47C2214yes—
38C24×D47C2312yes—
39C22×D188—22no—
40C22×D188—22no—
41C22×D188—22no—
42C22×D188—22no—
43C22×D188C2×C412yes—
44C22×D188C2×C412yes—
45C22×D188D412yes—
46C22×D188D412yes—
47C22×D188D412yes—
48C22×D188D412yes—
49C22×C22⋊D47C2×C418yescomplemented
50C22×C4×D47—24no—
51#13094020—24no—
52#13094020—24no—
53#13094013—24no—
54#13094013—24no—
55#13093998—28no—
56#13093998—28no—
57#13093992—28no—
58#13093992—28no—
59C4×D188—216no—
60#13093986—216no—
61#13093983—22no—
62#13093983—21no—
63#13093983—21no—
64#13093983—21no—
65#13093983—21no—
66#13093983—22no—
67#13093983—22no—
68#13093983D414yescomplemented
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2×C412yescomplemented
72C2×C22⋊Dic47C2312yes—
73C2×C22⋊Dic47C2312yes—
74C2×C22⋊Dic47C2312yes—
75C2×(C2×D47)⋊C4—24no—
76C2×(C2×D47)⋊C4—28no—
77C2×(C2×D47)⋊C4C2×C418yescomplemented
78C22⋊C4×D47—28no—
79C22⋊C4×D47—28no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2×C412yescomplemented
83#13093939—28no—
84C2×#41×C47—22no—
85C2×#41×C47—21no—
86C2×#41×C47—21no—
87C2×#41×C47—21no—
88C2×#41×C47—21no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C22×Q8×C47C2×C411yescharacteristic, complemented
93C22×D4×C47C2×C411yescharacteristic, complemented
94C22×D4×C47C2312yes—
95C23×C4×C47C2×C412yescomplemented
96C23×C4×C47C2311yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99#117×C47—22no—
100#117×C47D414yescomplemented
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C22⋊C4×C47C2312yes—
104C2×C22⋊C4×C47C2312yes—
105C2×C42×C47—21no—
106C2×C42×C47—21no—
107#99×C47—28no—
108C24×C47C22×C412yes—
109C24×C47C2411yescharacteristic
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—84no—
115C23×D47C22×C414yes—
116C23×D47C2×D414yes—
117C23×D47C2×D414yes—
118#41×C47—42no—
119#41×C47—42no—
120#41×C47—42no—
121#41×C47—42no—
122#41×C47—42no—
123#41×C47—42no—
124#41×C47—42no—
125#41×C47—42no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—22no—
129C2×Q8×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47—22no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—22no—
139C2×D4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—22no—
146C22×C4×C47C22×C412yes—
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D412yes—
154C22×C4×C47C2×D412yes—
155C22×C4×C47#4112yes—
156C22×C4×C47#4112yes—
157C2×C22⋊D47—216no—
158C2×C22⋊D47—88no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47C22×C414yes—
162C2×D188—24no—
163C2×D188—24no—
164C2×D188—24no—
165C2×D188—24no—
166C2×D188—24no—
167C2×D188—24no—
168C2×D188—24no—
169C2×D188—24no—
170C2×D188—24no—
171C2×D188—24no—
172C2×D188—24no—
173C2×D188—24no—
174C2×D188C22⋊C414yes—
175C2×D188C22⋊C414yes—
176C2×D188C22⋊C414yes—
177C2×D188C22⋊C414yes—
178C2×C4×D47—216no—
179C2×C4×D47—88no—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—42no—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47C2×D414yes—
186C42×C47—24no—
187C42×C47—42no—
188C22⋊Dic47—22no—
189C22⋊Dic47—22no—
190C22⋊Dic47—22no—
191C22⋊Dic47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47C2×D414yes—
194(C2×D47)⋊C4—216no—
195(C2×D47)⋊C4—216no—
196(C2×D47)⋊C4—88no—
197C4⋊Dic47—22no—
198C4⋊Dic47—22no—
199C4⋊Dic47—24no—
200C4⋊Dic47—42no—
201C4×Dic47—22no—
202C4×Dic47—22no—
203C4×Dic47—24no—
204C4×Dic47—42no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—42no—
209C23×C47—44no—
210C23×C47C23×C411yescharacteristic
211C23×C47C22×D411yescharacteristic
212C23×C47C22×D411yescharacteristic
213C23×C47C22×D411yescharacteristic
214C23×C47C22×D411yescharacteristic
215C23×C47C2×#4111yescharacteristic
216C23×C47C2×#4111yescharacteristic
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—24no—
221C22×D47—24no—
222C22×D47—24no—
223C22×D47—48no—
224C22×D47—48no—
225C22×D47—48no—
226C22×D47—84no—
227C22×D47—84no—
228C22×D47—84no—
229C22×D47—84no—
230C22×D47C2×C22⋊C418yes—
231Q8×C47—42no—
232Q8×C47—42no—
233D4×C47—42no—
234D4×C47—42no—
235D4×C47—42no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—42no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—44no—
244C2×C4×C47—44no—
245C2×C4×C47—44no—
246C2×C4×C47—44no—
247C2×C4×C47—42no—
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C2×C22⋊C411yescharacteristic
252C2×C4×C47C4×D412yes—
253C2×C4×C47C4×D412yes—
254C2×C4×C47C4×D412yes—
255C2×C4×C47C4×D412yes—
256C2×C4×C47#12012yes—
257C2×C4×C47#12012yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12112yes—
260C2×C4×C47#12112yes—
261C2×C4×C47#12112yes—
262C2×C4×C47#12312yes—
263C2×C4×C47#12312yes—
264C22⋊D47—816no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—44no—
268C2×Dic47—44no—
269C2×Dic47—44no—
270C2×Dic47—42no—
271D188—28no—
272D188—28no—
273D188—28no—
274D188—28no—
275C4×D47—816no—
276#20985—471nomaximal, Sylow
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—44no—
285C22×C47—42no—
286C22×C47—82no—
287C22×C47—82no—
288C22×C47C22×C22⋊C411yescharacteristic
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×C4×D411yescharacteristic
291C22×C47C2×#12011yescharacteristic
292C22×C47C2×#12111yescharacteristic
293C22×C47C2×#12111yescharacteristic
294C22×C47C2×#12311yescharacteristic
295C2×D47—216no—
296C2×D47—28no—
297C2×D47—48no—
298C2×D47—48no—
299C2×D47—84no—
300C2×D47—84no—
301C2×D47—84no—
302C2×D47—84no—
303C4×C47—44no—
304C4×C47—42no—
305C4×C47—82no—
306C4×C47#38612yes—
307C4×C47#38612yes—
308C4×C47#38612yes—
309C4×C47#38612yes—
310Dic47—44no—
311Dic47—44no—
312Dic47—82no—
313C2×#535—471no—
314C2×#518—471no—
315C2×#410—471no—
316C2×#410D4711yescharacteristic, complemented
317#2309—471no—
318#1898—478no—
319#1740—471no—
320#1740—471no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—21no—
324C2×C47—21no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—82no—
328C2×C47C2×#38611yescharacteristic
329C2×C47#288312yes—
330D47—48no—
331C22×#41C2×D4711yescharacteristic
332C23×D4—471no—
333#535—942no—
334#535—942no—
335#535—942no—
336#535—942no—
337C2×#127—472no—
338C2×#124—472no—
339C2×#121—474no—
340C2×#120—474no—
341#518—944no—
342#518—944no—
343C2×C4×D4—942no—
344C2×C4×D4—942no—
345C2×#117—471no—
346C2×#117—471no—
347C2×#117—471no—
348C2×#117C2×D4711yescharacteristic
349C2×#117C2×D4711yescharacteristic
350C22×C22⋊C4—474no—
351#410—24no—
352#410—944no—
353C2×#99—474no—
354C2×#99C2×D4714yes—
355#390—942no—
356#390—942no—
357#379—474no—
358#342—474no—
359#342—474no—
360C47#2098511yescharacteristic, complemented, Sylow
361C25—472no—
362C2×#41—22no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C22×Q8C4×D4711yescharacteristic, complemented
371C22×D4—478no—
372C22×D4—472no—
373C22×D4—472no—
374C22×D4—472no—
375C22×D4—472no—
376C22×D4—472no—
377C22×D4—472no—
378C22×D4—942no—
379C22×D4—942no—
380C22×D4—942no—
381C22×D4—942no—
382C22×D4C4×D4711yescharacteristic, complemented
383C22×D4C22×D4712yes—
384C23×C4—944no—
385C23×C4C4×D4712yescomplemented
386C23×C4C22×D4711yescharacteristic
387#127—944no—
388#127—944no—
389#124—944no—
390#124—944no—
391#121—948no—
392#121—948no—
393#120—948no—
394#120—948no—
395C4×D4—9416no—
396C4×D4—9416no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—474no—
401#117—941no—
402#117—941no—
403#117—941no—
404#117—941no—
405#117—942no—
406#117—942no—
407#117—942no—
408#117C22⋊D4714yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—472no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4—941no—
414C2×C22⋊C4—472no—
415C2×C22⋊C4—472no—
416C2×C22⋊C4—478no—
417C2×C22⋊C4—472no—
418C2×C22⋊C4—948no—
419C2×C22⋊C4—944no—
420C2×C22⋊C4—948no—
421C2×C22⋊C4—948no—
422C2×C22⋊C4C22×D4712yes—
423C2×C22⋊C4C22×D4712yes—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—472no—
427C2×C42—941no—
428C2×C42—941no—
429#99—28no—
430#99—948no—
431C24—474no—
432C24—474no—
433C24—474no—
434C24—942no—
435C24—942no—
436C24—942no—
437C24—942no—
438C24—3764no—
439C24C2×C4×D4712yes—
440C24C23×D4711yescharacteristic
441#41—42no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—42no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—474no—
464C2×D4—474no—
465C2×D4—474no—
466C2×D4—474no—
467C2×D4—9416no—
468C2×D4—944no—
469C2×D4—944no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—944no—
473C2×D4—944no—
474C2×D4—944no—
475C2×D4—944no—
476C2×D4—944no—
477C2×D4—944no—
478C2×D4—944no—
479C2×D4—944no—
480C2×D4—3768no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—474no—
488C22×C4—9416no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—3768no—
492C22×C4C2×C4×D4712yes—
493C22×C4C2×C4×D4711yescharacteristic
494C22×C4C2×C4×D4711yescharacteristic
495C22×C4#1062535012yes—
496C22×C4D4×D4711yescharacteristic
497C22×C4D4×D4711yescharacteristic
498C22×C4D4×D4711yescharacteristic
499C22×C4D4×D4711yescharacteristic
500C22×C4D4×D4712yes—
501C22×C4#1062535212yes—
502C22×C4C2×C22⋊D4712yes—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—944no—
508C4⋊C4—1882no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—474no—
513C22⋊C4—9416no—
514C22⋊C4—9416no—
515C22⋊C4—942no—
516C22⋊C4—942no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—3768no—
521C22⋊C4C2×C22⋊D4714yes—
522C42—24no—
523C42—42no—
524C42—942no—
525C42—942no—
526C42—944no—
527C42—1882no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—44no—
533C23—478no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—944no—
538C23—944no—
539C23—944no—
540C23—3764no—
541C23—3764no—
542C23—3764no—
543C23—3764no—
544C23—1888no—
545C23—1888no—
546C23—1888no—
547C23C22×C4×D4711yescharacteristic
548C23C2×D4×D4711yescharacteristic
549C23C2×D4×D4711yescharacteristic
550C23C2×D4×D4711yescharacteristic
551C23C22×C22⋊D4711yescharacteristic
552C23C2×#1062535211yescharacteristic
553C23C2×#1062535011yescharacteristic
554Q8—42no—
555Q8—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—948no—
563D4—948no—
564D4—948no—
565D4—948no—
566D4—37616no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—37616no—
579C2×C4—1884no—
580C2×C4—1882no—
581C2×C4—1884no—
582C2×C4—1884no—
583C2×C4C22⋊C4×D4711yescharacteristic
584C2×C4C22⋊C4×D4711yescharacteristic
585C2×C4C22⋊C4×D4711yescharacteristic
586C2×C4C22⋊C4×D4711yescharacteristic
587C2×C4(C22⋊D47)⋊C412yes—
588C2×C4(C22⋊D47)⋊C412yes—
589C2×C4C4×C22⋊D4712yes—
590C2×C4C4×C22⋊D4712yes—
591C2×C4#1309399212yes—
592C2×C4#1309399212yes—
593C2×C4#1309399512yes—
594C2×C4#1309399512yes—
595C2×C4#1309399712yes—
596C2×C4#1309399712yes—
597C2×C4#1309400612yes—
598C2×C4#1309400612yes—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—948no—
611C22—9416no—
612C22—3764no—
613C22—3764no—
614C22—3764no—
615C22—3764no—
616C22—1888no—
617C22—1888no—
618C22C2×C22⋊C4×D4711yescharacteristic
619C22C2×(C22⋊D47)⋊C411yescharacteristic
620C22C2×C4×C22⋊D4711yescharacteristic
621C22C2×#1309399211yescharacteristic
622C22C2×#1309399511yescharacteristic
623C22C2×#1309399711yescharacteristic
624C22C2×#1309400611yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—3762no—
629C4—1884no—
630C4—1884no—
631C4#1568899912yes—
632C4#1568899912yes—
633C4#1568899912yes—
634C4#1568899912yes—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—1888no—
643C2C2×#1568899911yescharacteristic
644C2#2025184712yes—
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC24C23×D47
FittingC2×C47×#410C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#15689687C2
cosocleC24×C47C24
hypercenterC2×#410D47
hypocenterC47#20985
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#15689597C2semidirect
3. C2×#15689097C2semidirect
4. #20245968C2semidirect
5. #20243838C2semidirect
6. #20243120C2semidirect
7. #20243120C2semidirect
8. C2×C47×#410C2semidirect
14. C2×#13094020C4semidirect
15. C2×#13094013C4semidirect
16. C2×#13093998C4semidirect
49. C22×C22⋊D47C2×C4semidirect
68. #13093983D4semidirect
71. C2×C4⋊Dic47C2×C4semidirect
77. C2×(C2×D47)⋊C4C2×C4semidirect
82. C2×C4×Dic47C2×C4semidirect
92. C22×Q8×C47C2×C4semidirect
93. C22×D4×C47C2×C4semidirect
95. C23×C4×C47C2×C4semidirect
100. #117×C47D4semidirect
316. C2×#410D47semidirect
360. C47#20985semidirect
370. C22×Q8C4×D47semidirect
382. C22×D4C4×D47semidirect
385. C23×C4C4×D47semidirect
408. #117C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896872
C2×#156895972
C2×#156890972
#202459682
#202438382
#202431202
C2×C47×#4102
#2098547