Finite Group Databasebeta

← #26121758

Subgroups — #26121758

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689604C211yesmaximal, characteristic
2C2×#15689141C211yesmaximal, characteristic
3C2×#15689139C211yesmaximal, characteristic
4#20243126C211yesmaximal, characteristic, complemented
5#20243120C211yesmaximal, characteristic, complemented
6#20243107C211yesmaximal, characteristic, complemented
7C47×#1736C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10#15689604—22no—
11#15689604—22no—
12#15689604—22no—
13#15689604—22no—
14C2×#13093989C412yescomplemented
15C2×#13093986C412yescomplemented
16C2×C4×C22⋊D47—22no—
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C2×#13093982—21no—
20C2×C4⋊C4×D47—21no—
21#15689141—22no—
22#15689141C414yescomplemented
23#15689139—22no—
24#15689139C414yescomplemented
25C2×#13093952—22no—
26C2×#13093950—22no—
27#15688891—21no—
28#15688891—21no—
29#15688889—22no—
30#15688885—22no—
31C22×M16×C47C2211yescharacteristic
32C2×#117×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C47×#340—22no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350C2×C412yes—
42C2×#10625350C2×C412yes—
43C2×#10625350D412yes—
44C2×#10625350D412yes—
45C2×#10625350D412yes—
46C2×#10625350Q812yes—
47C22×Dic94C2×C411yescharacteristic
48C22×D188C2×C411yescharacteristic
49C22×C22⋊D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—22no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329C2×C412yescomplemented
58C22×C47⋊C8—21no—
59#13093989—24no—
60#13093989—24no—
61#13093986—24no—
62#13093986—24no—
63C4×C22⋊D47—28no—
64C4×C22⋊D47—28no—
65#13093983—21no—
66#13093983—21no—
67#13093983—21no—
68#13093983—21no—
69#13093983—21no—
70#13093983—21no—
71#13093983—21no—
72#13093983C2×C412yescomplemented
73#13093982—24no—
74#13093982—24no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47D411yescharacteristic
77C2×C4⋊Dic47D411yescharacteristic
78C2×Dic47⋊C4—22no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47—24no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47D412yes—
83C2×(C2×D47)⋊C4—22no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2×C412yes—
87#13093952—28no—
88#13093950—28no—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C2×M16×C47—21no—
92C2×M16×C47—21no—
93C2×M16×C47—21no—
94C2×M16×C47—21no—
95C2×M16×C47—21no—
96C2×M16×C47C2×C412yescomplemented
97C22×C8×C47—21no—
98#117×C47—21no—
99#117×C47—21no—
100#117×C47—21no—
101#117×C47C2×C412yescomplemented
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47C2×C412yes—
107C2×C4⋊C4×C47D412yes—
108C2×C4⋊C4×C47D412yes—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C24×C47C2×D411yescharacteristic
112C23×D47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—41no—
117C22×C4×C47—82no—
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×Q811yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×C4×C47#4111yescharacteristic
130C22×C4×C47#4111yescharacteristic
131C22×C4×C47#4111yescharacteristic
132C2×C22⋊D47—22no—
133C2×C22⋊D47—22no—
134C2×C22⋊D47—22no—
135C2×C22⋊D47—22no—
136C2×C22⋊D47—22no—
137C2×C22⋊D47—22no—
138C22×Dic47—21no—
139C22×Dic47—41no—
140C22×Dic47C22⋊C412yes—
141C22×Dic47C22⋊C412yes—
142#10625350—22no—
143#10625350—22no—
144#10625350—22no—
145#10625350—22no—
146#10625350—22no—
147#10625350—22no—
148#10625350—22no—
149#10625350—22no—
150#10625350—22no—
151#10625350—22no—
152#10625350—22no—
153#10625350—22no—
154#10625350C22⋊C414yes—
155#10625350C22⋊C414yes—
156#10625350C4⋊C414yes—
157#10625350C4⋊C414yes—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188C4212yes—
162C2×D188C22⋊C412yes—
163C2×D188C4⋊C412yes—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47—22no—
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×C4×D47—28no—
171C2×C4×D47—84no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94C4212yes—
176C2×Dic94C22⋊C412yes—
177C2×Dic94C4⋊C412yes—
178M16×C47—22no—
179M16×C47—22no—
180M16×C47—41no—
181M16×C47—41no—
182C2×C8×C47—22no—
183C2×C8×C47—41no—
184C2×C8×C47—41no—
185C4⋊C4×C47—22no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47—22no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47—41no—
190C4⋊C4×C47C22⋊C414yes—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—41no—
193C42×C47—22no—
194C42×C47—41no—
195C22⋊Dic47—22no—
196C22⋊Dic47—22no—
197C22⋊Dic47—22no—
198C22⋊Dic47—41no—
199(C2×D47)⋊C4—28no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—22no—
205C4⋊Dic47—41no—
206Dic47⋊C4—28no—
207C4×Dic47—28no—
208C4×Dic47—22no—
209C4×Dic47—22no—
210C4×Dic47—22no—
211C4×Dic47—41no—
212#10625329—22no—
213#10625329—22no—
214#10625329—41no—
215#10625329—41no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—41no—
218C2×C47⋊C8—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47C2×C22⋊C411yescharacteristic
222C23×C47C4×D411yescharacteristic
223C23×C47C4×D411yescharacteristic
224C23×C47#12011yescharacteristic
225C23×C47#12011yescharacteristic
226C23×C47#12111yescharacteristic
227C23×C47C22⋊Q811yescharacteristic
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—42no—
232C22×D47—42no—
233C2×C4×C47—24no—
234C2×C4×C47—44no—
235C2×C4×C47—42no—
236C2×C4×C47—82no—
237C2×C4×C47—82no—
238C2×C4×C47—81no—
239C2×C4×C47—81no—
240C2×C4×C47C2×C4211yescharacteristic
241C2×C4×C47C2×C22⋊C411yescharacteristic
242C2×C4×C47C2×C22⋊C411yescharacteristic
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C2×C4⋊C411yescharacteristic
246C2×C4×C47C2×C4⋊C411yescharacteristic
247C2×C4×C47C2×C4⋊C411yescharacteristic
248C2×C4×C47#11711yescharacteristic
249C2×C4×C47#11711yescharacteristic
250C2×C4×C47C4×D411yescharacteristic
251C2×C4×C47C4×D411yescharacteristic
252C2×C4×C47C4×D411yescharacteristic
253C2×C4×C47C4×D411yescharacteristic
254C2×C4×C47C4×D411yescharacteristic
255C2×C4×C47C4×D411yescharacteristic
256C2×C4×C47#12011yescharacteristic
257C2×C4×C47#12011yescharacteristic
258C2×C4×C47#12111yescharacteristic
259C2×C4×C47#12111yescharacteristic
260C2×C4×C47#12111yescharacteristic
261C2×C4×C47C22⋊Q811yescharacteristic
262C2×C4×C47C22⋊Q811yescharacteristic
263C2×C4×C47C22⋊Q811yescharacteristic
264C2×C4×C47#12311yescharacteristic
265C2×C4×C47#12311yescharacteristic
266C2×C4×C47#12311yescharacteristic
267C2×C4×C47#12311yescharacteristic
268C22⋊D47—24no—
269C22⋊D47—24no—
270C22⋊D47—24no—
271C22⋊D47—24no—
272C2×Dic47—22no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—22no—
276C2×Dic47—22no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—81no—
281C2×Dic47—81no—
282D188—22no—
283D188—22no—
284D188—22no—
285D188#9514yes—
286C4×D47—24no—
287C4×D47—24no—
288C4×D47—24no—
289C4×D47—24no—
290C4×D47—88no—
291Dic94—22no—
292Dic94—22no—
293Dic94—22no—
294Dic94#9514yes—
295C8×C47—42no—
296C47⋊C8—42no—
297#10775—471nomaximal, Sylow
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—21no—
301C22×C47—21no—
302C22×C47—21no—
303C22×C47—21no—
304C22×C47—41no—
305C22×C47—41no—
306C22×C47—41no—
307C22×C47—41no—
308C22×C47C4×C22⋊C411yescharacteristic
309C22×C47#37911yescharacteristic
310C22×C47#38011yescharacteristic
311C22×C47#38511yescharacteristic
312C22×C47#38511yescharacteristic
313C22×C47#38611yescharacteristic
314C22×C47#38611yescharacteristic
315C2×D47—24no—
316C2×D47—44no—
317C2×D47—44no—
318C2×D47—42no—
319C2×D47—42no—
320C4×C47—44no—
321C4×C47—82no—
322C4×C47C2×#9511yescharacteristic
323C4×C47C4×C22⋊C411yescharacteristic
324C4×C47#38011yescharacteristic
325C4×C47#38111yescharacteristic
326C4×C47#38511yescharacteristic
327C4×C47#38511yescharacteristic
328C4×C47#38611yescharacteristic
329C4×C47#38611yescharacteristic
330Dic47—24no—
331Dic47—42no—
332Dic47—42no—
333Dic47—82no—
334C2×#520—471no—
335C2×#421—471no—
336C2×#421—471no—
337#1742—471no—
338#1740—471no—
339#1736—471no—
340#1736D4711yescharacteristic, complemented
341C2×C47—21no—
342C2×C47—21no—
343C2×C47—21no—
344C2×C47—21no—
345C2×C47—21no—
346C2×C47—21no—
347C2×C47C22⋊#9511yescharacteristic
348C2×C47#179511yescharacteristic
349C2×C47#179611yescharacteristic
350D47—44no—
351C22×#41—471no—
352C22×M16—471no—
353C22×M16C2×D4711yescharacteristic
354#520—942no—
355#520—942no—
356#520—942no—
357#520—942no—
358C2×C4×Q8—472no—
359C2×C4×D4—472no—
360C2×C4×D4—942no—
361C2×#117—471no—
362C2×#117—471no—
363C2×#117—941no—
364C2×#117C2×D4711yescharacteristic
365C22×C4⋊C4—941no—
366C22×C4⋊C4C2×D4711yescharacteristic
367#421—474no—
368#421—474no—
369#421—942no—
370#421—942no—
371C2×#104—942no—
372C2×#104—942no—
373#343—941no—
374#343—941no—
375#342—942no—
376#340—22no—
377#340—942no—
378C47#1077511yescharacteristic, complemented, Sylow
379C2×#41—472no—
380C2×#41—472no—
381C2×#41—472no—
382C2×#41—472no—
383C2×#41—472no—
384C2×#41—472no—
385C2×#41—941no—
386C2×#41—941no—
387C2×#41—941no—
388C2×#41—941no—
389C2×#41—941no—
390C2×#41—941no—
391C22×Q8—471no—
392C22×D4—471no—
393C22×D4—941no—
394C23×C4—21no—
395C23×C4—942no—
396C23×C4—941no—
397C23×C4C22×D4711yescharacteristic
398C2×M16—21no—
399C2×M16—21no—
400C2×M16—21no—
401C2×M16—21no—
402C2×M16—21no—
403C2×M16—472no—
404C2×M16—941no—
405C2×M16—941no—
406C2×M16—941no—
407C2×M16—941no—
408C2×M16—941no—
409C2×M16C4×D4712yescomplemented
410C22×C8—21no—
411C22×C8—941no—
412C4×Q8—944no—
413C4×Q8—944no—
414C4×D4—944no—
415C4×D4—948no—
416C4×D4—948no—
417C4×D4—944no—
418#117—21no—
419#117—21no—
420#117—21no—
421#117—472no—
422#117—944no—
423#117—944no—
424#117—941no—
425#117—941no—
426#117—941no—
427#117—941no—
428#117—941no—
429#117—941no—
430#117—941no—
431#117C4×D4712yescomplemented
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—471no—
437C2×C4⋊C4—471no—
438C2×C4⋊C4—944no—
439C2×C4⋊C4—944no—
440C2×C4⋊C4—942no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4C4×D4712yes—
443C2×C4⋊C4D18812yes—
444C2×C4⋊C4C22⋊D4712yes—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—472no—
447C2×C22⋊C4—942no—
448C2×C22⋊C4—941no—
449C2×C42—21no—
450C2×C42—472no—
451C2×C42—942no—
452C2×C42—941no—
453#104—948no—
454#104—948no—
455C24—941no—
456C24D4×D4711yescharacteristic
457#41—474no—
458#41—474no—
459#41—474no—
460#41—474no—
461#41—942no—
462#41—942no—
463#41—942no—
464#41—942no—
465#41—942no—
466#41—942no—
467#41—942no—
468#41—942no—
469#41—942no—
470#41—942no—
471#41—942no—
472#41—942no—
473C2×Q8—472no—
474C2×Q8—472no—
475C2×Q8—472no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×D4—472no—
480C2×D4—472no—
481C2×D4—472no—
482C2×D4—942no—
483C2×D4—942no—
484C2×D4—942no—
485C2×D4—942no—
486C2×D4—942no—
487C2×D4—942no—
488C2×D4—941no—
489C2×D4—941no—
490C2×D4—941no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—41no—
495C22×C4—82no—
496C22×C4—472no—
497C22×C4—472no—
498C22×C4—948no—
499C22×C4—941no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—3764no—
507C22×C4—1881no—
508C22×C4C2×C4×D4711yescharacteristic
509C22×C4C2×C4×D4711yescharacteristic
510C22×C4C2×C4×D4711yescharacteristic
511C22×C4C2×D18811yescharacteristic
512C22×C4#1062535011yescharacteristic
513C22×C4#1062535011yescharacteristic
514C22×C4D4×D4711yescharacteristic
515C22×C4D4×D4711yescharacteristic
516C22×C4D4×D4711yescharacteristic
517C22×C4D4×D4711yescharacteristic
518C22×C4#1062535211yescharacteristic
519C22×C4Q8×D4711yescharacteristic
520C22×C4#1062535411yescharacteristic
521C22×C4C2×C22⋊D4711yescharacteristic
522M16—22no—
523M16—22no—
524M16—41no—
525M16—41no—
526M16—942no—
527M16—942no—
528M16—1881no—
529M16—1881no—
530C2×C8—22no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—942no—
534C2×C8—1881no—
535C2×C8—1881no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—41no—
541C4⋊C4—948no—
542C4⋊C4—941no—
543C4⋊C4—941no—
544C4⋊C4—941no—
545C4⋊C4—941no—
546C4⋊C4—942no—
547C4⋊C4—1881no—
548C4⋊C4(C2×D47)⋊C414yes—
549C22⋊C4—22no—
550C22⋊C4—41no—
551C22⋊C4—948no—
552C22⋊C4—942no—
553C22⋊C4—942no—
554C22⋊C4—942no—
555C22⋊C4—1881no—
556C42—22no—
557C42—41no—
558C42—948no—
559C42—942no—
560C42—942no—
561C42—942no—
562C42—1881no—
563C23—41no—
564C23—41no—
565C23—942no—
566C23—942no—
567C23—942no—
568C23—1882no—
569C23—1882no—
570C23C22⋊C4×D4711yescharacteristic
571C23(C22⋊D47)⋊C411yescharacteristic
572C23#1309398611yescharacteristic
573C23#1309399211yescharacteristic
574C23#1309399411yescharacteristic
575C23#1309399711yescharacteristic
576C23#1309400211yescharacteristic
577Q8—474no—
578Q8—942no—
579Q8—942no—
580Q8—942no—
581D4—474no—
582D4—944no—
583D4—944no—
584D4—944no—
585D4—944no—
586D4—942no—
587D4—942no—
588D4—942no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—3768no—
606C2×C4—3761no—
607C2×C4—3761no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4C42×D4711yescharacteristic
612C2×C4C22⋊C4×D4711yescharacteristic
613C2×C4C22⋊C4×D4711yescharacteristic
614C2×C4C22⋊C4×D4711yescharacteristic
615C2×C4C2×(C2×D47)⋊C411yescharacteristic
616C2×C4C4⋊C4×D4711yescharacteristic
617C2×C4C4⋊C4×D4711yescharacteristic
618C2×C4C4⋊C4×D4711yescharacteristic
619C2×C4#1309398111yescharacteristic
620C2×C4#1309398211yescharacteristic
621C2×C4(C22⋊D47)⋊C411yescharacteristic
622C2×C4C4×C22⋊D4711yescharacteristic
623C2×C4C4×C22⋊D4711yescharacteristic
624C2×C4#1309398611yescharacteristic
625C2×C4C4×D18811yescharacteristic
626C2×C4C4×D18811yescharacteristic
627C2×C4#1309399211yescharacteristic
628C2×C4#1309399411yescharacteristic
629C2×C4#1309399511yescharacteristic
630C2×C4#1309399711yescharacteristic
631C2×C4#1309399911yescharacteristic
632C2×C4#1309400011yescharacteristic
633C2×C4#1309400211yescharacteristic
634C2×C4#1309400411yescharacteristic
635C2×C4#1309400611yescharacteristic
636C2×C4#1309400611yescharacteristic
637C2×C4#1309400911yescharacteristic
638C2×C4#1309400911yescharacteristic
639C8—42no—
640C8—1882no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—944no—
652C22—1884no—
653C22—1884no—
654C22—1882no—
655C22—1882no—
656C22#1568897011yescharacteristic
657C22#1568897811yescharacteristic
658C22#1568898211yescharacteristic
659C22#1568899411yescharacteristic
660C22#1568899711yescharacteristic
661C22#1568899911yescharacteristic
662C22#1568900111yescharacteristic
663C4—44no—
664C4—82no—
665C4—944no—
666C4—3762no—
667C4—1882no—
668C4—1882no—
669C4#95×D4711yescharacteristic
670C4C4×(C2×D47)⋊C411yescharacteristic
671C4#1568898011yescharacteristic
672C4#1568898511yescharacteristic
673C4#1568899411yescharacteristic
674C4#1568899711yescharacteristic
675C4#1568899911yescharacteristic
676C4#1568900111yescharacteristic
677C2—21no—
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—1884no—
684C2(C2×D47)⋊#9511yescharacteristic
685C2#2024335511yescharacteristic
686C2#2024336011yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1736C2
socleC22×C47#379
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#1736D47
hypocenterC47#10775
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20243126C2semidirect
5. #20243120C2semidirect
6. #20243107C2semidirect
7. C47×#1736C2semidirect
14. C2×#13093989C4semidirect
15. C2×#13093986C4semidirect
22. #15689141C4semidirect
24. #15689139C4semidirect
57. C2×#10625329C2×C4semidirect
72. #13093983C2×C4semidirect
96. C2×M16×C47C2×C4semidirect
101. #117×C47C2×C4semidirect
340. #1736D47semidirect
378. C47#10775semidirect
409. C2×M16C4×D47semidirect
431. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896042
C2×#156891412
C2×#156891392
#202431262
#202431202
#202431072
C47×#17362
#1077547