Finite Group Databasebeta

← #26131272

Subgroups — #26131272

Order 12032.

752 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689107C211yesmaximal, characteristic
2#20251833C211yesmaximal, characteristic
3#20241516C214yesmaximal, complemented
4C2×C47×#410C211yesmaximal, characteristic
5C22×#10625329—21no—
6C22×#10625329C411yescharacteristic, complemented
7C22×#10625329C411yescharacteristic, complemented
8#15689107—21no—
9#15689107—21no—
10#15689107C412yescomplemented
11#15689107C2212yes—
12C2×#13093943C412yescomplemented
13C2×#13093941—21no—
14#15689090—21no—
15#15689090—21no—
16#15689090C412yescomplemented
17#15689090C2212yes—
18#15689087—21no—
19#15689087—21no—
20C22×#41×C47C2211yescharacteristic
21C2×#117×C47—21no—
22C47×#410—21no—
23C47×#410—21no—
24C47×#410C412yes—
25C47×#410C2212yes—
26C2×#99×C47—21no—
27C2×#99×C47C412yes—
28C2×#10625329—21no—
29C2×#10625329—21no—
30C2×#10625329—21no—
31C2×#10625329—22no—
32C2×#10625329—22no—
33C2×#10625329—22no—
34C2×#10625329—41no—
35C2×#10625329—41no—
36C2×#10625329—41no—
37C2×#10625329—41no—
38C2×#10625329—41no—
39C2×#10625329—41no—
40C2×#10625329—41no—
41C2×#10625329C2×C412yescomplemented
42C22×C47⋊C8—21no—
43C22×C47⋊C8—41no—
44C22×C47⋊C8C2×C412yescomplemented
45#13093943—22no—
46#13093943C2×C414yescomplemented
47#13093941—22no—
48#13093941—22no—
49C22⋊(C47⋊C8)—22no—
50C22⋊(C47⋊C8)—22no—
51C22⋊(C47⋊C8)—22no—
52C22⋊(C47⋊C8)—22no—
53C2×#41×C47—41no—
54C2×#41×C47—41no—
55C2×#41×C47—41no—
56C2×#41×C47—41no—
57C2×#41×C47C2×C411yescharacteristic
58C2×#41×C47C2×C411yescharacteristic
59C2×#41×C47C2×C411yescharacteristic
60C2×#41×C47D411yescharacteristic
61C2×#41×C47D411yescharacteristic
62C2×#41×C47D411yescharacteristic
63C2×#41×C47Q811yescharacteristic
64C2×#41×C47C2311yescharacteristic
65C22×Q8×C47C2×C411yescharacteristic
66C22×D4×C47C2×C411yescharacteristic
67C22×D4×C47D411yescharacteristic
68C22×D4×C47Q811yescharacteristic
69C23×C4×C47C2×C411yescharacteristic
70C23×C4×C47D411yescharacteristic
71C23×C4×C47D411yescharacteristic
72#117×C47—22no—
73#117×C47—22no—
74#117×C47—41no—
75#117×C47—41no—
76C2×C4⋊C4×C47—41no—
77C2×C22⋊C4×C47—22no—
78C2×C42×C47—41no—
79#99×C47—22no—
80#99×C47—22no—
81#99×C47—22no—
82#99×C47C2×C414yes—
83C24×C47C22⋊C411yescharacteristic
84C24×C47C4⋊C411yescharacteristic
85C24×C47C4⋊C411yescharacteristic
86#41×C47—41no—
87#41×C47—41no—
88#41×C47—41no—
89#41×C47—41no—
90#41×C47—41no—
91#41×C47—41no—
92#41×C47—41no—
93#41×C47—41no—
94#41×C47—41no—
95#41×C47—41no—
96#41×C47—41no—
97#41×C47—41no—
98#41×C47—41no—
99#41×C47—41no—
100#41×C47—41no—
101#41×C47—41no—
102C2×Q8×C47—41no—
103C2×Q8×C47—41no—
104C2×Q8×C47C4211yescharacteristic
105C2×Q8×C47C22⋊C411yescharacteristic
106C2×Q8×C47C4⋊C411yescharacteristic
107C2×Q8×C47C22×C411yescharacteristic
108C2×D4×C47—41no—
109C2×D4×C47—41no—
110C2×D4×C47—41no—
111C2×D4×C47—41no—
112C2×D4×C47—41no—
113C2×D4×C47—41no—
114C2×D4×C47C4211yescharacteristic
115C2×D4×C47C22⋊C411yescharacteristic
116C2×D4×C47C22⋊C411yescharacteristic
117C2×D4×C47C22⋊C411yescharacteristic
118C2×D4×C47C4⋊C411yescharacteristic
119C2×D4×C47C4⋊C411yescharacteristic
120C2×D4×C47C4⋊C411yescharacteristic
121C2×D4×C47C4⋊C411yescharacteristic
122C2×D4×C47C4⋊C411yescharacteristic
123C2×D4×C47C22×C411yescharacteristic
124C2×D4×C47C2×D411yescharacteristic
125C2×D4×C47C2×Q811yescharacteristic
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—42no—
139C22×C4×C47C4211yescharacteristic
140C22×C4×C47C4211yescharacteristic
141C22×C4×C47C22⋊C411yescharacteristic
142C22×C4×C47C22⋊C411yescharacteristic
143C22×C4×C47C22⋊C411yescharacteristic
144C22×C4×C47C22⋊C411yescharacteristic
145C22×C4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22⋊C411yescharacteristic
147C22×C4×C47C22⋊C411yescharacteristic
148C22×C4×C47C4⋊C411yescharacteristic
149C22×C4×C47C4⋊C411yescharacteristic
150C22×C4×C47C4⋊C411yescharacteristic
151C22×C4×C47C4⋊C411yescharacteristic
152C22×C4×C47C22×C411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C4⋊C4×C47—42no—
156C4⋊C4×C47—42no—
157C22⋊C4×C47—24no—
158C22⋊C4×C47—42no—
159C42×C47—42no—
160C42×C47—42no—
161#10625329—24no—
162#10625329—42no—
163#10625329—42no—
164#10625329—42no—
165#10625329—42no—
166#10625329—41no—
167#10625329—41no—
168#10625329—41no—
169#10625329—41no—
170#10625329—42no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—42no—
174C2×C47⋊C8—42no—
175C2×C47⋊C8—42no—
176C2×C47⋊C8—42no—
177C2×C47⋊C8—42no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—81no—
185C23×C47—81no—
186C23×C47—81no—
187C23×C47#9511yescharacteristic
188C23×C47#9511yescharacteristic
189C23×C47#9511yescharacteristic
190C23×C47#9511yescharacteristic
191C23×C47C2×C22⋊C411yescharacteristic
192C23×C47C2×C4⋊C411yescharacteristic
193C23×C47C2×C4⋊C411yescharacteristic
194Q8×C47—41no—
195Q8×C47—41no—
196Q8×C47—41no—
197Q8×C47—41no—
198D4×C47—41no—
199D4×C47—41no—
200D4×C47—41no—
201D4×C47—41no—
202D4×C47—41no—
203D4×C47—41no—
204D4×C47—41no—
205D4×C47—41no—
206D4×C47—41no—
207D4×C47—41no—
208D4×C47—41no—
209D4×C47—41no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—44no—
241C2×C4×C47—84no—
242C2×C4×C47#9511yescharacteristic
243C2×C4×C47#9511yescharacteristic
244C2×C4×C47#9511yescharacteristic
245C2×C4×C47#9511yescharacteristic
246C2×C4×C47#9911yescharacteristic
247C2×C4×C47#9911yescharacteristic
248C2×C4×C47#9911yescharacteristic
249C2×C4×C47#9911yescharacteristic
250C2×C4×C47C2×C4211yescharacteristic
251C2×C4×C47C2×C22⋊C411yescharacteristic
252C2×C4×C47C2×C22⋊C411yescharacteristic
253C2×C4×C47C2×C4⋊C411yescharacteristic
254C47⋊C8—44no—
255C47⋊C8—42no—
256C47⋊C8—42no—
257#12589—471nomaximal, Sylow
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47#34211yescharacteristic
278C22×C47#34211yescharacteristic
279C22×C47C2×#9511yescharacteristic
280C4×C47—21no—
281C4×C47—21no—
282C4×C47—21no—
283C4×C47—21no—
284C4×C47—21no—
285C4×C47—21no—
286C4×C47—84no—
287C4×C47#34211yescharacteristic
288C4×C47#34211yescharacteristic
289C4×C47C2×#9911yescharacteristic
290C4×C47C2×#9911yescharacteristic
291C2×#413—471no—
292C2×#410D4711yescharacteristic
293#2879—471no—
294#1395—474no—
295C2×C47—21no—
296C2×C47—21no—
297C2×C47—21no—
298C2×C47—41no—
299C2×C47—41no—
300C2×C47—41no—
301C2×C47—41no—
302C2×C47—41no—
303C2×C47—41no—
304C2×C47C2×#34211yescharacteristic
305C22×#41C2×D4711yescharacteristic
306C22×M16—471no—
307C22×M16—471no—
308C22×M16—941no—
309C2×#117—21no—
310#413—472no—
311#413—472no—
312#413—941no—
313#413—941no—
314C2×#101—472no—
315C2×#100—941no—
316#410—21no—
317#410—21no—
318#410Dic4712yes—
319#410C2×D4712yes—
320C2×#99—21no—
321C2×#99Dic4712yes—
322#408—472no—
323#408—472no—
324#408—941no—
325#408—941no—
326#407—941no—
327#407—941no—
328C47#1258911yescharacteristic, complemented, Sylow
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic9411yescharacteristic
334C2×#41C4×D4711yescharacteristic
335C2×#41C4×D4711yescharacteristic
336C2×#41D18811yescharacteristic
337C2×#41C2×Dic4711yescharacteristic
338C2×#41C22⋊D4711yescharacteristic
339C2×#41C22⋊D4711yescharacteristic
340C2×#41C22×D4711yescharacteristic
341C22×Q8C4×D4711yescharacteristic
342C22×D4Dic9411yescharacteristic
343C22×D4C2×Dic4711yescharacteristic
344C22×D4C22⋊D4711yescharacteristic
345C23×C4C4×D4711yescharacteristic
346C23×C4D18811yescharacteristic
347C23×C4C22⋊D4711yescharacteristic
348C2×M16—472no—
349C2×M16—941no—
350C2×M16—941no—
351C2×M16—941no—
352C2×M16—942no—
353C2×M16—942no—
354C2×M16—942no—
355C2×M16—1881no—
356C2×M16—1881no—
357C2×M16—1881no—
358C2×M16—1881no—
359C2×M16—1881no—
360C2×M16—1881no—
361C2×M16—1881no—
362C22×C8—472no—
363C22×C8—941no—
364C22×C8—1881no—
365#117—22no—
366#117—22no—
367#117—41no—
368#117—41no—
369C2×C4⋊C4—41no—
370C2×C22⋊C4—22no—
371C2×C42—41no—
372#101—474no—
373#101—942no—
374#100—942no—
375#100—942no—
376#99—22no—
377#99—22no—
378#99—22no—
379#99C2×Dic4714yes—
380C22⋊C8—942no—
381C22⋊C8—942no—
382C22⋊C8—942no—
383C22⋊C8—942no—
384C24Dic47⋊C411yescharacteristic
385C24C4⋊Dic4711yescharacteristic
386C24C22⋊Dic4711yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic4711yescharacteristic
406C2×Q8Dic47⋊C411yescharacteristic
407C2×Q8(C2×D47)⋊C411yescharacteristic
408C2×Q8C2×C4×D4711yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic4711yescharacteristic
416C2×D4Dic47⋊C411yescharacteristic
417C2×D4Dic47⋊C411yescharacteristic
418C2×D4Dic47⋊C411yescharacteristic
419C2×D4C4⋊Dic4711yescharacteristic
420C2×D4C4⋊Dic4711yescharacteristic
421C2×D4(C2×D47)⋊C411yescharacteristic
422C2×D4C22⋊Dic4711yescharacteristic
423C2×D4C22⋊Dic4711yescharacteristic
424C2×D4C2×Dic9411yescharacteristic
425C2×D4C22×Dic4711yescharacteristic
426C2×D4C2×C22⋊D4711yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4C4×Dic4711yescharacteristic
441C22×C4C4×Dic4711yescharacteristic
442C22×C4Dic47⋊C411yescharacteristic
443C22×C4Dic47⋊C411yescharacteristic
444C22×C4Dic47⋊C411yescharacteristic
445C22×C4C4⋊Dic4711yescharacteristic
446C22×C4(C2×D47)⋊C411yescharacteristic
447C22×C4(C2×D47)⋊C411yescharacteristic
448C22×C4(C2×D47)⋊C411yescharacteristic
449C22×C4(C2×D47)⋊C411yescharacteristic
450C22×C4(C2×D47)⋊C411yescharacteristic
451C22×C4(C2×D47)⋊C411yescharacteristic
452C22×C4C22⋊Dic4711yescharacteristic
453C22×C4C2×C4×D4711yescharacteristic
454C22×C4C2×D18811yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456M16—944no—
457M16—1882no—
458M16—1882no—
459M16—1882no—
460M16—1882no—
461M16—1881no—
462M16—1881no—
463M16—1881no—
464M16—1881no—
465M16—1882no—
466C2×C8—942no—
467C2×C8—942no—
468C2×C8—1882no—
469C2×C8—1882no—
470C2×C8—1882no—
471C2×C8—1882no—
472C2×C8—1882no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C42—42no—
478C42—42no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23C47⋊#9511yescharacteristic
489C23C47⋊#9511yescharacteristic
490C23C47⋊#9511yescharacteristic
491C23C47⋊#9511yescharacteristic
492C23C2×C22⋊Dic4711yescharacteristic
493C23C2×Dic47⋊C411yescharacteristic
494C23C2×C4⋊Dic4711yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—84no—
543C2×C4C47⋊#9511yescharacteristic
544C2×C4C47⋊#9511yescharacteristic
545C2×C4C47⋊#9511yescharacteristic
546C2×C4C47⋊#9511yescharacteristic
547C2×C4#1309393811yescharacteristic
548C2×C4#1309393811yescharacteristic
549C2×C4#1309393811yescharacteristic
550C2×C4#1309393811yescharacteristic
551C2×C4C2×C4×Dic4711yescharacteristic
552C2×C4C2×(C2×D47)⋊C411yescharacteristic
553C2×C4C2×(C2×D47)⋊C411yescharacteristic
554C2×C4C2×Dic47⋊C411yescharacteristic
555C8—1884no—
556C8—1882no—
557C8—1882no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1568888911yescharacteristic
578C22#1568888911yescharacteristic
579C22C2×C47⋊#9511yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—84no—
587C4#1568888911yescharacteristic
588C4#1568888911yescharacteristic
589C4C2×#1309393811yescharacteristic
590C4C2×#1309393811yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2—41no—
600C2C2×#1568888911yescharacteristic
subgroupG/H
centerC4C2×#13093938
commutatorC2×C4×C47C2×C42
FrattiniC2×#41C22×D47
FittingC2×C47×#410C2
socleC2×C47C2×#342
prime radicalC22×#41×C47C22
cosocleC2×#41×C47C23
hypercenterC2×#410D47
hypocenterC47#12589
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #20241516C2semidirect
6. C22×#10625329C4semidirect
7. C22×#10625329C4semidirect
10. #15689107C4semidirect
12. C2×#13093943C4semidirect
16. #15689090C4semidirect
41. C2×#10625329C2×C4semidirect
44. C22×C47⋊C8C2×C4semidirect
46. #13093943C2×C4semidirect
328. C47#12589semidirect

Maximal subgroups

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

subgroupindex
C2×#156891072
#202518332
#202415162
C2×C47×#4102
#1258947