Finite Group Databasebeta

← #26376430

Subgroups — #26376430

Order 12032.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689591C211yesmaximal, characteristic
2#20246358C214yesmaximal, complemented
3#20246350C214yesmaximal, complemented
4#20246327C214yesmaximal, complemented
5#20246323C214yesmaximal, complemented
6#20245998C211yesmaximal, characteristic, complemented
7#20245997C211yesmaximal, characteristic, complemented
8#20245996C211yesmaximal, characteristic, complemented
9#20245977C212yesmaximal
10#20245793C211yesmaximal, characteristic, complemented
11#20245784C212yesmaximal, complemented
12#20245783C212yesmaximal, complemented
13#382×D47C211yesmaximal, characteristic, complemented
14#20245742C212yesmaximal, complemented
15C47×#2314C211yesmaximal, characteristic, complemented
16C22×#10625350C2211yescharacteristic
17#15689591C4116yescomplemented
18C2×#13093989C2212yes—
19C2×#13093986C2212yes—
20C2×(C22⋊D47)⋊C4C2212yes—
21C2×(C22⋊D47)⋊C4C2212yes—
22C2×#13093982C2212yes—
23C2×#13093981C2211yescharacteristic
24C2×#13093981C2211yescharacteristic
25C2×C4⋊C4×D47C2212yes—
26C22×C4×Dic47C2211yescharacteristic
27C2×C42×D47C2211yescharacteristic
28C2×C42×D47C2211yescharacteristic
29#15689027C2214yes—
30#15689023C2214yes—
31#15689016C2214yes—
32#15689015C2214yes—
33#15689013C2214yes—
34#15689010C2214yes—
35#15689009C2214yes—
36#15689004C2214yes—
37#15689001C2214yes—
38#15688999C2214yes—
39#15688993C2211yescharacteristic
40#15688990C2214yes—
41#15688990C2214yes—
42#15688989C2211yescharacteristic
43#15688988C2212yes—
44#15688988C2212yes—
45#15688987C2214yes—
46#15688985C2214yes—
47#15688984C2214yes—
48#15688983C2212yes—
49#15688980C2212yes—
50#15688980C2212yes—
51#15688980C2214yes—
52C4⋊C4×Dic47C2212yes—
53C4⋊C4×Dic47C2212yes—
54C4⋊C4×Dic47C2214yes—
55C4×Dic47⋊C4C2214yes—
56C4×C4⋊Dic47C2212yes—
57#15688974C2214yes—
58C22⋊C4×Dic47C2214yes—
59C4×(C2×D47)⋊C4C2214yes—
60C4×(C2×D47)⋊C4C2214yes—
61#15688970C2214yes—
62#15688969C2214yes—
63#15688968C2214yes—
64#95×D47C2214yes—
65C2×#117×C47C2211yescharacteristic
66C47×#388C2214yes—
67C47×#387C2214yes—
68C47×#384C2211yescharacteristic
69C47×#382C2211yescharacteristic
70C47×#380C2212yes—
71C4×C4⋊C4×C47C2212yes—
72C2×#10625350—24no—
73C2×#10625350—24no—
74C2×#10625350C2×C418yes—
75C22×Dic94C2311yescharacteristic
76C22×D188C2311yescharacteristic
77C22×C22⋊D47C2311yescharacteristic
78C22×C22⋊D47C2311yescharacteristic
79C23×Dic47C2312yes—
80C22×C4×D47C2311yescharacteristic
81C22×C4×D47C2314yes—
82C22×C4×D47C2312yes—
83C22×C4×D47C2312yes—
84C22×C4×D47C2311yescharacteristic
85#13093989C2×C4116yescomplemented
86#13093986C2×C4116yes—
87(C22⋊D47)⋊C4C2×C4116yes—
88(C22⋊D47)⋊C4C2×C4116yescomplemented
89#13093982—28no—
90#13093981C2×C418yes—
91#13093981C2×C418yescomplemented
92C2×C4⋊Dic47C2312yes—
93C2×C4⋊Dic47C2312yes—
94C2×C4⋊Dic47C2314yes—
95C2×Dic47⋊C4C2314yes—
96C2×Dic47⋊C4C2312yes—
97C2×Dic47⋊C4C2312yes—
98C4⋊C4×D47—28no—
99C2×C22⋊Dic47C2314yes—
100C2×(C2×D47)⋊C4C2314yes—
101C2×(C2×D47)⋊C4C2314yes—
102C2×(C2×D47)⋊C4C2314yes—
103C2×(C2×D47)⋊C4C2312yes—
104C2×(C2×D47)⋊C4C2312yes—
105C2×C4×Dic47C2×C418yescomplemented
106C2×C4×Dic47C2314yes—
107C2×C4×Dic47C2314yes—
108C2×C4×Dic47C2314yes—
109C2×C4×Dic47C2314yes—
110C2×C4×Dic47C2312yes—
111C2×C4×Dic47C2312yes—
112C2×C4×Dic47C2311yescharacteristic
113C2×C4×Dic47C2312yes—
114C2×C4×Dic47C2311yescharacteristic
115C42×D47C2×C418yes—
116C42×D47C2×C418yescomplemented
117C47⋊#95C2314yes—
118C47⋊#95C2314yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C47⋊#95C2314yes—
122C47⋊#95C2314yes—
123C47⋊#95C2314yes—
124C47⋊#95C2314yes—
125C47⋊#95C2314yes—
126C47⋊#95C2314yes—
127C47⋊#95C2314yes—
128C23×C4×C47C2311yescharacteristic
129#117×C47C2×C418yescomplemented
130C2×C4⋊C4×C47C2312yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2314yes—
133C2×C4⋊C4×C47C2312yes—
134C2×C22⋊C4×C47C2314yes—
135C2×C22⋊C4×C47C2312yes—
136C2×C42×C47C2314yes—
137C2×C42×C47C2311yescharacteristic
138C2×C42×C47C2311yescharacteristic
139#95×C47C2314yes—
140#95×C47C2314yes—
141#95×C47C2314yes—
142C24×C47C2411yescharacteristic
143C23×D47C2411yescharacteristic
144C23×D47C2411yescharacteristic
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47C22×C414yes—
148C22×C4×C47C2414yes—
149C22×C4×C47C2414yes—
150C22×C4×C47C2412yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2411yescharacteristic
153C22×C4×C47C2411yescharacteristic
154C2×C22⋊D47—22no—
155C2×C22⋊D47—22no—
156C2×C22⋊D47—22no—
157C2×C22⋊D47—22no—
158C2×C22⋊D47C22×C414yes—
159C2×C22⋊D47C22×C414yes—
160C22×Dic47—42no—
161C22×Dic47—42no—
162C22×Dic47C22×C418yes—
163C22×Dic47C2414yes—
164C22×Dic47C2414yes—
165C22×Dic47C2412yes—
166C22×Dic47C2411yescharacteristic
167C22×Dic47C2412yes—
168C22×Dic47C2411yescharacteristic
169#10625350—232no—
170C2×D188—22no—
171C2×D188—22no—
172C2×D188C22×C414yes—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—28no—
176C2×C4×D47—42no—
177C2×C4×D47—42no—
178C2×C4×D47—42no—
179C2×C4×D47—42no—
180C2×C4×D47—44no—
181C2×C4×D47—44no—
182C2×C4×D47C22×C414yes—
183C2×C4×D47C22×C418yes—
184C2×C4×D47C22×C418yes—
185C2×C4×D47C22×C414yes—
186C2×C4×D47#4114yes—
187C2×C4×D47#4114yes—
188C2×Dic94—22no—
189C2×Dic94—22no—
190C2×Dic94C22×C414yes—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—24no—
193C4⋊C4×C47—44no—
194C4⋊C4×C47C22×C418yes—
195C22⋊C4×C47—44no—
196C22⋊C4×C47C22×C418yes—
197C42×C47—44no—
198C42×C47C22×C414yes—
199C42×C47C22×C414yes—
200C22⋊Dic47—44no—
201(C2×D47)⋊C4—28no—
202(C2×D47)⋊C4—44no—
203(C2×D47)⋊C4—44no—
204(C2×D47)⋊C4C22×C418yes—
205(C2×D47)⋊C4C22×C418yes—
206C4⋊Dic47—24no—
207C4⋊Dic47—24no—
208C4⋊Dic47—44no—
209Dic47⋊C4—28no—
210Dic47⋊C4C22×C418yes—
211Dic47⋊C4C22×C418yes—
212C4×Dic47—28no—
213C4×Dic47—44no—
214C4×Dic47—44no—
215C4×Dic47—44no—
216C4×Dic47C22×C414yes—
217C4×Dic47C22×C418yes—
218C4×Dic47C22×C418yes—
219C4×Dic47C22×C414yes—
220C4×Dic47C22×C418yes—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47C23×C412yes—
226C23×C47C2511yescharacteristic
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—21no—
230C22×D47—21no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47C23×C412yes—
234C22×D47C23×C412yes—
235C22×D47C2×#4112yes—
236C22×D47C2×#4112yes—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—28no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—24no—
249C2×C4×C47—24no—
250C2×C4×C47C23×C412yes—
251C2×C4×C47C23×C414yes—
252C2×C4×C47C23×C414yes—
253C2×C4×C47C23×C412yes—
254C2×C4×C47C2×#4112yes—
255C2×C4×C47C2×#4112yes—
256C22⋊D47—28no—
257C22⋊D47—28no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—24no—
265C2×Dic47—24no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—48no—
271C2×Dic47C23×C414yes—
272C2×Dic47C23×C414yes—
273C2×Dic47C23×C412yes—
274C2×Dic47C23×C412yes—
275C2×Dic47C2×#4112yes—
276C2×Dic47C2×#4112yes—
277D188—28no—
278C4×D47—28no—
279C4×D47—48no—
280C4×D47—48no—
281C4×D47—88no—
282C4×D47#117116yes—
283Dic94—28no—
284#34189—471nomaximal, Sylow
285C22×C47—22no—
286C22×C47—24no—
287C22×C47—42no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47C24×C411yescharacteristic
291C22×C47C22×#4111yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47#58512yes—
294C22×C47#58512yes—
295C2×D47—24no—
296C2×D47—24no—
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—24no—
301C2×D47—42no—
302C2×D47—41no—
303C2×D47—41no—
304C2×D47C2×#11714yes—
305C4×C47—22no—
306C4×C47—24no—
307C4×C47—24no—
308C4×C47—44no—
309C4×C47—44no—
310C4×C47C2×#11714yes—
311Dic47—22no—
312Dic47—24no—
313Dic47—24no—
314Dic47—44no—
315Dic47—44no—
316Dic47C2×#11714yes—
317C2×C4×#41—471no—
318#2357—474no—
319#2356—474no—
320#2353—474no—
321#2353—474no—
322#2314—471no—
323#2314—471no—
324#2314—471no—
325#2314D4711yescharacteristic, complemented
326#2311—472no—
327#2285—471no—
328#2283—472no—
329#2283—472no—
330C2×#382—471no—
331#2274—472no—
332C2×C47—42no—
333C2×C47C22×#11711yescharacteristic
334C2×C47#342812yes—
335C2×C47#351012yes—
336C2×C47#351211yescharacteristic
337C2×C47#351211yescharacteristic
338D47—24no—
339D47—42no—
340C22×#41—471no—
341C4×#41—4716no—
342C2×C4×Q8—472no—
343C2×C4×D4—472no—
344C2×C4×D4—472no—
345C2×C4×D4—472no—
346C2×#117—472no—
347C2×#117—471no—
348C2×#117—471no—
349C2×#117C2×D4711yescharacteristic
350C22×C4⋊C4—472no—
351C22×C42—471no—
352C22×C42—471no—
353C22×C42—471no—
354#391—474no—
355#390—474no—
356#388—474no—
357#388—474no—
358#388—474no—
359#388C2×D4714yes—
360#387—474no—
361#387—474no—
362#387—474no—
363#387C2×D4714yes—
364#386—474no—
365#386—474no—
366#384—471no—
367#384C2×D4711yescharacteristic
368#383—474no—
369#383—474no—
370#382—472no—
371#382—472no—
372#382—471no—
373#382—474no—
374#382C2×D4711yescharacteristic
375#381—474no—
376#381—474no—
377#380—472no—
378#380—472no—
379#380—472no—
380#380—474no—
381#380C2×D4712yes—
382C4×C4⋊C4—472no—
383C4×C4⋊C4—472no—
384C4×C4⋊C4—474no—
385C4×C4⋊C4—472no—
386C4×C4⋊C4—474no—
387C4×C4⋊C4—474no—
388C4×C4⋊C4C2×D4712yes—
389C4×C22⋊C4—474no—
390C4×C22⋊C4—474no—
391C4×C22⋊C4—474no—
392C4×C22⋊C4—474no—
393#376—474no—
394#376—474no—
395C2×#95—474no—
396C47#3418911yescharacteristic, complemented, Sylow
397C2×#41—478no—
398C2×#41—944no—
399C2×#41—944no—
400C22×Q8—471no—
401C22×D4—471no—
402C22×D4—471no—
403C22×D4—471no—
404C23×C4—471no—
405C23×C4—472no—
406C23×C4—471no—
407C23×C4—472no—
408C23×C4—472no—
409C23×C4—474no—
410C23×C4C22×D4711yescharacteristic
411C4×Q8—4716no—
412C4×D4—4716no—
413C4×D4—4716no—
414C4×D4—4716no—
415#117—478no—
416#117—478no—
417#117—948no—
418#117C4×D4718yescomplemented
419C2×C4⋊C4—472no—
420C2×C4⋊C4—472no—
421C2×C4⋊C4—474no—
422C2×C4⋊C4—472no—
423C2×C4⋊C4—472no—
424C2×C4⋊C4—474no—
425C2×C4⋊C4—948no—
426C2×C4⋊C4C22×D4712yes—
427C2×C4⋊C4C22×D4712yes—
428C2×C4⋊C4C22×D4714yes—
429C2×C4⋊C4C22×D4712yes—
430C2×C22⋊C4—472no—
431C2×C22⋊C4—472no—
432C2×C22⋊C4—474no—
433C2×C22⋊C4—474no—
434C2×C22⋊C4—474no—
435C2×C22⋊C4—474no—
436C2×C22⋊C4C22×D4714yes—
437C2×C22⋊C4C22×D4712yes—
438C2×C42—478no—
439C2×C42—471no—
440C2×C42—472no—
441C2×C42—472no—
442C2×C42—472no—
443C2×C42—471no—
444C2×C42—478no—
445C2×C42—478no—
446C2×C42—474no—
447C2×C42—474no—
448C2×C42—474no—
449C2×C42—474no—
450C2×C42C22×D4714yes—
451C2×C42C22×D4711yescharacteristic
452C2×C42C22×D4711yescharacteristic
453#95—474no—
454#95—474no—
455#95—474no—
456#95—474no—
457#95—474no—
458#95—474no—
459#95—474no—
460#95—474no—
461#95—474no—
462#95—474no—
463#95—474no—
464#95C22×D4714yes—
465#95C22×D4714yes—
466#95C22×D4714yes—
467C24—471no—
468C24—471no—
469C24C23×D4711yescharacteristic
470#41—9432no—
471C2×Q8—474no—
472C2×Q8—942no—
473C2×Q8—942no—
474C2×D4—474no—
475C2×D4—474no—
476C2×D4—474no—
477C2×D4—942no—
478C2×D4—942no—
479C2×D4—942no—
480C2×D4—942no—
481C2×D4—942no—
482C2×D4—942no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—471no—
486C22×C4—474no—
487C22×C4—474no—
488C22×C4—474no—
489C22×C4—472no—
490C22×C4—471no—
491C22×C4—472no—
492C22×C4—478no—
493C22×C4—474no—
494C22×C4—478no—
495C22×C4—478no—
496C22×C4—474no—
497C22×C4—474no—
498C22×C4—948no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—1884no—
502C22×C4—1884no—
503C22×C4—1882no—
504C22×C4—1882no—
505C22×C4—1882no—
506C22×C4—1882no—
507C22×C4—1882no—
508C22×C4—1882no—
509C22×C4C2×C4×D4714yes—
510C22×C4C23×D4714yes—
511C22×C4C23×D4714yes—
512C22×C4C23×D4711yescharacteristic
513C22×C4C23×D4712yes—
514C22×C4C23×D4712yes—
515C22×C4C23×D4711yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—478no—
520C4⋊C4—478no—
521C4⋊C4—948no—
522C4⋊C4—944no—
523C4⋊C4—944no—
524C4⋊C4—1884no—
525C4⋊C4C2×C4×D4718yes—
526C22⋊C4—44no—
527C22⋊C4—478no—
528C22⋊C4—478no—
529C22⋊C4—948no—
530C22⋊C4—1884no—
531C22⋊C4—1884no—
532C22⋊C4—1884no—
533C22⋊C4C2×C4×D4718yes—
534C42—44no—
535C42—474no—
536C42—474no—
537C42—478no—
538C42—478no—
539C42—478no—
540C42—948no—
541C42—1884no—
542C42—1884no—
543C42—1884no—
544C42C2×C4×D4714yes—
545C42C2×C4×D4714yes—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—472no—
551C23—472no—
552C23—472no—
553C23—472no—
554C23—942no—
555C23—942no—
556C23—941no—
557C23—941no—
558C23—942no—
559C23—942no—
560C23C22×C4×D4712yes—
561C23C24×D4711yescharacteristic
562Q8—948no—
563D4—948no—
564D4—948no—
565D4—948no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—472no—
580C2×C4—472no—
581C2×C4—472no—
582C2×C4—4716no—
583C2×C4—474no—
584C2×C4—474no—
585C2×C4—472no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—942no—
592C2×C4—948no—
593C2×C4—944no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—944no—
599C2×C4—3768no—
600C2×C4—1888no—
601C2×C4—1888no—
602C2×C4—1888no—
603C2×C4C22×C4×D4712yes—
604C2×C4C22×C4×D4714yes—
605C2×C4C22×C4×D4714yes—
606C2×C4C22×C4×D4712yes—
607C2×C4#41×D4712yes—
608C2×C4#41×D4712yes—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—474no—
615C22—944no—
616C22—942no—
617C22—942no—
618C22—944no—
619C22—942no—
620C22—944no—
621C22—1882no—
622C22—1881no—
623C22—1881no—
624C22C23×C4×D4711yescharacteristic
625C22C2×#41×D4711yescharacteristic
626C22C2×#41×D4711yescharacteristic
627C22#1568995812yes—
628C22#1568996012yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—474no—
635C4—942no—
636C4—944no—
637C4—944no—
638C4—1884no—
639C4—1884no—
640C4#117×D4714yes—
641C2—42no—
642C2—944no—
643C2—1882no—
644C2C2×#117×D4711yescharacteristic
645C2#2025658812yes—
646C2#2025728012yes—
647C2#2025730711yescharacteristic
648C2#2025731111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2314C2
socle, cosocleC23×C47C25
prime radicalC22×#10625350C22
hypercenter#2314D47
hypocenterC47#34189
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #20246358C2semidirect
3. #20246350C2semidirect
4. #20246327C2semidirect
5. #20246323C2semidirect
6. #20245998C2semidirect
7. #20245997C2semidirect
8. #20245996C2semidirect
10. #20245793C2semidirect
11. #20245784C2semidirect
12. #20245783C2semidirect
13. #382×D47C2semidirect
14. #20245742C2semidirect
15. C47×#2314C2semidirect
17. #15689591C4semidirect
85. #13093989C2×C4semidirect
88. (C22⋊D47)⋊C4C2×C4semidirect
91. #13093981C2×C4semidirect
105. C2×C4×Dic47C2×C4semidirect
116. C42×D47C2×C4semidirect
129. #117×C47C2×C4semidirect
325. #2314D47semidirect
396. C47#34189semidirect
418. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895912
#202463582
#202463502
#202463272
#202463232
#202459982
#202459972
#202459962
#202459772
#202457932
#202457842
#202457832
#382×D472
#202457422
C47×#23142
#3418947