Finite Group Databasebeta

← #26144771

Subgroups — #26144771

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C2×#15689006C212yesmaximal, complemented
3#20241658C212yesmaximal, complemented
4#20241654C212yesmaximal, complemented
5#20241631C212yesmaximal
6#20241625C212yesmaximal
7Dic47⋊#95C212yesmaximal, complemented
8#20241611C211yesmaximal, characteristic
9C47×C22⋊#95C211yesmaximal, characteristic
10C23×Dic94C2211yescharacteristic
11C2×#13094003—28no—
12C2×#13094003C418yescomplemented
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16C22×C22⋊Dic47C2212yes—
17C22×C22⋊Dic47C2212yes—
18C22×C4×Dic47C2212yes—
19#15689006—28no—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C24×C4×C47C2211yescharacteristic
30C22×C22⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2211yescharacteristic
32C2×#95×C47C2212yes—
33C2×#95×C47C2212yes—
34C22×Dic94—24no—
35C22×Dic94—41no—
36C22×Dic94—41no—
37C22×Dic94—42no—
38C22×Dic94C2×C414yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43#13094003—232no—
44C2×C4⋊Dic47C2×C414yes—
45C2×C4⋊Dic47D418yes—
46C2×Dic47⋊C4—24no—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—24no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—42no—
51C2×Dic47⋊C4C2×C418yes—
52C2×C22⋊Dic47—24no—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—42no—
57C2×C22⋊Dic47C2×C418yes—
58C2×C4×Dic47—24no—
59C2×C4×Dic47—24no—
60C2×C4×Dic47—42no—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—24no—
64C47⋊#95—41no—
65C47⋊#95—42no—
66C47⋊#95—42no—
67C47⋊#95—42no—
68C47⋊#95—42no—
69C47⋊#95—41no—
70C25×C47C2311yescharacteristic
71C23×C4×C47—24no—
72C23×C4×C47—41no—
73C23×C4×C47—41no—
74C23×C4×C47—42no—
75C23×C4×C47C2×C414yes—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2312yes—
79C23×C4×C47C2311yescharacteristic
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47—41no—
82C2×C22⋊C4×C47—41no—
83C2×C22⋊C4×C47—41no—
84C2×C22⋊C4×C47—41no—
85C2×C22⋊C4×C47Q814yescomplemented
86#95×C47—24no—
87#95×C47—42no—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—21no—
91C24×C47—21no—
92C24×C47—21no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—21no—
96C24×C47C22×C412yes—
97C24×C47C2×Q812yes—
98C24×C47#4112yes—
99C24×C47C2411yescharacteristic
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—28no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—44no—
122C22×C4×C47—44no—
123C22×C4×C47—48no—
124C22×C4×C47—48no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C2×D414yes—
128C22×C4×C47C2×Q814yes—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47C22×C414yes—
153C22×Dic47C22×C414yes—
154C22×Dic47C2×D414yes—
155C22×Dic47C2×D414yes—
156C22×Dic47#4114yes—
157C2×Dic94—28no—
158C2×Dic94—44no—
159C2×Dic94—44no—
160C2×Dic94—48no—
161C2×Dic94—48no—
162C22⋊C4×C47—44no—
163C22⋊C4×C47—44no—
164C22⋊Dic47—216no—
165C22⋊Dic47—48no—
166C4⋊Dic47C22⋊C4116yescomplemented
167Dic47⋊C4—216no—
168Dic47⋊C4—48no—
169C4×Dic47—48no—
170C23×C47—22no—
171C23×C47—22no—
172C23×C47—22no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47C4×Q814yes—
193C23×C47C23×C411yescharacteristic
194C23×C47C22×D412yes—
195C23×C47C22×Q811yescharacteristic
196C23×C47C2×#4112yes—
197C23×C47C2×#4111yescharacteristic
198C23×C47H(5,𝔽2)11yescharacteristic
199C23×C47H(5,𝔽2)11yescharacteristic
200C23×C47H(5,𝔽2)12yes—
201C23×C47H(5,𝔽2)11yescharacteristic
202C23×C472−1+412yes—
203C23×C472−1+411yescharacteristic
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—416no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47C2×C22⋊C414yes—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47C2×C22⋊C418yes—
257Dic94—416no—
258#14448—471nomaximal, Sylow
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—44no—
267C22×C47—44no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47C22×C22⋊C411yescharacteristic
271C22×C47C2×C4×Q811yescharacteristic
272C22×C47C4×#4111yescharacteristic
273C22×C47#51811yescharacteristic
274C22×C47#51811yescharacteristic
275C22×C47#51911yescharacteristic
276C22×C47#52011yescharacteristic
277C22×C47#53512yes—
278C22×C47#53512yes—
279C22×C47#53811yescharacteristic
280C22×C47#53912yes—
281C22×C47#53911yescharacteristic
282C22×C47#54212yes—
283C22×C47#54311yescharacteristic
284C22×C47#54411yescharacteristic
285C22×C47#54612yes—
286C22×C47#54712yes—
287C22×C47#54812yes—
288C22×C47D4×Q812yes—
289C22×C47#55112yes—
290C22×C47#55412yes—
291C22×C47#56212yes—
292C22×C47#56312yes—
293C4×C47—24no—
294C4×C47—42no—
295C4×C47—44no—
296C4×C47—82no—
297Dic47—28no—
298Dic47—44no—
299Dic47—44no—
300Dic47—82no—
301C22×C22⋊Q8—471no—
302C2×#387—472no—
303#1446—472no—
304#1445—472no—
305#1439—472no—
306#1437—472no—
307C4⋊#95—472no—
308C22⋊#95—471no—
309C22⋊#95D4711yescharacteristic
310C2×C47—44no—
311C2×C47#230911yescharacteristic
312C2×C47#232911yescharacteristic
313C2×C47#233211yescharacteristic
314C2×C47#233911yescharacteristic
315C2×C47Q8×C22⋊C411yescharacteristic
316C2×C47#236811yescharacteristic
317C2×C47#237811yescharacteristic
318C2×C47#269212yes—
319C2×C47#269612yes—
320C2×C47#272012yes—
321C2×C47#272312yes—
322C23×Q8—471no—
323C24×C4C2×D4711yescharacteristic
324C2×C22⋊Q8—478no—
325C2×C22⋊Q8—948no—
326C22×C4⋊C4—471no—
327C22×C4⋊C4—472no—
328C22×C4⋊C4—472no—
329C22×C22⋊C4—472no—
330C22×C22⋊C4—472no—
331C22×C22⋊C4C2×D4711yescharacteristic
332C22×C22⋊C4C2×D4711yescharacteristic
333C22×C42—472no—
334#387—948no—
335C2×#95—472no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95—471no—
339C2×#95—471no—
340C2×#95—472no—
341C2×#95—472no—
342C2×#95—472no—
343C2×#95—472no—
344C2×#95C2×D4712yes—
345C2×#95C2×D4712yes—
346C47#1444811yescharacteristic, complemented, Sylow
347C25C22×D4711yescharacteristic
348C22×Q8—474no—
349C22×Q8—944no—
350C22×Q8—1881no—
351C22×Q8—1881no—
352C22×Q8—1882no—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4—41no—
356C23×C4—41no—
357C23×C4—472no—
358C23×C4—472no—
359C23×C4—472no—
360C23×C4—472no—
361C23×C4C4×D4714yes—
362C23×C4C22×D4712yes—
363C23×C4C22×D4712yes—
364C23×C4C22×D4711yescharacteristic
365C23×C4C22×D4711yescharacteristic
366C22⋊Q8—9432no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—478no—
369C2×C4⋊C4—478no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—944no—
374C2×C4⋊C4—1882no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—478no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—944no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—944no—
385C2×C22⋊C4—1882no—
386C2×C22⋊C4Dic9414yescomplemented
387C2×C42—944no—
388C2×C42—944no—
389C2×C42—1882no—
390#95—24no—
391#95—42no—
392#95—944no—
393#95—944no—
394#95—944no—
395#95—1882no—
396#95—1882no—
397#95—1881no—
398#95—1881no—
399#95—1882no—
400#95—1882no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24C2×Dic9412yes—
410C24C2×C4×D4712yes—
411C24#1062535012yes—
412C24C23×D4711yescharacteristic
413C2×Q8—948no—
414C2×Q8—1884no—
415C2×Q8—1884no—
416C2×Q8—1888no—
417C2×Q8—1888no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—474no—
444C22×C4—474no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4C2×Dic9414yes—
472C22×C4C2×C4×D4712yes—
473C22×C4C2×C4×D4712yes—
474C22×C4D4×D4714yes—
475C4⋊C4—4716no—
476C4⋊C4—9416no—
477C4⋊C4—1888no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—9416no—
481C22⋊C4—1888no—
482C42—1888no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C4×Dic9414yes—
506C23C22×C4×D4711yescharacteristic
507C23C2×D4×D4712yes—
508C23C22×Dic9411yescharacteristic
509C23#41×D4712yes—
510C23C2×#1062535011yescharacteristic
511C23#1309406911yescharacteristic
512C23#1309406911yescharacteristic
513C23#1309406911yescharacteristic
514C23#1309406912yes—
515C23#1309407112yes—
516C23#1309407211yescharacteristic
517Q8—18816no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—478no—
543C2×C4—944no—
544C2×C4—944no—
545C2×C4—944no—
546C2×C4—944no—
547C2×C4—944no—
548C2×C4—944no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—944no—
552C2×C4—944no—
553C2×C4—944no—
554C2×C4—944no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4—1882no—
560C2×C4—1882no—
561C2×C4—1882no—
562C2×C4—1882no—
563C2×C4—1882no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4C22⋊C4×D4714yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22C2×C22⋊C4×D4711yescharacteristic
583C22C2×C4×Dic9411yescharacteristic
584C22#1568959111yescharacteristic
585C22#1568959611yescharacteristic
586C22#1568959611yescharacteristic
587C22#1568960011yescharacteristic
588C22#1568960411yescharacteristic
589C22#1568968812yes—
590C22#1568968812yes—
591C22#1568971111yescharacteristic
592C22#1568971612yes—
593C22#1568972011yescharacteristic
594C22#1568974112yes—
595C22#1568974911yescharacteristic
596C22#1568975411yescharacteristic
597C22#1568977012yes—
598C22#1568977512yes—
599C22#1568978212yes—
600C22D4×Dic9412yes—
601C22#1568980212yes—
602C22#1568983212yes—
603C22#1568986812yes—
604C22#1568987412yes—
605C4—24no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—948no—
610C4—3762no—
611C4—1884no—
612C4—1884no—
613C2—44no—
614C2#2024596911yescharacteristic
615C2#2024612111yescharacteristic
616C2#2024614111yescharacteristic
617C2#2024621611yescharacteristic
618C2C22⋊C4×Dic9411yescharacteristic
619C2#2024647111yescharacteristic
620C2#2024655111yescharacteristic
621C2#2024988512yes—
622C2#2024993912yes—
623C2#2025026912yes—
624C2#2025030812yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C22⋊#95C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC22⋊#95D47
hypocenterC47#14448
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#15689006C2semidirect
3. #20241658C2semidirect
4. #20241654C2semidirect
7. Dic47⋊#95C2semidirect
12. C2×#13094003C4semidirect
85. C2×C22⋊C4×C47Q8semidirect
166. C4⋊Dic47C22⋊C4semidirect
346. C47#14448semidirect
386. C2×C22⋊C4Dic94semidirect

Maximal subgroups

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

subgroupindex
C22×#130940032
C2×#156890062
#202416582
#202416542
#202416312
#202416252
Dic47⋊#952
#202416112
C47×C22⋊#952
#1444847