Finite Group Databasebeta

← #26216360

Subgroups — #26216360

Order 12032.

1382 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic, complemented
3#20245793C211yesmaximal, characteristic, complemented
4#20243481C212yesmaximal, complemented
5#20243393C211yesmaximal, characteristic, complemented
6#20243389C211yesmaximal, characteristic, complemented
7#20243385C211yesmaximal, characteristic, complemented
8#20243327C212yesmaximal
9#20243220C212yesmaximal, complemented
10#20243212C212yesmaximal, complemented
11C47×#1803C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C4×#10625354—24no—
14C4×#10625354C418yescomplemented
15C2×Q8×Dic47C2211yescharacteristic
16C2×C4×D188—21no—
17C2×C4×D188C2211yescharacteristic
18C2×#13093986—21no—
19C2×#13093986C2211yescharacteristic
20C2×#13093982—21no—
21C2×#13093982—21no—
22C2×#13093982C2211yescharacteristic
23C2×C42×D47—21no—
24C2×C42×D47C2211yescharacteristic
25#15689179—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2212yes—
28C2×#13093946C2212yes—
29C2×#13093944C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×#13093934C2212yes—
32#15688993C2211yescharacteristic
33#15688988—21no—
34#15688980—21no—
35#15688980C2212yes—
36C4⋊C4×Dic47—21no—
37C4⋊C4×Dic47C2212yes—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#384C2211yescharacteristic
45C47×#340C2212yes—
46C2×#10625354—22no—
47C2×#10625354—22no—
48C2×#10625354—22no—
49C2×#10625354—22no—
50C2×#10625354—22no—
51C2×#10625354C2×C414yes—
52C22×D188—21no—
53C22×D188C2311yescharacteristic
54C22×C4×D47—21no—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47D411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C22×C47⋊C8C2312yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yes—
63C4×D188—28no—
64C4×D188—22no—
65C4×D188C2×C414yes—
66#13093986—28no—
67#13093986—22no—
68#13093986C2×C414yes—
69#13093982—22no—
70#13093982—28no—
71#13093982—44no—
72#13093982C2×C414yes—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2312yes—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—22no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—22no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C42×D47—28no—
89C42×D47C2×C414yes—
90C42×D47D414yes—
91C4⋊(C47⋊C8)—22no—
92#13093948—24no—
93#13093946—24no—
94#13093944—24no—
95(C2×D47)⋊C8—24no—
96#13093934—24no—
97C47⋊#95—22no—
98C47⋊#95—22no—
99C22×Q8×C47C2311yescharacteristic
100C22×C8×C47C2312yes—
101C4×Q8×C47—22no—
102C4×Q8×C47C2×C414yes—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47C2312yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C42×C47—21no—
109C2×C42×C47C2311yescharacteristic
110C4⋊C8×C47—22no—
111#103×C47—24no—
112C23×D47—21no—
113C23×D47C2×D411yescharacteristic
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×Q8×C47C22×C412yes—
118C2×Q8×C47#4112yes—
119C2×Q8×C47#4112yes—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2411yescharacteristic
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47C2×D411yescharacteristic
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131#10625354—28no—
132#10625354—28no—
133#10625354—28no—
134#10625354—28no—
135C2×D188—24no—
136C2×D188—24no—
137C2×D188—24no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188C22×C412yes—
142C2×D188#4112yes—
143C2×D188#4112yes—
144C2×C4×D47—24no—
145C2×C4×D47—24no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—21no—
149C2×C4×D47—21no—
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—24no—
155C2×C4×D47—41no—
156C2×C4×D47—41no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×C4×D47—42no—
160C2×C4×D47—42no—
161C2×C4×D47C22⋊C414yes—
162C2×C4×D47C22⋊C414yes—
163C2×C4×D47C22×C412yes—
164C2×C4×D47C2×D412yes—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—42no—
173C4⋊C4×C47—42no—
174C4⋊C4×C47C22×C412yes—
175C42×C47—24no—
176C42×C47C22×C412yes—
177C42×C47C2×D412yes—
178(C2×D47)⋊C4—24no—
179(C2×D47)⋊C4—24no—
180(C2×D47)⋊C4—24no—
181(C2×D47)⋊C4—44no—
182C4⋊Dic47—21no—
183C4⋊Dic47—24no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—42no—
187C4⋊Dic47—42no—
188C4⋊Dic47C22×C412yes—
189C4×Dic47—24no—
190C4×Dic47—24no—
191C4×Dic47—44no—
192C4×Dic47C22⋊C414yes—
193C4×Dic47C22⋊C414yes—
194C4×Dic47C22×C412yes—
195C4×Dic47C2×D412yes—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C23×C47C22×D411yescharacteristic
200C22×D47—22no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—41no—
210C22×D47—41no—
211C22×D47—41no—
212C22×D47—41no—
213C22×D47C2×C22⋊C412yes—
214Q8×C47—22no—
215Q8×C47—22no—
216Q8×C47—22no—
217Q8×C47#11714yescomplemented
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47C2×C22⋊C412yes—
230C2×C4×C47C2×C22⋊C412yes—
231C2×C4×C47AGL(1,ℤ/8)12yes—
232C2×C4×C47#13712yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—22no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47C2×C22⋊C412yes—
252C2×Dic47C2×C22⋊C412yes—
253C2×Dic47C2×C22⋊C412yes—
254D188—216no—
255D188—22no—
256D188—22no—
257D188—22no—
258D188#11714yescomplemented
259C4×D47—216no—
260C4×D47—22no—
261C4×D47—22no—
262C4×D47—22no—
263C4×D47—22no—
264C4×D47—28no—
265C4×D47—44no—
266C4×D47—48no—
267C4×D47—48no—
268C8×C47—42no—
269C47⋊C8—42no—
270#20743—471nomaximal, Sylow
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47#52511yescharacteristic
273C22×C47#52511yescharacteristic
274C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C47C2×#13711yescharacteristic
276C22×C47#57511yescharacteristic
277C22×C47#57511yescharacteristic
278C2×D47—22no—
279C2×D47—22no—
280C2×D47—22no—
281C2×D47—24no—
282C2×D47—41no—
283C2×D47—41no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—41no—
287C2×D47—41no—
288C2×D47—42no—
289C2×D47—42no—
290C2×D47—42no—
291C2×D47—42no—
292C4×C47—22no—
293C4×C47—24no—
294C4×C47—42no—
295C4×C47—82no—
296C4×C47#41714yes—
297C4×C47C2×#11711yescharacteristic
298C4×C47C2×#11711yescharacteristic
299C4×C47#52511yescharacteristic
300C4×C47#52511yescharacteristic
301Dic47—22no—
302Dic47—22no—
303Dic47—22no—
304Dic47—42no—
305Dic47—82no—
306C2×C4×#41—471no—
307C2×#432—471no—
308#2285—471no—
309#1825—472no—
310#1803—471no—
311#1803D4711yescharacteristic, complemented
312#1802—471no—
313#1802—471no—
314#1788—472no—
315#1763—472no—
316#1762—472no—
317C2×C47#228011yescharacteristic
318C2×C47C2×#41711yescharacteristic
319C2×C47#289611yescharacteristic
320C2×C47#311511yescharacteristic
321C2×C47#311511yescharacteristic
322C2×C47#311611yescharacteristic
323C2×C47#311611yescharacteristic
324D47—42no—
325D47—44no—
326C22×#41—471no—
327C4×#41—478no—
328C4×#41—944no—
329C2×C4×Q8—471no—
330C2×C4×Q8C2×D4711yescharacteristic
331C2×C4×D4—471no—
332C2×C4×D4—471no—
333C2×C4×D4—941no—
334C2×C4×D4—941no—
335C2×#117—471no—
336C2×#117—941no—
337C2×#117—941no—
338C22×C42—471no—
339C22×C42—941no—
340#432—944no—
341C2×C4⋊C8—471no—
342C2×C4⋊C8C2×D4711yescharacteristic
343C2×#103—472no—
344C2×#103C2×D4712yes—
345C2×#102—472no—
346C2×#102—472no—
347C2×C22⋊C8—472no—
348C2×#97—472no—
349#384—471no—
350#384C2×D4711yescharacteristic
351#382—941no—
352#380—472no—
353#380—941no—
354C4×C4⋊C4—472no—
355C4×C4⋊C4—941no—
356#340—472no—
357#340—472no—
358#340—472no—
359#340C2×D4712yes—
360C47#2074311yescharacteristic, complemented, Sylow
361C2×#41—474no—
362C2×#41—942no—
363C2×#41—942no—
364C2×#41—942no—
365C2×#41—942no—
366C2×#41—942no—
367C22×Q8C22×D4711yescharacteristic
368C22×D4—471no—
369C22×D4—941no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—941no—
374C23×C4—941no—
375C23×C4—941no—
376C22×C8—472no—
377C22×C8C22×D4712yes—
378C4×Q8—22no—
379C4×Q8—474no—
380C4×Q8—942no—
381C4×Q8C4×D4714yes—
382C4×D4—474no—
383C4×D4—474no—
384C4×D4—948no—
385C4×D4—942no—
386C4×D4—948no—
387C4×D4—942no—
388#117—474no—
389#117—948no—
390#117—942no—
391#117—1884no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—472no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4C22×D4712yes—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—942no—
406C2×C42—21no—
407C2×C42—471no—
408C2×C42—471no—
409C2×C42—471no—
410C2×C42—474no—
411C2×C42—474no—
412C2×C42—948no—
413C2×C42—941no—
414C2×C42—941no—
415C2×C42—942no—
416C2×C42C22×D4711yescharacteristic
417C4⋊C8—22no—
418C4⋊C8—942no—
419#103—24no—
420#103—944no—
421#102—944no—
422#102—944no—
423C22⋊C8—944no—
424#97—944no—
425#95—942no—
426#95—942no—
427C24—471no—
428C24—941no—
429#41—948no—
430#41—948no—
431#41—948no—
432#41—948no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D4712yes—
437C2×Q8#1062535212yes—
438C2×Q8#1062535412yes—
439C2×D4—472no—
440C2×D4—472no—
441C2×D4—472no—
442C2×D4—944no—
443C2×D4—944no—
444C2×D4—944no—
445C2×D4—941no—
446C2×D4—941no—
447C2×D4—941no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—474no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—474no—
458C22×C4—944no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—944no—
470C22×C4—942no—
471C22×C4—1881no—
472C22×C4—1881no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4—1882no—
476C22×C4—1882no—
477C22×C4D4×D4711yescharacteristic
478C22×C4D4×D4711yescharacteristic
479C22×C4C23×D4711yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—942no—
484C2×C8—942no—
485C2×C8—942no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—472no—
493C4⋊C4—944no—
494C4⋊C4—941no—
495C4⋊C4—942no—
496C4⋊C4—942no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4C2×C4×D4712yes—
500C22⋊C4—944no—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C22⋊C4—1884no—
504C42—24no—
505C42—474no—
506C42—474no—
507C42—472no—
508C42—472no—
509C42—944no—
510C42—944no—
511C42—1884no—
512C42C2×C4×D4712yes—
513C42D4×D4712yes—
514C23—472no—
515C23—942no—
516C23—942no—
517C23—941no—
518C23—942no—
519C23—941no—
520C23—941no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23—1881no—
525C23—1881no—
526C23—1881no—
527C23—1881no—
528C23C2×D4×D4711yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8#1309398214yescomplemented
533D4—474no—
534D4—9416no—
535D4—942no—
536D4—942no—
537D4—942no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—9416no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—948no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—1884no—
566C2×C4—1888no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1888no—
573C2×C4C22⋊C4×D4712yes—
574C2×C4C22⋊C4×D4712yes—
575C2×C4#1309405112yes—
576C2×C4#1309405612yes—
577C2×C4C22×C4×D4711yescharacteristic
578C2×C4C2×D4×D4711yescharacteristic
579C2×C4#41×D4711yescharacteristic
580C2×C4#41×D4711yescharacteristic
581C2×C4C2×#1062535211yescharacteristic
582C2×C4C2×#1062535411yescharacteristic
583C8—42no—
584C8—1882no—
585C22—942no—
586C22—944no—
587C22—942no—
588C22—942no—
589C22—1882no—
590C22—1881no—
591C22—1882no—
592C22—1881no—
593C22—1882no—
594C22—1881no—
595C22—1882no—
596C22—1881no—
597C22—1882no—
598C22—1882no—
599C22C2×C22⋊C4×D4711yescharacteristic
600C22#1568963311yescharacteristic
601C22#1568963611yescharacteristic
602C22C2×#1309405111yescharacteristic
603C22C2×#1309405611yescharacteristic
604C22#1568991911yescharacteristic
605C22#1568992211yescharacteristic
606C4—22no—
607C4—24no—
608C4—42no—
609C4—82no—
610C4—942no—
611C4—942no—
612C4—942no—
613C4—3762no—
614C4—1882no—
615C4#1568912314yes—
616C4#117×D4711yescharacteristic
617C4C2×#1309398211yescharacteristic
618C4#1568963311yescharacteristic
619C4#1568963611yescharacteristic
620C2—1882no—
621C2—1884no—
622C2#2024577011yescharacteristic
623C2C2×#1568912311yescharacteristic
624C2#2025190411yescharacteristic
625C2#2025353811yescharacteristic
626C2#2025354211yescharacteristic
627C2#2025354911yescharacteristic
628C2#2025355311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1803C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenter#1803D47
hypocenterC47#20743
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#15689179C2semidirect
3. #20245793C2semidirect
4. #20243481C2semidirect
5. #20243393C2semidirect
6. #20243389C2semidirect
7. #20243385C2semidirect
9. #20243220C2semidirect
10. #20243212C2semidirect
11. C47×#1803C2semidirect
14. C4×#10625354C4semidirect
217. Q8×C47#117semidirect
258. D188#117semidirect
311. #1803D47semidirect
360. C47#20743semidirect
532. Q8#13093982semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891792
#202457932
#202434812
#202433932
#202433892
#202433852
#202433272
#202432202
#202432122
C47×#18032
#2074347