Finite Group Databasebeta

← #26225203

Subgroups — #26225203

Order 12032.

1502 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2C2×#15689088C212yesmaximal, complemented
3#20243682C212yesmaximal, complemented
4#20243682C212yesmaximal, complemented
5#20243677C212yesmaximal, complemented
6#20243677C212yesmaximal, complemented
7#20243457C212yesmaximal, complemented
8#20243298C211yesmaximal, characteristic, complemented
9C47×#1779C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625324C2212yescomplemented
12C22×C8×D47C2212yescomplemented
13#15689634—28no—
14C2×#13094007C412yes—
15C2×#13094003C411yescharacteristic
16C2×#13094003C411yescharacteristic
17C2×#13093998C411yescharacteristic
18C2×#13093998C411yescharacteristic
19C2×#13093993C412yes—
20C2×C4×C22⋊D47C2212yes—
21C2×C4×C22⋊D47C2212yes—
22C2×#13093983C2211yescharacteristic
23#15689088—28no—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×C22⋊(C47⋊C8)C2212yes—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×(C2×D47)⋊C8C2212yescomplemented
29C2×(C2×D47)⋊C8C2212yescomplemented
30#15688877C2212yescomplemented
31#15688877C2212yescomplemented
32#15688877C2212yescomplemented
33#15688877C2212yescomplemented
34C24×C4×C47C2211yescharacteristic
35C2×C22⋊C8×C47C2212yes—
36C2×C22⋊C8×C47C2211yescharacteristic
37C2×C22⋊C8×C47C2212yes—
38C2×C22⋊C8×C47C2211yescharacteristic
39C2×#10625350—41no—
40C2×#10625350—42no—
41C2×#10625350—41no—
42C2×#10625350D418yes—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47C2×C412yes—
46C22×C22⋊D47C2×C412yes—
47C22×C22⋊D47C2×C412yes—
48C22×C4×D47C2312yes—
49C2×#10625324—24no—
50C2×#10625324—42no—
51C2×#10625324—42no—
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2312yes—
54C2×C8×D47—24no—
55C2×C8×D47—42no—
56C2×C8×D47—42no—
57#13094007—28no—
58#13094003—24no—
59#13094003—24no—
60#13093998—24no—
61#13093998—24no—
62#13093993—28no—
63C4×C22⋊D47—24no—
64C4×C22⋊D47—24no—
65C4×C22⋊D47—24no—
66C4×C22⋊D47—24no—
67#13093983—24no—
68C2×C4⋊Dic47C2×C411yescharacteristic
69C2×C4⋊Dic47C2×C411yescharacteristic
70C2×Dic47⋊C4C2×C412yes—
71C2×Dic47⋊C4C2×C412yes—
72C2×C22⋊Dic47C2×C412yes—
73C2×C22⋊Dic47C2×C412yes—
74C2×C22⋊Dic47C2×C412yes—
75C2×(C2×D47)⋊C4C2×C412yes—
76C2×(C2×D47)⋊C4C2×C412yes—
77C2×C4×Dic47C2312yes—
78C22⋊(C47⋊C8)—24no—
79C22⋊(C47⋊C8)—41no—
80C22⋊(C47⋊C8)—42no—
81C22⋊(C47⋊C8)—41no—
82(C2×D47)⋊C8—24no—
83(C2×D47)⋊C8—24no—
84C25×C47C2×C411yescharacteristic
85C23×C4×C47—24no—
86C23×C4×C47—42no—
87C23×C4×C47—41no—
88C23×C4×C47—41no—
89C23×C4×C47C2×C411yescharacteristic
90C23×C4×C47C2×C411yescharacteristic
91C23×C4×C47C2×C411yescharacteristic
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C22×C8×C47C2312yes—
96C22×C8×C47C2312yes—
97C22⋊C8×C47—24no—
98C22⋊C8×C47—41no—
99C22⋊C8×C47—42no—
100C22⋊C8×C47—41no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—41no—
105C24×C47—41no—
106C24×C47—41no—
107C24×C47—41no—
108C24×C47C22×C411yescharacteristic
109C24×C47C22×C411yescharacteristic
110C24×C47C22×C411yescharacteristic
111C23×D47C22×C412yes—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
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—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—41no—
127C22×C4×C47—41no—
128C22×C4×C47—41no—
129C22×C4×C47—41no—
130C22×C4×C47—41no—
131C22×C4×C47—41no—
132C22×C4×C47—48no—
133C22×C4×C47—48no—
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—24no—
140C2×C22⋊D47—24no—
141C2×C22⋊D47—24no—
142C2×C22⋊D47—24no—
143C2×C22⋊D47—42no—
144C2×C22⋊D47—42no—
145C2×C22⋊D47—42no—
146C2×C22⋊D47—42no—
147C2×C22⋊D47C22⋊C418yes—
148C22×Dic47C22×C412yes—
149C22×Dic47C22×C412yes—
150C22×Dic47C22×C412yes—
151#10625350—48no—
152#10625350—48no—
153C2×D188—41no—
154C2×D188—41no—
155C2×D188C22⋊C414yes—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47C2×D414yes—
161C2×C4×D47C2×D414yes—
162C2×Dic94—41no—
163C2×Dic94—41no—
164C2×Dic94C22⋊C414yes—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C22⋊Dic47—24no—
172C22⋊Dic47—24no—
173C22⋊Dic47—24no—
174(C2×D47)⋊C4—24no—
175(C2×D47)⋊C4—24no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178Dic47⋊C4—24no—
179Dic47⋊C4—24no—
180C4×Dic47—22no—
181C4×Dic47—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188#10625324—48no—
189C8×D47—48no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47—44no—
211C23×C47—44no—
212C23×C47—44no—
213C23×C47—44no—
214C23×C47—44no—
215C23×C47C2×C22⋊C412yes—
216C23×C47#13112yes—
217C23×C47#13112yes—
218C23×C47C23×C411yescharacteristic
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47C2×C22⋊C414yes—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
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—44no—
237C2×C4×C47—44no—
238C2×C4×C47—44no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47C22×D412yes—
245C2×C4×C47H(5,𝔽2)11yescharacteristic
246C2×C4×C47H(5,𝔽2)11yescharacteristic
247C2×C4×C47H(5,𝔽2)12yes—
248C22⋊D47—48no—
249C22⋊D47—48no—
250C22⋊D47—48no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47C2×C22⋊C414yes—
260D188—44no—
261C4×D47—24no—
262C4×D47—24no—
263C4×D47—24no—
264C4×D47—24no—
265Dic94—44no—
266C8×C47—42no—
267C8×C47—42no—
268C47⋊C8—42no—
269C47⋊C8—42no—
270#21744—471nomaximal, Sylow
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—416no—
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—41no—
293C22×C47C22×C22⋊C411yescharacteristic
294C22×C47#51811yescharacteristic
295C22×C47#51811yescharacteristic
296C22×C47C2×#13111yescharacteristic
297C22×C47C2×#13111yescharacteristic
298C22×C47#56811yescharacteristic
299C22×C47#56811yescharacteristic
300C2×D47—24no—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—42no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C4×C47—22no—
308C4×C47—41no—
309C4×C47—41no—
310C4×C47#53412yes—
311C4×C47#53412yes—
312Dic47—24no—
313Dic47—42no—
314Dic47—42no—
315C2×#525—471no—
316C2×#408—472no—
317#1870—472no—
318#1870—472no—
319#1869—472no—
320#1869—472no—
321#1818—472no—
322#1779—471no—
323#1779D4711yescharacteristic, complemented
324C2×C47—42no—
325C2×C47—42no—
326C2×C47—42no—
327C2×C47#230811yescharacteristic
328C2×C47#288011yescharacteristic
329C2×C47#288011yescharacteristic
330C2×C47#297611yescharacteristic
331C2×C47#297611yescharacteristic
332C2×C47#297711yescharacteristic
333C2×C47#297811yescharacteristic
334D47—44no—
335C22×#41—471no—
336C24×C4C2×D4711yescharacteristic
337C22×M16—472no—
338C23×C8—472no—
339#525—948no—
340C2×#123—472no—
341C2×C22⋊Q8—471no—
342C2×C22⋊Q8—471no—
343C2×#121—471no—
344C2×#121—471no—
345C2×#120—472no—
346C2×C4×D4—472no—
347C2×C4×D4—472no—
348C2×#117—471no—
349#408—948no—
350C2×C22⋊C8—472no—
351C2×C22⋊C8—472no—
352C2×C22⋊C8—472no—
353C2×C22⋊C8—471no—
354C2×C22⋊C8—472no—
355C2×C22⋊C8—471no—
356C2×C22⋊C8C2×D4712yes—
357C2×C22⋊C8C2×D4711yescharacteristic
358C2×C22⋊C8C2×D4712yes—
359C2×C22⋊C8C2×D4711yescharacteristic
360#336—472no—
361#336—472no—
362#336—472no—
363#336—472no—
364C47#2174411yescharacteristic, complemented, Sylow
365C25C4×D4711yescharacteristic
366C2×#41—478no—
367C2×#41—1881no—
368C2×#41—1882no—
369C2×#41—1881no—
370C22×Q8—471no—
371C22×D4—472no—
372C22×D4—472no—
373C22×D4—472no—
374C22×D4—471no—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4—472no—
380C23×C4C4×D4711yescharacteristic
381C23×C4C4×D4711yescharacteristic
382C23×C4C4×D4711yescharacteristic
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C2×M16—944no—
387C2×M16—1882no—
388C2×M16—1882no—
389C22×C8—472no—
390C22×C8—472no—
391C22×C8—944no—
392C22×C8—1882no—
393C22×C8—1882no—
394C22×C8C22×D4712yes—
395C22×C8C22×D4712yes—
396#123—948no—
397C22⋊Q8—944no—
398C22⋊Q8—944no—
399#121—944no—
400#121—944no—
401#120—948no—
402C4×D4—944no—
403C4×D4—944no—
404C4×D4—944no—
405C4×D4—944no—
406#117—944no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—472no—
410C2×C4⋊C4—472no—
411C2×C22⋊C4—472no—
412C2×C22⋊C4—472no—
413C2×C22⋊C4—472no—
414C2×C22⋊C4—472no—
415C2×C22⋊C4—472no—
416C2×C42—472no—
417C22⋊C8—24no—
418C22⋊C8—41no—
419C22⋊C8—42no—
420C22⋊C8—41no—
421C22⋊C8—944no—
422C22⋊C8—944no—
423C22⋊C8—944no—
424C22⋊C8—1881no—
425C22⋊C8—1882no—
426C22⋊C8—1881no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—41no—
431C24—41no—
432C24—41no—
433C24—41no—
434C24—472no—
435C24C2×C4×D4711yescharacteristic
436C24C2×C4×D4711yescharacteristic
437C24C2×C4×D4711yescharacteristic
438#41—1888no—
439#41—1888no—
440C2×Q8—474no—
441C2×Q8—1881no—
442C2×Q8—1881no—
443C2×D4—478no—
444C2×D4—474no—
445C2×D4—944no—
446C2×D4—944no—
447C2×D4—944no—
448C2×D4—944no—
449C2×D4—1881no—
450C2×D4—1881no—
451C2×D4—1882no—
452C2×D4—1882no—
453C2×D4—1882no—
454C2×D4—1882no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—472no—
478C22×C4—474no—
479C22×C4—474no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4C2×C4×D4711yescharacteristic
487C22×C4C2×C4×D4711yescharacteristic
488C22×C4C2×C4×D4711yescharacteristic
489C22×C4D4×D4714yes—
490C22×C4C23×D4711yescharacteristic
491M16—1888no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—942no—
499C2×C8—942no—
500C2×C8—942no—
501C2×C8—942no—
502C2×C8—942no—
503C2×C8—942no—
504C2×C8—1888no—
505C4⋊C4—944no—
506C4⋊C4—944no—
507C4⋊C4—942no—
508C4⋊C4—942no—
509C22⋊C4—944no—
510C22⋊C4—944no—
511C22⋊C4—944no—
512C22⋊C4—944no—
513C22⋊C4—944no—
514C42—942no—
515C42—942no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—474no—
542C23—942no—
543C23—942no—
544C23—942no—
545C23—942no—
546C23—942no—
547C23—942no—
548C23C22⋊C4×D4712yes—
549C23#1309403012yes—
550C23#1309403212yes—
551C23C22×C4×D4711yescharacteristic
552Q8—1884no—
553D4—1884no—
554D4—1888no—
555D4—1888no—
556D4—1888no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—474no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4C22⋊C4×D4712yes—
588C2×C4C2×D4×D4712yes—
589C2×C4#1309406911yescharacteristic
590C2×C4#1309406911yescharacteristic
591C2×C4#1309406912yes—
592C8—42no—
593C8—42no—
594C8—1882no—
595C8—1882no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—416no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—944no—
619C22—944no—
620C22—944no—
621C22—1882no—
622C22—1882no—
623C22—1882no—
624C22—1882no—
625C22C2×C22⋊C4×D4711yescharacteristic
626C22#1568959611yescharacteristic
627C22#1568959611yescharacteristic
628C22C2×#1309403011yescharacteristic
629C22C2×#1309403211yescharacteristic
630C22#1568988911yescharacteristic
631C22#1568988911yescharacteristic
632C4—22no—
633C4—41no—
634C4—41no—
635C4—944no—
636C4—1882no—
637C4—1882no—
638C4#1568968312yes—
639C4#1568968312yes—
640C2—42no—
641C2—42no—
642C2—42no—
643C2—1884no—
644C2#2024596411yescharacteristic
645C2#2025183411yescharacteristic
646C2#2025183811yescharacteristic
647C2#2025243311yescharacteristic
648C2#2025243811yescharacteristic
649C2#2025244211yescharacteristic
650C2#2025244811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1779C2
prime radicalC2×#15689634C2
cosocleC22×C4×C47C24
hypercenter#1779D47
hypocenterC47#21744
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#15689088C2semidirect
3. #20243682C2semidirect
4. #20243682C2semidirect
5. #20243677C2semidirect
6. #20243677C2semidirect
7. #20243457C2semidirect
8. #20243298C2semidirect
9. C47×#1779C2semidirect
11. C22×#10625324C22semidirect
12. C22×C8×D47C22semidirect
28. C2×(C2×D47)⋊C8C22semidirect
29. C2×(C2×D47)⋊C8C22semidirect
30. #15688877C22semidirect
31. #15688877C22semidirect
32. #15688877C22semidirect
33. #15688877C22semidirect
323. #1779D47semidirect
364. C47#21744semidirect

Maximal subgroups

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

subgroupindex
C2×#156896342
C2×#156890882
#202436822
#202436772
#202434572
#202432982
C47×#17792
#2174447