Finite Group Databasebeta

← #26146964

Subgroups — #26146964

Order 12032.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689035C211yesmaximal, characteristic
2C2×#15689028C211yesmaximal, characteristic, complemented
3C2×#15689016C211yesmaximal, characteristic
4#20241658C211yesmaximal, characteristic, complemented
5#20241651C211yesmaximal, characteristic, complemented
6#20241646C212yesmaximal, complemented
7#20241644C212yesmaximal, complemented
8#20241635C212yesmaximal
9#20241631C212yesmaximal
10#20241613C211yesmaximal, characteristic, complemented
11C47×#1439C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×C22⋊Dic47C2211yescharacteristic
16C22×C22⋊Dic47C2211yescharacteristic
17C22×C22⋊Dic47C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19#15689035C418yescomplemented
20#15689028—24no—
21#15689016—24no—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2212yes—
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94—42no—
41C22×Dic94—42no—
42C22×Dic94C2×C414yes—
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2312yes—
46C23×Dic47C2312yes—
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2311yescharacteristic
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—42no—
51C2×Dic47⋊C4—42no—
52C2×Dic47⋊C4—42no—
53C2×Dic47⋊C4—42no—
54C2×C22⋊Dic47—22no—
55C2×C22⋊Dic47—22no—
56C2×C22⋊Dic47—42no—
57C2×C22⋊Dic47—42no—
58C2×C22⋊Dic47—42no—
59C2×C22⋊Dic47—42no—
60C2×C22⋊Dic47C2×C418yes—
61C2×C4×Dic47—42no—
62C2×C4×Dic47D414yes—
63C47⋊#95—22no—
64C47⋊#95—22no—
65C47⋊#95—24no—
66C47⋊#95—24no—
67C47⋊#95—42no—
68C47⋊#95—42no—
69C47⋊#95—42no—
70C47⋊#95—42no—
71C47⋊#95—42no—
72C47⋊#95C2×C414yes—
73C47⋊#95C2×C418yes—
74C25×C47C2311yescharacteristic
75C23×C4×C47C2312yes—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—42no—
81C2×C22⋊C4×C47—22no—
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47—42no—
84C2×C22⋊C4×C47—42no—
85C2×C22⋊C4×C47—42no—
86C2×C22⋊C4×C47C2×C414yes—
87#95×C47—22no—
88#95×C47—42no—
89C24×C47—21no—
90C24×C47—21no—
91C24×C47—21no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—21no—
96C24×C47—21no—
97C24×C47—21no—
98C24×C47—22no—
99C24×C47C22×C412yes—
100C24×C47C2411yescharacteristic
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
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×C47C22×C412yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47#4112yes—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—21no—
128C22×Dic47—22no—
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—21no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47C22×C414yes—
156C22×Dic47C22×C414yes—
157C22×Dic47C2×D412yes—
158C22×Dic47C2×D412yes—
159C2×Dic94—28no—
160C2×Dic94—48no—
161C2×Dic94—44no—
162C2×Dic94—44no—
163C2×Dic94—48no—
164C4⋊C4×C47—44no—
165C22⋊C4×C47—28no—
166C22⋊C4×C47—48no—
167C22⋊Dic47—48no—
168C22⋊Dic47—44no—
169C22⋊Dic47—44no—
170C4⋊Dic47—44no—
171Dic47⋊C4—48no—
172C4×Dic47—44no—
173C23×C47—22no—
174C23×C47—24no—
175C23×C47—22no—
176C23×C47—24no—
177C23×C47—22no—
178C23×C47—41no—
179C23×C47—41no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47—41no—
183C23×C47—41no—
184C23×C47—42no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47C23×C411yescharacteristic
197C23×C47C22×D411yescharacteristic
198C23×C47C2×#4111yescharacteristic
199C23×C47H(5,𝔽2)12yes—
200C23×C47H(5,𝔽2)12yes—
201C23×C47H(5,𝔽2)11yescharacteristic
202C23×C47H(5,𝔽2)12yes—
203C23×C472−1+411yescharacteristic
204C23×C472−1+411yescharacteristic
205C23×C472−1+412yes—
206C23×C472−1+411yescharacteristic
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—24no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
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×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47C4×D414yes—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
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—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265Dic94—416no—
266#14625—471nomaximal, Sylow
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—44no—
271C22×C47—44no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—82no—
276C22×C47—82no—
277C22×C47—81no—
278C22×C47C2×C4×D411yescharacteristic
279C22×C47#51812yes—
280C22×C47#51911yescharacteristic
281C22×C47#52012yes—
282C22×C47#52011yescharacteristic
283C22×C47#53411yescharacteristic
284C22×C47#53511yescharacteristic
285C22×C47#53511yescharacteristic
286C22×C47#53611yescharacteristic
287C22×C47#53611yescharacteristic
288C22×C47#53711yescharacteristic
289C22×C47#53812yes—
290C22×C47#54111yescharacteristic
291C22×C47#54212yes—
292C22×C47#54211yescharacteristic
293C22×C47H(3,𝔽4)12yes—
294C22×C47#56212yes—
295C22×C47#56212yes—
296C22×C47#56212yes—
297C22×C47#56212yes—
298C22×C47#56312yes—
299C22×C47#56312yes—
300C22×C47#56312yes—
301C4×C47—24no—
302C4×C47—42no—
303C4×C47—82no—
304C4×C47—82no—
305Dic47—42no—
306Dic47—44no—
307Dic47—44no—
308Dic47—42no—
309Dic47—81no—
310Dic47—81no—
311C2×#393—471no—
312C2×#391—471no—
313C2×#388—471no—
314#1446—471no—
315#1444—471no—
316#1442—472no—
317#1442—472no—
318#1440—472no—
319#1439—472no—
320#1439D4711yescharacteristic
321#1434—471no—
322C2×C47—82no—
323C2×C47#232111yescharacteristic
324C2×C47#232311yescharacteristic
325C2×C47#232611yescharacteristic
326C2×C47#237812yes—
327C2×C47#237912yes—
328C2×C47#281411yescharacteristic
329C2×C47#281611yescharacteristic
330C2×C47#281911yescharacteristic
331C2×C47#282111yescharacteristic
332C2×C47#283112yes—
333C2×C47#283212yes—
334C23×Q8—471no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4C2×D4711yescharacteristic
338C22×C22⋊C4—472no—
339C22×C22⋊C4—471no—
340C22×C22⋊C4—471no—
341C22×C22⋊C4C2×D4711yescharacteristic
342C22×C22⋊C4C2×D4712yes—
343C22×C42—471no—
344#393—478no—
345#391—944no—
346#388—944no—
347C2×#95—471no—
348C2×#95—472no—
349C2×#95—472no—
350C2×#95—472no—
351C2×#95—471no—
352C2×#95—471no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95C2×D4711yescharacteristic
359C2×#95C2×D4712yes—
360C47#1462511yescharacteristic, complemented, Sylow
361C25C22×D4711yescharacteristic
362C22×Q8—474no—
363C22×Q8—942no—
364C22×Q8—942no—
365C22×Q8—1882no—
366C22×Q8—1882no—
367C23×C4—472no—
368C23×C4—471no—
369C23×C4—472no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4C22×D4712yes—
374C23×C4C22×D4712yes—
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—1882no—
381C2×C4⋊C4—1882no—
382C2×C4⋊C4—1882no—
383C2×C4⋊C4—1882no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—478no—
390C2×C22⋊C4—942no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—1882no—
393C2×C22⋊C4—1882no—
394C2×C22⋊C4—1882no—
395C2×C22⋊C4—1882no—
396C2×C22⋊C4C4×D4714yes—
397C2×C42—474no—
398C2×C42—1882no—
399#95—22no—
400#95—42no—
401#95—474no—
402#95—478no—
403#95—942no—
404#95—944no—
405#95—944no—
406#95—942no—
407#95—1882no—
408#95—1882no—
409#95—1882no—
410#95—1882no—
411#95—1882no—
412C24—21no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—21no—
422C24C2×C4×D4712yes—
423C24C23×D4711yescharacteristic
424C2×Q8—948no—
425C2×Q8—1888no—
426C2×Q8—1884no—
427C2×Q8—1884no—
428C2×Q8—1888no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—474no—
450C22×C4—474no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—941no—
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—941no—
470C22×C4—941no—
471C22×C4—942no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—941no—
479C22×C4—942no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4C2×C4×D4712yes—
484C22×C4C2×C4×D4712yes—
485C22×C4D4×D4712yes—
486C22×C4#1062535212yes—
487C4⋊C4—44no—
488C4⋊C4—1884no—
489C4⋊C4—1888no—
490C22⋊C4—28no—
491C22⋊C4—48no—
492C22⋊C4—1888no—
493C22⋊C4—1884no—
494C22⋊C4—1884no—
495C42—1884no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23C22×C4×D4711yescharacteristic
520C23C2×D4×D4711yescharacteristic
521C23C2×#1062535211yescharacteristic
522C23#1309406912yes—
523C23#1309406911yescharacteristic
524C23#1309406912yes—
525C23#1309407012yes—
526C23#1309407111yescharacteristic
527C23#1309407111yescharacteristic
528C23#1309407112yes—
529C23#1309407211yescharacteristic
530Q8—18816no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—944no—
555C2×C4—944no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—944no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—1882no—
567C2×C4—1882no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4(C22⋊D47)⋊C414yes—
589C22—44no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22C2×(C22⋊D47)⋊C411yescharacteristic
601C22#1568959512yes—
602C22#1568960011yescharacteristic
603C22#1568960212yes—
604C22#1568960411yescharacteristic
605C22#1568968411yescharacteristic
606C22#1568968611yescharacteristic
607C22#1568968811yescharacteristic
608C22#1568969311yescharacteristic
609C22#1568969611yescharacteristic
610C22#1568970211yescharacteristic
611C22#1568971012yes—
612C22#1568973311yescharacteristic
613C22#1568974011yescharacteristic
614C22#1568974012yes—
615C22#1568986612yes—
616C22#1568986812yes—
617C22#1568986912yes—
618C22#1568987012yes—
619C22#1568987012yes—
620C22#1568987312yes—
621C22#1568987412yes—
622C22#1568987512yes—
623C4—24no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—3761no—
628C4—3761no—
629C4—1882no—
630C4—1882no—
631C4—1884no—
632C4—1884no—
633C2—82no—
634C2#2024605111yescharacteristic
635C2#2024606611yescharacteristic
636C2#2024609811yescharacteristic
637C2#2024654812yes—
638C2#2024656112yes—
639C2#2025140211yescharacteristic
640C2#2025141711yescharacteristic
641C2#2025143511yescharacteristic
642C2#2025145011yescharacteristic
643C2#2025154612yes—
644C2#2025155912yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14625
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#15689028C2semidirect
4. #20241658C2semidirect
5. #20241651C2semidirect
6. #20241646C2semidirect
7. #20241644C2semidirect
10. #20241613C2semidirect
19. #15689035C4semidirect
360. C47#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#156890352
C2×#156890282
C2×#156890162
#202416582
#202416512
#202416462
#202416442
#202416352
#202416312
#202416132
C47×#14392
#1462547