Finite Group Databasebeta

← #26147408

Subgroups — #26147408

Order 12032.

1486 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×#15688985C211yesmaximal, characteristic
2C2×#15688982C211yesmaximal, characteristic
3C2×#15688980C211yesmaximal, characteristic
4#20241651C211yesmaximal, characteristic, complemented
5#20241651C212yesmaximal, complemented
6#20241645C211yesmaximal, characteristic, complemented
7#20241644C211yesmaximal, characteristic, complemented
8#20241643C212yesmaximal, complemented
9#20241632C212yesmaximal
10#20241630C212yesmaximal
11C47×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2211yescharacteristic
15C22×Dic47⋊C4C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2211yescharacteristic
18C22×(C2×D47)⋊C4C2211yescharacteristic
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2212yes—
21#15688985C418yescomplemented
22#15688982—24no—
23#15688980—24no—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2212yes—
35C2×#95×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C24×D47C2311yescharacteristic
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2312yes—
43C23×Dic47C2311yescharacteristic
44C22×C4×D47—42no—
45C22×C4×D47—42no—
46C22×C4×D47C2×C414yes—
47C22×C4×D47D414yes—
48C22×C4×D47Q814yes—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—42no—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—42no—
54C2×Dic47⋊C4—42no—
55C2×Dic47⋊C4—42no—
56C2×Dic47⋊C4—42no—
57C2×Dic47⋊C4—42no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—42no—
62C2×(C2×D47)⋊C4—42no—
63C2×(C2×D47)⋊C4—42no—
64C2×(C2×D47)⋊C4—42no—
65C2×(C2×D47)⋊C4—42no—
66C2×(C2×D47)⋊C4—42no—
67C2×(C2×D47)⋊C4C2×C418yes—
68C47⋊#95—22no—
69C47⋊#95—24no—
70C47⋊#95—42no—
71C47⋊#95—42no—
72C47⋊#95—42no—
73C47⋊#95—42no—
74C47⋊#95—42no—
75C47⋊#95C2×C414yes—
76C47⋊#95C2×C418yes—
77C23×C4×C47C2312yes—
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—42no—
84C2×C4⋊C4×C47—42no—
85C2×C4⋊C4×C47—42no—
86C2×C4⋊C4×C47—42no—
87#95×C47—22no—
88#95×C47—42no—
89#95×C47C2×C414yes—
90C24×C47C2411yescharacteristic
91C23×D47—21no—
92C23×D47—22no—
93C23×D47—21no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—21no—
97C23×D47—22no—
98C23×D47—21no—
99C23×D47C22×C412yes—
100C23×D47C2×D412yes—
101C23×D47C2×Q812yes—
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—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—21no—
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—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47C22×C414yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×Q812yes—
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—21no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—21no—
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—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47C22×C414yes—
157C22×Dic47C22×C412yes—
158C22×Dic47C2×D412yes—
159C22×Dic47C2×Q812yes—
160C2×C4×D47—48no—
161C2×C4×D47—44no—
162C2×C4×D47—44no—
163C2×C4×D47—48no—
164C2×C4×D47C4⋊C4116yescomplemented
165C4⋊C4×C47—44no—
166C4⋊C4×C47—48no—
167(C2×D47)⋊C4—48no—
168(C2×D47)⋊C4—48no—
169(C2×D47)⋊C4—44no—
170(C2×D47)⋊C4—44no—
171C4⋊Dic47—44no—
172Dic47⋊C4—44no—
173Dic47⋊C4—44no—
174Dic47⋊C4—48no—
175C23×C47C23×C411yescharacteristic
176C23×C47C22×D411yescharacteristic
177C23×C47C22×Q811yescharacteristic
178C23×C47H(5,𝔽2)12yes—
179C23×C47H(5,𝔽2)11yescharacteristic
180C23×C47H(5,𝔽2)12yes—
181C23×C47H(5,𝔽2)12yes—
182C23×C47H(5,𝔽2)11yescharacteristic
183C23×C472−1+411yescharacteristic
184C23×C472−1+412yes—
185C23×C472−1+411yescharacteristic
186C22×D47—24no—
187C22×D47—22no—
188C22×D47—24no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—42no—
201C22×D47—42no—
202C22×D47—42no—
203C22×D47—42no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47C2×C4⋊C414yes—
209C2×C4×C47—24no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47C2×C4⋊C414yes—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—42no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47C2×C4⋊C414yes—
267C4×D47—416no—
268#14674—471nomaximal, Sylow
269C22×C47C22×C4⋊C411yescharacteristic
270C22×C47#51812yes—
271C22×C47#52011yescharacteristic
272C22×C47#52012yes—
273C22×C47#52011yescharacteristic
274C22×C47#53411yescharacteristic
275C22×C47#53411yescharacteristic
276C22×C47#53511yescharacteristic
277C22×C47#53511yescharacteristic
278C22×C47#53611yescharacteristic
279C22×C47#53611yescharacteristic
280C22×C47#54312yes—
281C22×C47#54412yes—
282C22×C47#54411yescharacteristic
283C22×C47#54411yescharacteristic
284C22×C47#56012yes—
285C22×C47#56012yes—
286C22×C47#56212yes—
287C22×C47#56212yes—
288C22×C47#56212yes—
289C22×C47#56212yes—
290C22×C47#56312yes—
291C22×C47#56312yes—
292C2×D47—24no—
293C2×D47—24no—
294C2×D47—24no—
295C2×D47—42no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—44no—
299C2×D47—44no—
300C2×D47—44no—
301C2×D47—44no—
302C2×D47—42no—
303C4×C47—24no—
304C4×C47—44no—
305C4×C47—81no—
306C4×C47—81no—
307C4×C47—82no—
308Dic47—24no—
309Dic47—44no—
310Dic47—81no—
311Dic47—82no—
312Dic47—81no—
313C2×#381—471no—
314C2×#380—471no—
315C2×#380—471no—
316#1444—472no—
317#1444—471no—
318#1444D4711yescharacteristic, complemented
319#1442—472no—
320#1442—471no—
321#1442—471no—
322#1439—472no—
323#1439—472no—
324C2×C47#231511yescharacteristic
325C2×C47#231711yescharacteristic
326C2×C47#231711yescharacteristic
327C2×C47#237612yes—
328C2×C47#237612yes—
329C2×C47#281511yescharacteristic
330C2×C47#281611yescharacteristic
331C2×C47#281611yescharacteristic
332C2×C47#281711yescharacteristic
333C2×C47#284112yes—
334C2×C47#284112yes—
335D47—44no—
336C24×C4—471no—
337C22×C4⋊C4—471no—
338C22×C4⋊C4—471no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—472no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4712yes—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—471no—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—472no—
347#381—478no—
348#380—944no—
349#380—944no—
350C2×#95—472no—
351C2×#95—471no—
352C2×#95—472no—
353C2×#95—471no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—472no—
359C2×#95C2×D4711yescharacteristic
360C2×#95C2×D4712yes—
361C2×#95C2×D4711yescharacteristic
362C47#1467411yescharacteristic, complemented, Sylow
363C25—471no—
364C23×C4—471no—
365C23×C4—474no—
366C23×C4—474no—
367C23×C4—474no—
368C23×C4—472no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—472no—
372C23×C4—1882no—
373C23×C4—1882no—
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4712yes—
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4712yes—
378C23×C4C22×D4711yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—1882no—
388C2×C4⋊C4—1882no—
389C2×C4⋊C4—1882no—
390C2×C4⋊C4—1882no—
391C2×C4⋊C4—1882no—
392C2×C4⋊C4—1882no—
393C2×C22⋊C4—478no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—942no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—1882no—
398C2×C22⋊C4—1882no—
399C2×C22⋊C4—1882no—
400C2×C22⋊C4—1882no—
401C2×C22⋊C4—1882no—
402C2×C22⋊C4—1882no—
403#95—22no—
404#95—42no—
405#95—478no—
406#95—474no—
407#95—944no—
408#95—942no—
409#95—1882no—
410#95—1882no—
411#95—1882no—
412#95—1882no—
413#95—1882no—
414#95C4×D4714yes—
415C24—472no—
416C24—472no—
417C24—472no—
418C24—941no—
419C24—942no—
420C24—942no—
421C24—941no—
422C24—942no—
423C24—941no—
424C24—942no—
425C24—941no—
426C24C23×D4711yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—472no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—4716no—
456C22×C4—474no—
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—941no—
466C22×C4—941no—
467C22×C4—942no—
468C22×C4—941no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—1888no—
483C22×C4—1888no—
484C22×C4—1884no—
485C22×C4—1884no—
486C22×C4C2×C4×D4714yes—
487C22×C4C2×C4×D4712yes—
488C22×C4D4×D4712yes—
489C22×C4Q8×D4712yes—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—1884no—
493C4⋊C4—1884no—
494C4⋊C4—1884no—
495C4⋊C4—1888no—
496C22⋊C4—1888no—
497C22⋊C4—1884no—
498C22⋊C4—1884no—
499C22⋊C4—1888no—
500C23—474no—
501C23—944no—
502C23—944no—
503C23—942no—
504C23—942no—
505C23—944no—
506C23—942no—
507C23—944no—
508C23—942no—
509C23—942no—
510C23—942no—
511C23—942no—
512C23—942no—
513C23—942no—
514C23—942no—
515C23—1882no—
516C23—1882no—
517C23—1882no—
518C23—1882no—
519C23—1882no—
520C23—1882no—
521C23—1882no—
522C23—1882no—
523C23C22×C4×D4711yescharacteristic
524C23C2×D4×D4711yescharacteristic
525C23C2×Q8×D4711yescharacteristic
526C23#1309406912yes—
527C23#1309406911yescharacteristic
528C23#1309406912yes—
529C23#1309407012yes—
530C23#1309407011yescharacteristic
531C23#1309407111yescharacteristic
532C23#1309407212yes—
533C23#1309407211yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
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—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—474no—
563C2×C4—944no—
564C2×C4—944no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—18816no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1882no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1882no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4C4⋊C4×D4714yes—
593C22—944no—
594C22—944no—
595C22—944no—
596C22—1884no—
597C22—1884no—
598C22—1882no—
599C22—1882no—
600C22—1882no—
601C22—1882no—
602C22—1884no—
603C22—1884no—
604C22C2×C4⋊C4×D4711yescharacteristic
605C22#1568959512yes—
606C22#1568960211yescharacteristic
607C22#1568960412yes—
608C22#1568960511yescharacteristic
609C22#1568968311yescharacteristic
610C22#1568968411yescharacteristic
611C22#1568968611yescharacteristic
612C22#1568968911yescharacteristic
613C22#1568969611yescharacteristic
614C22#1568969811yescharacteristic
615C22#1568974812yes—
616C22#1568975012yes—
617C22#1568975011yescharacteristic
618C22#1568975311yescharacteristic
619C22#1568986112yes—
620C22#1568986212yes—
621C22#1568986912yes—
622C22#1568986912yes—
623C22#1568987012yes—
624C22#1568987112yes—
625C22#1568987212yes—
626C22#1568987412yes—
627C4—24no—
628C4—44no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—944no—
633C4—3761no—
634C4—3761no—
635C4—3762no—
636C4—1884no—
637C2—1884no—
638C2#2024599911yescharacteristic
639C2#2024600911yescharacteristic
640C2#2024601011yescharacteristic
641C2#2024653312yes—
642C2#2024653712yes—
643C2#2025140911yescharacteristic
644C2#2025141711yescharacteristic
645C2#2025142111yescharacteristic
646C2#2025142211yescharacteristic
647C2#2025164212yes—
648C2#2025164412yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1444C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1444D47
hypocenterC47#14674
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
4. #20241651C2semidirect
5. #20241651C2semidirect
6. #20241645C2semidirect
7. #20241644C2semidirect
8. #20241643C2semidirect
11. C47×#1444C2semidirect
21. #15688985C4semidirect
164. C2×C4×D47C4⋊C4semidirect
318. #1444D47semidirect
362. C47#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#156889852
C2×#156889822
C2×#156889802
#202416512
#202416452
#202416442
#202416432
#202416322
#202416302
C47×#14442
#1467447