Finite Group Databasebeta

← #26144209

Subgroups — #26144209

Order 12032.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689018C212yesmaximal, complemented
2C2×#15689017C212yesmaximal
3C2×#15688995C211yesmaximal, characteristic
4C2×#15688981C212yesmaximal
5#20241652C212yesmaximal, complemented
6#20241639C212yesmaximal, complemented
7#20241633C211yesmaximal, characteristic
8#20241619C212yesmaximal, complemented
9C47×#1439C211yesmaximal, characteristic
10C24×Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16C22×C22⋊Dic47C2212yes—
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C4×Dic47C2212yes—
19#15689018—28no—
20#15689017—28no—
21#15688995C418yescomplemented
22#15688981—28no—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2212yes—
32C22×C22⋊C4×C47C2211yescharacteristic
33C2×#95×C47C2212yes—
34C2×#95×C47C2211yescharacteristic
35C23×Dic47—24no—
36C23×Dic47—24no—
37C23×Dic47—41no—
38C23×Dic47—41no—
39C23×Dic47C2×C414yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C23×Dic47C2312yes—
44C2×C4⋊Dic47—42no—
45C2×C4⋊Dic47—41no—
46C2×Dic47⋊C4—24no—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—24no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—42no—
55C2×Dic47⋊C4—42no—
56C2×Dic47⋊C4—42no—
57C2×Dic47⋊C4C2×C418yes—
58C2×C22⋊Dic47—24no—
59C2×C22⋊Dic47—24no—
60C2×C22⋊Dic47—42no—
61C2×C22⋊Dic47C2×C414yes—
62C2×C22⋊Dic47D414yes—
63C2×C22⋊Dic47Q814yes—
64C2×C4×Dic47—24no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—42no—
67C47⋊#95—24no—
68C47⋊#95—24no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—41no—
72C47⋊#95C2×C418yes—
73C47⋊#95D418yes—
74C25×C47C2311yescharacteristic
75C23×C4×C47C2312yes—
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2312yes—
79C2×C4⋊C4×C47—24no—
80C2×C4⋊C4×C47—41no—
81C2×C22⋊C4×C47—24no—
82C2×C22⋊C4×C47—24no—
83C2×C22⋊C4×C47—41no—
84C2×C22⋊C4×C47—41no—
85C2×C22⋊C4×C47—42no—
86C2×C22⋊C4×C47C2×C414yes—
87#95×C47—41no—
88#95×C47—42no—
89C24×C47—22no—
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—21no—
93C24×C47—21no—
94C24×C47—21no—
95C24×C47—21no—
96C24×C47—22no—
97C24×C47C22×C412yes—
98C24×C47C2×D412yes—
99C24×C47C2×Q812yes—
100C24×C47C2411yescharacteristic
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
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—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×C47—22no—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D414yes—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
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—216no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—44no—
148C22×Dic47—44no—
149C22×Dic47—48no—
150C22×Dic47—44no—
151C22×Dic47C22×C414yes—
152C22×Dic47C22×C414yes—
153C22×Dic47C2×D414yes—
154C22×Dic47C2×D414yes—
155C22×Dic47C2×D414yes—
156C22×Dic47C2×Q814yes—
157C22×Dic47#4114yes—
158C4⋊C4×C47—44no—
159C22⋊C4×C47—44no—
160C22⋊C4×C47—48no—
161C22⋊Dic47—48no—
162C22⋊Dic47C4⋊C4116yescomplemented
163C4⋊Dic47—44no—
164Dic47⋊C4—216no—
165Dic47⋊C4—48no—
166Dic47⋊C4—48no—
167Dic47⋊C4—48no—
168C4×Dic47—48no—
169C23×C47—24no—
170C23×C47—22no—
171C23×C47—22no—
172C23×C47—24no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—41no—
186C23×C47—42no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—42no—
190C23×C47—41no—
191C23×C47C2×C4⋊C414yes—
192C23×C47C23×C411yescharacteristic
193C23×C47C22×D411yescharacteristic
194C23×C47C22×D412yes—
195C23×C47C22×Q811yescharacteristic
196C23×C47C2×#4112yes—
197C23×C47H(5,𝔽2)11yescharacteristic
198C23×C47H(5,𝔽2)11yescharacteristic
199C23×C47H(5,𝔽2)12yes—
200C23×C472−1+411yescharacteristic
201C23×C472−1+411yescharacteristic
202C23×C472−1+412yes—
203C2×C4×C47—24no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—41no—
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×Dic47—24no—
228C2×Dic47—24no—
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—24no—
242C2×Dic47—416no—
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×Dic47C2×C4⋊C418yes—
255C2×Dic47C4×D418yes—
256#14402—471nomaximal, Sylow
257C22×C47—24no—
258C22×C47—28no—
259C22×C47—44no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—44no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47C22×C4⋊C411yescharacteristic
269C22×C47C2×C4×D412yes—
270C22×C47#51811yescharacteristic
271C22×C47#51911yescharacteristic
272C22×C47#52012yes—
273C22×C47#53612yes—
274C22×C47#53711yescharacteristic
275C22×C47#53711yescharacteristic
276C22×C47#53712yes—
277C22×C47#54112yes—
278C22×C47#54212yes—
279C22×C47#54411yescharacteristic
280C22×C47#54411yescharacteristic
281C22×C47D4212yes—
282C22×C47D4211yescharacteristic
283C22×C47#54612yes—
284C22×C47#54712yes—
285C22×C47D4×Q812yes—
286C22×C47#55111yescharacteristic
287C22×C47#55412yes—
288C22×C47#56011yescharacteristic
289C22×C47#56212yes—
290C22×C47#56311yescharacteristic
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—44no—
294C4×C47—82no—
295Dic47—28no—
296Dic47—28no—
297Dic47—44no—
298Dic47—82no—
299C2×#389—472no—
300C2×#388—472no—
301C2×#385—471no—
302C2×#380—472no—
303#1444—472no—
304#1441—472no—
305#1439—471no—
306#1439D4711yescharacteristic
307#1436—472no—
308C2×C47—44no—
309C2×C47#231611yescharacteristic
310C2×C47#232312yes—
311C2×C47D4×C4⋊C412yes—
312C2×C47#235811yescharacteristic
313C2×C47#237611yescharacteristic
314C2×C47#267212yes—
315C2×C47#267511yescharacteristic
316C2×C47#267711yescharacteristic
317C2×C47#270412yes—
318C2×C47#272211yescharacteristic
319C2×C47#274211yescharacteristic
320C24×C4—471no—
321C22×C4⋊C4—472no—
322C22×C4⋊C4—472no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—471no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4C2×D4712yes—
330C22×C22⋊C4C2×D4711yescharacteristic
331C22×C42—472no—
332#389—948no—
333#388—948no—
334#385—478no—
335#380—948no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95—472no—
339C2×#95—472no—
340C2×#95—472no—
341C2×#95—472no—
342C2×#95—471no—
343C2×#95C2×D4712yes—
344C2×#95C2×D4711yescharacteristic
345C47#1440211yescharacteristic, complemented, Sylow
346C25C22×D4711yescharacteristic
347C23×C4—472no—
348C23×C4—472no—
349C23×C4—472no—
350C23×C4—474no—
351C23×C4—472no—
352C23×C4—944no—
353C23×C4—944no—
354C23×C4—1881no—
355C23×C4—1881no—
356C23×C4C22×D4712yes—
357C23×C4C22×D4711yescharacteristic
358C23×C4C22×D4711yescharacteristic
359C23×C4C22×D4712yes—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—41no—
362C2×C4⋊C4—478no—
363C2×C4⋊C4—944no—
364C2×C4⋊C4—944no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—944no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—1882no—
372C2×C4⋊C4—1882no—
373C2×C4⋊C4—1882no—
374C2×C4⋊C4—1882no—
375C2×C4⋊C4—1881no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—474no—
382C2×C22⋊C4—474no—
383C2×C22⋊C4—474no—
384C2×C22⋊C4—944no—
385C2×C22⋊C4—944no—
386C2×C22⋊C4—1882no—
387C2×C22⋊C4C4×D4714yes—
388C2×C42—944no—
389C2×C42—944no—
390C2×C42—1882no—
391#95—42no—
392#95—41no—
393#95—478no—
394#95—478no—
395#95—944no—
396#95—944no—
397#95—944no—
398#95—944no—
399#95—1881no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—22no—
408C24C2×C4×D4712yes—
409C24D4×D4712yes—
410C24Q8×D4712yes—
411C24C23×D4711yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—474no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—9416no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
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—1884no—
463C22×C4—1884no—
464C22×C4—1888no—
465C22×C4—1884no—
466C22×C4C2×C4×D4712yes—
467C22×C4C2×C4×D4712yes—
468C22×C4D4×D4714yes—
469C4⋊C4—44no—
470C4⋊C4—9416no—
471C4⋊C4—1888no—
472C4⋊C4—1888no—
473C4⋊C4—1888no—
474C4⋊C4—1884no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—4716no—
478C22⋊C4—1888no—
479C42—1888no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23C4⋊C4×D4714yes—
503C23C22×C4×D4711yescharacteristic
504C23C2×D4×D4712yes—
505C23C2×D4×D4711yescharacteristic
506C23C2×Q8×D4711yescharacteristic
507C23#41×D4712yes—
508C23#1309406912yes—
509C23#1309406911yescharacteristic
510C23#1309406911yescharacteristic
511C23#1309407111yescharacteristic
512C23#1309407112yes—
513C23#1309407111yescharacteristic
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
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—478no—
539C2×C4—478no—
540C2×C4—944no—
541C2×C4—944no—
542C2×C4—944no—
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—18816no—
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—
567C22—28no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22C2×C4⋊C4×D4711yescharacteristic
579C22C4×D4×D4712yes—
580C22#1568959611yescharacteristic
581C22#1568960011yescharacteristic
582C22#1568960212yes—
583C22#1568969312yes—
584C22#1568970011yescharacteristic
585C22#1568970012yes—
586C22#1568970011yescharacteristic
587C22#1568973012yes—
588C22#1568974112yes—
589C22#1568975211yescharacteristic
590C22#1568975211yescharacteristic
591C22#1568975512yes—
592C22#1568975511yescharacteristic
593C22#1568977012yes—
594C22#1568977112yes—
595C22#1568978612yes—
596C22#1568980111yescharacteristic
597C22#1568982712yes—
598C22#1568986011yescharacteristic
599C22#1568986812yes—
600C22#1568987311yescharacteristic
601C4—42no—
602C4—44no—
603C4—42no—
604C4—82no—
605C4—948no—
606C4—948no—
607C4—3762no—
608C4—1884no—
609C2—44no—
610C2#2024600511yescharacteristic
611C2#2024606312yes—
612C2#2024628612yes—
613C2#2024637111yescharacteristic
614C2#2024653811yescharacteristic
615C2#2024963212yes—
616C2#2024967711yescharacteristic
617C2#2024969711yescharacteristic
618C2#2025003912yes—
619C2#2025029111yescharacteristic
620C2#2025051811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14402
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#15689018C2semidirect
5. #20241652C2semidirect
6. #20241639C2semidirect
8. #20241619C2semidirect
21. #15688995C4semidirect
162. C22⋊Dic47C4⋊C4semidirect
345. C47#14402semidirect

Maximal subgroups

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

subgroupindex
C2×#156890182
C2×#156890172
C2×#156889952
C2×#156889812
#202416522
#202416392
#202416332
#202416192
C47×#14392
#1440247