Finite Group Databasebeta

← #26147412

Subgroups — #26147412

Order 12032.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688984C211yesmaximal, characteristic
2C2×#15688981C212yesmaximal
3#20241652C212yesmaximal, complemented
4#20241651C212yesmaximal, complemented
5#20241646C212yesmaximal, complemented
6#20241643C212yesmaximal, complemented
7#20241633C211yesmaximal, characteristic
8#20241631C212yesmaximal
9C47×#1439C211yesmaximal, characteristic
10C24×Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×C22⋊Dic47C2212yes—
16C22×C22⋊Dic47C2211yescharacteristic
17#15688984C418yescomplemented
18#15688981—28no—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C22×C4⋊C4×C47C2211yescharacteristic
29C22×C22⋊C4×C47C2212yes—
30C22×C22⋊C4×C47C2211yescharacteristic
31C2×#95×C47C2212yes—
32C2×#95×C47C2211yescharacteristic
33C23×Dic47—41no—
34C23×Dic47—41no—
35C23×Dic47—41no—
36C23×Dic47—41no—
37C23×Dic47C2×C414yes—
38C23×Dic47D414yes—
39C23×Dic47Q814yes—
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—42no—
49C2×Dic47⋊C4—42no—
50C2×Dic47⋊C4—42no—
51C2×Dic47⋊C4—42no—
52C2×Dic47⋊C4—42no—
53C2×Dic47⋊C4—42no—
54C2×Dic47⋊C4—42no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—41no—
57C2×C22⋊Dic47—41no—
58C2×C22⋊Dic47—42no—
59C2×C22⋊Dic47—42no—
60C2×C22⋊Dic47C2×C414yes—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95—42no—
66C47⋊#95—42no—
67C47⋊#95—41no—
68C47⋊#95C2×C418yes—
69C47⋊#95C2×C418yes—
70C25×C47C2311yescharacteristic
71C23×C4×C47C2312yes—
72C23×C4×C47C2312yes—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75C2×C4⋊C4×C47—41no—
76C2×C4⋊C4×C47—42no—
77C2×C22⋊C4×C47—24no—
78C2×C22⋊C4×C47—41no—
79C2×C22⋊C4×C47—41no—
80C2×C22⋊C4×C47—42no—
81C2×C22⋊C4×C47—42no—
82C2×C22⋊C4×C47C2×C414yes—
83#95×C47—41no—
84#95×C47—42no—
85C24×C47—22no—
86C24×C47—22no—
87C24×C47—21no—
88C24×C47—21no—
89C24×C47—21no—
90C24×C47—21no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47C22×C412yes—
94C24×C47C2×D412yes—
95C24×C47C2×Q812yes—
96C24×C47C2411yescharacteristic
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
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—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×C47C22×C412yes—
120C22×C4×C47C22×C412yes—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
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—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—44no—
146C22×Dic47—44no—
147C22×Dic47—44no—
148C22×Dic47—44no—
149C22×Dic47—44no—
150C22×Dic47—44no—
151C22×Dic47C4⋊C4116yescomplemented
152C22×Dic47C22×C414yes—
153C22×Dic47C22×C414yes—
154C22×Dic47C2×D414yes—
155C22×Dic47C2×Q814yes—
156C4⋊C4×C47—44no—
157C22⋊C4×C47—44no—
158C22⋊C4×C47—48no—
159C22⋊Dic47—44no—
160C22⋊Dic47—48no—
161C4⋊Dic47—44no—
162Dic47⋊C4—48no—
163Dic47⋊C4—48no—
164Dic47⋊C4—48no—
165C23×C47—22no—
166C23×C47—22no—
167C23×C47—24no—
168C23×C47—24no—
169C23×C47—22no—
170C23×C47—22no—
171C23×C47—24no—
172C23×C47—24no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—24no—
176C23×C47—24no—
177C23×C47—41no—
178C23×C47—42no—
179C23×C47—41no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—41no—
185C23×C47—42no—
186C23×C47—41no—
187C23×C47C2×C4⋊C414yes—
188C23×C47C23×C411yescharacteristic
189C23×C47C22×D411yescharacteristic
190C23×C47C22×Q811yescharacteristic
191C23×C47H(5,𝔽2)11yescharacteristic
192C23×C47H(5,𝔽2)11yescharacteristic
193C23×C47H(5,𝔽2)12yes—
194C23×C47H(5,𝔽2)12yes—
195C23×C47H(5,𝔽2)11yescharacteristic
196C23×C47H(5,𝔽2)11yescharacteristic
197C23×C472−1+412yes—
198C23×C472−1+412yes—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
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×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—416no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
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×Dic47C2×C4⋊C418yes—
252#14674—471nomaximal, Sylow
253C22×C47—24no—
254C22×C47—28no—
255C22×C47—44no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—44no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—44no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47C22×C4⋊C411yescharacteristic
265C22×C47#51811yescharacteristic
266C22×C47#51811yescharacteristic
267C22×C47#52012yes—
268C22×C47#52012yes—
269C22×C47#53411yescharacteristic
270C22×C47#53411yescharacteristic
271C22×C47#53512yes—
272C22×C47#53612yes—
273C22×C47#54311yescharacteristic
274C22×C47#54311yescharacteristic
275C22×C47#54412yes—
276C22×C47#54412yes—
277C22×C47#56011yescharacteristic
278C22×C47#56012yes—
279C22×C47#56011yescharacteristic
280C22×C47#56212yes—
281C22×C47#56212yes—
282C22×C47#56212yes—
283C22×C47#56212yes—
284C22×C47#56311yescharacteristic
285C22×C47#56312yes—
286C22×C47#56311yescharacteristic
287C4×C47—42no—
288C4×C47—42no—
289C4×C47—82no—
290C4×C47—82no—
291Dic47—28no—
292Dic47—44no—
293Dic47—82no—
294Dic47—82no—
295C2×#381—471no—
296C2×#380—472no—
297#1444—472no—
298#1444—472no—
299#1442—472no—
300#1442—472no—
301#1439—472no—
302#1439—471no—
303#1439D4711yescharacteristic
304C2×C47—44no—
305C2×C47#231511yescharacteristic
306C2×C47#231712yes—
307C2×C47#237611yescharacteristic
308C2×C47#237612yes—
309C2×C47#237611yescharacteristic
310C2×C47#281511yescharacteristic
311C2×C47#281612yes—
312C2×C47#281711yescharacteristic
313C2×C47#284111yescharacteristic
314C2×C47#284112yes—
315C2×C47#284111yescharacteristic
316C24×C4—471no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4C2×D4711yescharacteristic
322C22×C22⋊C4—471no—
323C22×C22⋊C4—472no—
324C22×C22⋊C4C2×D4712yes—
325C22×C22⋊C4C2×D4711yescharacteristic
326#381—478no—
327#380—948no—
328C2×#95—472no—
329C2×#95—472no—
330C2×#95—472no—
331C2×#95—472no—
332C2×#95—472no—
333C2×#95—471no—
334C2×#95—472no—
335C2×#95—472no—
336C2×#95—472no—
337C2×#95C2×D4711yescharacteristic
338C2×#95C2×D4712yes—
339C47#1467411yescharacteristic, complemented, Sylow
340C25C22×D4711yescharacteristic
341C23×C4—472no—
342C23×C4—474no—
343C23×C4—474no—
344C23×C4—474no—
345C23×C4—472no—
346C23×C4—472no—
347C23×C4—472no—
348C23×C4—1881no—
349C23×C4—1881no—
350C23×C4—1881no—
351C23×C4—1881no—
352C23×C4C22×D4712yes—
353C23×C4C22×D4712yes—
354C23×C4C22×D4711yescharacteristic
355C23×C4C22×D4711yescharacteristic
356C2×C4⋊C4—42no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—944no—
360C2×C4⋊C4—1882no—
361C2×C4⋊C4—1882no—
362C2×C4⋊C4—1882no—
363C2×C4⋊C4—1882no—
364C2×C4⋊C4—1882no—
365C2×C4⋊C4—1882no—
366C2×C4⋊C4—1882no—
367C2×C4⋊C4—1882no—
368C2×C4⋊C4—1881no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—41no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—474no—
375C2×C22⋊C4—944no—
376C2×C22⋊C4—1881no—
377C2×C22⋊C4—1881no—
378C2×C22⋊C4—1882no—
379C2×C22⋊C4—1882no—
380C2×C22⋊C4C4×D4714yes—
381#95—41no—
382#95—42no—
383#95—478no—
384#95—478no—
385#95—944no—
386#95—944no—
387#95—1881no—
388#95—1882no—
389#95—1882no—
390#95—1882no—
391#95—1882no—
392C24—22no—
393C24—22no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×C4×D4712yes—
401C24D4×D4712yes—
402C24Q8×D4712yes—
403C24C23×D4711yescharacteristic
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—474no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—4716no—
430C22×C4—474no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
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—1884no—
456C22×C4—1884no—
457C22×C4—1884no—
458C22×C4—1884no—
459C22×C4—1884no—
460C22×C4—1884no—
461C22×C4C2×C4×D4712yes—
462C22×C4C2×C4×D4712yes—
463C4⋊C4—44no—
464C4⋊C4—1888no—
465C4⋊C4—1888no—
466C4⋊C4—1888no—
467C4⋊C4—1884no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—1884no—
471C22⋊C4—1888no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23C4⋊C4×D4714yes—
495C23C22×C4×D4711yescharacteristic
496C23C2×D4×D4711yescharacteristic
497C23C2×Q8×D4711yescharacteristic
498C23#1309406911yescharacteristic
499C23#1309406911yescharacteristic
500C23#1309406912yes—
501C23#1309406912yes—
502C23#1309407011yescharacteristic
503C23#1309407011yescharacteristic
504C23#1309407112yes—
505C23#1309407112yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—478no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—18816no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4—1882no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1882no—
559C22—24no—
560C22—28no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22C2×C4⋊C4×D4711yescharacteristic
571C22#1568959611yescharacteristic
572C22#1568959711yescharacteristic
573C22#1568960212yes—
574C22#1568960212yes—
575C22#1568968111yescharacteristic
576C22#1568968111yescharacteristic
577C22#1568968812yes—
578C22#1568969512yes—
579C22#1568974811yescharacteristic
580C22#1568974811yescharacteristic
581C22#1568975212yes—
582C22#1568975212yes—
583C22#1568986011yescharacteristic
584C22#1568986112yes—
585C22#1568986211yescharacteristic
586C22#1568986712yes—
587C22#1568986812yes—
588C22#1568986812yes—
589C22#1568987012yes—
590C22#1568987311yescharacteristic
591C22#1568987512yes—
592C22#1568987611yescharacteristic
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—948no—
598C4—3762no—
599C4—3762no—
600C4—1884no—
601C2—44no—
602C2#2024600111yescharacteristic
603C2#2024600712yes—
604C2#2024653111yescharacteristic
605C2#2024653412yes—
606C2#2024653811yescharacteristic
607C2#2025141311yescharacteristic
608C2#2025142012yes—
609C2#2025142611yescharacteristic
610C2#2025164811yescharacteristic
611C2#2025164912yes—
612C2#2025165011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14674
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #20241652C2semidirect
4. #20241651C2semidirect
5. #20241646C2semidirect
6. #20241643C2semidirect
17. #15688984C4semidirect
151. C22×Dic47C4⋊C4semidirect
339. C47#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#156889842
C2×#156889812
#202416522
#202416512
#202416462
#202416432
#202416332
#202416312
C47×#14392
#1467447