Finite Group Databasebeta

← #26143045

Subgroups — #26143045

Order 12032.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093989C211yesmaximal, characteristic, complemented
2C2×#15689035C211yesmaximal, characteristic
3C2×#15689017C212yesmaximal
4C2×#15689016C211yesmaximal, characteristic
5#20241658C211yesmaximal, characteristic, complemented
6#20241652C212yesmaximal, complemented
7#20241651C211yesmaximal, characteristic, complemented
8#20241631C212yesmaximal
9#20241614C212yesmaximal, complemented
10#20241613C211yesmaximal, characteristic, complemented
11C47×#1439C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C2×#13093989—24no—
14C2×#13093989—28no—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2212yes—
18C22×C22⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47C2212yes—
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2212yes—
23#15689035C418yescomplemented
24#15689017—28no—
25#15689016—24no—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2211yescharacteristic
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C22×C22⋊C4×C47C2212yes—
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94—24no—
43C22×Dic94—24no—
44C22×Dic94C2×C414yes—
45C23×Dic47C2311yescharacteristic
46C23×Dic47C2311yescharacteristic
47C23×Dic47C2312yes—
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2312yes—
50C23×Dic47C2311yescharacteristic
51#13093989—232no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—42no—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—42no—
59C2×Dic47⋊C4—42no—
60C2×C22⋊Dic47—24no—
61C2×C22⋊Dic47—24no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—24no—
65C2×C22⋊Dic47—42no—
66C2×C22⋊Dic47C2×C418yes—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47—24no—
70C2×C4×Dic47D414yes—
71C2×C4×Dic47D418yes—
72C47⋊#95—22no—
73C47⋊#95—22no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—42no—
78C47⋊#95—42no—
79C47⋊#95C2×C414yes—
80C47⋊#95C2×C418yes—
81C25×C47C2311yescharacteristic
82C23×C4×C47C2312yes—
83C23×C4×C47C2311yescharacteristic
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47—24no—
87C2×C4⋊C4×C47—22no—
88C2×C22⋊C4×C47—22no—
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—24no—
91C2×C22⋊C4×C47—42no—
92C2×C22⋊C4×C47—42no—
93C2×C22⋊C4×C47C2×C414yes—
94#95×C47—22no—
95#95×C47—42no—
96C24×C47—22no—
97C24×C47—21no—
98C24×C47—21no—
99C24×C47—22no—
100C24×C47—21no—
101C24×C47—22no—
102C24×C47—21no—
103C24×C47—21no—
104C24×C47—22no—
105C24×C47—21no—
106C24×C47C22×C412yes—
107C24×C47C2411yescharacteristic
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
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—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D414yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47#4112yes—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47C22×C414yes—
160C22×Dic47C22×C414yes—
161C22×Dic47C2×D412yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×D414yes—
165C22×Dic47#4114yes—
166C2×Dic94—28no—
167C2×Dic94—216no—
168C2×Dic94—28no—
169C2×Dic94—28no—
170C2×Dic94—216no—
171C4⋊C4×C47—28no—
172C22⋊C4×C47—28no—
173C22⋊C4×C47—48no—
174C22⋊Dic47—216no—
175C22⋊Dic47—44no—
176C22⋊Dic47—44no—
177C4⋊Dic47—44no—
178Dic47⋊C4—216no—
179Dic47⋊C4—48no—
180C4×Dic47—216no—
181C4×Dic47—28no—
182C23×C47—24no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—24no—
186C23×C47—22no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—42no—
190C23×C47—41no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—41no—
198C23×C47—42no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47C23×C411yescharacteristic
206C23×C47C22×D412yes—
207C23×C47C22×D411yescharacteristic
208C23×C47C2×#4112yes—
209C23×C47C2×#4111yescharacteristic
210C23×C47H(5,𝔽2)11yescharacteristic
211C23×C47H(5,𝔽2)12yes—
212C23×C472−1+411yescharacteristic
213C23×C472−1+411yescharacteristic
214C23×C472−1+412yes—
215C23×C472−1+411yescharacteristic
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—41no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
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×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47C4×D414yes—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—42no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—42no—
269C2×Dic47—41no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47C4×D418yes—
273C2×Dic47#12014yes—
274Dic94—232no—
275#14301—471nomaximal, Sylow
276C22×C47—44no—
277C22×C47—42no—
278C22×C47—44no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—81no—
282C22×C47—81no—
283C22×C47—81no—
284C22×C47—82no—
285C22×C47—81no—
286C22×C47—82no—
287C22×C47C2×C4×D412yes—
288C22×C47C2×C4×D411yescharacteristic
289C22×C47#51911yescharacteristic
290C22×C47#52012yes—
291C22×C47#52011yescharacteristic
292C22×C47C2×#12011yescharacteristic
293C22×C47#52511yescharacteristic
294C22×C47#52512yes—
295C22×C47#53511yescharacteristic
296C22×C47#53512yes—
297C22×C47#53612yes—
298C22×C47#53611yescharacteristic
299C22×C47#53611yescharacteristic
300C22×C47#53612yes—
301C22×C47#53712yes—
302C22×C47#53711yescharacteristic
303C22×C47#54112yes—
304C22×C47#54111yescharacteristic
305C22×C47#54211yescharacteristic
306C22×C47#54212yes—
307C22×C47#54212yes—
308C22×C47#54212yes—
309C22×C47#54212yes—
310C4×C47—24no—
311C4×C47—42no—
312C4×C47—44no—
313C4×C47—82no—
314Dic47—24no—
315Dic47—28no—
316Dic47—44no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C22×C4×Q8—471no—
321C2×#393—471no—
322C2×#388—471no—
323C2×#388—472no—
324#1446—471no—
325#1444—472no—
326#1444—471no—
327#1439—472no—
328#1439D4711yescharacteristic
329#1434—471no—
330#1434—472no—
331C2×C47—82no—
332C2×C47C4×#12011yescharacteristic
333C2×C47#232312yes—
334C2×C47#232311yescharacteristic
335C2×C47#232612yes—
336C2×C47#232611yescharacteristic
337C2×C47#261411yescharacteristic
338C2×C47#261511yescharacteristic
339C2×C47#263211yescharacteristic
340C2×C47#263212yes—
341C2×C47#263711yescharacteristic
342C2×C47#263712yes—
343C23×Q8—471no—
344C2×C4×Q8—944no—
345C2×C4×Q8—948no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4C2×D4711yescharacteristic
350C22×C22⋊C4—472no—
351C22×C22⋊C4—471no—
352C22×C22⋊C4—471no—
353C22×C22⋊C4C2×D4711yescharacteristic
354C22×C22⋊C4C2×D4712yes—
355C22×C42—472no—
356C22×C42—471no—
357#393—478no—
358#388—944no—
359#388—948no—
360C2×#95—472no—
361C2×#95—471no—
362C2×#95—472no—
363C2×#95—471no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95—471no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95C2×D4711yescharacteristic
370C2×#95C2×D4712yes—
371C47#1430111yescharacteristic, complemented, Sylow
372C25C22×D4711yescharacteristic
373C22×Q8—474no—
374C22×Q8—944no—
375C22×Q8—944no—
376C22×Q8—942no—
377C22×Q8—942no—
378C23×C4—472no—
379C23×C4—471no—
380C23×C4—471no—
381C23×C4—472no—
382C23×C4—471no—
383C23×C4—471no—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4712yes—
387C23×C4C22×D4711yescharacteristic
388C4×Q8—9432no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—944no—
394C2×C4⋊C4—942no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—1882no—
397C2×C4⋊C4—1882no—
398C2×C4⋊C4—1882no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—478no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—944no—
409C2×C22⋊C4—942no—
410C2×C22⋊C4—1882no—
411C2×C22⋊C4C4×D4714yes—
412C2×C42—478no—
413C2×C42—474no—
414C2×C42—944no—
415C2×C42—944no—
416C2×C42—944no—
417#95—22no—
418#95—42no—
419#95—478no—
420#95—474no—
421#95—944no—
422#95—942no—
423#95—944no—
424#95—944no—
425#95—942no—
426#95—1882no—
427#95—1882no—
428C24—22no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—21no—
433C24—22no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—21no—
438C24C2×C4×D4712yes—
439C24C23×D4711yescharacteristic
440C2×Q8—9416no—
441C2×Q8—948no—
442C2×Q8—948no—
443C2×Q8—9416no—
444C2×Q8—948no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—472no—
467C22×C4—474no—
468C22×C4—474no—
469C22×C4—474no—
470C22×C4—472no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—941no—
487C22×C4—942no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—941no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4C2×C4×D4712yes—
499C22×C4C2×C4×D4712yes—
500C22×C4D4×D4714yes—
501C22×C4D4×D4712yes—
502C22×C4#1062535212yes—
503C4⋊C4—28no—
504C4⋊C4—9416no—
505C4⋊C4—1888no—
506C4⋊C4—1884no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—9416no—
510C22⋊C4—1884no—
511C22⋊C4—1884no—
512C42—948no—
513C42—9416no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23C22×C4×D4711yescharacteristic
538C23C2×D4×D4712yes—
539C23C2×D4×D4711yescharacteristic
540C23#41×D4712yes—
541C23C2×#1062535211yescharacteristic
542C23#1309406911yescharacteristic
543C23#1309406912yes—
544C23#1309407111yescharacteristic
545C23#1309407112yes—
546C23#1309407111yescharacteristic
547C23#1309407211yescharacteristic
548Q8—9432no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—478no—
573C2×C4—474no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—944no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1882no—
597C2×C4—1881no—
598C2×C4—1882no—
599C2×C4—1881no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4(C22⋊D47)⋊C414yes—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—81no—
617C22—82no—
618C22C4×D4×D4712yes—
619C22C2×(C22⋊D47)⋊C411yescharacteristic
620C22#1568960011yescharacteristic
621C22#1568960212yes—
622C22#1568960411yescharacteristic
623C22#120×D4711yescharacteristic
624C22#1568963111yescharacteristic
625C22#1568963312yes—
626C22#1568968611yescharacteristic
627C22#1568968812yes—
628C22#1568969312yes—
629C22#1568969311yescharacteristic
630C22#1568969611yescharacteristic
631C22#1568969612yes—
632C22#1568970012yes—
633C22#1568970211yescharacteristic
634C22#1568973012yes—
635C22#1568973311yescharacteristic
636C22#1568974011yescharacteristic
637C22#1568974012yes—
638C22#1568974112yes—
639C22#1568974112yes—
640C22#1568974312yes—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—82no—
645C4—944no—
646C4—948no—
647C4—3761no—
648C4—3761no—
649C4—1882no—
650C4—1884no—
651C2—82no—
652C2#2024589311yescharacteristic
653C2#2024606312yes—
654C2#2024606611yescharacteristic
655C2#2024609412yes—
656C2#2024609811yescharacteristic
657C2#2024910611yescharacteristic
658C2#2024911111yescharacteristic
659C2#2024925011yescharacteristic
660C2#2024925312yes—
661C2#2024929711yescharacteristic
662C2#2024930112yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1439C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1439D47
hypocenterC47#14301
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#13093989C2semidirect
5. #20241658C2semidirect
6. #20241652C2semidirect
7. #20241651C2semidirect
9. #20241614C2semidirect
10. #20241613C2semidirect
23. #15689035C4semidirect
371. C47#14301semidirect

Maximal subgroups

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

subgroupindex
C22×#130939892
C2×#156890352
C2×#156890172
C2×#156890162
#202416582
#202416522
#202416512
#202416312
#202416142
#202416132
C47×#14392
#1430147