Finite Group Databasebeta

← #26145458

Subgroups — #26145458

Order 12032.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal, complemented
2#20241656C212yesmaximal, complemented
3#20241633C212yesmaximal, complemented
4#20241629C211yesmaximal, characteristic
5#20241629C211yesmaximal, characteristic
6#20241625C212yesmaximal, complemented
7#20241617C212yesmaximal, complemented
8#20241611C212yesmaximal, complemented
9C22×#123×C47C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C22⋊Dic47C2211yescharacteristic
14C22×C22⋊Dic47C2212yes—
15C22×C22⋊Dic47C2211yescharacteristic
16C22×C4×Dic47C2212yes—
17#15689016—28no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C23×D4×C47C2211yescharacteristic
28C24×C4×C47C2211yescharacteristic
29C2×#123×C47—28no—
30C2×#123×C47C418yescomplemented
31C22×C4⋊C4×C47C2212yes—
32C22×C22⋊C4×C47C2212yes—
33C22×C22⋊C4×C47C2211yescharacteristic
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C23×Dic47C2312yes—
38C2×C4⋊Dic47—24no—
39C2×C4⋊Dic47—24no—
40C2×C4⋊Dic47—42no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—24no—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47—24no—
47C2×C22⋊Dic47—41no—
48C2×C22⋊Dic47—42no—
49C2×C22⋊Dic47—42no—
50C2×C22⋊Dic47—42no—
51C2×C22⋊Dic47—42no—
52C2×C22⋊Dic47—41no—
53C2×C22⋊Dic47—42no—
54C2×C4×Dic47—24no—
55C2×C4×Dic47—24no—
56C2×C4×Dic47—42no—
57C47⋊#95—24no—
58C47⋊#95—24no—
59C47⋊#95—41no—
60C47⋊#95—42no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—42no—
64C47⋊#95—41no—
65C47⋊#95—42no—
66C25×C47C2311yescharacteristic
67C25×C47C2311yescharacteristic
68C22×D4×C47—24no—
69C22×D4×C47—42no—
70C22×D4×C47—41no—
71C22×D4×C47—41no—
72C22×D4×C47C2×C414yes—
73C23×C4×C47—24no—
74C23×C4×C47—42no—
75C23×C4×C47—41no—
76C23×C4×C47—41no—
77C23×C4×C47C2×C414yes—
78C23×C4×C47C2312yes—
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2312yes—
81#123×C47—232no—
82C2×C4⋊C4×C47—24no—
83C2×C4⋊C4×C47—24no—
84C2×C4⋊C4×C47C2×C418yes—
85C2×C22⋊C4×C47—24no—
86C2×C22⋊C4×C47—24no—
87C2×C22⋊C4×C47C2×C418yes—
88C2×C22⋊C4×C47C2×C414yes—
89C2×C22⋊C4×C47D418yes—
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47—21no—
94C24×C47—21no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—21no—
102C24×C47—21no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47C22×C412yes—
106C24×C47C22×C412yes—
107C24×C47C2×D414yes—
108C24×C47C2411yescharacteristic
109C2×D4×C47—28no—
110C2×D4×C47—44no—
111C2×D4×C47—44no—
112C2×D4×C47—48no—
113C2×D4×C47—48no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—28no—
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×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—44no—
130C22×C4×C47—44no—
131C22×C4×C47—48no—
132C22×C4×C47—48no—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C414yes—
136C22×C4×C47C2×D414yes—
137C22×C4×C47C2×D414yes—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
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—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47#4114yes—
165C22×Dic47#4114yes—
166C4⋊C4×C47—216no—
167C22⋊C4×C47—216no—
168C22⋊C4×C47C22⋊C4116yescomplemented
169C22⋊Dic47—44no—
170C22⋊Dic47—48no—
171C22⋊Dic47—48no—
172C22⋊Dic47—48no—
173C22⋊Dic47—44no—
174C4⋊Dic47—48no—
175C4×Dic47—48no—
176C23×C47—24no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—24no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—22no—
187C23×C47—24no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—24no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—41no—
196C23×C47—42no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—42no—
200C23×C47—41no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—41no—
212C23×C47—42no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—42no—
216C23×C47—41no—
217C23×C47C2×C22⋊C414yes—
218C23×C47C23×C411yescharacteristic
219C23×C47C22×D412yes—
220C23×C47C2×#4112yes—
221C23×C47C2×#4112yes—
222C23×C47H(5,𝔽2)12yes—
223C23×C47H(5,𝔽2)11yescharacteristic
224C23×C47H(5,𝔽2)11yescharacteristic
225C23×C47H(5,𝔽2)12yes—
226C23×C472−1+412yes—
227D4×C47—416no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—416no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47C2×C22⋊C418yes—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
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×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276#14506—471nomaximal, Sylow
277C22×C47—24no—
278C22×C47—28no—
279C22×C47—44no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—44no—
283C22×C47—44no—
284C22×C47—44no—
285C22×C47—44no—
286C22×C47—42no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—82no—
292C22×C47—82no—
293C22×C47—82no—
294C22×C47—82no—
295C22×C47C22×C22⋊C411yescharacteristic
296C22×C47C4×#4112yes—
297C22×C47#51811yescharacteristic
298C22×C47#51811yescharacteristic
299C22×C47#52012yes—
300C22×C47#53512yes—
301C22×C47#53512yes—
302C22×C47#53812yes—
303C22×C47#53912yes—
304C22×C47#54012yes—
305C22×C47#54212yes—
306C22×C47#54612yes—
307C22×C47#54612yes—
308C22×C47#54712yes—
309C22×C47#54712yes—
310C22×C47#55111yescharacteristic
311C22×C47#55312yes—
312C22×C47#55911yescharacteristic
313C22×C47#56011yescharacteristic
314C22×C47#56212yes—
315C22×C47#56311yescharacteristic
316C4×C47—28no—
317C4×C47—44no—
318C4×C47—42no—
319Dic47—44no—
320Dic47—44no—
321Dic47—82no—
322Dic47—82no—
323C22×#123D4711yescharacteristic
324C2×#388—472no—
325#1445—472no—
326#1439—472no—
327#1438—471no—
328#1438—471no—
329#1437—472no—
330C4⋊#95—472no—
331C22⋊#95—472no—
332C2×C47—44no—
333C2×C47—82no—
334C2×C47#230911yescharacteristic
335C2×C47#233512yes—
336C2×C47#233912yes—
337C2×C47#237411yescharacteristic
338C2×C47#237611yescharacteristic
339C2×C47#270312yes—
340C2×C47#270612yes—
341C2×C47#271012yes—
342C2×C47#271812yes—
343C23×D4C2×D4711yescharacteristic
344C24×C4C2×D4711yescharacteristic
345C2×#123—28no—
346C2×#123Dic4718yescomplemented
347C22×C4⋊C4—472no—
348C22×C4⋊C4C2×D4712yes—
349C22×C22⋊C4—472no—
350C22×C22⋊C4—472no—
351C22×C22⋊C4—471no—
352C22×C22⋊C4—472no—
353C22×C22⋊C4—471no—
354C22×C22⋊C4C2×D4712yes—
355C22×C22⋊C4C2×D4711yescharacteristic
356C22×C42—472no—
357#388—948no—
358C2×#95—472no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C2×#95—472no—
364C2×#95—471no—
365C2×#95—472no—
366C2×#95—471no—
367C47#1450611yescharacteristic, complemented, Sylow
368C25C22×D4711yescharacteristic
369C25C22×D4711yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×Dic4714yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4—472no—
380C23×C4—472no—
381C23×C4—472no—
382C23×C4—472no—
383C23×C4C2×Dic4714yes—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4712yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—1882no—
393C2×C4⋊C4C2×Dic4718yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—944no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—1881no—
403C2×C22⋊C4—1882no—
404C2×C22⋊C4—1882no—
405C2×C22⋊C4—1882no—
406C2×C22⋊C4—1882no—
407C2×C22⋊C4—1881no—
408C2×C22⋊C4—1882no—
409C2×C22⋊C4C2×Dic4718yes—
410C2×C22⋊C4C2×Dic4714yes—
411C2×C22⋊C4C22⋊D4718yes—
412C2×C42—944no—
413C2×C42—944no—
414C2×C42—1882no—
415#95—944no—
416#95—944no—
417#95—1882no—
418#95—1882no—
419#95—1882no—
420#95—1882no—
421#95—1881no—
422#95—1882no—
423#95—1881no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24C22×Dic4712yes—
440C24C22×Dic4712yes—
441C24C2×C22⋊D4714yes—
442C24C23×D4711yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
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—21no—
462C22×C4—21no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—474no—
468C22×C4—474no—
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—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4C22×Dic4714yes—
496C22×C4C22×Dic4712yes—
497C22×C4C22×Dic4714yes—
498C22×C4C2×C22⋊D4714yes—
499C22×C4C2×C22⋊D4714yes—
500C4⋊C4—216no—
501C4⋊C4—1888no—
502C22⋊C4—216no—
503C22⋊C4—1884no—
504C22⋊C4—1888no—
505C22⋊C4—1888no—
506C22⋊C4—1888no—
507C22⋊C4—1884no—
508C22⋊C4C22⋊Dic47116yescomplemented
509C42—1888no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊Dic4714yes—
552C23C23×Dic4711yescharacteristic
553C23C22×C22⋊D4712yes—
554C23#41×D4712yes—
555C23#41×D4712yes—
556C23#1309406912yes—
557C23#1309406911yescharacteristic
558C23#1309406911yescharacteristic
559C23#1309407012yes—
560C23#1309407112yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—944no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4C2×C22⋊Dic4718yes—
610C22—28no—
611C22—24no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊Dic4711yescharacteristic
629C22#41×Dic4712yes—
630C22#1568959811yescharacteristic
631C22#1568959811yescharacteristic
632C22#1568960612yes—
633C22#1568969012yes—
634C22#1568969012yes—
635C22#1568970912yes—
636C22#1568971612yes—
637C22#1568972512yes—
638C22#1568974112yes—
639C22#1568976112yes—
640C22#1568976112yes—
641C22#1568977212yes—
642C22#1568977212yes—
643C22#1568980111yescharacteristic
644C22#1568981812yes—
645C22#1568985711yescharacteristic
646C22#1568986311yescharacteristic
647C22#1568987012yes—
648C22#1568987611yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—3762no—
653C4—3762no—
654C4—1884no—
655C4—1884no—
656C2—44no—
657C2—82no—
658C2#2024597011yescharacteristic
659C2#2024618212yes—
660C2#2024621812yes—
661C2#2024652411yescharacteristic
662C2#2024653611yescharacteristic
663C2#2025002512yes—
664C2#2025007012yes—
665C2#2025012612yes—
666C2#2025023412yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×#123×C47C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC22×#123D47
hypocenterC47#14506
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#15689016C2semidirect
2. #20241656C2semidirect
3. #20241633C2semidirect
6. #20241625C2semidirect
7. #20241617C2semidirect
8. #20241611C2semidirect
30. C2×#123×C47C4semidirect
168. C22⋊C4×C47C22⋊C4semidirect
346. C2×#123Dic47semidirect
367. C47#14506semidirect
508. C22⋊C4C22⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890162
#202416562
#202416332
#202416292
#202416252
#202416172
#202416112
C22×#123×C472
#1450647