Finite Group Databasebeta

← #26374877

Subgroups — #26374877

Order 12032.

2390 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D47C211yesmaximal, characteristic
2#20245998C211yesmaximal, characteristic, complemented
3#20245997C211yesmaximal, characteristic, complemented
4#20245996C211yesmaximal, characteristic, complemented
5#20245995C212yesmaximal, complemented
6#20245993C212yesmaximal, complemented
7#20245987C214yesmaximal, complemented
8#20245982C214yesmaximal
9#20245882C214yesmaximal, complemented
10#20245880C214yesmaximal, complemented
11#20245794C211yesmaximal, characteristic, complemented
12#20245793C211yesmaximal, characteristic, complemented
13#20245770C212yesmaximal
14C4×#13093982C212yesmaximal, complemented
15C47×#2314C211yesmaximal, characteristic, complemented
16C23×C4×D47C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C2×#13093982C2212yes—
19C2×#13093982C2212yes—
20C2×#13093982C2212yes—
21C2×#13093981C2211yescharacteristic
22C2×#13093981C2211yescharacteristic
23C2×#13093980C2212yes—
24#117×D47C4116yescomplemented
25C2×C4⋊C4×D47C2212yes—
26C2×C22⋊C4×D47C2212yes—
27C2×C42×D47C2211yescharacteristic
28C2×C42×D47C2211yescharacteristic
29#15689017C2214yes—
30#15689016C2214yes—
31#15689015C2214yes—
32#15689014C2214yes—
33#15689013C2214yes—
34#15689012C2214yes—
35#15689011C2214yes—
36#15689010C2214yescomplemented
37#15689009C2214yescomplemented
38#15689008C2214yescomplemented
39#15689007C2214yescomplemented
40#15689006C2214yescomplemented
41#15689005C2214yescomplemented
42#15689004C2214yescomplemented
43#15688993C2211yescharacteristic, complemented
44#15688992C2212yescomplemented
45#15688989C2211yescharacteristic, complemented
46#15688988C2212yescomplemented
47#15688985C2214yes—
48#15688983C2212yes—
49#15688982C2214yes—
50#15688980C2212yes—
51#15688980C2212yes—
52#15688980C2214yes—
53#15688978C2214yes—
54C4⋊C4×Dic47C2214yescomplemented
55C4⋊C4×Dic47C2212yescomplemented
56C4⋊C4×Dic47C2212yescomplemented
57C4×C4⋊Dic47C2212yescomplemented
58#15688974C2214yescomplemented
59C22⋊C4×Dic47C2214yes—
60C4×(C2×D47)⋊C4C2214yes—
61#15688970C2214yes—
62#15688970C2214yes—
63#15688968C2214yescomplemented
64C42×Dic47C2214yescomplemented
65C2×#117×C47C2211yescharacteristic
66C47×#388C2214yes—
67C47×#387C2214yescomplemented
68C47×#384C2211yescharacteristic, complemented
69C47×#382C2211yescharacteristic, complemented
70C47×#380C2212yes—
71C4×C4⋊C4×C47C2212yescomplemented
72C24×D47C2311yescharacteristic
73C23×Dic47C2311yescharacteristic
74C22×C4×D47—24no—
75C22×C4×D47—24no—
76C22×C4×D47C2×C418yes—
77C22×C4×D47C2311yescharacteristic
78C22×C4×D47C2311yescharacteristic
79C22×C4×D47C2311yescharacteristic
80C22×C4×D47C2312yes—
81C22×C4×D47C2312yes—
82C22×C4×D47C2311yescharacteristic
83#13093983C2×C418yes—
84#13093982—28no—
85#13093982—28no—
86#13093982C2×C4116yes—
87#13093981C2×C418yes—
88#13093981C2×C418yes—
89#13093980C2×C4116yes—
90C2×C4⋊Dic47C2312yes—
91C2×C4⋊Dic47C2312yes—
92C2×C4⋊Dic47C2314yes—
93C2×C4⋊Dic47C2312yes—
94C2×Dic47⋊C4C2314yes—
95C2×Dic47⋊C4C2314yes—
96C2×Dic47⋊C4C2312yes—
97C2×Dic47⋊C4C2312yes—
98C4⋊C4×D47C2×C4116yes—
99C2×C22⋊Dic47C2314yes—
100C2×C22⋊Dic47C2312yes—
101C2×(C2×D47)⋊C4C2314yes—
102C2×(C2×D47)⋊C4C2314yes—
103C2×(C2×D47)⋊C4C2314yes—
104C2×(C2×D47)⋊C4C2314yes—
105C2×(C2×D47)⋊C4C2312yes—
106C2×(C2×D47)⋊C4C2312yes—
107C22⋊C4×D47C2×C4116yes—
108C2×C4×Dic47C2314yes—
109C2×C4×Dic47C2314yes—
110C2×C4×Dic47C2314yes—
111C2×C4×Dic47C2314yes—
112C2×C4×Dic47C2314yes—
113C2×C4×Dic47C2312yes—
114C2×C4×Dic47C2312yes—
115C2×C4×Dic47C2311yescharacteristic
116C2×C4×Dic47C2311yescharacteristic
117C42×D47C2×C418yes—
118C42×D47C2×C418yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C47⋊#95C2314yes—
122C47⋊#95C2314yes—
123C47⋊#95C2314yes—
124C47⋊#95C2314yes—
125C47⋊#95C2314yes—
126C47⋊#95C2314yes—
127C47⋊#95C2314yes—
128C23×C4×C47C2311yescharacteristic
129#117×C47C2×C418yes—
130C2×C4⋊C4×C47C2312yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2314yes—
133C2×C4⋊C4×C47C2312yes—
134C2×C22⋊C4×C47C2314yes—
135C2×C22⋊C4×C47C2312yes—
136C2×C42×C47C2314yes—
137C2×C42×C47C2311yescharacteristic
138C2×C42×C47C2311yescharacteristic
139#95×C47C2314yes—
140#95×C47C2314yes—
141#95×C47C2314yes—
142C24×C47C2411yescharacteristic
143C23×D47—22no—
144C23×D47—22no—
145C23×D47—42no—
146C23×D47—42no—
147C23×D47C22×C414yes—
148C23×D47C2411yescharacteristic
149C23×D47C2411yescharacteristic
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47C22×C414yes—
153C22×C4×C47C2414yes—
154C22×C4×C47C2414yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2412yes—
157C22×C4×C47C2411yescharacteristic
158C22×C4×C47C2411yescharacteristic
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47C22×C414yes—
162C22×Dic47C2414yes—
163C22×Dic47C2414yes—
164C22×Dic47C2411yescharacteristic
165C22×Dic47C2412yes—
166C22×Dic47C2412yes—
167C22×Dic47C2411yescharacteristic
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×C4×D47—22no—
171C2×C4×D47—22no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—232no—
175C2×C4×D47—42no—
176C2×C4×D47—42no—
177C2×C4×D47—42no—
178C2×C4×D47—42no—
179C2×C4×D47C22×C414yes—
180C2×C4×D47C22×C414yes—
181C2×C4×D47C22×C414yes—
182C2×C4×D47C22×C418yes—
183C2×C4×D47C22×C418yes—
184C2×C4×D47C22×C414yes—
185C2×C4×D47#4114yes—
186C2×C4×D47#4114yes—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—24no—
189C4⋊C4×C47—44no—
190C4⋊C4×C47C22×C418yes—
191C22⋊C4×C47—44no—
192C22⋊C4×C47C22×C418yes—
193C42×C47—44no—
194C42×C47C22×C414yes—
195C42×C47C22×C414yes—
196C22⋊Dic47—44no—
197C22⋊Dic47C22×C418yes—
198(C2×D47)⋊C4—28no—
199(C2×D47)⋊C4—28no—
200(C2×D47)⋊C4—44no—
201(C2×D47)⋊C4—44no—
202(C2×D47)⋊C4C22×C418yes—
203(C2×D47)⋊C4C22×C418yes—
204C4⋊Dic47—24no—
205C4⋊Dic47—24no—
206C4⋊Dic47—44no—
207C4⋊Dic47C22×C418yes—
208Dic47⋊C4—44no—
209Dic47⋊C4—44no—
210Dic47⋊C4C22×C418yes—
211Dic47⋊C4C22×C418yes—
212C4×Dic47—28no—
213C4×Dic47—28no—
214C4×Dic47—44no—
215C4×Dic47—44no—
216C4×Dic47—44no—
217C4×Dic47C22×C418yes—
218C4×Dic47C22×C418yes—
219C4×Dic47C22×C414yes—
220C4×Dic47C22×C414yes—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47C23×C412yes—
226C23×C47C2511yescharacteristic
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—21no—
230C22×D47—21no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—28no—
234C22×D47—48no—
235C22×D47—44no—
236C22×D47—48no—
237C22×D47—44no—
238C22×D47C23×C412yes—
239C22×D47C23×C412yes—
240C22×D47C2×#4112yes—
241C22×D47C2×#4112yes—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—28no—
249C2×C4×C47—24no—
250C2×C4×C47—24no—
251C2×C4×C47—24no—
252C2×C4×C47—24no—
253C2×C4×C47—24no—
254C2×C4×C47—24no—
255C2×C4×C47C23×C414yes—
256C2×C4×C47C23×C414yes—
257C2×C4×C47C23×C412yes—
258C2×C4×C47C23×C412yes—
259C2×C4×C47C2×#4112yes—
260C2×C4×C47C2×#4112yes—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—28no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47C23×C414yes—
275C2×Dic47C23×C414yes—
276C2×Dic47C23×C412yes—
277C2×Dic47C23×C412yes—
278C2×Dic47C2×#4112yes—
279C2×Dic47C2×#4112yes—
280C4×D47—28no—
281C4×D47—28no—
282C4×D47—28no—
283C4×D47—48no—
284C4×D47—48no—
285C4×D47#117116yes—
286#34048—471nomaximal, Sylow
287C22×C47—22no—
288C22×C47—24no—
289C22×C47—41no—
290C22×C47—42no—
291C22×C47—41no—
292C22×C47C24×C411yescharacteristic
293C22×C47C22×#4111yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47#58512yes—
296C22×C47#58512yes—
297C2×D47—24no—
298C2×D47—22no—
299C2×D47—24no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—24no—
303C2×D47—416no—
304C2×D47—41no—
305C2×D47—42no—
306C2×D47—41no—
307C2×D47C2×#11714yes—
308C4×C47—22no—
309C4×C47—24no—
310C4×C47—24no—
311C4×C47—44no—
312C4×C47—44no—
313C4×C47C2×#11714yes—
314Dic47—22no—
315Dic47—24no—
316Dic47—24no—
317Dic47—44no—
318Dic47—44no—
319Dic47C2×#11714yes—
320C22×#117—471no—
321#2314—471no—
322#2314—471no—
323#2314—472no—
324#2314—472no—
325#2314—471no—
326#2314D4711yescharacteristic, complemented
327#2313—474no—
328#2312—474no—
329#2298—474no—
330#2298—474no—
331#2285—471no—
332#2285—471no—
333#2280—472no—
334C4×#117—472no—
335C2×C47—42no—
336C2×C47C22×#11711yescharacteristic
337C2×C47#342812yes—
338C2×C47#345511yescharacteristic
339C2×C47#345512yes—
340C2×C47#345511yescharacteristic
341D47—24no—
342D47—42no—
343C24×C4—471no—
344C2×#117—471no—
345C2×#117—4716no—
346C2×#117—472no—
347C2×#117—472no—
348C2×#117—472no—
349C2×#117—472no—
350C2×#117—471no—
351C2×#117—471no—
352C2×#117C2×D4711yescharacteristic
353C22×C4⋊C4—472no—
354C22×C22⋊C4—472no—
355C22×C42—471no—
356C22×C42—471no—
357#388—474no—
358#388—474no—
359#388—474no—
360#388—474no—
361#388—474no—
362#388—474no—
363#388—474no—
364#388C2×D4714yes—
365#387—474no—
366#387—474no—
367#387—474no—
368#387—474no—
369#387—474no—
370#387—474no—
371#387—474no—
372#387C2×D4714yescomplemented
373#384—472no—
374#384—471no—
375#384C2×D4711yescharacteristic, complemented
376#382—472no—
377#382—471no—
378#382C2×D4711yescharacteristic, complemented
379#381—474no—
380#380—472no—
381#380—474no—
382#380—472no—
383#380—472no—
384#380—474no—
385#380C2×D4712yes—
386#379—474no—
387C4×C4⋊C4—472no—
388C4×C4⋊C4—472no—
389C4×C4⋊C4—474no—
390C4×C4⋊C4—472no—
391C4×C4⋊C4—474no—
392C4×C4⋊C4C2×D4712yescomplemented
393C4×C22⋊C4—474no—
394C4×C22⋊C4—474no—
395C4×C22⋊C4—474no—
396C4×C22⋊C4—474no—
397#376—474no—
398C43—474no—
399C47#3404811yescharacteristic, complemented, Sylow
400C25—471no—
401C23×C4—471no—
402C23×C4—471no—
403C23×C4—478no—
404C23×C4—471no—
405C23×C4—471no—
406C23×C4—471no—
407C23×C4—472no—
408C23×C4—472no—
409C23×C4—944no—
410C23×C4—944no—
411C23×C4C22×D4711yescharacteristic
412#117—478no—
413#117—4716no—
414#117—4716no—
415#117—478no—
416#117—478no—
417#117—948no—
418#117—948no—
419#117C4×D4718yes—
420C2×C4⋊C4—472no—
421C2×C4⋊C4—472no—
422C2×C4⋊C4—472no—
423C2×C4⋊C4—4716no—
424C2×C4⋊C4—472no—
425C2×C4⋊C4—472no—
426C2×C4⋊C4—474no—
427C2×C4⋊C4—474no—
428C2×C4⋊C4—474no—
429C2×C4⋊C4C22×D4712yes—
430C2×C4⋊C4C22×D4712yes—
431C2×C4⋊C4C22×D4714yes—
432C2×C4⋊C4C22×D4712yes—
433C2×C22⋊C4—472no—
434C2×C22⋊C4—472no—
435C2×C22⋊C4—472no—
436C2×C22⋊C4—4716no—
437C2×C22⋊C4—474no—
438C2×C22⋊C4—474no—
439C2×C22⋊C4—474no—
440C2×C22⋊C4—474no—
441C2×C22⋊C4—474no—
442C2×C22⋊C4C22×D4714yes—
443C2×C22⋊C4C22×D4712yes—
444C2×C42—472no—
445C2×C42—472no—
446C2×C42—471no—
447C2×C42—471no—
448C2×C42—478no—
449C2×C42—478no—
450C2×C42—474no—
451C2×C42—474no—
452C2×C42—474no—
453C2×C42—474no—
454C2×C42—474no—
455C2×C42C22×D4714yes—
456C2×C42C22×D4711yescharacteristic
457C2×C42C22×D4711yescharacteristic
458#95—474no—
459#95—474no—
460#95—474no—
461#95—474no—
462#95—474no—
463#95—474no—
464#95—474no—
465#95—474no—
466#95—474no—
467#95C22×D4714yes—
468#95C22×D4714yes—
469#95C22×D4714yes—
470C24—471no—
471C24—471no—
472C24—474no—
473C24—942no—
474C24—942no—
475C24—1882no—
476C24—1882no—
477C24C23×D4711yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—471no—
481C22×C4—474no—
482C22×C4—474no—
483C22×C4—474no—
484C22×C4—474no—
485C22×C4—471no—
486C22×C4—472no—
487C22×C4—472no—
488C22×C4—474no—
489C22×C4—474no—
490C22×C4—474no—
491C22×C4—478no—
492C22×C4—478no—
493C22×C4—474no—
494C22×C4—474no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—9432no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—1882no—
505C22×C4—1882no—
506C22×C4—1882no—
507C22×C4—1882no—
508C22×C4C2×C4×D4714yes—
509C22×C4C23×D4714yes—
510C22×C4C23×D4714yes—
511C22×C4C23×D4711yescharacteristic
512C22×C4C23×D4712yes—
513C22×C4C23×D4712yes—
514C22×C4C23×D4711yescharacteristic
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—478no—
519C4⋊C4—478no—
520C4⋊C4—478no—
521C4⋊C4—944no—
522C4⋊C4—944no—
523C4⋊C4—1884no—
524C4⋊C4—1884no—
525C4⋊C4—1884no—
526C4⋊C4C2×C4×D4718yes—
527C22⋊C4—44no—
528C22⋊C4—478no—
529C22⋊C4—478no—
530C22⋊C4—478no—
531C22⋊C4—948no—
532C22⋊C4—948no—
533C22⋊C4—1884no—
534C22⋊C4—1884no—
535C22⋊C4—1884no—
536C22⋊C4C2×C4×D4718yes—
537C42—44no—
538C42—478no—
539C42—478no—
540C42—474no—
541C42—474no—
542C42—948no—
543C42—948no—
544C42—1884no—
545C42—1884no—
546C42—1884no—
547C42C2×C4×D4714yes—
548C42C2×C4×D4714yes—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—472no—
554C23—472no—
555C23—472no—
556C23—472no—
557C23—948no—
558C23—942no—
559C23—942no—
560C23—941no—
561C23—941no—
562C23—942no—
563C23—942no—
564C23—1888no—
565C23—1884no—
566C23—1888no—
567C23—1884no—
568C23C22×C4×D4712yes—
569C23C24×D4711yescharacteristic
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—472no—
584C2×C4—472no—
585C2×C4—472no—
586C2×C4—4716no—
587C2×C4—474no—
588C2×C4—474no—
589C2×C4—472no—
590C2×C4—948no—
591C2×C4—948no—
592C2×C4—948no—
593C2×C4—941no—
594C2×C4—941no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—948no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—1888no—
607C2×C4—1888no—
608C2×C4C22×C4×D4714yes—
609C2×C4C22×C4×D4714yes—
610C2×C4C22×C4×D4712yes—
611C2×C4C22×C4×D4712yes—
612C2×C4#41×D4712yes—
613C2×C4#41×D4712yes—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—474no—
620C22—942no—
621C22—944no—
622C22—942no—
623C22—944no—
624C22—942no—
625C22—944no—
626C22—18816no—
627C22—1881no—
628C22—1882no—
629C22—1881no—
630C22C23×C4×D4711yescharacteristic
631C22C2×#41×D4711yescharacteristic
632C22C2×#41×D4711yescharacteristic
633C22#1568995812yes—
634C22#1568996012yes—
635C4—22no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—474no—
641C4—942no—
642C4—944no—
643C4—944no—
644C4—1884no—
645C4—1884no—
646C4#117×D4714yes—
647C2—42no—
648C2—944no—
649C2—1882no—
650C2C2×#117×D4711yescharacteristic
651C2#2025658812yes—
652C2#2025677211yescharacteristic
653C2#2025677512yes—
654C2#2025677911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2314C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenter#2314D47
hypocenterC47#34048
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #20245998C2semidirect
3. #20245997C2semidirect
4. #20245996C2semidirect
5. #20245995C2semidirect
6. #20245993C2semidirect
7. #20245987C2semidirect
9. #20245882C2semidirect
10. #20245880C2semidirect
11. #20245794C2semidirect
12. #20245793C2semidirect
14. C4×#13093982C2semidirect
15. C47×#2314C2semidirect
24. #117×D47C4semidirect
36. #15689010C22semidirect
37. #15689009C22semidirect
38. #15689008C22semidirect
39. #15689007C22semidirect
40. #15689006C22semidirect
41. #15689005C22semidirect
42. #15689004C22semidirect
43. #15688993C22semidirect
44. #15688992C22semidirect
45. #15688989C22semidirect
46. #15688988C22semidirect
54. C4⋊C4×Dic47C22semidirect
55. C4⋊C4×Dic47C22semidirect
56. C4⋊C4×Dic47C22semidirect
57. C4×C4⋊Dic47C22semidirect
58. #15688974C22semidirect
63. #15688968C22semidirect
64. C42×Dic47C22semidirect
67. C47×#387C22semidirect
68. C47×#384C22semidirect
69. C47×#382C22semidirect
71. C4×C4⋊C4×C47C22semidirect
326. #2314D47semidirect
372. #387C2×D47semidirect
375. #384C2×D47semidirect
378. #382C2×D47semidirect
392. C4×C4⋊C4C2×D47semidirect
399. C47#34048semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D472
#202459982
#202459972
#202459962
#202459952
#202459932
#202459872
#202459822
#202458822
#202458802
#202457942
#202457932
#202457702
C4×#130939822
C47×#23142
#3404847