Finite Group Databasebeta

← #26123311

Subgroups — #26123311

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689127C211yesmaximal, characteristic
2C2×#15689127C211yesmaximal, characteristic
3C2×#15689024C211yesmaximal, characteristic
4C2×#15688892C211yesmaximal, characteristic, complemented
5C2×#15688882C211yesmaximal, characteristic, complemented
6C2×#15688882C211yesmaximal, characteristic, complemented
7C47×#2317C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×(C2×D47)⋊C4—22no—
12C22×C4×Dic47C2211yescharacteristic
13#15689127—22no—
14#15689127—22no—
15#15689127—22no—
16#15689127C414yescomplemented
17C2×#13093944—21no—
18C2×#13093944—21no—
19C2×#13093944—22no—
20#15689024—22no—
21#15689024—22no—
22#15688892—22no—
23#15688882—22no—
24#15688882—22no—
25C2×#117×C47C2211yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C22×C4⋊C4×C47C2211yescharacteristic
28C47×#389—22no—
29C47×#385—22no—
30C47×#384—21no—
31C47×#382—21no—
32C24×D47—21no—
33C22×D188—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C22×D188—21no—
41C22×D188—21no—
42C22×D188—21no—
43C22×D188—21no—
44C22×D188C2×C412yes—
45C22×D188D412yes—
46C22×D188Q812yes—
47C23×Dic47—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329C2×C412yes—
59C2×#10625329D412yes—
60C22×C47⋊C8—21no—
61C22×C47⋊C8—21no—
62C2×(C2×D47)⋊C4—28no—
63C2×(C2×D47)⋊C4—48no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—21no—
66C2×C4×Dic47—21no—
67C2×C4×Dic47—21no—
68C2×C4×Dic47—21no—
69C2×C4×Dic47D412yes—
70#13093944—24no—
71#13093944—24no—
72#13093944—28no—
73C23×C4×C47D412yes—
74C23×C4×C47C2311yescharacteristic
75#117×C47—21no—
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47C2×C412yes—
82#117×C47D412yes—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—24no—
90C2×C4⋊C4×C47D411yescharacteristic
91C2×C4⋊C4×C47D411yescharacteristic
92C2×C4⋊C4×C47D412yes—
93C2×C22⋊C4×C47—21no—
94C2×C22⋊C4×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97#95×C47—24no—
98C24×C47C2×D411yescharacteristic
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—44no—
103C23×D47—44no—
104C23×D47—41no—
105C23×D47—41no—
106C23×D47—41no—
107C23×D47—41no—
108C22×C4×C47—22no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—44no—
115C22×C4×C47C22×C411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×Q811yescharacteristic
123C22×C4×C47#4111yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C22×C4×C47#4111yescharacteristic
126C22×C4×C47#4111yescharacteristic
127C22×C4×C47#4111yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×Dic47—24no—
130C22×Dic47—22no—
131C22×Dic47—42no—
132C22×Dic47—41no—
133C22×Dic47—41no—
134C2×D188—22no—
135C2×D188—22no—
136C2×D188—22no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—22no—
148C2×D188—22no—
149C2×D188—41no—
150C2×D188—41no—
151C2×D188—41no—
152C2×D188—41no—
153C2×D188—41no—
154C2×D188—41no—
155C2×D188—41no—
156C2×D188—41no—
157C2×D188—41no—
158C2×D188—41no—
159C2×D188—41no—
160C2×D188—41no—
161C2×D188C4⋊C414yescomplemented
162C4⋊C4×C47—22no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—41no—
168C4⋊C4×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47—41no—
171C4⋊C4×C47—84no—
172C4⋊C4×C47C22⋊C414yescomplemented
173C22⋊C4×C47—22no—
174C22⋊C4×C47—22no—
175C22⋊C4×C47—41no—
176C22⋊C4×C47—41no—
177C42×C47—22no—
178C42×C47—22no—
179C42×C47—41no—
180C42×C47—41no—
181(C2×D47)⋊C4—416no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—41no—
185C4×Dic47—41no—
186#10625329—22no—
187#10625329—22no—
188#10625329—22no—
189#10625329—22no—
190#10625329—41no—
191#10625329—41no—
192#10625329—41no—
193#10625329—41no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—41no—
197C2×C47⋊C8—41no—
198C2×C47⋊C8—41no—
199C2×C47⋊C8—41no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47C4×D411yescharacteristic
203C23×C47#12011yescharacteristic
204C23×C47#12011yescharacteristic
205C23×C47#12111yescharacteristic
206C23×C47#12111yescharacteristic
207C23×C47C22⋊Q811yescharacteristic
208C23×C47#12411yescharacteristic
209C22×D47—24no—
210C22×D47—44no—
211C22×D47—48no—
212C22×D47—44no—
213C22×D47—48no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—84no—
221C22×D47—82no—
222C22×D47—82no—
223C22×D47—84no—
224C22×D47—82no—
225C22×D47—82no—
226C2×C4×C47—24no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—44no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—44no—
235C2×C4×C47—42no—
236C2×C4×C47—82no—
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C2×C4⋊C411yescharacteristic
239C2×C4×C47#11711yescharacteristic
240C2×C4×C47C4×D411yescharacteristic
241C2×C4×C47C4×D411yescharacteristic
242C2×C4×C47C4×D411yescharacteristic
243C2×C4×C47#12011yescharacteristic
244C2×C4×C47#12111yescharacteristic
245C2×C4×C47#12111yescharacteristic
246C2×C4×C47#12111yescharacteristic
247C2×C4×C47#12111yescharacteristic
248C2×C4×C47#12111yescharacteristic
249C2×C4×C47#12111yescharacteristic
250C2×C4×C47C22⋊Q811yescharacteristic
251C2×C4×C47C22⋊Q811yescharacteristic
252C2×C4×C47C22⋊Q811yescharacteristic
253C2×C4×C47C22⋊Q811yescharacteristic
254C2×C4×C47C22⋊Q811yescharacteristic
255C2×C4×C47#12311yescharacteristic
256C2×C4×C47#12311yescharacteristic
257C2×C4×C47#12311yescharacteristic
258C2×C4×C47#12311yescharacteristic
259C2×C4×C47#12311yescharacteristic
260C2×C4×C47#12411yescharacteristic
261C2×C4×C47#12411yescharacteristic
262C2×C4×C47#12611yescharacteristic
263C2×C4×C47#12611yescharacteristic
264C2×C4×C47#12711yescharacteristic
265C2×Dic47—24no—
266C2×Dic47—44no—
267C2×Dic47—42no—
268C2×Dic47—44no—
269C2×Dic47—42no—
270D188—24no—
271D188—24no—
272D188—42no—
273D188—42no—
274D188—42no—
275D188—42no—
276D188—42no—
277D188—42no—
278C47⋊C8—42no—
279C47⋊C8—42no—
280#11063—471nomaximal, Sylow
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—21no—
286C22×C47—21no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47#38511yescharacteristic
290C22×C47#38611yescharacteristic
291C22×C47#38811yescharacteristic
292C22×C47#39211yescharacteristic
293C22×C47#39311yescharacteristic
294C22×C47#39411yescharacteristic
295C22×C47#39611yescharacteristic
296C22×C47#45312yes—
297C22×C47#45412yes—
298C2×D47—48no—
299C2×D47—44no—
300C2×D47—84no—
301C2×D47—84no—
302C2×D47—84no—
303C2×D47—84no—
304C2×D47—82no—
305C2×D47—82no—
306C2×D47—84no—
307C4×C47—44no—
308C4×C47—81no—
309C4×C47—82no—
310C4×C47—81no—
311C4×C47#38011yescharacteristic
312C4×C47#38511yescharacteristic
313C4×C47#38811yescharacteristic
314C4×C47#39011yescharacteristic
315C4×C47#39411yescharacteristic
316C4×C47#39611yescharacteristic
317C4×C47#39711yescharacteristic
318C4×C47#39911yescharacteristic
319Dic47—44no—
320C2×#418—471no—
321C2×#418—471no—
322#2317D4711yescharacteristic, complemented
323C2×#390—471no—
324C2×#343—471no—
325C2×#339—471no—
326C2×#339—471no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47—21no—
330C2×C47—21no—
331C2×C47#143911yescharacteristic
332C2×C47#188312yes—
333C2×C47#200211yescharacteristic
334C2×C47#200311yescharacteristic
335C2×C47#202712yes—
336D47—84no—
337C23×D4—471no—
338C22×M16—471no—
339C22×M16—471no—
340C2×#117C2×D4711yescharacteristic
341C2×#117C2×D4711yescharacteristic
342C22×C4⋊C4C2×D4711yescharacteristic
343C22×C22⋊C4—942no—
344C22×C42—471no—
345#418—474no—
346#418—942no—
347#418—942no—
348#418—942no—
349C2×#102—941no—
350C2×#102—941no—
351C2×#102—942no—
352#390—942no—
353#390—942no—
354#389—22no—
355#385—22no—
356#384—21no—
357#382—21no—
358#343—942no—
359#339—942no—
360#339—942no—
361C47#1106311yescharacteristic, complemented, Sylow
362C25—941no—
363C22×D4—472no—
364C22×D4—472no—
365C22×D4—472no—
366C22×D4—941no—
367C22×D4—941no—
368C22×D4—941no—
369C22×D4—941no—
370C22×D4—941no—
371C22×D4—941no—
372C22×D4—941no—
373C22×D4—941no—
374C22×D4—941no—
375C22×D4—941no—
376C22×D4—941no—
377C23×C4—941no—
378C23×C4C22⋊D4712yes—
379C23×C4C22×D4711yescharacteristic
380C2×M16—472no—
381C2×M16—472no—
382C2×M16—941no—
383C2×M16—941no—
384C2×M16—941no—
385C2×M16—941no—
386C2×M16—941no—
387C2×M16—941no—
388C2×M16—941no—
389C2×M16—941no—
390C2×M16—941no—
391C2×M16—941no—
392C22×C8—941no—
393C22×C8—941no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117C4×D4712yes—
401#117D18812yes—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4C22⋊D4711yescharacteristic
410C2×C4⋊C4C22⋊D4711yescharacteristic
411C2×C4⋊C4C22⋊D4712yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—948no—
415C2×C22⋊C4—1888no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—472no—
419C2×C42—941no—
420C2×C42—941no—
421C2×C42—941no—
422C2×C42—941no—
423C2×C42—941no—
424#102—944no—
425#102—944no—
426#102—948no—
427#95—24no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24—1884no—
432C24—1884no—
433C24—1881no—
434C24—1881no—
435C24—1881no—
436C24—1881no—
437C24C2×C22⋊D4711yescharacteristic
438C2×D4—474no—
439C2×D4—942no—
440C2×D4—942no—
441C2×D4—942no—
442C2×D4—942no—
443C2×D4—942no—
444C2×D4—942no—
445C2×D4—942no—
446C2×D4—942no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—942no—
451C2×D4—942no—
452C2×D4—942no—
453C2×D4—942no—
454C2×D4—1881no—
455C2×D4—1881no—
456C2×D4—1881no—
457C2×D4—1881no—
458C2×D4—1881no—
459C2×D4—1881no—
460C2×D4—1881no—
461C2×D4—1881no—
462C2×D4—1881no—
463C2×D4—1881no—
464C2×D4—1881no—
465C2×D4—1881no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—944no—
474C22×C4—942no—
475C22×C4—1882no—
476C22×C4—1881no—
477C22×C4—1881no—
478C22×C4C2×C4×D4711yescharacteristic
479C22×C4C2×D18811yescharacteristic
480C22×C4#1062535011yescharacteristic
481C22×C4#1062535011yescharacteristic
482C22×C4D4×D4711yescharacteristic
483C22×C4D4×D4711yescharacteristic
484C22×C4D4×D4711yescharacteristic
485C22×C4#1062535211yescharacteristic
486C22×C4Q8×D4711yescharacteristic
487C22×C4#1062535411yescharacteristic
488C22×C4#1062535411yescharacteristic
489C22×C4#1062535411yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C2×C22⋊D4711yescharacteristic
492M16—942no—
493M16—942no—
494M16—942no—
495M16—942no—
496M16—1881no—
497M16—1881no—
498M16—1881no—
499M16—1881no—
500C2×C8—942no—
501C2×C8—942no—
502C2×C8—1881no—
503C2×C8—1881no—
504C2×C8—1881no—
505C2×C8—1881no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—84no—
516C4⋊C4(C2×D47)⋊C414yescomplemented
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—18816no—
522C42—22no—
523C42—22no—
524C42—41no—
525C42—41no—
526C42—942no—
527C42—942no—
528C42—1881no—
529C42—1881no—
530C23—22no—
531C23—22no—
532C23—944no—
533C23—3764no—
534C23—3764no—
535C23—3762no—
536C23—3762no—
537C23—3762no—
538C23—3762no—
539C23—1884no—
540C23—1888no—
541C23—1888no—
542C23—1884no—
543C23—1882no—
544C23—1882no—
545C23—1882no—
546C23—1882no—
547C23—1882no—
548C23—1882no—
549C23C4×C22⋊D4711yescharacteristic
550C23#1309399211yescharacteristic
551C23#1309399311yescharacteristic
552C23#1309399611yescharacteristic
553C23#1309399811yescharacteristic
554C23#1309400011yescharacteristic
555C23#1309401311yescharacteristic
556D4—944no—
557D4—944no—
558D4—1882no—
559D4—1882no—
560D4—1882no—
561D4—1882no—
562D4—1882no—
563D4—1882no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—944no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4C2×(C2×D47)⋊C411yescharacteristic
581C2×C4C4⋊C4×D4711yescharacteristic
582C2×C4#1309398211yescharacteristic
583C2×C4C4×C22⋊D4711yescharacteristic
584C2×C4#1309398611yescharacteristic
585C2×C4#1309398611yescharacteristic
586C2×C4#1309399211yescharacteristic
587C2×C4#1309399511yescharacteristic
588C2×C4#1309399611yescharacteristic
589C2×C4#1309399811yescharacteristic
590C2×C4#1309399911yescharacteristic
591C2×C4#1309399911yescharacteristic
592C2×C4#1309399911yescharacteristic
593C2×C4#1309400011yescharacteristic
594C2×C4#1309400011yescharacteristic
595C2×C4#1309400211yescharacteristic
596C2×C4#1309400211yescharacteristic
597C2×C4#1309400411yescharacteristic
598C2×C4#1309400711yescharacteristic
599C2×C4#1309400911yescharacteristic
600C2×C4#1309400911yescharacteristic
601C2×C4#1309400911yescharacteristic
602C2×C4#1309400911yescharacteristic
603C2×C4#1309401311yescharacteristic
604C2×C4#1309401311yescharacteristic
605C2×C4#1309401911yescharacteristic
606C2×C4#1309401911yescharacteristic
607C2×C4#1309402011yescharacteristic
608C8—1882no—
609C8—1882no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—3764no—
619C22—3764no—
620C22—3764no—
621C22—3764no—
622C22—3764no—
623C22—3762no—
624C22—3762no—
625C22—1888no—
626C22—1884no—
627C22#1568899411yescharacteristic
628C22#1568900211yescharacteristic
629C22#1568901311yescharacteristic
630C22#1568903111yescharacteristic
631C22#1568903411yescharacteristic
632C22#1568904011yescharacteristic
633C22#1568904511yescharacteristic
634C22#1568929112yes—
635C22#1568929612yes—
636C4—44no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—1884no—
641C4#1568898011yescharacteristic
642C4#1568899411yescharacteristic
643C4#1568901311yescharacteristic
644C4#1568902411yescharacteristic
645C4#1568904011yescharacteristic
646C4#1568904511yescharacteristic
647C4#1568904911yescharacteristic
648C4#1568905811yescharacteristic
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—3764no—
654C2#2024163011yescharacteristic
655C2#2024376412yes—
656C2#2024440111yescharacteristic
657C2#2024440811yescharacteristic
658C2#2024454112yes—
subgroupG/H
centerC23#13093993
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2317C2
socleC23×C47#120
prime radicalC23×D188C22
cosocleC23×C4×C47C23
hypercenter#2317D47
hypocenterC47#11063
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. C2×#15688892C2semidirect
5. C2×#15688882C2semidirect
6. C2×#15688882C2semidirect
7. C47×#2317C2semidirect
16. #15689127C4semidirect
161. C2×D188C4⋊C4semidirect
172. C4⋊C4×C47C22⋊C4semidirect
322. #2317D47semidirect
361. C47#11063semidirect
516. C4⋊C4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156891272
C2×#156890242
C2×#156888922
C2×#156888822
C47×#23172
#1106347