Finite Group Databasebeta

← #26215491

Subgroups — #26215491

Order 12032.

3966 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625354C211yesmaximal, characteristic
2C2×#15689132C211yesmaximal, characteristic, complemented
3C2×#15689129C211yesmaximal, characteristic, complemented
4C2×#15689127C211yesmaximal, characteristic, complemented
5C2×#15689086C211yesmaximal, characteristic, complemented
6#20245767C211yesmaximal, characteristic, complemented
7#20243327C218yesmaximal, complemented
8C2×C47×#419C211yesmaximal, characteristic, complemented
9C22×#10625354—24no—
10C22×#10625354—24no—
11C22×#10625354—24no—
12C22×#10625354—22no—
13C22×#10625354—24no—
14C22×#10625354—24no—
15C22×#10625354C418yescomplemented
16C22×#10625354C2214yes—
17C23×D188—21no—
18C23×D188C2211yescharacteristic
19C23×C4×D47—21no—
20C23×C4×D47C2211yescharacteristic
21C22×#10625329C2211yescharacteristic
22C2×#13093983C2211yescharacteristic
23#15689132—24no—
24#15689129—24no—
25#15689127—24no—
26C2×#13093948C2214yescomplemented
27C2×#13093946C2214yescomplemented
28C2×#13093944C2214yescomplemented
29#15689086—24no—
30C2×(C2×D47)⋊C8C2214yescomplemented
31#15688984—21no—
32#15688980C2214yescomplemented
33#15688978—21no—
34C4×(C2×D47)⋊C4—22no—
35C23×Q8×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#419—24no—
39C2×#103×C47C2214yescomplemented
40C24×D47—21no—
41C24×D47D411yescharacteristic
42C2×#10625354—216no—
43C2×#10625354—216no—
44C2×#10625354—232no—
45C2×#10625354—232no—
46C2×#10625354—232no—
47C2×#10625354—28no—
48C2×#10625354—28no—
49C2×#10625354—232no—
50C2×#10625354—28no—
51C2×#10625354—28no—
52C2×#10625354—216no—
53C2×#10625354—28no—
54C2×#10625354C2×C4116yes—
55C22×D188—28no—
56C22×D188—28no—
57C22×D188—28no—
58C22×D188—24no—
59C22×D188—21no—
60C22×D188—22no—
61C22×D188—22no—
62C22×D188—22no—
63C22×D188C2×C414yes—
64C22×D188D414yes—
65C22×D188D414yes—
66C22×D188C2312yes—
67C23×Dic47D411yescharacteristic
68C22×C4×D47—28no—
69C22×C4×D47—28no—
70C22×C4×D47—22no—
71C22×C4×D47—22no—
72C22×C4×D47—22no—
73C22×C4×D47—22no—
74C22×C4×D47—28no—
75C22×C4×D47—24no—
76C22×C4×D47C2×C414yes—
77C22×C4×D47D412yes—
78C22×C4×D47D414yes—
79C22×C4×D47C2312yes—
80C2×#10625329—22no—
81C2×#10625329—22no—
82C2×#10625329—22no—
83C22×C47⋊C8C2312yes—
84#13093983—22no—
85#13093983—22no—
86C2×C4⋊Dic47C2312yes—
87C2×C22⋊Dic47—21no—
88C2×(C2×D47)⋊C4—24no—
89C2×C4×Dic47—21no—
90#13093948—28no—
91#13093946—28no—
92#13093944—28no—
93(C2×D47)⋊C8—28no—
94C47⋊#95—24no—
95C22×Q8×C47—22no—
96C22×Q8×C47—22no—
97C22×Q8×C47—22no—
98C22×Q8×C47—21no—
99C22×Q8×C47C2×C414yes—
100C22×Q8×C47D414yes—
101C22×Q8×C47D414yes—
102C22×Q8×C47C2312yes—
103C23×C4×C47—21no—
104C23×C4×C47C2311yescharacteristic
105C2×M16×C47—22no—
106C2×M16×C47—22no—
107C2×M16×C47—22no—
108C22×C8×C47C2312yes—
109#117×C47—22no—
110#117×C47—22no—
111C2×C4⋊C4×C47C2312yes—
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114#103×C47—28no—
115C24×C47C2×D411yescharacteristic
116C23×D47—24no—
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—24no—
120C23×D47—24no—
121C23×D47—22no—
122C23×D47—44no—
123C23×D47—48no—
124C23×D47C22⋊C414yes—
125C23×D47C2×D412yes—
126C2×Q8×C47—24no—
127C2×Q8×C47—28no—
128C2×Q8×C47—24no—
129C2×Q8×C47—24no—
130C2×Q8×C47—28no—
131C2×Q8×C47—28no—
132C2×Q8×C47—22no—
133C2×Q8×C47—22no—
134C2×Q8×C47—22no—
135C2×Q8×C47C22⋊C4116yescomplemented
136C2×Q8×C47C22×C414yes—
137C2×Q8×C47C2×D414yes—
138C2×Q8×C47C2×D414yes—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2411yescharacteristic
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47C22⋊C414yes—
156C22×Dic47C2×D412yes—
157#10625354—232no—
158#10625354—232no—
159#10625354—2128no—
160#10625354—232no—
161#10625354—232no—
162C2×D188—216no—
163C2×D188—216no—
164C2×D188—264no—
165C2×D188—216no—
166C2×D188—28no—
167C2×D188—24no—
168C2×D188—28no—
169C2×D188—28no—
170C2×D188—24no—
171C2×D188—24no—
172C2×D188—22no—
173C2×D188—22no—
174C2×D188—22no—
175C2×D188C22⋊C4116yescomplemented
176C2×D188C22×C414yes—
177C2×D188C2×D414yes—
178C2×D188C2×D414yes—
179C2×C4×D47—216no—
180C2×C4×D47—216no—
181C2×C4×D47—264no—
182C2×C4×D47—28no—
183C2×C4×D47—28no—
184C2×C4×D47—28no—
185C2×C4×D47—28no—
186C2×C4×D47—24no—
187C2×C4×D47—24no—
188C2×C4×D47—22no—
189C2×C4×D47—22no—
190C2×C4×D47—22no—
191C2×C4×D47—22no—
192C2×C4×D47—216no—
193C2×C4×D47C22⋊C418yes—
194C2×C4×D47C22×C414yes—
195C2×C4×D47C2×D414yes—
196M16×C47—44no—
197C2×C8×C47—22no—
198C2×C8×C47—22no—
199C2×C8×C47—22no—
200C4⋊C4×C47—22no—
201C4⋊C4×C47—22no—
202C22⋊C4×C47—42no—
203C42×C47—42no—
204C22⋊Dic47—42no—
205(C2×D47)⋊C4—48no—
206C4⋊Dic47—22no—
207C4⋊Dic47—22no—
208C4×Dic47—42no—
209#10625329—44no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—22no—
213C23×C47—24no—
214C23×C47C2×C22⋊C412yes—
215C23×C47#12012yes—
216C23×C47#12012yes—
217C23×C47C22×D411yescharacteristic
218C22×D47—24no—
219C22×D47—22no—
220C22×D47—22no—
221C22×D47—24no—
222C22×D47—24no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47—28no—
226C22×D47—216no—
227C22×D47—48no—
228C22×D47—416no—
229C22×D47—48no—
230C22×D47—48no—
231C22×D47—416no—
232C22×D47—416no—
233C22×D47—48no—
234C22×D47C2×C22⋊C414yes—
235Q8×C47—24no—
236Q8×C47—24no—
237Q8×C47—24no—
238Q8×C47—216no—
239Q8×C47C2×C22⋊C418yes—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—216no—
244C2×C4×C47—42no—
245C2×C4×C47—44no—
246C2×C4×C47C2×C22⋊C414yes—
247C2×C4×C47C2×C22⋊C412yes—
248C2×C4×C47C2×C22⋊C414yes—
249C2×C4×C47#12014yes—
250C2×C4×C47#12012yes—
251C2×C4×C47#12014yes—
252C2×C4×C47#12012yes—
253C2×C4×C47C23×C411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—28no—
262C2×Dic47—42no—
263C2×Dic47—44no—
264C2×Dic47C2×C22⋊C414yes—
265D188—264no—
266D188—24no—
267D188—24no—
268D188—24no—
269D188—216no—
270D188C2×C22⋊C418yes—
271C4×D47—264no—
272C4×D47—24no—
273C4×D47—24no—
274C4×D47—24no—
275C4×D47—24no—
276C4×D47—216no—
277C8×C47—42no—
278C47⋊C8—42no—
279#20600—471nomaximal, Sylow
280C22×C47—24no—
281C22×C47—24no—
282C22×C47—24no—
283C22×C47#37914yes—
284C22×C47C22×C22⋊C411yescharacteristic
285C22×C47C2×#12011yescharacteristic
286C22×C47C2×#12011yescharacteristic
287C22×C47#57514yes—
288C2×D47—24no—
289C2×D47—24no—
290C2×D47—24no—
291C2×D47—28no—
292C2×D47—416no—
293C2×D47—432no—
294C2×D47—48no—
295C2×D47—44no—
296C2×D47—44no—
297C2×D47—48no—
298C2×D47—48no—
299C4×C47—28no—
300C4×C47—82no—
301C4×C47#37914yes—
302C4×C47C22×C22⋊C411yescharacteristic
303C4×C47C22×C22⋊C411yescharacteristic
304C4×C47C2×#12011yescharacteristic
305C4×C47C2×#12011yescharacteristic
306Dic47—24no—
307Dic47—82no—
308C23×#41—471no—
309C2×#419—471no—
310C2×#419D4711yescharacteristic, complemented
311C2×#418—471no—
312C2×#418—471no—
313C2×#407—471no—
314#2280—471no—
315#1788—478no—
316C2×C47—24no—
317C2×C47C2×#37911yescharacteristic
318C2×C47#289612yes—
319C2×C47#300512yes—
320C2×C47#300512yes—
321D47—44no—
322D47—48no—
323C22×#41—478no—
324C22×#41—474no—
325C22×#41—942no—
326C22×#41—944no—
327C22×#41—944no—
328C22×#41—944no—
329C22×#41—944no—
330C22×#41—944no—
331C23×Q8C2×D4711yescharacteristic
332C23×D4—471no—
333C23×D4—941no—
334C24×C4—471no—
335C24×C4—941no—
336C22×M16—471no—
337C22×M16C2×D4711yescharacteristic
338C2×#117—471no—
339C2×#117C2×D4711yescharacteristic
340#419—24no—
341#419—944no—
342#418—944no—
343#418—944no—
344C2×#103—474no—
345C2×#103C2×D4714yescomplemented
346C2×#102—474no—
347C2×#102—474no—
348#407—944no—
349C2×C22⋊C8—474no—
350#381—941no—
351#380—474no—
352#379—941no—
353C4×C22⋊C4—942no—
354C47#2060011yescharacteristic, complemented, Sylow
355C25—471no—
356C25—941no—
357C2×#41—4716no—
358C2×#41—9416no—
359C2×#41—948no—
360C2×#41—9432no—
361C2×#41—948no—
362C2×#41—948no—
363C2×#41—9432no—
364C2×#41—9416no—
365C2×#41—9432no—
366C2×#41—9416no—
367C2×#41—9432no—
368C2×#41—948no—
369C2×#41—948no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—21no—
374C22×Q8C4×D4714yes—
375C22×Q8D18814yes—
376C22×Q8C22⋊D4714yes—
377C22×Q8C22×D4712yes—
378C22×D4—474no—
379C22×D4—474no—
380C22×D4—474no—
381C22×D4—472no—
382C22×D4—948no—
383C22×D4—944no—
384C22×D4—948no—
385C22×D4—948no—
386C22×D4—942no—
387C22×D4—941no—
388C22×D4—942no—
389C22×D4—942no—
390C23×C4—21no—
391C23×C4—471no—
392C23×C4—474no—
393C23×C4—472no—
394C23×C4—474no—
395C23×C4—472no—
396C23×C4—948no—
397C23×C4—944no—
398C23×C4—948no—
399C23×C4—948no—
400C23×C4—942no—
401C23×C4—942no—
402C23×C4—942no—
403C23×C4—942no—
404C23×C4C22×D4711yescharacteristic
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—942no—
409C2×M16—942no—
410C2×M16—942no—
411C22×C8—472no—
412C22×C8C22×D4712yes—
413#117—22no—
414#117—22no—
415#117—942no—
416#117—942no—
417C2×C4⋊C4—472no—
418C2×C4⋊C4C22×D4712yes—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—944no—
421C2×C22⋊C4—941no—
422C2×C42—21no—
423C2×C42—941no—
424#103—28no—
425#103—948no—
426#102—948no—
427#102—948no—
428C22⋊C8—948no—
429#95—944no—
430C24—472no—
431C24—474no—
432C24—944no—
433C24—944no—
434C24—942no—
435C24—944no—
436C24—942no—
437C24—942no—
438C24—1884no—
439C24—1888no—
440C24D4×D4711yescharacteristic
441#41—9432no—
442#41—94128no—
443#41—9432no—
444#41—9432no—
445#41—9432no—
446C2×Q8—28no—
447C2×Q8—24no—
448C2×Q8—28no—
449C2×Q8—24no—
450C2×Q8—28no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8(C2×D47)⋊C4116yescomplemented
456C2×Q8C2×C4×D4714yes—
457C2×Q8C2×D18814yes—
458C2×Q8C2×C22⋊D4714yes—
459C2×D4—4716no—
460C2×D4—474no—
461C2×D4—474no—
462C2×D4—474no—
463C2×D4—9416no—
464C2×D4—9464no—
465C2×D4—9416no—
466C2×D4—9416no—
467C2×D4—944no—
468C2×D4—948no—
469C2×D4—944no—
470C2×D4—948no—
471C2×D4—944no—
472C2×D4—948no—
473C2×D4—942no—
474C2×D4—942no—
475C2×D4—942no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—474no—
482C22×C4—472no—
483C22×C4—478no—
484C22×C4—474no—
485C22×C4—474no—
486C22×C4—9416no—
487C22×C4—9464no—
488C22×C4—9416no—
489C22×C4—9416no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—948no—
493C22×C4—948no—
494C22×C4—948no—
495C22×C4—944no—
496C22×C4—948no—
497C22×C4—944no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4C2×C4×D4712yes—
504C22×C4C2×D18812yes—
505C22×C4D4×D4711yescharacteristic
506C22×C4D4×D4712yes—
507C22×C4D4×D4712yes—
508C22×C4D4×D4712yes—
509C22×C4C2×C22⋊D4712yes—
510C22×C4C23×D4711yescharacteristic
511M16—44no—
512M16—1884no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—942no—
517C2×C8—942no—
518C2×C8—942no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—942no—
522C4⋊C4—942no—
523C22⋊C4—42no—
524C22⋊C4—1888no—
525C22⋊C4—1882no—
526C42—42no—
527C42—1882no—
528C23—24no—
529C23—474no—
530C23—9416no—
531C23—948no—
532C23—944no—
533C23—944no—
534C23—942no—
535C23—944no—
536C23—942no—
537C23—944no—
538C23—944no—
539C23—1888no—
540C23—18816no—
541C23—1888no—
542C23—18816no—
543C23—1888no—
544C23—18816no—
545C23—1888no—
546C23C22⋊C4×D4712yes—
547C23#1309399212yes—
548C23#1309399412yes—
549C23C2×D4×D4711yescharacteristic
550Q8—216no—
551Q8—24no—
552Q8—24no—
553Q8—24no—
554Q8C2×(C2×D47)⋊C418yes—
555D4—478no—
556D4—9464no—
557D4—9416no—
558D4—944no—
559D4—944no—
560D4—944no—
561C2×C4—216no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—474no—
568C2×C4—9464no—
569C2×C4—948no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—9416no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—1882no—
578C2×C4—1884no—
579C2×C4C22⋊C4×D4712yes—
580C2×C4C22⋊C4×D4714yes—
581C2×C4C2×(C2×D47)⋊C414yes—
582C2×C4#1309399212yes—
583C2×C4#1309399214yes—
584C2×C4#1309399412yes—
585C2×C4#1309399414yes—
586C2×C4C22×C4×D4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C2×D4×D4711yescharacteristic
589C2×C4C2×D4×D4711yescharacteristic
590C2×C4C22×C22⋊D4711yescharacteristic
591C2×C4C22×D18811yescharacteristic
592C8—42no—
593C8—1882no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—944no—
598C22—944no—
599C22—948no—
600C22—944no—
601C22—18816no—
602C22—18832no—
603C22—1888no—
604C22—1884no—
605C22—1888no—
606C22—1884no—
607C22—1888no—
608C22#1568897814yes—
609C22C2×C22⋊C4×D4711yescharacteristic
610C22C2×#1309399211yescharacteristic
611C22C2×#1309399411yescharacteristic
612C22#1568992214yes—
613C4—28no—
614C4—82no—
615C4—944no—
616C4—3762no—
617C4#1568897814yes—
618C4C2×C22⋊C4×D4711yescharacteristic
619C4C22×(C2×D47)⋊C411yescharacteristic
620C4C2×#1309399211yescharacteristic
621C4C2×#1309399411yescharacteristic
622C2—24no—
623C2—1884no—
624C2—1888no—
625C2C2×#1568897811yescharacteristic
626C2#2025190612yes—
627C2#2025262912yes—
628C2#2025263412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC23×#10625354C2
cosocleC22×C4×C47C24
hypercenterC2×#419D47
hypocenterC47#20600
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#15689132C2semidirect
3. C2×#15689129C2semidirect
4. C2×#15689127C2semidirect
5. C2×#15689086C2semidirect
6. #20245767C2semidirect
7. #20243327C2semidirect
8. C2×C47×#419C2semidirect
15. C22×#10625354C4semidirect
26. C2×#13093948C22semidirect
27. C2×#13093946C22semidirect
28. C2×#13093944C22semidirect
30. C2×(C2×D47)⋊C8C22semidirect
32. #15688980C22semidirect
39. C2×#103×C47C22semidirect
135. C2×Q8×C47C22⋊C4semidirect
175. C2×D188C22⋊C4semidirect
310. C2×#419D47semidirect
345. C2×#103C2×D47semidirect
354. C47#20600semidirect
455. C2×Q8(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#106253542
C2×#156891322
C2×#156891292
C2×#156891272
C2×#156890862
#202457672
#202433272
C2×C47×#4192
#2060047