Finite Group Databasebeta

← #26243142

Subgroups — #26243142

Order 12032.

2014 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×#15689701C211yesmaximal, characteristic
2C2×#15689697C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×#13093948C211yesmaximal, characteristic, complemented
5C2×#15689091C211yesmaximal, characteristic, complemented
6#20244395C214yesmaximal, complemented
7#20244388C214yesmaximal, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C22×#137×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12#15689701—24no—
13#15689701—24no—
14#15689697—24no—
15#15689697—24no—
16C2×#13094022—21no—
17C2×#13094013—21no—
18C2×#13094007—22no—
19C2×#13094003—21no—
20C2×#13094000C2214yes—
21C2×#13094000C2212yes—
22C2×#13093998—21no—
23C2×#13093996—22no—
24C2×#13093995C2212yes—
25C2×#13093983C2211yescharacteristic
26C22×C4⋊Dic47C2211yescharacteristic
27#15689119—24no—
28C2×#13093948—24no—
29C2×#13093948C2214yes—
30C2×#13093948C2212yes—
31C2×#13093945C2212yes—
32#15689091—24no—
33C2×(C2×D47)⋊C8C2214yes—
34#15688886C2214yes—
35C22×#41×C47C2211yescharacteristic
36C23×Q8×C47C2211yescharacteristic
37C2×#137×C47—24no—
38C2×#137×C47—24no—
39C2×#137×C47—24no—
40C22×Q16×C47C2212yes—
41C22×SD16×C47C2212yes—
42C22×M16×C47C2211yescharacteristic
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C22×Dic94D411yescharacteristic
50C22×D188D411yescharacteristic
51C22×C22⋊D47—22no—
52C22×C22⋊D47D412yes—
53C23×Dic47—21no—
54C22×C4×D47C2312yes—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8C2312yes—
59#13094022—28no—
60#13094013—28no—
61#13094007—216no—
62#13094003—28no—
63#13094000—28no—
64#13094000—24no—
65#13094000—28no—
66#13094000—24no—
67#13093998—28no—
68#13093996—216no—
69#13093995—24no—
70#13093995—24no—
71#13093983—22no—
72#13093983D414yescomplemented
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47D414yes—
76C2×C4⋊Dic47C2312yes—
77C2×C4⋊Dic47C2312yes—
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—24no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—24no—
84C2×C4×Dic47—21no—
85#13093948—28no—
86#13093948—24no—
87#13093945—24no—
88(C2×D47)⋊C8—28no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47D414yes—
95C22×Q8×C47—21no—
96C22×Q8×C47—22no—
97C22×Q8×C47—22no—
98C22×Q8×C47—22no—
99C22×Q8×C47—22no—
100C22×Q8×C47—22no—
101C22×Q8×C47D414yescomplemented
102C22×Q8×C47C2312yes—
103C22×Q8×C47C2311yescharacteristic
104C22×D4×C47—21no—
105C22×D4×C47C2311yescharacteristic
106C23×C4×C47—21no—
107C23×C4×C47—21no—
108C23×C4×C47C2311yescharacteristic
109#137×C47—232no—
110C2×Q16×C47—24no—
111C2×Q16×C47—24no—
112C2×Q16×C47—24no—
113C2×SD16×C47—24no—
114C2×SD16×C47—24no—
115C2×SD16×C47—24no—
116C2×M16×C47—22no—
117C2×M16×C47—22no—
118C2×M16×C47D414yescomplemented
119C22×C8×C47C2312yes—
120C24×C47—21no—
121C24×C47C2×D411yescharacteristic
122C23×D47C2×D412yes—
123#41×C47—28no—
124#41×C47—28no—
125#41×C47—44no—
126#41×C47—44no—
127C2×Q8×C47—24no—
128C2×Q8×C47—22no—
129C2×Q8×C47—21no—
130C2×Q8×C47—28no—
131C2×Q8×C47—28no—
132C2×Q8×C47—22no—
133C2×Q8×C47—22no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—22no—
137C2×Q8×C47—21no—
138C2×Q8×C47—22no—
139C2×Q8×C47—21no—
140C2×Q8×C47—42no—
141C2×Q8×C47—42no—
142C2×Q8×C47—44no—
143C2×Q8×C47—44no—
144C2×Q8×C47C2×D414yes—
145C2×Q8×C47C2×D412yes—
146C2×D4×C47—24no—
147C2×D4×C47—21no—
148C2×D4×C47—21no—
149C2×D4×C47—21no—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—42no—
153C2×D4×C47—42no—
154C2×D4×C47C2×D412yes—
155C22×C4×C47—24no—
156C22×C4×C47—21no—
157C22×C4×C47—22no—
158C22×C4×C47—24no—
159C22×C4×C47—42no—
160C22×C4×C47—42no—
161C22×C4×C47—42no—
162C22×C4×C47—42no—
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2×D412yes—
165C22×C4×C47C2×D412yes—
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2×D412yes—
168C22×C4×C47C2×D412yes—
169C22×C4×C47#4112yes—
170C22×C4×C47C2411yescharacteristic
171C2×C22⋊D47—24no—
172C2×C22⋊D47—24no—
173C2×C22⋊D47—24no—
174C2×C22⋊D47—28no—
175C2×C22⋊D47—84no—
176C22×Dic47—22no—
177C22×Dic47—22no—
178C22×Dic47—24no—
179C22×Dic47—82no—
180C22×Dic47C2×D412yes—
181#10625350—44no—
182#10625350—44no—
183#10625350—44no—
184#10625350—44no—
185C2×D188—22no—
186C2×D188—22no—
187C2×D188—22no—
188C2×C4×D47—22no—
189C2×C4×D47—22no—
190C2×C4×D47—22no—
191C2×C4×D47—22no—
192C2×C4×D47—22no—
193C2×C4×D47—22no—
194C2×Dic94—22no—
195C2×Dic94—22no—
196C2×Dic94—22no—
197Q16×C47—216no—
198SD16×C47—216no—
199M16×C47—28no—
200C2×C8×C47—22no—
201C2×C8×C47—22no—
202C2×C8×C47C2×D414yes—
203C22⋊Dic47—28no—
204C22⋊Dic47—24no—
205(C2×D47)⋊C4—28no—
206(C2×D47)⋊C4—216no—
207C4⋊Dic47—24no—
208C4⋊Dic47—22no—
209C4⋊Dic47—22no—
210C4⋊Dic47C2×D414yes—
211C4⋊Dic47C2×D414yes—
212Dic47⋊C4—28no—
213Dic47⋊C4—216no—
214C4×Dic47—24no—
215C2×C47⋊C8—22no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—44no—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—42no—
225C23×C47—42no—
226C23×C47#12012yes—
227C23×C47#12012yes—
228C23×C47#12112yes—
229C23×C47C22×D411yescharacteristic
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—24no—
234C22×D47—24no—
235Q8×C47—24no—
236Q8×C47—22no—
237Q8×C47—24no—
238Q8×C47—22no—
239Q8×C47—42no—
240Q8×C47—41no—
241Q8×C47—42no—
242Q8×C47—41no—
243Q8×C47—48no—
244D4×C47—22no—
245D4×C47—22no—
246D4×C47—48no—
247D4×C47—41no—
248D4×C47—41no—
249C2×C4×C47—22no—
250C2×C4×C47—24no—
251C2×C4×C47—41no—
252C2×C4×C47—42no—
253C2×C4×C47—41no—
254C2×C4×C47—42no—
255C2×C4×C47—48no—
256C2×C4×C47—48no—
257C2×C4×C47#12012yes—
258C2×C4×C47#12012yes—
259C2×C4×C47#12014yes—
260C2×C4×C47#12112yes—
261C2×C4×C47#12114yes—
262C2×C4×C47#12114yes—
263C2×C4×C47#12714yes—
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C22⋊D47—48no—
271C22⋊D47—88no—
272C2×Dic47—24no—
273C2×Dic47—22no—
274C2×Dic47—22no—
275C2×Dic47—24no—
276C2×Dic47—22no—
277C2×Dic47—42no—
278C2×Dic47—44no—
279C2×Dic47—42no—
280C2×Dic47—84no—
281D188—44no—
282C4×D47—42no—
283C4×D47—42no—
284C4×D47—42no—
285C4×D47—42no—
286Dic94—44no—
287C8×C47—24no—
288C47⋊C8—42no—
289#23394—471nomaximal, Sylow
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—24no—
293C22×C47—24no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—82no—
298C22×C47—82no—
299C22×C47#39214yes—
300C22×C47C2×#12011yescharacteristic
301C22×C47C2×#12011yescharacteristic
302C22×C47C2×#12111yescharacteristic
303C22×C47#57712yes—
304C22×C47#57812yes—
305C2×D47—42no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—44no—
309C2×D47—44no—
310C4×C47—42no—
311C4×C47—44no—
312C4×C47#39214yes—
313C4×C47C2×#12011yescharacteristic
314C4×C47C2×#12111yescharacteristic
315C4×C47C2×#12111yescharacteristic
316C4×C47C2×#12711yescharacteristic
317Dic47—42no—
318Dic47—44no—
319Dic47—44no—
320C22×#137D4711yescharacteristic, complemented
321C2×#537—471no—
322C2×#536—471no—
323C2×#416—471no—
324C22×#103—471no—
325C2×#408—471no—
326#2001—474no—
327#2000—474no—
328#1736—471no—
329C2×C47—24no—
330C2×C47—82no—
331C2×C47C2×#39211yescharacteristic
332C2×C47#300911yescharacteristic
333C2×C47#301111yescharacteristic
334C2×C47#301312yes—
335C2×C47#306312yes—
336D47—82no—
337C22×#41—471no—
338C22×#41C2×D4711yescharacteristic
339C23×Q8C2×D4711yescharacteristic
340C2×#137—24no—
341C2×#137—24no—
342C2×#137—24no—
343C22×Q16C2×D4712yes—
344C22×SD16C2×D4712yes—
345C22×M16C2×D4711yescharacteristic
346C23×C8—471no—
347#537—944no—
348#537—944no—
349#536—944no—
350#536—944no—
351C2×C4⋊Q8—941no—
352C2×#124—941no—
353C2×#123—942no—
354C2×C22⋊Q8—474no—
355C2×C22⋊Q8—472no—
356C2×C22⋊Q8—941no—
357C2×#121—472no—
358C2×#121—941no—
359C2×#121—942no—
360C2×#117—471no—
361C22×C4⋊C4—471no—
362#416—944no—
363C2×#103—474no—
364C2×#103—472no—
365C2×#103—944no—
366C2×#102—472no—
367#408—944no—
368C2×C22⋊C8—474no—
369#340—474no—
370C47#2339411yescharacteristic, complemented, Sylow
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—942no—
377C2×#41—942no—
378C2×#41—942no—
379C2×#41—942no—
380C2×#41—942no—
381C2×#41—942no—
382C2×#41C22⋊D4714yes—
383C22×Q8—21no—
384C22×Q8—22no—
385C22×Q8—22no—
386C22×Q8—22no—
387C22×Q8—22no—
388C22×Q8—22no—
389C22×Q8—471no—
390C22×Q8C22⋊D4714yescomplemented
391C22×Q8C22×D4712yes—
392C22×Q8C22×D4711yescharacteristic
393C22×D4—21no—
394C22×D4—472no—
395C22×D4—471no—
396C22×D4—942no—
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—472no—
401C23×C4—941no—
402C23×C4C22×D4711yescharacteristic
403#137—232no—
404C2×Q16—24no—
405C2×Q16—24no—
406C2×Q16—24no—
407C2×SD16—24no—
408C2×SD16—24no—
409C2×SD16—24no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16C22⋊D4714yescomplemented
413C22×C8—472no—
414C22×C8—942no—
415C22×C8—942no—
416C22×C8—942no—
417C22×C8C22×D4712yes—
418C4⋊Q8—948no—
419#124—948no—
420#123—9416no—
421C22⋊Q8—948no—
422C22⋊Q8—944no—
423C22⋊Q8—948no—
424C22⋊Q8—948no—
425C22⋊Q8—944no—
426#121—944no—
427#121—948no—
428#121—9416no—
429#121—944no—
430#117—474no—
431#117—942no—
432C2×C4⋊C4—474no—
433C2×C4⋊C4—472no—
434C2×C4⋊C4—472no—
435C2×C4⋊C4—942no—
436C2×C4⋊C4—944no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—942no—
439C2×C22⋊C4—942no—
440C2×C22⋊C4—944no—
441C2×C22⋊C4—942no—
442C2×C22⋊C4—941no—
443C2×C42—941no—
444#103—948no—
445#103—944no—
446#102—944no—
447C22⋊C8—948no—
448C24—21no—
449C24—472no—
450C24D4×D4711yescharacteristic
451#41—28no—
452#41—28no—
453#41—44no—
454#41—44no—
455#41—1884no—
456#41—1884no—
457#41—1884no—
458#41—1884no—
459C2×Q8—24no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—28no—
463C2×Q8—28no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—942no—
477C2×Q8—942no—
478C2×Q8—942no—
479C2×Q8C2×C22⋊D4714yes—
480C2×Q8C2×C22⋊D4712yes—
481C2×D4—24no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—948no—
490C2×D4—944no—
491C2×D4—944no—
492C2×D4—944no—
493C2×D4—942no—
494C2×D4—942no—
495C2×D4—942no—
496C2×D4—3764no—
497C2×D4C2×C22⋊D4712yes—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—472no—
507C22×C4—944no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—3762no—
517C22×C4D4×D4711yescharacteristic
518C22×C4D4×D4712yes—
519C22×C4D4×D4712yes—
520C22×C4#1062535212yes—
521C22×C4C2×C22⋊D4712yes—
522C22×C4C2×C22⋊D4712yes—
523C22×C4C2×C22⋊D4712yes—
524C22×C4C23×D4711yescharacteristic
525Q16—216no—
526SD16—216no—
527M16—28no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—942no—
531C2×C8—942no—
532C2×C8—942no—
533C2×C8—1884no—
534C2×C8C2×C22⋊D4714yes—
535C4⋊C4—474no—
536C4⋊C4—474no—
537C4⋊C4—948no—
538C4⋊C4—9416no—
539C4⋊C4—944no—
540C4⋊C4—942no—
541C4⋊C4—942no—
542C22⋊C4—948no—
543C22⋊C4—9416no—
544C22⋊C4—948no—
545C22⋊C4—944no—
546C42—944no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—942no—
555C23—942no—
556C23—942no—
557C23—944no—
558C23—944no—
559C23#1309399212yes—
560C23#1309399212yes—
561C23#1309399512yes—
562C23C2×D4×D4711yescharacteristic
563Q8—24no—
564Q8—22no—
565Q8—24no—
566Q8—22no—
567Q8—48no—
568Q8—42no—
569Q8—41no—
570Q8—42no—
571Q8—41no—
572Q8—1884no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—41no—
577D4—41no—
578D4—3768no—
579D4—1888no—
580D4—1884no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—944no—
590C2×C4—944no—
591C2×C4—942no—
592C2×C4—942no—
593C2×C4—942no—
594C2×C4—3764no—
595C2×C4—1882no—
596C2×C4—1884no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4#1309399212yes—
603C2×C4#1309399212yes—
604C2×C4#1309399314yes—
605C2×C4#1309399512yes—
606C2×C4#1309399614yes—
607C2×C4#1309399614yes—
608C2×C4#1309402014yes—
609C2×C4C2×D4×D4711yescharacteristic
610C2×C4C2×D4×D4711yescharacteristic
611C2×C4C22×C22⋊D4711yescharacteristic
612C2×C4C22×C22⋊D4711yescharacteristic
613C2×C4C22×C22⋊D4711yescharacteristic
614C2×C4C2×#1062535211yescharacteristic
615C8—24no—
616C8—1882no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—1882no—
627C22—1884no—
628C22—1882no—
629C22—1884no—
630C22—1882no—
631C22#1568903114yes—
632C22C2×#1309399211yescharacteristic
633C22C2×#1309399211yescharacteristic
634C22C2×#1309399511yescharacteristic
635C22#1568993212yes—
636C22#1568993912yes—
637C4—42no—
638C4—44no—
639C4—1884no—
640C4—1882no—
641C4—1884no—
642C4#1568903114yes—
643C4C2×#1309399311yescharacteristic
644C4C2×#1309399611yescharacteristic
645C4C2×#1309399611yescharacteristic
646C4C2×#1309402011yescharacteristic
647C2—24no—
648C2—82no—
649C2—3762no—
650C2C2×#1568903111yescharacteristic
651C2#2025266411yescharacteristic
652C2#2025268211yescharacteristic
653C2#2025270212yes—
654C2#2025307312yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689701C2
hypercenterC22×#137D47
hypocenterC47#23394
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
2. C2×#15689697C2semidirect
3. C2×#15689119C2semidirect
4. C22×#13093948C2semidirect
5. C2×#15689091C2semidirect
6. #20244395C2semidirect
7. #20244388C2semidirect
8. #20243108C2semidirect
9. C22×#137×C47C2semidirect
72. #13093983D4semidirect
101. C22×Q8×C47D4semidirect
118. C2×M16×C47D4semidirect
320. C22×#137D47semidirect
370. C47#23394semidirect
390. C22×Q8C22⋊D47semidirect
412. C2×M16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897012
C2×#156896972
C2×#156891192
C22×#130939482
C2×#156890912
#202443952
#202443882
#202431082
C22×#137×C472
#2339447