Finite Group Databasebeta

← #26243125

Subgroups — #26243125

Order 12032.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094054C211yesmaximal, characteristic, complemented
2C22×#13094003C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic, complemented
4C2×#15689131C211yesmaximal, characteristic, complemented
5C2×#15689087C211yesmaximal, characteristic, complemented
6#20244396C214yesmaximal, complemented
7#20244391C214yesmaximal, complemented
8#20243109C211yesmaximal, characteristic, complemented
9C2×C47×#525C211yesmaximal, characteristic
10C23×Dic94C2211yescharacteristic
11C2×#13094054—24no—
12C2×#13094054—24no—
13C2×#13094054—24no—
14C22×#10625337C2212yescomplemented
15C22×#10625335C2212yescomplemented
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C2×#13094003—24no—
19C2×#13094003—24no—
20C2×#13094003—22no—
21C2×#13094003C2214yes—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×Dic47⋊C4—21no—
24C22×C22⋊Dic47—21no—
25#15689131—24no—
26#15689124—24no—
27C2×#13093948C2212yescomplemented
28C2×#13093947C2214yescomplemented
29C2×#13093945C2212yescomplemented
30#15689087—24no—
31C2×C22⋊(C47⋊C8)C2214yes—
32#15688885C2214yescomplemented
33C22×#41×C47C2211yescharacteristic
34C24×C4×C47C2211yescharacteristic
35C47×#525—24no—
36C47×#525—24no—
37C2×#123×C47—21no—
38C2×C22⋊Q8×C47C2212yes—
39C2×#121×C47C2212yes—
40C2×#120×C47—21no—
41C2×C4×D4×C47—22no—
42C2×#117×C47C2211yescharacteristic
43C22×Dic94—22no—
44C22×Dic94—22no—
45C22×Dic94—22no—
46C22×Dic94—22no—
47C22×Dic94—21no—
48C22×Dic94—22no—
49C22×Dic94D414yescomplemented
50C22×Dic94C2312yes—
51C23×Dic47—21no—
52C23×Dic47—21no—
53#13094054—232no—
54C2×#10625337—24no—
55C2×#10625337—24no—
56C2×#10625337—24no—
57C2×#10625335—24no—
58C2×#10625335—24no—
59C2×#10625335—24no—
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329D414yescomplemented
66C22×C47⋊C8C2312yes—
67C22×C47⋊C8C2312yes—
68#13094003—216no—
69#13094003—28no—
70#13094003—28no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47D414yescomplemented
74C2×C4⋊Dic47C2312yes—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—24no—
77C2×C22⋊Dic47—28no—
78C2×C22⋊Dic47—24no—
79#13093948—24no—
80#13093947—28no—
81#13093945—24no—
82C22⋊(C47⋊C8)—28no—
83C25×C47D411yescharacteristic
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47D414yes—
90C22×Q8×C47C2311yescharacteristic
91C22×D4×C47—22no—
92C22×D4×C47—21no—
93C22×D4×C47C2311yescharacteristic
94C23×C4×C47—21no—
95C23×C4×C47—22no—
96C23×C4×C47—22no—
97C23×C4×C47—22no—
98C23×C4×C47—22no—
99C23×C4×C47—22no—
100C23×C4×C47—22no—
101C23×C4×C47D412yes—
102C23×C4×C47D411yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2311yescharacteristic
105#123×C47—28no—
106C22⋊Q8×C47—24no—
107C22⋊Q8×C47—24no—
108#121×C47—24no—
109#121×C47—24no—
110#120×C47—28no—
111C4×D4×C47—216no—
112#117×C47—22no—
113#117×C47D414yes—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47C2312yes—
116C2×C22⋊C4×C47—21no—
117C2×C22⋊C4×C47—22no—
118C2×C22⋊C4×C47—22no—
119C2×C42×C47—21no—
120C24×C47—22no—
121C24×C47—24no—
122C24×C47—22no—
123C24×C47—22no—
124C24×C47—21no—
125C24×C47—42no—
126C24×C47C2×D412yes—
127C24×C47C2×D411yescharacteristic
128#41×C47—28no—
129#41×C47—28no—
130#41×C47—44no—
131#41×C47—44no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47C2×D412yes—
138C2×D4×C47—28no—
139C2×D4×C47—24no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—21no—
143C2×D4×C47—21no—
144C2×D4×C47—21no—
145C2×D4×C47—44no—
146C2×D4×C47—42no—
147C2×D4×C47—42no—
148C2×D4×C47—44no—
149C2×D4×C47C2×D412yes—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—21no—
155C22×C4×C47—24no—
156C22×C4×C47—24no—
157C22×C4×C47—24no—
158C22×C4×C47—24no—
159C22×C4×C47—22no—
160C22×C4×C47—22no—
161C22×C4×C47—22no—
162C22×C4×C47—22no—
163C22×C4×C47—22no—
164C22×C4×C47—22no—
165C22×C4×C47—44no—
166C22×C4×C47—42no—
167C22×C4×C47—44no—
168C22×C4×C47—44no—
169C22×C4×C47—42no—
170C22×C4×C47—44no—
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D412yes—
174C22×C4×C47C2×D412yes—
175C22×C4×C47C2×D412yes—
176C22×C4×C47C2×D412yes—
177C22×C4×C47C2×D412yes—
178C22×C4×C47#4112yes—
179C22×C4×C47C2411yescharacteristic
180C22×Dic47—24no—
181C22×Dic47—24no—
182C22×Dic47—22no—
183C22×Dic47—22no—
184C22×Dic47—44no—
185C22×Dic47—42no—
186C22×Dic47—42no—
187C2×Dic94—28no—
188C2×Dic94—28no—
189C2×Dic94—22no—
190C2×Dic94—22no—
191C2×Dic94—22no—
192C2×Dic94—22no—
193C2×Dic94—24no—
194C2×Dic94—22no—
195C2×Dic94—42no—
196C2×Dic94—44no—
197C2×Dic94—42no—
198C2×Dic94—44no—
199C2×Dic94C2×D414yes—
200C4⋊C4×C47—28no—
201C4⋊C4×C47—22no—
202C4⋊C4×C47C2×D414yes—
203C22⋊C4×C47—28no—
204C22⋊C4×C47—28no—
205C22⋊C4×C47—24no—
206C42×C47—24no—
207C22⋊Dic47—216no—
208#10625337—216no—
209#10625335—216no—
210C4⋊Dic47—24no—
211C4⋊Dic47—22no—
212C4⋊Dic47C2×D414yes—
213Dic47⋊C4—216no—
214#10625329—28no—
215#10625329—44no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—22no—
221C2×C47⋊C8C2×D414yes—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—28no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47—22no—
229C23×C47—22no—
230C23×C47—22no—
231C23×C47—22no—
232C23×C47—48no—
233C23×C47—48no—
234C23×C47—44no—
235C23×C47—44no—
236C23×C47—41no—
237C23×C47—41no—
238C23×C47—42no—
239C23×C47—42no—
240C23×C47#12012yes—
241C23×C47#12012yes—
242C23×C47#12112yes—
243C23×C47#13714yescomplemented
244C23×C47C22×D411yescharacteristic
245Q8×C47—22no—
246Q8×C47—22no—
247Q8×C47—41no—
248Q8×C47—41no—
249D4×C47—22no—
250D4×C47—22no—
251D4×C47—416no—
252D4×C47—48no—
253D4×C47—41no—
254D4×C47—41no—
255C2×C4×C47—22no—
256C2×C4×C47—24no—
257C2×C4×C47—22no—
258C2×C4×C47—22no—
259C2×C4×C47—22no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—42no—
263C2×C4×C47—42no—
264C2×C4×C47—44no—
265C2×C4×C47—42no—
266C2×C4×C47—42no—
267C2×C4×C47—42no—
268C2×C4×C47—42no—
269C2×C4×C47—48no—
270C2×C4×C47—48no—
271C2×C4×C47#12014yes—
272C2×C4×C47#12012yes—
273C2×C4×C47#12012yes—
274C2×C4×C47#12114yes—
275C2×C4×C47#12114yes—
276C2×C4×C47#12112yes—
277C2×C4×C47#12714yes—
278C2×C4×C47C22×D411yescharacteristic
279C2×C4×C47C22×D411yescharacteristic
280C2×C4×C47C22×D411yescharacteristic
281C2×C4×C47C22×D411yescharacteristic
282C2×C4×C47C22×D411yescharacteristic
283C2×C4×C47C2×#4111yescharacteristic
284C2×Dic47—24no—
285C2×Dic47—24no—
286C2×Dic47—48no—
287C2×Dic47—42no—
288C2×Dic47—42no—
289C2×Dic47—44no—
290C2×Dic47—48no—
291Dic94—24no—
292Dic94—24no—
293Dic94—42no—
294Dic94—42no—
295Dic94—48no—
296C47⋊C8—24no—
297C47⋊C8—42no—
298#23392—471nomaximal, Sylow
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—24no—
302C22×C47—24no—
303C22×C47—24no—
304C22×C47—24no—
305C22×C47—24no—
306C22×C47—416no—
307C22×C47—42no—
308C22×C47—42no—
309C22×C47—42no—
310C22×C47—42no—
311C22×C47—42no—
312C22×C47—44no—
313C22×C47—81no—
314C22×C47—81no—
315C22×C47—82no—
316C22×C47#39214yes—
317C22×C47C2×#12011yescharacteristic
318C22×C47C2×#12011yescharacteristic
319C22×C47C2×#12111yescharacteristic
320C22×C47C2×#13712yes—
321C22×C47#57512yes—
322C4×C47—42no—
323C4×C47—42no—
324C4×C47—44no—
325C4×C47#39214yes—
326C4×C47C2×#12011yescharacteristic
327C4×C47C2×#12111yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12711yescharacteristic
330Dic47—44no—
331Dic47—44no—
332C22×#137—471no—
333C2×#525D4711yescharacteristic
334C22×C22⋊Q8—471no—
335C2×#417—471no—
336C2×#419—471no—
337C2×#407—471no—
338#2001—474no—
339#2000—474no—
340#1736—471no—
341C2×C47—24no—
342C2×C47—44no—
343C2×C47—82no—
344C2×C47C2×#39211yescharacteristic
345C2×C47#300411yescharacteristic
346C2×C47#300511yescharacteristic
347C2×C47#300712yes—
348C2×C47#305612yes—
349C22×#41C2×D4711yescharacteristic
350C23×Q8—471no—
351C24×C4C2×D4711yescharacteristic
352C2×#137—944no—
353C2×#137—944no—
354C2×#137—944no—
355C22×Q16—472no—
356C22×SD16—472no—
357C22×M16—471no—
358C22×M16—471no—
359#525—24no—
360#525—24no—
361C2×#123—21no—
362C2×C22⋊Q8—474no—
363C2×C22⋊Q8—944no—
364C2×C22⋊Q8—944no—
365C2×C22⋊Q8—942no—
366C2×C22⋊Q8C2×D4712yes—
367C2×#121C2×D4712yes—
368C2×#120—21no—
369C2×C4×D4—22no—
370C2×#117C2×D4711yescharacteristic
371C22×C4⋊C4—471no—
372C22×C4⋊C4—941no—
373C22×C22⋊C4—941no—
374#419—944no—
375#417—944no—
376C2×#103—472no—
377C2×#103—474no—
378C2×#102—472no—
379#407—944no—
380C2×C22⋊C8—474no—
381#340—474no—
382C47#2339211yescharacteristic, complemented, Sylow
383C25C22⋊D4711yescharacteristic
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D4714yes—
390C22×Q8—474no—
391C22×Q8—472no—
392C22×Q8—942no—
393C22×Q8—941no—
394C22×Q8—942no—
395C22×Q8—942no—
396C22×Q8—942no—
397C22×Q8—942no—
398C22×Q8C22×D4711yescharacteristic
399C22×D4—22no—
400C22×D4—21no—
401C22×D4C22×D4711yescharacteristic
402C23×C4—22no—
403C23×C4—22no—
404C23×C4—22no—
405C23×C4—22no—
406C23×C4—21no—
407C23×C4—22no—
408C23×C4—22no—
409C23×C4—941no—
410C23×C4—941no—
411C23×C4C22⋊D4711yescharacteristic
412C23×C4C22⋊D4712yes—
413C23×C4C22×D4712yes—
414C23×C4C22×D4711yescharacteristic
415#137—9432no—
416C2×Q16—944no—
417C2×Q16—944no—
418C2×Q16—944no—
419C2×SD16—944no—
420C2×SD16—944no—
421C2×SD16—944no—
422C2×M16—474no—
423C2×M16—942no—
424C2×M16—942no—
425C2×M16—942no—
426C2×M16—942no—
427C2×M16—942no—
428C22×C8—472no—
429C22×C8—472no—
430#123—28no—
431C22⋊Q8—24no—
432C22⋊Q8—24no—
433C22⋊Q8—948no—
434C22⋊Q8—948no—
435C22⋊Q8—9416no—
436#121—24no—
437#121—24no—
438#120—28no—
439C4×D4—216no—
440#117—22no—
441#117C22⋊D4714yes—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—474no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—948no—
446C2×C4⋊C4—942no—
447C2×C4⋊C4—941no—
448C2×C4⋊C4—944no—
449C2×C4⋊C4C22×D4712yes—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—948no—
454C2×C22⋊C4—944no—
455C2×C42—21no—
456#103—944no—
457#103—948no—
458#102—944no—
459C22⋊C8—948no—
460C24—22no—
461C24—24no—
462C24—22no—
463C24—22no—
464C24—21no—
465C24—42no—
466C24C2×C22⋊D4712yes—
467C24C2×C22⋊D4711yescharacteristic
468#41—28no—
469#41—28no—
470#41—44no—
471#41—44no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—474no—
478C2×Q8—944no—
479C2×Q8—942no—
480C2×Q8—948no—
481C2×Q8—948no—
482C2×Q8—942no—
483C2×Q8—942no—
484C2×Q8—942no—
485C2×Q8—942no—
486C2×Q8—1882no—
487C2×Q8—1884no—
488C2×Q8—1882no—
489C2×Q8—1884no—
490C2×Q8C2×C22⋊D4712yes—
491C2×D4—28no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—44no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—44no—
502C2×D4C2×C22⋊D4712yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—44no—
519C22×C4—42no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—44no—
524C22×C4—944no—
525C22×C4—944no—
526C22×C4—942no—
527C22×C4—942no—
528C22×C4—1884no—
529C22×C4—1882no—
530C22×C4—1882no—
531C22×C4D4×D4712yes—
532C22×C4D4×D4712yes—
533C22×C4D4×D4712yes—
534C22×C4#1062535212yes—
535C22×C4C2×C22⋊D4712yes—
536C22×C4C2×C22⋊D4711yescharacteristic
537C22×C4C2×C22⋊D4712yes—
538C22×C4C2×C22⋊D4712yes—
539C22×C4C23×D4711yescharacteristic
540Q16—9416no—
541SD16—9416no—
542M16—948no—
543M16—1884no—
544C2×C8—474no—
545C2×C8—942no—
546C2×C8—942no—
547C2×C8—942no—
548C2×C8—942no—
549C2×C8—942no—
550C4⋊C4—28no—
551C4⋊C4—22no—
552C4⋊C4—474no—
553C4⋊C4—9416no—
554C4⋊C4—944no—
555C4⋊C4—942no—
556C4⋊C4C2×C22⋊D4714yes—
557C22⋊C4—28no—
558C22⋊C4—28no—
559C22⋊C4—24no—
560C22⋊C4—9416no—
561C42—24no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—48no—
573C23—48no—
574C23—44no—
575C23—44no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23#1309399212yes—
581C23#1309399312yes—
582C23#1309399612yes—
583C23#1309405414yescomplemented
584C23C22×C22⋊D4711yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589Q8—944no—
590Q8—944no—
591Q8—1888no—
592Q8—1882no—
593Q8—1882no—
594D4—22no—
595D4—22no—
596D4—416no—
597D4—48no—
598D4—41no—
599D4—41no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—944no—
617C2×C4—944no—
618C2×C4—1888no—
619C2×C4—1888no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1884no—
623C2×C4#1309399212yes—
624C2×C4#1309399214yes—
625C2×C4#1309399312yes—
626C2×C4#1309399514yes—
627C2×C4#1309399612yes—
628C2×C4#1309399614yes—
629C2×C4#1309402014yes—
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4C2×D4×D4711yescharacteristic
632C2×C4C2×D4×D4711yescharacteristic
633C2×C4C22×C22⋊D4711yescharacteristic
634C2×C4C22×C22⋊D4711yescharacteristic
635C2×C4C2×#1062535211yescharacteristic
636C8—944no—
637C8—1882no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22#1568903114yes—
656C22C2×#1309399211yescharacteristic
657C22C2×#1309399311yescharacteristic
658C22C2×#1309399611yescharacteristic
659C22C2×#1309405412yes—
660C22#1568992112yes—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—1884no—
665C4—1884no—
666C4#1568903114yes—
667C4C2×#1309399211yescharacteristic
668C4C2×#1309399511yescharacteristic
669C4C2×#1309399611yescharacteristic
670C4C2×#1309402011yescharacteristic
671C2—24no—
672C2—44no—
673C2—82no—
674C2C2×#1568903111yescharacteristic
675C2#2025262111yescharacteristic
676C2#2025262811yescharacteristic
677C2#2025264812yes—
678C2#2025302012yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#525C2
socleC23×C47C22×D4
hypercenterC2×#525D47
hypocenterC47#23392
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×#13094054C2semidirect
2. C22×#13094003C2semidirect
3. C2×#15689124C2semidirect
4. C2×#15689131C2semidirect
5. C2×#15689087C2semidirect
6. #20244396C2semidirect
7. #20244391C2semidirect
8. #20243109C2semidirect
14. C22×#10625337C22semidirect
15. C22×#10625335C22semidirect
27. C2×#13093948C22semidirect
28. C2×#13093947C22semidirect
29. C2×#13093945C22semidirect
32. #15688885C22semidirect
49. C22×Dic94D4semidirect
65. C2×#10625329D4semidirect
73. C2×C4⋊Dic47D4semidirect
243. C23×C47#137semidirect
382. C47#23392semidirect
583. C23#13094054semidirect

Maximal subgroups

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

subgroupindex
C22×#130940542
C22×#130940032
C2×#156891242
C2×#156891312
C2×#156890872
#202443962
#202443912
#202431092
C2×C47×#5252
#2339247