Finite Group Databasebeta

← #26243127

Subgroups — #26243127

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×#13094055C211yesmaximal, characteristic, complemented
2C2×#15689634C211yesmaximal, characteristic
3C2×#15689122C211yesmaximal, characteristic, complemented
4C2×#15689132C211yesmaximal, characteristic, complemented
5C2×#15689087C211yesmaximal, characteristic, complemented
6#20244396C214yesmaximal, complemented
7#20244389C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C2×#13094055—24no—
12C2×#13094055—24no—
13C2×#13094055—24no—
14C22×#10625337C2212yescomplemented
15C22×#10625336C2212yescomplemented
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18#15689634—24no—
19#15689634—24no—
20C2×#13094007—21no—
21C2×#13094003C2212yes—
22C2×#13093998C2212yes—
23C2×#13093993—21no—
24C2×C4×C22⋊D47—22no—
25C2×#13093983C2211yescharacteristic
26#15689132—24no—
27#15689122—24no—
28C2×#13093948C2214yescomplemented
29C2×#13093947C2212yescomplemented
30C2×#13093944C2212yescomplemented
31#15689087—24no—
32C2×C22⋊(C47⋊C8)C2214yes—
33#15688885C2214yescomplemented
34C23×Q8×C47C2211yescharacteristic
35C24×C4×C47C2211yescharacteristic
36C2×C22⋊Q8×C47—24no—
37C2×C22⋊Q8×C47—22no—
38C2×C22⋊Q8×C47—24no—
39C2×C22⋊Q8×C47C2214yes—
40C22×C4⋊C4×C47—21no—
41C22×C4⋊C4×C47C2211yescharacteristic
42C22×C22⋊C4×C47—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350D414yes—
49C22×Dic94C2311yescharacteristic
50C22×D188C2311yescharacteristic
51C22×C22⋊D47—22no—
52C22×C22⋊D47—21no—
53C22×C4×D47—21no—
54#13094055—232no—
55C2×#10625337—24no—
56C2×#10625337—24no—
57C2×#10625337—24no—
58C2×#10625336—24no—
59C2×#10625336—24no—
60C2×#10625336—24no—
61C2×#10625329—22no—
62C2×#10625329—22no—
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329—22no—
66C2×#10625329D414yescomplemented
67C22×C47⋊C8C2312yes—
68C22×C47⋊C8C2312yes—
69#13094007—28no—
70#13094003—24no—
71#13094003—24no—
72#13093998—24no—
73#13093998—24no—
74#13093993—28no—
75C4×C22⋊D47—216no—
76#13093983—22no—
77#13093983D414yes—
78C2×C4⋊Dic47C2312yes—
79C2×Dic47⋊C4—22no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×C4×Dic47—21no—
84#13093948—28no—
85#13093947—24no—
86#13093944—24no—
87C22⋊(C47⋊C8)—28no—
88C25×C47D411yescharacteristic
89C22×Q8×C47—22no—
90C22×Q8×C47—21no—
91C22×Q8×C47—22no—
92C22×Q8×C47—22no—
93C22×Q8×C47—22no—
94C22×Q8×C47—22no—
95C22×Q8×C47D414yescomplemented
96C22×Q8×C47C2312yes—
97C23×C4×C47—21no—
98C23×C4×C47—21no—
99C23×C4×C47—22no—
100C23×C4×C47—22no—
101C23×C4×C47—22no—
102C23×C4×C47—22no—
103C23×C4×C47—22no—
104C23×C4×C47—22no—
105C23×C4×C47D411yescharacteristic
106C23×C4×C47D412yes—
107C23×C4×C47C2312yes—
108C23×C4×C47C2311yescharacteristic
109C22⋊Q8×C47—216no—
110C22⋊Q8×C47—28no—
111C22⋊Q8×C47—28no—
112C2×C4⋊C4×C47—28no—
113C2×C4⋊C4×C47—24no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47—22no—
116C2×C4⋊C4×C47D414yescomplemented
117C2×C4⋊C4×C47C2312yes—
118C2×C22⋊C4×C47—28no—
119C2×C22⋊C4×C47—24no—
120C24×C47—22no—
121C24×C47—24no—
122C24×C47—22no—
123C24×C47—22no—
124C24×C47—42no—
125C24×C47C2×D412yes—
126C24×C47C2×D411yescharacteristic
127C23×D47—21no—
128C2×Q8×C47—24no—
129C2×Q8×C47—22no—
130C2×Q8×C47—28no—
131C2×Q8×C47—28no—
132C2×Q8×C47—22no—
133C2×Q8×C47—22no—
134C2×Q8×C47—22no—
135C2×Q8×C47—22no—
136C2×Q8×C47—42no—
137C2×Q8×C47—42no—
138C2×Q8×C47—44no—
139C2×Q8×C47—44no—
140C2×Q8×C47C2×D414yes—
141C22×C4×C47—24no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—24no—
150C22×C4×C47—24no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—22no—
155C22×C4×C47—22no—
156C22×C4×C47—22no—
157C22×C4×C47—44no—
158C22×C4×C47—42no—
159C22×C4×C47—42no—
160C22×C4×C47—44no—
161C22×C4×C47—44no—
162C22×C4×C47—44no—
163C22×C4×C47—44no—
164C22×C4×C47C2×D412yes—
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D412yes—
167C22×C4×C47C2×D412yes—
168C22×C4×C47C2×D412yes—
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2×D412yes—
171C22×C4×C47#4112yes—
172C22×C4×C47C2411yescharacteristic
173C2×C22⋊D47—28no—
174C2×C22⋊D47—24no—
175C2×C22⋊D47—44no—
176C2×C22⋊D47—42no—
177C2×C22⋊D47—42no—
178C2×C22⋊D47—44no—
179C22×Dic47—22no—
180C22×Dic47—21no—
181#10625350—28no—
182#10625350—28no—
183#10625350—44no—
184#10625350—44no—
185C2×D188—21no—
186C2×D188—21no—
187C2×D188—21no—
188C2×D188—21no—
189C2×D188—21no—
190C2×D188C2×D412yes—
191C2×C4×D47—24no—
192C2×C4×D47—42no—
193C2×C4×D47—42no—
194C2×Dic94—21no—
195C2×Dic94—21no—
196C2×Dic94—21no—
197C2×Dic94—21no—
198C2×Dic94—21no—
199C2×Dic94C2×D412yes—
200C4⋊C4×C47—216no—
201C4⋊C4×C47—24no—
202C4⋊C4×C47—22no—
203C4⋊C4×C47C2×D414yes—
204C22⋊C4×C47—216no—
205C22⋊Dic47—28no—
206C22⋊Dic47—24no—
207#10625337—216no—
208#10625336—216no—
209(C2×D47)⋊C4—28no—
210C4⋊Dic47—22no—
211C4⋊Dic47C2×D414yes—
212Dic47⋊C4—28no—
213C4×Dic47—24no—
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—48no—
232C23×C47—48no—
233C23×C47—44no—
234C23×C47—44no—
235C23×C47#12012yes—
236C23×C47#12012yes—
237C23×C47#12112yes—
238C23×C47#13714yescomplemented
239C23×C47C22×D411yescharacteristic
240C22×D47—22no—
241C22×D47—41no—
242C22×D47—41no—
243C22×D47—42no—
244C22×D47—42no—
245Q8×C47—24no—
246Q8×C47—24no—
247Q8×C47—42no—
248Q8×C47—42no—
249Q8×C47—48no—
250C2×C4×C47—22no—
251C2×C4×C47—24no—
252C2×C4×C47—24no—
253C2×C4×C47—22no—
254C2×C4×C47—22no—
255C2×C4×C47—48no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47—44no—
259C2×C4×C47—44no—
260C2×C4×C47—48no—
261C2×C4×C47—42no—
262C2×C4×C47—42no—
263C2×C4×C47—42no—
264C2×C4×C47—42no—
265C2×C4×C47—48no—
266C2×C4×C47#12014yes—
267C2×C4×C47#12012yes—
268C2×C4×C47#12012yes—
269C2×C4×C47#12114yes—
270C2×C4×C47#12114yes—
271C2×C4×C47#12112yes—
272C2×C4×C47#12714yes—
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C22×D411yescharacteristic
275C2×C4×C47C22×D411yescharacteristic
276C2×C4×C47C22×D411yescharacteristic
277C2×C4×C47C22×D411yescharacteristic
278C2×C4×C47C2×#4111yescharacteristic
279C22⋊D47—416no—
280C22⋊D47—48no—
281C2×Dic47—24no—
282C2×Dic47—22no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287D188—22no—
288D188—22no—
289D188—41no—
290D188—41no—
291C4×D47—48no—
292Dic94—22no—
293Dic94—22no—
294Dic94—41no—
295Dic94—41no—
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#39214yes—
312C22×C47C2×#12011yescharacteristic
313C22×C47C2×#12011yescharacteristic
314C22×C47C2×#12111yescharacteristic
315C22×C47C2×#13712yes—
316C22×C47#57512yes—
317C2×D47—42no—
318C2×D47—44no—
319C2×D47—81no—
320C2×D47—81no—
321C2×D47—82no—
322C4×C47—42no—
323C4×C47—44no—
324C4×C47—44no—
325C4×C47#39214yes—
326C4×C47C2×#12011yescharacteristic
327C4×C47C2×#12111yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12711yescharacteristic
330Dic47—42no—
331Dic47—44no—
332C22×#137—471no—
333C2×#525—471no—
334C22×C22⋊Q8D4711yescharacteristic, complemented
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×C47C2×#39211yescharacteristic
344C2×C47#300411yescharacteristic
345C2×C47#300511yescharacteristic
346C2×C47#300712yes—
347C2×C47#305612yes—
348D47—82no—
349C22×#41—471no—
350C23×Q8C2×D4711yescharacteristic
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—944no—
360#525—944no—
361C2×#123—941no—
362C2×C22⋊Q8—24no—
363C2×C22⋊Q8—22no—
364C2×C22⋊Q8—24no—
365C2×C22⋊Q8—472no—
366C2×C22⋊Q8C2×D4714yes—
367C2×#121—472no—
368C2×#120—941no—
369C2×C4×D4—942no—
370C2×#117—471no—
371C22×C4⋊C4—21no—
372C22×C4⋊C4C2×D4711yescharacteristic
373C22×C22⋊C4—21no—
374#419—944no—
375#417—944no—
376C2×#103—474no—
377C2×#103—472no—
378C2×#102—472no—
379#407—944no—
380C2×C22⋊C8—474no—
381#340—474no—
382C47#2339211yescharacteristic, complemented, Sylow
383C25C22⋊D4711yescharacteristic
384C2×#41—474no—
385C2×#41—942no—
386C2×#41—942no—
387C2×#41—942no—
388C2×#41—942no—
389C2×#41—942no—
390C22×Q8—22no—
391C22×Q8—21no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—471no—
397C22×Q8C22⋊D4714yescomplemented
398C22×Q8C22×D4712yes—
399C22×D4—471no—
400C22×D4—942no—
401C22×D4—941no—
402C23×C4—21no—
403C23×C4—21no—
404C23×C4—22no—
405C23×C4—22no—
406C23×C4—22no—
407C23×C4—22no—
408C23×C4—22no—
409C23×C4—22no—
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—948no—
431C22⋊Q8—216no—
432C22⋊Q8—28no—
433C22⋊Q8—28no—
434C22⋊Q8—944no—
435C22⋊Q8—944no—
436#121—944no—
437#121—944no—
438#120—948no—
439C4×D4—9416no—
440#117—474no—
441#117—942no—
442C2×C4⋊C4—28no—
443C2×C4⋊C4—24no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—22no—
446C2×C4⋊C4—472no—
447C2×C4⋊C4—942no—
448C2×C4⋊C4C22⋊D4714yescomplemented
449C2×C4⋊C4C22×D4712yes—
450C2×C22⋊C4—28no—
451C2×C22⋊C4—24no—
452C2×C22⋊C4—941no—
453C2×C22⋊C4—942no—
454C2×C22⋊C4—942no—
455C2×C42—941no—
456#103—948no—
457#103—944no—
458#102—944no—
459C22⋊C8—948no—
460C24—22no—
461C24—24no—
462C24—22no—
463C24—22no—
464C24—42no—
465C24—941no—
466C24C2×C22⋊D4712yes—
467C24C2×C22⋊D4711yescharacteristic
468#41—948no—
469#41—948no—
470#41—1884no—
471#41—1884no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—28no—
475C2×Q8—28no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—42no—
481C2×Q8—44no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—472no—
485C2×Q8—941no—
486C2×Q8—941no—
487C2×Q8—941no—
488C2×Q8—941no—
489C2×Q8—941no—
490C2×Q8C2×C22⋊D4714yes—
491C2×D4—472no—
492C2×D4—948no—
493C2×D4—944no—
494C2×D4—941no—
495C2×D4—941no—
496C2×D4—941no—
497C2×D4—941no—
498C2×D4—941no—
499C2×D4—1884no—
500C2×D4—1882no—
501C2×D4—1882no—
502C2×D4—1884no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—942no—
527C22×C4—941no—
528C22×C4—944no—
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—216no—
551C4⋊C4—24no—
552C4⋊C4—22no—
553C4⋊C4—474no—
554C4⋊C4—948no—
555C4⋊C4—942no—
556C4⋊C4C2×C22⋊D4714yes—
557C22⋊C4—216no—
558C22⋊C4—948no—
559C22⋊C4—948no—
560C22⋊C4—944no—
561C42—944no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—48no—
572C23—48no—
573C23—44no—
574C23—44no—
575C23—942no—
576C23—1881no—
577C23—1881no—
578C23—1882no—
579C23—1882no—
580C23#1309399212yes—
581C23#1309399312yes—
582C23#1309399612yes—
583C23#1309405514yescomplemented
584C23C22×C22⋊D4711yescharacteristic
585Q8—24no—
586Q8—24no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—942no—
591Q8—942no—
592Q8—1881no—
593Q8—1881no—
594D4—942no—
595D4—942no—
596D4—18816no—
597D4—1888no—
598D4—1881no—
599D4—1881no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—944no—
617C2×C4—942no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1888no—
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—3762no—
651C22—3761no—
652C22—3761no—
653C22—1882no—
654C22—1884no—
655C22#1568903114yes—
656C22C2×#1309399211yescharacteristic
657C22C2×#1309399311yescharacteristic
658C22C2×#1309399611yescharacteristic
659C22C2×#1309405512yes—
660C22#1568992112yes—
661C4—42no—
662C4—44no—
663C4—44no—
664C4—1882no—
665C4—1884no—
666C4#1568903114yes—
667C4C2×#1309399211yescharacteristic
668C4C2×#1309399511yescharacteristic
669C4C2×#1309399611yescharacteristic
670C4C2×#1309402011yescharacteristic
671C2—24no—
672C2—44no—
673C2—3762no—
674C2C2×#1568903111yescharacteristic
675C2#2025262211yescharacteristic
676C2#2025263011yescharacteristic
677C2#2025264612yes—
678C2#2025301412yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×C22⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
hypercenterC22×C22⋊Q8D47
hypocenterC47#23392
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13094055C2semidirect
3. C2×#15689122C2semidirect
4. C2×#15689132C2semidirect
5. C2×#15689087C2semidirect
6. #20244396C2semidirect
7. #20244389C2semidirect
8. #20243107C2semidirect
9. C22×C22⋊Q8×C47C2semidirect
14. C22×#10625337C22semidirect
15. C22×#10625336C22semidirect
28. C2×#13093948C22semidirect
29. C2×#13093947C22semidirect
30. C2×#13093944C22semidirect
33. #15688885C22semidirect
66. C2×#10625329D4semidirect
95. C22×Q8×C47D4semidirect
116. C2×C4⋊C4×C47D4semidirect
238. C23×C47#137semidirect
334. C22×C22⋊Q8D47semidirect
382. C47#23392semidirect
397. C22×Q8C22⋊D47semidirect
448. C2×C4⋊C4C22⋊D47semidirect
583. C23#13094055semidirect

Maximal subgroups

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

subgroupindex
C22×#130940552
C2×#156896342
C2×#156891222
C2×#156891322
C2×#156890872
#202443962
#202443892
#202431072
C22×C22⋊Q8×C472
#2339247