Finite Group Databasebeta

← C2×#20244421 #26243175

Subgroups — C2×#20244421 #26243175

Order 12032.

1998 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
1C2×#15689694C211yesmaximal, characteristic
2C22×#13094022C211yesmaximal, characteristic, complemented
3C22×#13093951C211yesmaximal, characteristic
4C2×#15689132C211yesmaximal, characteristic, complemented
5C22×#13093942C211yesmaximal, characteristic, complemented
6#20244421C218yesmaximal, direct product factor
7C2×#15688884C211yesmaximal, characteristic, complemented
8C22×#137×C47C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689694—22no—
12#15689694—22no—
13#15689694—22no—
14#15689694C2214yes—
15C2×#13094022—22no—
16C2×#13094022—24no—
17C2×#13094022—24no—
18C2×#13094022—21no—
19C2×#13094022C2214yescomplemented
20C2×#13094011—21no—
21C2×#13094005—22no—
22C2×#13094003—22no—
23C2×#13093983C2211yescharacteristic
24C22×Dic47⋊C4—22no—
25C22×C4×Dic47C2211yescharacteristic
26C2×#13093951—24no—
27C2×#13093951C2214yes—
28#15689132—22no—
29#15689132C2214yescomplemented
30C2×#13093948—22no—
31C2×#13093942—24no—
32C2×#13093942C2214yescomplemented
33#15688884C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C23×Q8×C47C2211yescharacteristic
36C2×#137×C47—22no—
37C2×#137×C47—24no—
38C2×#137×C47—24no—
39C2×#137×C47C2214yes—
40C22×Q16×C47—21no—
41C22×SD16×C47—21no—
42C22×M16×C47C2211yescharacteristic
43C22×Dic94—22no—
44C22×Dic94—21no—
45C22×Dic94—21no—
46C22×Dic94—22no—
47C22×Dic94—22no—
48C22×Dic94—21no—
49C22×Dic94—21no—
50C22×Dic94—22no—
51C22×Dic94D412yes—
52C22×Dic94C2312yes—
53C23×Dic47—21no—
54C23×Dic47D412yes—
55C2×#10625329—21no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2312yes—
59C22×C47⋊C8—21no—
60#13094022—216no—
61#13094022—24no—
62#13094022—28no—
63#13094022—24no—
64#13094022—28no—
65#13094011—24no—
66#13094011—24no—
67#13094005—28no—
68#13094005—28no—
69#13094003—28no—
70#13094003—28no—
71#13093983—21no—
72#13093983—21no—
73#13093983D412yes—
74#13093983C2312yes—
75C2×C4⋊Dic47D412yes—
76C2×Dic47⋊C4—216no—
77C2×Dic47⋊C4—24no—
78C2×Dic47⋊C4—28no—
79C2×C22⋊Dic47—24no—
80C2×C22⋊Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47D414yes—
85C2×C4×Dic47C2312yes—
86#13093951—28no—
87#13093948—28no—
88#13093942—28no—
89C2×#41×C47—21no—
90C2×#41×C47—21no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47D412yes—
96C2×#41×C47C2312yes—
97C22×Q8×C47—22no—
98C22×Q8×C47—21no—
99C22×Q8×C47—21no—
100C22×Q8×C47—21no—
101C22×Q8×C47—22no—
102C22×Q8×C47—21no—
103C22×Q8×C47—21no—
104C22×Q8×C47—22no—
105C22×Q8×C47D411yescharacteristic
106C22×Q8×C47D414yes—
107C22×Q8×C47C2312yes—
108C22×D4×C47—21no—
109C22×D4×C47D411yescharacteristic
110C23×C4×C47—21no—
111C23×C4×C47—21no—
112C23×C4×C47C2311yescharacteristic
113#137×C47—216no—
114#137×C47—28no—
115#137×C47—28no—
116C2×Q16×C47—28no—
117C2×Q16×C47—24no—
118C2×SD16×C47—28no—
119C2×SD16×C47—24no—
120C2×M16×C47—22no—
121C2×M16×C47—21no—
122C2×M16×C47D414yes—
123C2×M16×C47C2312yes—
124C22×C8×C47—21no—
125C24×C47—21no—
126C24×C47C2×D411yescharacteristic
127#41×C47—24no—
128#41×C47—24no—
129#41×C47—22no—
130#41×C47—22no—
131#41×C47—22no—
132#41×C47—22no—
133#41×C47—42no—
134#41×C47—42no—
135#41×C47—42no—
136#41×C47—42no—
137C2×Q8×C47—24no—
138C2×Q8×C47—21no—
139C2×Q8×C47—22no—
140C2×Q8×C47—24no—
141C2×Q8×C47—22no—
142C2×Q8×C47—24no—
143C2×Q8×C47—24no—
144C2×Q8×C47—24no—
145C2×Q8×C47—22no—
146C2×Q8×C47—22no—
147C2×Q8×C47—22no—
148C2×Q8×C47—22no—
149C2×Q8×C47—22no—
150C2×Q8×C47—41no—
151C2×Q8×C47—41no—
152C2×Q8×C47—42no—
153C2×Q8×C47—42no—
154C2×Q8×C47—42no—
155C2×Q8×C47—42no—
156C2×Q8×C47—42no—
157C2×Q8×C47C2×D412yes—
158C2×Q8×C47C2×D414yes—
159C2×D4×C47—22no—
160C2×D4×C47—22no—
161C2×D4×C47—22no—
162C2×D4×C47—21no—
163C2×D4×C47—22no—
164C2×D4×C47—42no—
165C2×D4×C47—42no—
166C2×D4×C47C2×D412yes—
167C22×C4×C47—21no—
168C22×C4×C47—24no—
169C22×C4×C47—22no—
170C22×C4×C47—22no—
171C22×C4×C47—22no—
172C22×C4×C47—41no—
173C22×C4×C47—41no—
174C22×C4×C47—42no—
175C22×C4×C47—42no—
176C22×C4×C47—42no—
177C22×C4×C47C2×D411yescharacteristic
178C22×C4×C47C2×D411yescharacteristic
179C22×C4×C47C2×D411yescharacteristic
180C22×C4×C47C2×D412yes—
181C22×C4×C47C2×D412yes—
182C22×C4×C47C2×D411yescharacteristic
183C22×C4×C47C2×D412yes—
184C22×C4×C47C2×D411yescharacteristic
185C22×C4×C47#4112yes—
186C22×C4×C47C2411yescharacteristic
187C22×Dic47—28no—
188C22×Dic47—24no—
189C22×Dic47—22no—
190C22×Dic47—22no—
191C22×Dic47—22no—
192C22×Dic47—24no—
193C22×Dic47—22no—
194C22×Dic47—24no—
195C22×Dic47C2×D414yes—
196C2×Dic94—22no—
197C2×Dic94—22no—
198C2×Dic94—24no—
199C2×Dic94—24no—
200C2×Dic94—22no—
201C2×Dic94—22no—
202C2×Dic94—22no—
203C2×Dic94—22no—
204C2×Dic94—22no—
205C2×Dic94—22no—
206C2×Dic94—42no—
207C2×Dic94—42no—
208C2×Dic94—42no—
209C2×Dic94—42no—
210C2×Dic94—42no—
211C2×Dic94—42no—
212C2×Dic94—42no—
213C2×Dic94—42no—
214Q16×C47—216no—
215SD16×C47—216no—
216M16×C47—24no—
217M16×C47—22no—
218M16×C47C2×D414yes—
219C2×C8×C47—24no—
220C2×C8×C47—22no—
221C22⋊Dic47—216no—
222C22⋊Dic47—22no—
223C22⋊Dic47—22no—
224C4⋊Dic47—22no—
225C4⋊Dic47C2×D414yes—
226Dic47⋊C4—232no—
227Dic47⋊C4—216no—
228C4×Dic47—24no—
229C4×Dic47—22no—
230C4×Dic47—22no—
231C4×Dic47—22no—
232C4×Dic47C2×D414yes—
233#10625329—22no—
234#10625329—22no—
235#10625329—42no—
236C2×C47⋊C8—22no—
237C2×C47⋊C8—42no—
238C23×C47—24no—
239C23×C47—22no—
240C23×C47—42no—
241C23×C47—44no—
242C23×C47#12011yescharacteristic
243C23×C47#12012yes—
244C23×C47#12011yescharacteristic
245C23×C47#12112yes—
246C23×C47C22×D411yescharacteristic
247Q8×C47—24no—
248Q8×C47—22no—
249Q8×C47—24no—
250Q8×C47—22no—
251Q8×C47—42no—
252Q8×C47—42no—
253Q8×C47—44no—
254Q8×C47—44no—
255Q8×C47—42no—
256D4×C47—22no—
257D4×C47—22no—
258D4×C47—42no—
259D4×C47—42no—
260D4×C47—44no—
261D4×C47—42no—
262C2×C4×C47—22no—
263C2×C4×C47—24no—
264C2×C4×C47—42no—
265C2×C4×C47—42no—
266C2×C4×C47—42no—
267C2×C4×C47—44no—
268C2×C4×C47—44no—
269C2×C4×C47—42no—
270C2×C4×C47—42no—
271C2×C4×C47—44no—
272C2×C4×C47#12011yescharacteristic
273C2×C4×C47#12012yes—
274C2×C4×C47#12011yescharacteristic
275C2×C4×C47#12012yes—
276C2×C4×C47#12012yes—
277C2×C4×C47#12112yes—
278C2×C4×C47#12112yes—
279C2×C4×C47#12112yes—
280C2×C4×C47#12112yes—
281C2×C4×C47#12112yes—
282C2×C4×C47#12712yes—
283C2×C4×C47#12712yes—
284C2×C4×C47C22×D411yescharacteristic
285C2×C4×C47C22×D411yescharacteristic
286C2×C4×C47C22×D411yescharacteristic
287C2×C4×C47C22×D411yescharacteristic
288C2×C4×C47C22×D411yescharacteristic
289C2×C4×C47C2×#4111yescharacteristic
290C2×Dic47—28no—
291C2×Dic47—216no—
292C2×Dic47—24no—
293C2×Dic47—24no—
294C2×Dic47—24no—
295C2×Dic47—24no—
296C2×Dic47—24no—
297C2×Dic47—44no—
298C2×Dic47—44no—
299C2×Dic47—44no—
300Dic94—42no—
301Dic94—42no—
302Dic94—42no—
303Dic94—42no—
304Dic94—44no—
305Dic94—44no—
306C8×C47—24no—
307C47⋊C8—42no—
308C2×#2005—471nomaximal, Sylow
309C22×C47—22no—
310C22×C47—22no—
311C22×C47—24no—
312C22×C47—24no—
313C22×C47—42no—
314C22×C47—44no—
315C22×C47—84no—
316C22×C47#39212yes—
317C22×C47#39212yes—
318C22×C47#45614yes—
319C22×C47C2×#12011yescharacteristic
320C22×C47C2×#12011yescharacteristic
321C22×C47C2×#12111yescharacteristic
322C4×C47—42no—
323C4×C47—44no—
324C4×C47#39212yes—
325C4×C47#39212yes—
326C4×C47C2×#12011yescharacteristic
327C4×C47C2×#12111yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12711yescharacteristic
330Dic47—28no—
331Dic47—44no—
332Dic47—44no—
333C22×#137D4711yescharacteristic
334C2×#536—471no—
335C22×C4⋊Q8—471no—
336C22×#104—471no—
337C2×#419—471no—
338C22×#101—471no—
339#2005—478no—
340C2×#339—471no—
341C2×C47—21no—
342C2×C47—21no—
343C2×C47—22no—
344C2×C47—82no—
345C2×C47#200514yescomplemented
346C2×C47C2×#39211yescharacteristic
347C2×C47C2×#45612yes—
348C22×#41C2×D4711yescharacteristic
349C23×Q8—471no—
350C23×Q8C2×D4711yescharacteristic
351C2×#137—22no—
352C2×#137—24no—
353C2×#137—24no—
354C2×#137C2×D4714yes—
355C22×Q16—21no—
356C22×SD16—21no—
357C22×M16—471no—
358C22×M16C2×D4711yescharacteristic
359#536—474no—
360#536—942no—
361#536—942no—
362#536—942no—
363C2×C4⋊Q8—474no—
364C2×C4⋊Q8—944no—
365C2×C4⋊Q8—942no—
366C2×C4⋊Q8—944no—
367C2×C4⋊Q8—941no—
368C2×#124—941no—
369C2×#123—942no—
370C2×C22⋊Q8—942no—
371C2×#117—471no—
372C22×C4⋊C4—942no—
373C22×C42—471no—
374C2×#104—474no—
375C2×#104—944no—
376#419—474no—
377#419—942no—
378C2×#103—942no—
379C2×#101—474no—
380C2×#101—944no—
381#339—474no—
382C47C2×#200511yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D4712yes—
390C2×#41C22×D4712yes—
391C22×Q8—22no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—22no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—472no—
400C22×Q8—472no—
401C22×Q8—941no—
402C22×Q8—942no—
403C22×Q8—941no—
404C22×Q8—942no—
405C22×Q8—941no—
406C22×Q8—942no—
407C22×Q8—941no—
408C22×Q8—942no—
409C22×Q8C22⋊D4711yescharacteristic
410C22×Q8C22⋊D4714yes—
411C22×Q8C22×D4712yes—
412C22×D4—21no—
413C22×D4C22⋊D4711yescharacteristic
414C23×C4—21no—
415C23×C4—21no—
416C23×C4—472no—
417C23×C4—941no—
418C23×C4C22×D4711yescharacteristic
419#137—216no—
420#137—28no—
421#137—28no—
422C2×Q16—28no—
423C2×Q16—24no—
424C2×SD16—28no—
425C2×SD16—24no—
426C2×M16—22no—
427C2×M16—21no—
428C2×M16—472no—
429C2×M16—941no—
430C2×M16—942no—
431C2×M16—942no—
432C2×M16C22⋊D4714yes—
433C2×M16C22×D4712yes—
434C22×C8—21no—
435C22×C8—941no—
436C4⋊Q8—948no—
437C4⋊Q8—9416no—
438C4⋊Q8—944no—
439C4⋊Q8—948no—
440C4⋊Q8—944no—
441#124—944no—
442#124—944no—
443#123—948no—
444#123—948no—
445C22⋊Q8—948no—
446C22⋊Q8—948no—
447#117—472no—
448#117—472no—
449#117—941no—
450#117—941no—
451C2×C4⋊C4—472no—
452C2×C4⋊C4—948no—
453C2×C4⋊C4—9416no—
454C2×C4⋊C4—944no—
455C2×C22⋊C4—944no—
456C2×C22⋊C4—941no—
457C2×C42—474no—
458C2×C42—472no—
459C2×C42—941no—
460C2×C42—941no—
461C2×C42—942no—
462#104—948no—
463#103—948no—
464#101—948no—
465C24—21no—
466C24D4×D4711yescharacteristic
467#41—24no—
468#41—24no—
469#41—22no—
470#41—22no—
471#41—22no—
472#41—22no—
473#41—42no—
474#41—42no—
475#41—42no—
476#41—42no—
477C2×Q8—24no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—24no—
481C2×Q8—22no—
482C2×Q8—24no—
483C2×Q8—24no—
484C2×Q8—24no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—41no—
491C2×Q8—41no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—942no—
498C2×Q8—942no—
499C2×Q8—944no—
500C2×Q8—944no—
501C2×Q8—942no—
502C2×Q8—942no—
503C2×Q8—942no—
504C2×Q8—942no—
505C2×Q8—942no—
506C2×Q8—942no—
507C2×Q8—1882no—
508C2×Q8—1882no—
509C2×Q8—1882no—
510C2×Q8—1882no—
511C2×Q8—1882no—
512C2×Q8—1882no—
513C2×Q8—1882no—
514C2×Q8—1882no—
515C2×Q8C2×C22⋊D4712yes—
516C2×Q8C2×C22⋊D4714yes—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—21no—
520C2×D4—22no—
521C2×D4—22no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4C2×C22⋊D4712yes—
525C22×C4—24no—
526C22×C4—21no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—41no—
531C22×C4—41no—
532C22×C4—42no—
533C22×C4—42no—
534C22×C4—42no—
535C22×C4—474no—
536C22×C4—948no—
537C22×C4—944no—
538C22×C4—942no—
539C22×C4—944no—
540C22×C4—942no—
541C22×C4—942no—
542C22×C4—942no—
543C22×C4—944no—
544C22×C4D4×D4711yescharacteristic
545C22×C4D4×D4712yes—
546C22×C4D4×D4711yescharacteristic
547C22×C4D4×D4711yescharacteristic
548C22×C4#1062535212yes—
549C22×C4C2×C22⋊D4712yes—
550C22×C4C2×C22⋊D4711yescharacteristic
551C22×C4C2×C22⋊D4711yescharacteristic
552C22×C4C2×C22⋊D4712yes—
553C22×C4C23×D4711yescharacteristic
554Q16—216no—
555SD16—216no—
556M16—24no—
557M16—22no—
558M16—942no—
559M16—942no—
560M16—1882no—
561M16C2×C22⋊D4714yes—
562C2×C8—24no—
563C2×C8—22no—
564C2×C8—942no—
565C2×C8—1882no—
566C4⋊C4—474no—
567C4⋊C4—9432no—
568C4⋊C4—9416no—
569C4⋊C4—942no—
570C22⋊C4—9416no—
571C22⋊C4—942no—
572C22⋊C4—942no—
573C42—474no—
574C42—944no—
575C42—942no—
576C42—942no—
577C42—942no—
578C23—24no—
579C23—22no—
580C23—42no—
581C23—44no—
582C23#1309399211yescharacteristic
583C23#1309399212yes—
584C23#1309399211yescharacteristic
585C23#1309399512yes—
586C23C2×D4×D4711yescharacteristic
587Q8—24no—
588Q8—22no—
589Q8—24no—
590Q8—22no—
591Q8—44no—
592Q8—44no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—1884no—
597Q8—1884no—
598Q8—1882no—
599Q8—1882no—
600Q8—1882no—
601Q8—1882no—
602D4—22no—
603D4—22no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—948no—
619C2×C4—9416no—
620C2×C4—944no—
621C2×C4—944no—
622C2×C4—944no—
623C2×C4—944no—
624C2×C4—944no—
625C2×C4—1884no—
626C2×C4—1884no—
627C2×C4—1884no—
628C2×C4#1309399211yescharacteristic
629C2×C4#1309399212yes—
630C2×C4#1309399211yescharacteristic
631C2×C4#1309399312yes—
632C2×C4#1309399312yes—
633C2×C4#1309399512yes—
634C2×C4#1309399612yes—
635C2×C4#1309399612yes—
636C2×C4#1309399612yes—
637C2×C4#1309399612yes—
638C2×C4#1309402012yes—
639C2×C4#1309402012yes—
640C2×C4C2×D4×D4711yescharacteristic
641C2×C4C2×D4×D4711yescharacteristic
642C2×C4C22×C22⋊D4711yescharacteristic
643C2×C4C22×C22⋊D4711yescharacteristic
644C2×C4C22×C22⋊D4711yescharacteristic
645C2×C4C2×#1062535211yescharacteristic
646C8—24no—
647C8—1882no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—44no—
654C22—84no—
655C22#1568903112yes—
656C22#1568903112yes—
657C22#1568930414yes—
658C22C2×#1309399211yescharacteristic
659C22C2×#1309399211yescharacteristic
660C22C2×#1309399511yescharacteristic
661C4—42no—
662C4—44no—
663C4—948no—
664C4—1884no—
665C4—1884no—
666C4#1568903112yes—
667C4#1568903112yes—
668C4C2×#1309399311yescharacteristic
669C4C2×#1309399611yescharacteristic
670C4C2×#1309399611yescharacteristic
671C4C2×#1309402011yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—82no—
676C2#2024442114yesdirect product factor
677C2C2×#1568903111yescharacteristic
678C2C2×#1568930412yes—
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47#120
prime radicalC22×#41×C47C22
hypercenterC22×#137D47
hypocenterC47C2×#2005
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. C22×#13094022C2semidirect
4. C2×#15689132C2semidirect
5. C22×#13093942C2semidirect
6. #20244421C2direct (N × H)
7. C2×#15688884C2semidirect
19. C2×#13094022C22semidirect
29. #15689132C22semidirect
32. C2×#13093942C22semidirect
33. #15688884C22semidirect
345. C2×C47#2005semidirect
382. C47C2×#2005semidirect
676. C2#20244421direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896942
C22×#130940222
C22×#130939512
C2×#156891322
C22×#130939422
#202444212
C2×#156888842
C22×#137×C472
C2×#200547