Finite Group Databasebeta

← C2×#20244422 #26243176

Subgroups — C2×#20244422 #26243176

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
1C22×#13094054C211yesmaximal, characteristic
2C22×#13094022C211yesmaximal, characteristic, complemented
3C22×#13093951C211yesmaximal, characteristic
4C2×#15689131C211yesmaximal, characteristic, complemented
5C22×#13093943C211yesmaximal, characteristic, complemented
6#20244422C218yesmaximal, direct product factor
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×#536C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C2×#13094054—24no—
11C2×#13094054—24no—
12C2×#13094054—22no—
13C2×#13094054C2214yes—
14C22×#10625337—21no—
15C22×#10625335—21no—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C2×#13094022—24no—
19C2×#13094022—24no—
20C2×#13094022—22no—
21C2×#13094022C2214yescomplemented
22C22×Dic47⋊C4—22no—
23C22×C4×Dic47C2211yescharacteristic
24C2×#13093951—24no—
25C2×#13093951C2214yes—
26#15689131—22no—
27#15689131C2214yescomplemented
28C2×#13093947—22no—
29C2×#13093943—24no—
30C2×#13093943C2214yescomplemented
31#15688882C2214yescomplemented
32C22×#41×C47C2211yescharacteristic
33C23×Q8×C47C2211yescharacteristic
34C47×#536—22no—
35C47×#536—22no—
36C47×#536—22no—
37C47×#536C2214yes—
38C2×C4⋊Q8×C47—21no—
39C2×#124×C47—21no—
40C2×#123×C47—22no—
41C2×C22⋊Q8×C47—22no—
42C2×#117×C47C2211yescharacteristic
43C22×Dic94—21no—
44C22×Dic94—21no—
45C22×Dic94—21no—
46C22×Dic94—22no—
47C22×Dic94—21no—
48C22×Dic94—22no—
49C22×Dic94—21no—
50C22×Dic94—22no—
51C22×Dic94D414yes—
52C22×Dic94C2312yes—
53C23×Dic47—21no—
54C23×Dic47—21no—
55#13094054—216no—
56#13094054—28no—
57#13094054—28no—
58C2×#10625337—28no—
59C2×#10625337—24no—
60C2×#10625335—28no—
61C2×#10625335—24no—
62C2×#10625329—22no—
63C2×#10625329—21no—
64C2×#10625329—21no—
65C2×#10625329—22no—
66C2×#10625329—22no—
67C2×#10625329D414yes—
68C2×#10625329C2312yes—
69C2×#10625329C2312yes—
70C22×C47⋊C8—21no—
71C22×C47⋊C8—21no—
72#13094022—216no—
73#13094022—28no—
74#13094022—28no—
75C2×Dic47⋊C4—216no—
76C2×Dic47⋊C4—28no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47D414yes—
80C2×C4×Dic47C2312yes—
81#13093951—28no—
82#13093947—28no—
83#13093943—28no—
84C2×#41×C47—21no—
85C2×#41×C47—21no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47—22no—
90C2×#41×C47D412yes—
91C2×#41×C47C2312yes—
92C22×Q8×C47—21no—
93C22×Q8×C47—21no—
94C22×Q8×C47—21no—
95C22×Q8×C47—21no—
96C22×Q8×C47—22no—
97C22×Q8×C47—22no—
98C22×Q8×C47—22no—
99C22×Q8×C47—22no—
100C22×Q8×C47D411yescharacteristic
101C22×Q8×C47D412yes—
102C22×Q8×C47C2312yes—
103C22×D4×C47—21no—
104C22×D4×C47D411yescharacteristic
105C23×C4×C47—21no—
106C23×C4×C47D412yes—
107C23×C4×C47C2311yescharacteristic
108C4⋊Q8×C47—24no—
109C4⋊Q8×C47—24no—
110#124×C47—24no—
111#124×C47—24no—
112#123×C47—28no—
113#123×C47—28no—
114C22⋊Q8×C47—28no—
115C22⋊Q8×C47—28no—
116#117×C47—21no—
117#117×C47—21no—
118#117×C47D412yes—
119#117×C47C2312yes—
120C2×C4⋊C4×C47—24no—
121C2×C4⋊C4×C47D412yes—
122C2×C22⋊C4×C47—24no—
123C2×C22⋊C4×C47—21no—
124C2×C42×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—22no—
138C2×Q8×C47—22no—
139C2×Q8×C47—21no—
140C2×Q8×C47—24no—
141C2×Q8×C47—24no—
142C2×Q8×C47—22no—
143C2×Q8×C47—22no—
144C2×Q8×C47—22no—
145C2×Q8×C47—22no—
146C2×Q8×C47—22no—
147C2×Q8×C47—22no—
148C2×Q8×C47—22no—
149C2×Q8×C47—22no—
150C2×Q8×C47—42no—
151C2×Q8×C47—42no—
152C2×Q8×C47—42no—
153C2×Q8×C47—42no—
154C2×Q8×C47—42no—
155C2×Q8×C47—42no—
156C2×Q8×C47—42no—
157C2×Q8×C47—42no—
158C2×Q8×C47C2×D412yes—
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—22no—
168C22×C4×C47—22no—
169C22×C4×C47—21no—
170C22×C4×C47—22no—
171C22×C4×C47—24no—
172C22×C4×C47—24no—
173C22×C4×C47—22no—
174C22×C4×C47—22no—
175C22×C4×C47—42no—
176C22×C4×C47—42no—
177C22×C4×C47C2×D414yes—
178C22×C4×C47C2×D411yescharacteristic
179C22×C4×C47C2×D412yes—
180C22×C4×C47C2×D411yescharacteristic
181C22×C4×C47C2×D411yescharacteristic
182C22×C4×C47C2×D412yes—
183C22×C4×C47C2×D412yes—
184C22×C4×C47C2×D411yescharacteristic
185C22×C4×C47C2×D411yescharacteristic
186C22×C4×C47#4112yes—
187C22×C4×C47C2411yescharacteristic
188C22×Dic47—28no—
189C22×Dic47—24no—
190C22×Dic47—22no—
191C22×Dic47—24no—
192C22×Dic47—22no—
193C22×Dic47—41no—
194C22×Dic47—41no—
195C22×Dic47—42no—
196C2×Dic94—24no—
197C2×Dic94—24no—
198C2×Dic94—24no—
199C2×Dic94—24no—
200C2×Dic94—22no—
201C2×Dic94—22no—
202C2×Dic94—22no—
203C2×Dic94—22no—
204C2×Dic94—24no—
205C2×Dic94—22no—
206C2×Dic94—41no—
207C2×Dic94—41no—
208C2×Dic94—42no—
209C2×Dic94—42no—
210C2×Dic94—42no—
211C2×Dic94—42no—
212C2×Dic94—42no—
213C2×Dic94C2×D414yes—
214C4⋊C4×C47—216no—
215C4⋊C4×C47—22no—
216C4⋊C4×C47C2×D414yes—
217C22⋊C4×C47—216no—
218C22⋊C4×C47—22no—
219C22⋊C4×C47—22no—
220C42×C47—22no—
221C42×C47—22no—
222#10625337—216no—
223#10625335—216no—
224Dic47⋊C4—232no—
225C4×Dic47—24no—
226C4×Dic47—22no—
227C4×Dic47C2×D414yes—
228#10625329—24no—
229#10625329—22no—
230#10625329—22no—
231#10625329—22no—
232#10625329—42no—
233#10625329C2×D414yes—
234C2×C47⋊C8—24no—
235C2×C47⋊C8—22no—
236C2×C47⋊C8—22no—
237C2×C47⋊C8—42no—
238C23×C47—24no—
239C23×C47—22no—
240C23×C47—42no—
241C23×C47—44no—
242C23×C47#12012yes—
243C23×C47#12011yescharacteristic
244C23×C47#12011yescharacteristic
245C23×C47#12112yes—
246C23×C47C22×D411yescharacteristic
247Q8×C47—22no—
248Q8×C47—22no—
249Q8×C47—42no—
250Q8×C47—42no—
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—24no—
263C2×C4×C47—24no—
264C2×C4×C47—24no—
265C2×C4×C47—22no—
266C2×C4×C47—24no—
267C2×C4×C47—44no—
268C2×C4×C47—44no—
269C2×C4×C47—42no—
270C2×C4×C47—44no—
271C2×C4×C47—42no—
272C2×C4×C47—42no—
273C2×C4×C47—44no—
274C2×C4×C47#12012yes—
275C2×C4×C47#12012yes—
276C2×C4×C47#12012yes—
277C2×C4×C47#12011yescharacteristic
278C2×C4×C47#12011yescharacteristic
279C2×C4×C47#12112yes—
280C2×C4×C47#12112yes—
281C2×C4×C47#12112yes—
282C2×C4×C47#12112yes—
283C2×C4×C47#12112yes—
284C2×C4×C47#12712yes—
285C2×C4×C47#12712yes—
286C2×C4×C47C22×D411yescharacteristic
287C2×C4×C47C22×D411yescharacteristic
288C2×C4×C47C22×D411yescharacteristic
289C2×C4×C47C22×D411yescharacteristic
290C2×C4×C47C22×D411yescharacteristic
291C2×C4×C47C2×#4111yescharacteristic
292C2×Dic47—28no—
293C2×Dic47—216no—
294C2×Dic47—24no—
295C2×Dic47—24no—
296C2×Dic47—42no—
297C2×Dic47—42no—
298C2×Dic47—44no—
299C2×Dic47—44no—
300Dic94—24no—
301Dic94—24no—
302Dic94—42no—
303Dic94—42no—
304Dic94—44no—
305Dic94—44no—
306C47⋊C8—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—44no—
323C4×C47—42no—
324C4×C47—44no—
325C4×C47#39212yes—
326C4×C47#39212yes—
327C4×C47C2×#12011yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12111yescharacteristic
330C4×C47C2×#12711yescharacteristic
331Dic47—28no—
332Dic47—44no—
333C22×#137—471no—
334C2×#536D4711yescharacteristic
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—474no—
352C2×#137—944no—
353C2×#137—944no—
354C2×#137—942no—
355C22×Q16—941no—
356C22×SD16—941no—
357C22×M16—471no—
358C22×M16—471no—
359#536—22no—
360#536—22no—
361#536—22no—
362#536C2×D4714yes—
363C2×C4⋊Q8—21no—
364C2×C4⋊Q8—474no—
365C2×C4⋊Q8—944no—
366C2×C4⋊Q8—944no—
367C2×C4⋊Q8—942no—
368C2×#124—21no—
369C2×#123—22no—
370C2×C22⋊Q8—22no—
371C2×#117C2×D4711yescharacteristic
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—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—474no—
400C22×Q8—472no—
401C22×Q8—942no—
402C22×Q8—941no—
403C22×Q8—942no—
404C22×Q8—942no—
405C22×Q8—941no—
406C22×Q8—941no—
407C22×Q8—941no—
408C22×Q8—941no—
409C22×Q8C22⋊D4711yescharacteristic
410C22×Q8C22⋊D4712yes—
411C22×Q8C22×D4712yes—
412C22×D4—21no—
413C22×D4C22⋊D4711yescharacteristic
414C23×C4—21no—
415C23×C4—941no—
416C23×C4—941no—
417C23×C4C22⋊D4712yes—
418C23×C4C22×D4711yescharacteristic
419#137—948no—
420#137—948no—
421#137—9416no—
422C2×Q16—948no—
423C2×Q16—944no—
424C2×SD16—948no—
425C2×SD16—944no—
426C2×M16—474no—
427C2×M16—472no—
428C2×M16—472no—
429C2×M16—942no—
430C2×M16—941no—
431C2×M16—941no—
432C2×M16—942no—
433C2×M16—942no—
434C22×C8—941no—
435C22×C8—941no—
436C4⋊Q8—24no—
437C4⋊Q8—24no—
438C4⋊Q8—948no—
439C4⋊Q8—948no—
440C4⋊Q8—9416no—
441#124—24no—
442#124—24no—
443#123—28no—
444#123—28no—
445C22⋊Q8—28no—
446C22⋊Q8—28no—
447#117—21no—
448#117—21no—
449#117C22⋊D4712yes—
450#117C22×D4712yes—
451C2×C4⋊C4—24no—
452C2×C4⋊C4—9416no—
453C2×C4⋊C4—948no—
454C2×C4⋊C4C22⋊D4712yes—
455C2×C22⋊C4—24no—
456C2×C22⋊C4—21no—
457C2×C42—21no—
458C2×C42—474no—
459C2×C42—472no—
460C2×C42—942no—
461C2×C42—941no—
462#104—948no—
463#103—948no—
464#101—948no—
465C24—21no—
466C24C2×C22⋊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—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—474no—
499C2×Q8—944no—
500C2×Q8—942no—
501C2×Q8—944no—
502C2×Q8—944no—
503C2×Q8—944no—
504C2×Q8—944no—
505C2×Q8—942no—
506C2×Q8—942no—
507C2×Q8—942no—
508C2×Q8—942no—
509C2×Q8—1881no—
510C2×Q8—1881no—
511C2×Q8—1882no—
512C2×Q8—1882no—
513C2×Q8—1882no—
514C2×Q8—1882no—
515C2×Q8—1882no—
516C2×Q8C2×C22⋊D4712yes—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—22no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4C2×C22⋊D4712yes—
525C22×C4—24no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—24no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—42no—
534C22×C4—42no—
535C22×C4—948no—
536C22×C4—944no—
537C22×C4—942no—
538C22×C4—944no—
539C22×C4—942no—
540C22×C4—1881no—
541C22×C4—1881no—
542C22×C4—1882no—
543C22×C4D4×D4712yes—
544C22×C4D4×D4712yes—
545C22×C4D4×D4712yes—
546C22×C4#1062535212yes—
547C22×C4C2×C22⋊D4714yes—
548C22×C4C2×C22⋊D4711yescharacteristic
549C22×C4C2×C22⋊D4711yescharacteristic
550C22×C4C2×C22⋊D4711yescharacteristic
551C22×C4C2×C22⋊D4711yescharacteristic
552C22×C4C2×C22⋊D4711yescharacteristic
553C22×C4C23×D4711yescharacteristic
554Q16—9416no—
555SD16—9416no—
556M16—474no—
557M16—944no—
558M16—942no—
559M16—942no—
560M16—942no—
561M16—1882no—
562C2×C8—944no—
563C2×C8—942no—
564C2×C8—942no—
565C2×C8—1882no—
566C4⋊C4—216no—
567C4⋊C4—22no—
568C4⋊C4—9432no—
569C4⋊C4C2×C22⋊D4714yes—
570C22⋊C4—216no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C42—22no—
574C42—22no—
575C42—474no—
576C42—944no—
577C42—942no—
578C23—24no—
579C23—22no—
580C23—42no—
581C23—44no—
582C23#1309399212yes—
583C23#1309399311yescharacteristic
584C23#1309399311yescharacteristic
585C23#1309399612yes—
586C23C22×C22⋊D4711yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—44no—
590Q8—44no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—944no—
597Q8—944no—
598Q8—1884no—
599Q8—1884no—
600Q8—1882no—
601Q8—1882no—
602D4—22no—
603D4—22no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—948no—
621C2×C4—9416no—
622C2×C4—944no—
623C2×C4—944no—
624C2×C4—1884no—
625C2×C4—1884no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4#1309399212yes—
629C2×C4#1309399212yes—
630C2×C4#1309399212yes—
631C2×C4#1309399311yescharacteristic
632C2×C4#1309399311yescharacteristic
633C2×C4#1309399512yes—
634C2×C4#1309399512yes—
635C2×C4#1309399612yes—
636C2×C4#1309399612yes—
637C2×C4#1309399612yes—
638C2×C4#1309402012yes—
639C2×C4#1309402012yes—
640C2×C4C2×D4×D4711yescharacteristic
641C2×C4C2×D4×D4711yescharacteristic
642C2×C4C2×D4×D4711yescharacteristic
643C2×C4C22×C22⋊D4711yescharacteristic
644C2×C4C22×C22⋊D4711yescharacteristic
645C2×C4C2×#1062535211yescharacteristic
646C8—944no—
647C8—1882no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—44no—
654C22—84no—
655C22#1568903112yes—
656C22#1568903112yes—
657C22#1568930614yes—
658C22C2×#1309399211yescharacteristic
659C22C2×#1309399311yescharacteristic
660C22C2×#1309399611yescharacteristic
661C4—44no—
662C4—42no—
663C4—44no—
664C4—948no—
665C4—1884no—
666C4#1568903112yes—
667C4#1568903112yes—
668C4C2×#1309399211yescharacteristic
669C4C2×#1309399511yescharacteristic
670C4C2×#1309399611yescharacteristic
671C4C2×#1309402011yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—82no—
676C2#2024442214yesdirect product factor
677C2C2×#1568903111yescharacteristic
678C2C2×#1568930612yes—
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#536C2
socleC23×C47#120
prime radicalC22×#41×C47C22
hypercenterC2×#536D47
hypocenterC47C2×#2005
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C22×#13094022C2semidirect
4. C2×#15689131C2semidirect
5. C22×#13093943C2semidirect
6. #20244422C2direct (N × H)
7. C2×#15688882C2semidirect
21. C2×#13094022C22semidirect
27. #15689131C22semidirect
30. C2×#13093943C22semidirect
31. #15688882C22semidirect
345. C2×C47#2005semidirect
382. C47C2×#2005semidirect
676. C2#20244422direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940542
C22×#130940222
C22×#130939512
C2×#156891312
C22×#130939432
#202444222
C2×#156888822
C2×C47×#5362
C2×#200547