Finite Group Databasebeta

← C2×#20244424 #26243178

Subgroups — C2×#20244424 #26243178

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×#13094055C211yesmaximal, characteristic
2C2×#15689697C211yesmaximal, characteristic
3C22×#13093952C211yesmaximal, characteristic
4C2×#15689132C211yesmaximal, characteristic, complemented
5C22×#13093943C211yesmaximal, characteristic, complemented
6#20244424C218yesmaximal, direct product factor
7C2×#15688883C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8×C47C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C2×#13094055—22no—
11C2×#13094055—24no—
12C2×#13094055—24no—
13C2×#13094055C2214yes—
14C22×#10625337—21no—
15C22×#10625336—21no—
16C22×#10625329C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18#15689697—22no—
19#15689697—22no—
20#15689697—22no—
21#15689697C2214yes—
22C2×#13094022—21no—
23C2×#13094013—21no—
24C2×#13094007—22no—
25C2×#13094000—22no—
26C2×#13093983C2211yescharacteristic
27C2×#13093952—24no—
28C2×#13093952C2214yes—
29#15689132—22no—
30#15689132C2214yescomplemented
31C2×#13093948—22no—
32C2×#13093943—24no—
33C2×#13093943C2214yescomplemented
34#15688883C2214yescomplemented
35C23×Q8×C47C2211yescharacteristic
36C23×Q8×C47C2211yescharacteristic
37C2×C4⋊Q8×C47—24no—
38C2×C4⋊Q8×C47—22no—
39C2×C4⋊Q8×C47—24no—
40C2×C4⋊Q8×C47C2214yescomplemented
41C22×C4⋊C4×C47—22no—
42C22×C42×C47C2211yescharacteristic
43C2×#10625350—21no—
44C2×#10625350—21no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350D412yes—
50C2×#10625350C2312yes—
51C22×Dic94D411yescharacteristic
52C22×D188D411yescharacteristic
53C22×C22⋊D47—21no—
54C22×C4×D47—21no—
55#13094055—216no—
56#13094055—28no—
57#13094055—28no—
58C2×#10625337—28no—
59C2×#10625337—24no—
60C2×#10625336—28no—
61C2×#10625336—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—24no—
73#13094022—24no—
74#13094013—24no—
75#13094013—24no—
76#13094007—28no—
77#13094007—28no—
78#13094000—28no—
79#13094000—28no—
80#13093983—21no—
81#13093983—21no—
82#13093983D412yes—
83#13093983C2312yes—
84C2×C4⋊Dic47D412yes—
85C2×Dic47⋊C4—24no—
86C2×C22⋊Dic47—21no—
87C2×(C2×D47)⋊C4—24no—
88C2×C4×Dic47—21no—
89#13093952—28no—
90#13093948—28no—
91#13093943—28no—
92C22×Q8×C47—22no—
93C22×Q8×C47—22no—
94C22×Q8×C47—22no—
95C22×Q8×C47—21no—
96C22×Q8×C47—22no—
97C22×Q8×C47—21no—
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—21no—
105C22×Q8×C47—21no—
106C22×Q8×C47—22no—
107C22×Q8×C47—22no—
108C22×Q8×C47D414yes—
109C22×Q8×C47D412yes—
110C22×Q8×C47C2312yes—
111C22×Q8×C47C2312yes—
112C23×C4×C47—21no—
113C23×C4×C47—21no—
114C23×C4×C47D412yes—
115C23×C4×C47C2311yescharacteristic
116C4⋊Q8×C47—216no—
117C4⋊Q8×C47—28no—
118C4⋊Q8×C47—28no—
119C2×C4⋊C4×C47—216no—
120C2×C4⋊C4×C47—28no—
121C2×C42×C47—21no—
122C2×C42×C47—22no—
123C2×C42×C47D414yes—
124C2×C42×C47C2312yes—
125C24×C47C2×D411yescharacteristic
126C23×D47—21no—
127C2×Q8×C47—24no—
128C2×Q8×C47—24no—
129C2×Q8×C47—22no—
130C2×Q8×C47—22no—
131C2×Q8×C47—24no—
132C2×Q8×C47—22no—
133C2×Q8×C47—22no—
134C2×Q8×C47—22no—
135C2×Q8×C47—22no—
136C2×Q8×C47—22no—
137C2×Q8×C47—22no—
138C2×Q8×C47—22no—
139C2×Q8×C47—24no—
140C2×Q8×C47—24no—
141C2×Q8×C47—24no—
142C2×Q8×C47—24no—
143C2×Q8×C47—22no—
144C2×Q8×C47—22no—
145C2×Q8×C47—22no—
146C2×Q8×C47—22no—
147C2×Q8×C47—41no—
148C2×Q8×C47—41no—
149C2×Q8×C47—42no—
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×C47—42no—
159C2×Q8×C47—42no—
160C2×Q8×C47—42no—
161C2×Q8×C47—42no—
162C2×Q8×C47C2×D414yes—
163C22×C4×C47—28no—
164C22×C4×C47—24no—
165C22×C4×C47—22no—
166C22×C4×C47—24no—
167C22×C4×C47—24no—
168C22×C4×C47—22no—
169C22×C4×C47—22no—
170C22×C4×C47—24no—
171C22×C4×C47—22no—
172C22×C4×C47—41no—
173C22×C4×C47—41no—
174C22×C4×C47—42no—
175C22×C4×C47C2×D414yes—
176C22×C4×C47C2×D411yescharacteristic
177C22×C4×C47C2×D411yescharacteristic
178C22×C4×C47C2×D411yescharacteristic
179C22×C4×C47C2×D412yes—
180C22×C4×C47C2×D412yes—
181C22×C4×C47C2×D411yescharacteristic
182C22×C4×C47C2×D411yescharacteristic
183C22×C4×C47C2×D412yes—
184C22×C4×C47#4112yes—
185C22×C4×C47C2411yescharacteristic
186C2×C22⋊D47—22no—
187C2×C22⋊D47—22no—
188C2×C22⋊D47—42no—
189C2×C22⋊D47—42no—
190C22×Dic47—22no—
191C22×Dic47—21no—
192#10625350—24no—
193#10625350—24no—
194#10625350—22no—
195#10625350—22no—
196#10625350—22no—
197#10625350—22no—
198#10625350—42no—
199#10625350—42no—
200#10625350—42no—
201#10625350—42no—
202C2×D188—22no—
203C2×D188—22no—
204C2×D188—21no—
205C2×D188C2×D412yes—
206C2×C4×D47—22no—
207C2×C4×D47—22no—
208C2×C4×D47—42no—
209C2×C4×D47—42no—
210C2×Dic94—22no—
211C2×Dic94—22no—
212C2×Dic94—21no—
213C2×Dic94C2×D412yes—
214C4⋊C4×C47—232no—
215C42×C47—24no—
216C42×C47—22no—
217C42×C47C2×D414yes—
218C22⋊Dic47—22no—
219C22⋊Dic47—22no—
220#10625337—216no—
221#10625336—216no—
222(C2×D47)⋊C4—216no—
223C4⋊Dic47—22no—
224C4⋊Dic47C2×D414yes—
225Dic47⋊C4—216no—
226C4×Dic47—22no—
227C4×Dic47—22no—
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#12011yescharacteristic
240C23×C47#12011yescharacteristic
241C23×C47#12012yes—
242C23×C47#12112yes—
243C23×C47C22×D411yescharacteristic
244C22×D47—22no—
245C22×D47—42no—
246C22×D47—44no—
247Q8×C47—24no—
248Q8×C47—24no—
249Q8×C47—42no—
250Q8×C47—42no—
251Q8×C47—42no—
252Q8×C47—42no—
253Q8×C47—42no—
254Q8×C47—42no—
255Q8×C47—44no—
256Q8×C47—44no—
257Q8×C47—44no—
258Q8×C47—44no—
259C2×C4×C47—28no—
260C2×C4×C47—216no—
261C2×C4×C47—24no—
262C2×C4×C47—24no—
263C2×C4×C47—24no—
264C2×C4×C47—24no—
265C2×C4×C47—24no—
266C2×C4×C47—42no—
267C2×C4×C47—42no—
268C2×C4×C47—44no—
269C2×C4×C47—44no—
270C2×C4×C47—44no—
271C2×C4×C47—44no—
272C2×C4×C47#12012yes—
273C2×C4×C47#12012yes—
274C2×C4×C47#12011yescharacteristic
275C2×C4×C47#12011yescharacteristic
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
290C22⋊D47—42no—
291C22⋊D47—42no—
292C22⋊D47—44no—
293C2×Dic47—24no—
294C2×Dic47—22no—
295C2×Dic47—42no—
296C2×Dic47—44no—
297D188—22no—
298D188—22no—
299D188—42no—
300C4×D47—42no—
301C4×D47—42no—
302C4×D47—44no—
303Dic94—22no—
304Dic94—22no—
305Dic94—42no—
306C47⋊C8—24no—
307C47⋊C8—42no—
308C2×#2005—471nomaximal, Sylow
309C22×C47—22no—
310C22×C47—22no—
311C22×C47—24no—
312C22×C47—24no—
313C22×C47#39212yes—
314C22×C47#39212yes—
315C22×C47#45614yes—
316C22×C47C2×#12011yescharacteristic
317C22×C47C2×#12011yescharacteristic
318C22×C47C2×#12111yescharacteristic
319C2×D47—42no—
320C2×D47—44no—
321C2×D47—84no—
322C4×C47—28no—
323C4×C47—44no—
324C4×C47—44no—
325C4×C47#39212yes—
326C4×C47#39212yes—
327C4×C47C2×#12011yescharacteristic
328C4×C47C2×#12111yescharacteristic
329C4×C47C2×#12111yescharacteristic
330C4×C47C2×#12711yescharacteristic
331Dic47—42no—
332Dic47—44no—
333C22×#137—471no—
334C2×#536—471no—
335C22×C4⋊Q8D4711yescharacteristic, complemented
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#200514yescomplemented
345C2×C47C2×#39211yescharacteristic
346C2×C47C2×#45612yes—
347D47—82no—
348C22×#41—471no—
349C23×Q8C2×D4711yescharacteristic
350C23×Q8C2×D4711yescharacteristic
351C2×#137—474no—
352C2×#137—944no—
353C2×#137—942no—
354C2×#137—944no—
355C22×Q16—941no—
356C22×SD16—941no—
357C22×M16—471no—
358C22×M16—471no—
359#536—474no—
360#536—942no—
361#536—942no—
362#536—942no—
363C2×C4⋊Q8—24no—
364C2×C4⋊Q8—22no—
365C2×C4⋊Q8—24no—
366C2×C4⋊Q8—941no—
367C2×C4⋊Q8C2×D4714yescomplemented
368C2×#124—941no—
369C2×#123—942no—
370C2×C22⋊Q8—942no—
371C2×#117—471no—
372C22×C4⋊C4—22no—
373C22×C42C2×D4711yescharacteristic
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—472no—
384C2×#41—472no—
385C2×#41—941no—
386C2×#41—941no—
387C2×#41—942no—
388C2×#41—942no—
389C2×#41—942no—
390C2×#41—942no—
391C22×Q8—21no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—22no—
396C22×Q8—22no—
397C22×Q8—22no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—21no—
401C22×Q8—21no—
402C22×Q8—21no—
403C22×Q8—22no—
404C22×Q8—21no—
405C22×Q8—21no—
406C22×Q8—22no—
407C22×Q8—471no—
408C22×Q8C22⋊D4714yes—
409C22×Q8C22⋊D4712yes—
410C22×Q8C22×D4712yes—
411C22×Q8C22×D4712yes—
412C22×D4—471no—
413C22×D4—941no—
414C23×C4—21no—
415C23×C4—21no—
416C23×C4—941no—
417C23×C4C22⋊D4712yes—
418C23×C4C22×D4711yescharacteristic
419#137—948no—
420#137—9416no—
421#137—948no—
422C2×Q16—944no—
423C2×Q16—948no—
424C2×SD16—944no—
425C2×SD16—948no—
426C2×M16—474no—
427C2×M16—472no—
428C2×M16—472no—
429C2×M16—941no—
430C2×M16—942no—
431C2×M16—941no—
432C2×M16—942no—
433C2×M16—942no—
434C22×C8—941no—
435C22×C8—941no—
436C4⋊Q8—216no—
437C4⋊Q8—28no—
438C4⋊Q8—28no—
439C4⋊Q8—944no—
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—216no—
452C2×C4⋊C4—28no—
453C2×C4⋊C4—472no—
454C2×C4⋊C4—944no—
455C2×C22⋊C4—944no—
456C2×C22⋊C4—941no—
457C2×C42—21no—
458C2×C42—22no—
459C2×C42—941no—
460C2×C42C22⋊D4714yes—
461C2×C42C22×D4712yes—
462#104—948no—
463#103—948no—
464#101—948no—
465C24—941no—
466C24C2×C22⋊D4711yescharacteristic
467#41—944no—
468#41—944no—
469#41—942no—
470#41—942no—
471#41—942no—
472#41—942no—
473#41—1882no—
474#41—1882no—
475#41—1882no—
476#41—1882no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—24no—
480C2×Q8—24no—
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—24no—
490C2×Q8—24no—
491C2×Q8—24no—
492C2×Q8—24no—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—41no—
500C2×Q8—41no—
501C2×Q8—42no—
502C2×Q8—42no—
503C2×Q8—42no—
504C2×Q8—42no—
505C2×Q8—42no—
506C2×Q8—42no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—42no—
512C2×Q8—472no—
513C2×Q8—941no—
514C2×Q8—942no—
515C2×Q8—942no—
516C2×Q8C2×C22⋊D4714yes—
517C2×D4—472no—
518C2×D4—942no—
519C2×D4—942no—
520C2×D4—942no—
521C2×D4—941no—
522C2×D4—942no—
523C2×D4—1882no—
524C2×D4—1882no—
525C22×C4—28no—
526C22×C4—24no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—24no—
531C22×C4—24no—
532C22×C4—24no—
533C22×C4—22no—
534C22×C4—41no—
535C22×C4—41no—
536C22×C4—42no—
537C22×C4—942no—
538C22×C4—941no—
539C22×C4—942no—
540C22×C4—942no—
541C22×C4—1882no—
542C22×C4—1882no—
543C22×C4D4×D4711yescharacteristic
544C22×C4D4×D4711yescharacteristic
545C22×C4D4×D4712yes—
546C22×C4D4×D4711yescharacteristic
547C22×C4D4×D4711yescharacteristic
548C22×C4#1062535212yes—
549C22×C4C2×C22⋊D4714yes—
550C22×C4C2×C22⋊D4711yescharacteristic
551C22×C4C2×C22⋊D4712yes—
552C22×C4C2×C22⋊D4712yes—
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—232no—
567C4⋊C4—474no—
568C4⋊C4—9416no—
569C4⋊C4—942no—
570C22⋊C4—9416no—
571C22⋊C4—942no—
572C22⋊C4—942no—
573C42—24no—
574C42—22no—
575C42—942no—
576C42—942no—
577C42C2×C22⋊D4714yes—
578C23—24no—
579C23—942no—
580C23—1882no—
581C23—1884no—
582C23#1309399211yescharacteristic
583C23#1309399211yescharacteristic
584C23#1309399312yes—
585C23#1309399612yes—
586C23C22×C22⋊D4711yescharacteristic
587Q8—24no—
588Q8—24no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—942no—
600Q8—942no—
601Q8—1882no—
602D4—942no—
603D4—942no—
604D4—1884no—
605D4—1882no—
606D4—1882no—
607D4—1882no—
608C2×C4—28no—
609C2×C4—216no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—944no—
622C2×C4—942no—
623C2×C4—1882no—
624C2×C4—1884no—
625C2×C4—1884no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4#1309399211yescharacteristic
629C2×C4#1309399211yescharacteristic
630C2×C4#1309399212yes—
631C2×C4#1309399212yes—
632C2×C4#1309399312yes—
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—3764no—
653C22—1882no—
654C22—1884no—
655C22#1568903112yes—
656C22#1568903112yes—
657C22#1568930514yes—
658C22C2×#1309399211yescharacteristic
659C22C2×#1309399311yescharacteristic
660C22C2×#1309399611yescharacteristic
661C4—28no—
662C4—44no—
663C4—44no—
664C4—1882no—
665C4—1884no—
666C4#1568903112yes—
667C4#1568903112yes—
668C4C2×#1309399211yescharacteristic
669C4C2×#1309399511yescharacteristic
670C4C2×#1309399611yescharacteristic
671C4C2×#1309402011yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—3762no—
676C2#2024442414yesdirect product factor
677C2C2×#1568903111yescharacteristic
678C2C2×#1568930512yes—
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×C4⋊Q8×C47C2
socleC23×C47#120
prime radicalC22×#10625350C22
hypercenterC22×C4⋊Q8D47
hypocenterC47C2×#2005
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C2×#15689132C2semidirect
5. C22×#13093943C2semidirect
6. #20244424C2direct (N × H)
7. C2×#15688883C2semidirect
8. C22×C4⋊Q8×C47C2semidirect
30. #15689132C22semidirect
33. C2×#13093943C22semidirect
34. #15688883C22semidirect
40. C2×C4⋊Q8×C47C22semidirect
335. C22×C4⋊Q8D47semidirect
344. C2×C47#2005semidirect
367. C2×C4⋊Q8C2×D47semidirect
382. C47C2×#2005semidirect
676. C2#20244424direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940552
C2×#156896972
C22×#130939522
C2×#156891322
C22×#130939432
#202444242
C2×#156888832
C22×C4⋊Q8×C472
C2×#200547