Finite Group Databasebeta

← C2×#20244414 #26243159

Subgroups — C2×#20244414 #26243159

Order 12032.

2190 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689687C211yesmaximal, characteristic
2C22×#13094013C211yesmaximal, characteristic, complemented
3C22×#13093951C211yesmaximal, characteristic, complemented
4C2×#15689132C211yesmaximal, characteristic, complemented
5C22×#13093940C211yesmaximal, characteristic, complemented
6#20244414C218yesmaximal, direct product factor
7C2×#15688884C211yesmaximal, characteristic, complemented
8C22×#137×C47C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689687—22no—
12#15689687—22no—
13#15689687—22no—
14#15689687C2214yes—
15C2×#13094020—21no—
16C2×#13094013—22no—
17C2×#13094013—24no—
18C2×#13094013—21no—
19C2×#13094013—24no—
20C2×#13094013C2214yescomplemented
21C2×#13093998—22no—
22C2×#13093992—22no—
23C2×#13093983C2211yescharacteristic
24C22×(C2×D47)⋊C4—22no—
25C22×C4×Dic47C2211yescharacteristic
26C2×#13093951—24no—
27C2×#13093951C2214yescomplemented
28#15689132—22no—
29#15689132C2214yescomplemented
30C2×#13093948—22no—
31C2×#13093940—24no—
32C2×#13093940C2214yescomplemented
33#15688884C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C23×Q8×C47C2211yescharacteristic
36C2×#137×C47—22no—
37C2×#137×C47—24no—
38C2×#137×C47—24no—
39C2×#137×C47C2214yescomplemented
40C22×Q16×C47—21no—
41C22×SD16×C47—21no—
42C22×M16×C47C2211yescharacteristic
43C24×D47D412yes—
44C22×D188—22no—
45C22×D188—21no—
46C22×D188—22no—
47C22×D188—22no—
48C22×D188—21no—
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—22no—
52C22×D188D412yes—
53C22×D188C2312yes—
54C22×C22⋊D47—24no—
55C23×Dic47—21no—
56C2×#10625329—21no—
57C2×#10625329—22no—
58C2×#10625329—22no—
59C2×#10625329C2312yes—
60C22×C47⋊C8—21no—
61#13094020—24no—
62#13094020—24no—
63#13094013—216no—
64#13094013—24no—
65#13094013—24no—
66#13094013—28no—
67#13094013—28no—
68#13093998—28no—
69#13093998—28no—
70#13093992—28no—
71#13093992—28no—
72#13093983—21no—
73#13093983—21no—
74#13093983D412yes—
75#13093983C2312yes—
76C2×C4⋊Dic47D412yes—
77C2×C22⋊Dic47—21no—
78C2×(C2×D47)⋊C4—216no—
79C2×(C2×D47)⋊C4—24no—
80C2×(C2×D47)⋊C4—28no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47D414yescomplemented
85C2×C4×Dic47C2312yes—
86#13093951—28no—
87#13093948—28no—
88#13093940—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×C47D414yescomplemented
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×C47D414yescomplemented
123C2×M16×C47C2312yes—
124C22×C8×C47—21no—
125C24×C47—21no—
126C24×C47C2×D411yescharacteristic
127C23×D47—24no—
128C23×D47—22no—
129C23×D47—22no—
130C23×D47—24no—
131C23×D47—48no—
132C23×D47C2×D414yes—
133#41×C47—24no—
134#41×C47—24no—
135#41×C47—22no—
136#41×C47—22no—
137#41×C47—22no—
138#41×C47—22no—
139#41×C47—42no—
140#41×C47—42no—
141#41×C47—42no—
142#41×C47—42no—
143C2×Q8×C47—24no—
144C2×Q8×C47—21no—
145C2×Q8×C47—22no—
146C2×Q8×C47—24no—
147C2×Q8×C47—22no—
148C2×Q8×C47—24no—
149C2×Q8×C47—24no—
150C2×Q8×C47—24no—
151C2×Q8×C47—22no—
152C2×Q8×C47—22no—
153C2×Q8×C47—22no—
154C2×Q8×C47—22no—
155C2×Q8×C47—22no—
156C2×Q8×C47—41no—
157C2×Q8×C47—41no—
158C2×Q8×C47—42no—
159C2×Q8×C47—42no—
160C2×Q8×C47—42no—
161C2×Q8×C47—42no—
162C2×Q8×C47—42no—
163C2×Q8×C47C2×D412yes—
164C2×Q8×C47C2×D414yescomplemented
165C2×D4×C47—22no—
166C2×D4×C47—22no—
167C2×D4×C47—22no—
168C2×D4×C47—21no—
169C2×D4×C47—22no—
170C2×D4×C47—42no—
171C2×D4×C47—42no—
172C2×D4×C47C2×D412yes—
173C22×C4×C47—21no—
174C22×C4×C47—24no—
175C22×C4×C47—22no—
176C22×C4×C47—22no—
177C22×C4×C47—22no—
178C22×C4×C47—41no—
179C22×C4×C47—41no—
180C22×C4×C47—42no—
181C22×C4×C47—42no—
182C22×C4×C47—42no—
183C22×C4×C47C2×D411yescharacteristic
184C22×C4×C47C2×D411yescharacteristic
185C22×C4×C47C2×D411yescharacteristic
186C22×C4×C47C2×D412yes—
187C22×C4×C47C2×D412yes—
188C22×C4×C47C2×D411yescharacteristic
189C22×C4×C47C2×D412yes—
190C22×C4×C47C2×D411yescharacteristic
191C22×C4×C47#4112yes—
192C22×C4×C47C2411yescharacteristic
193C2×C22⋊D47—216no—
194C2×C22⋊D47—88no—
195C22×Dic47—24no—
196C22×Dic47—22no—
197C22×Dic47—22no—
198C22×Dic47—44no—
199C2×D188—22no—
200C2×D188—24no—
201C2×D188—22no—
202C2×D188—24no—
203C2×D188—22no—
204C2×D188—22no—
205C2×D188—22no—
206C2×D188—22no—
207C2×D188—22no—
208C2×D188—22no—
209C2×D188—42no—
210C2×D188—42no—
211C2×D188—42no—
212C2×D188—42no—
213C2×D188—42no—
214C2×D188—42no—
215C2×D188—42no—
216C2×D188—42no—
217Q16×C47—216no—
218SD16×C47—216no—
219M16×C47—24no—
220M16×C47—22no—
221M16×C47C2×D414yescomplemented
222C2×C8×C47—24no—
223C2×C8×C47—22no—
224C22⋊Dic47—22no—
225C22⋊Dic47—22no—
226(C2×D47)⋊C4—232no—
227(C2×D47)⋊C4—216no—
228C4⋊Dic47—22no—
229C4⋊Dic47C2×D414yes—
230C4×Dic47—24no—
231C4×Dic47—22no—
232C4×Dic47—22no—
233C4×Dic47—22no—
234C4×Dic47C2×D414yescomplemented
235#10625329—22no—
236#10625329—22no—
237#10625329—42no—
238C2×C47⋊C8—22no—
239C2×C47⋊C8—42no—
240C23×C47—24no—
241C23×C47—22no—
242C23×C47—42no—
243C23×C47—44no—
244C23×C47#12011yescharacteristic
245C23×C47#12012yes—
246C23×C47#12011yescharacteristic
247C23×C47#12112yes—
248C23×C47C22×D411yescharacteristic
249C22×D47—24no—
250C22×D47—24no—
251C22×D47—24no—
252C22×D47—48no—
253C22×D47—416no—
254C22×D47—44no—
255C22×D47—44no—
256C22×D47—48no—
257C22×D47—84no—
258C22×D47—88no—
259Q8×C47—24no—
260Q8×C47—22no—
261Q8×C47—24no—
262Q8×C47—22no—
263Q8×C47—42no—
264Q8×C47—42no—
265Q8×C47—44no—
266Q8×C47—44no—
267Q8×C47—42no—
268D4×C47—22no—
269D4×C47—22no—
270D4×C47—42no—
271D4×C47—42no—
272D4×C47—44no—
273D4×C47—42no—
274C2×C4×C47—22no—
275C2×C4×C47—24no—
276C2×C4×C47—42no—
277C2×C4×C47—42no—
278C2×C4×C47—42no—
279C2×C4×C47—44no—
280C2×C4×C47—44no—
281C2×C4×C47—42no—
282C2×C4×C47—42no—
283C2×C4×C47—44no—
284C2×C4×C47#12011yescharacteristic
285C2×C4×C47#12012yes—
286C2×C4×C47#12011yescharacteristic
287C2×C4×C47#12012yes—
288C2×C4×C47#12012yes—
289C2×C4×C47#12112yes—
290C2×C4×C47#12112yes—
291C2×C4×C47#12112yes—
292C2×C4×C47#12112yes—
293C2×C4×C47#12112yes—
294C2×C4×C47#12712yes—
295C2×C4×C47#12712yes—
296C2×C4×C47C22×D411yescharacteristic
297C2×C4×C47C22×D411yescharacteristic
298C2×C4×C47C22×D411yescharacteristic
299C2×C4×C47C22×D411yescharacteristic
300C2×C4×C47C22×D411yescharacteristic
301C2×C4×C47C2×#4111yescharacteristic
302C22⋊D47—816no—
303C2×Dic47—24no—
304C2×Dic47—24no—
305C2×Dic47—44no—
306C2×Dic47—48no—
307C2×Dic47—44no—
308D188—42no—
309D188—42no—
310D188—42no—
311D188—42no—
312D188—44no—
313D188—44no—
314C8×C47—24no—
315C47⋊C8—42no—
316C2×#2004—471nomaximal, Sylow
317C22×C47—22no—
318C22×C47—22no—
319C22×C47—24no—
320C22×C47—24no—
321C22×C47—42no—
322C22×C47—44no—
323C22×C47—84no—
324C22×C47#39212yes—
325C22×C47#39212yes—
326C22×C47#45514yes—
327C22×C47C2×#12011yescharacteristic
328C22×C47C2×#12011yescharacteristic
329C22×C47C2×#12111yescharacteristic
330C2×D47—48no—
331C2×D47—44no—
332C2×D47—48no—
333C2×D47—88no—
334C2×D47—88no—
335C2×D47—84no—
336C4×C47—42no—
337C4×C47—44no—
338C4×C47#39212yes—
339C4×C47#39212yes—
340C4×C47C2×#12011yescharacteristic
341C4×C47C2×#12111yescharacteristic
342C4×C47C2×#12111yescharacteristic
343C4×C47C2×#12711yescharacteristic
344Dic47—44no—
345Dic47—44no—
346C22×#137D4711yescharacteristic, complemented
347C2×#535—471no—
348C22×#124—471no—
349C22×#104—471no—
350C2×#419—471no—
351C22×#100—471no—
352#2004—478no—
353C2×#339—471no—
354C2×C47—21no—
355C2×C47—21no—
356C2×C47—22no—
357C2×C47—82no—
358C2×C47#200414yescomplemented
359C2×C47C2×#39211yescharacteristic
360C2×C47C2×#45512yes—
361D47—84no—
362C22×#41C2×D4711yescharacteristic
363C23×Q8C2×D4711yescharacteristic
364C23×D4—471no—
365C2×#137—22no—
366C2×#137—24no—
367C2×#137—24no—
368C2×#137C2×D4714yescomplemented
369C22×Q16—21no—
370C22×SD16—21no—
371C22×M16—471no—
372C22×M16C2×D4711yescharacteristic
373#535—474no—
374#535—942no—
375#535—942no—
376#535—942no—
377C2×#127—941no—
378C2×#124—474no—
379C2×#124—944no—
380C2×#124—942no—
381C2×#124—941no—
382C2×#124—944no—
383C2×#121—942no—
384C2×#120—942no—
385C2×#117—471no—
386C22×C22⋊C4—942no—
387C22×C42—471no—
388C2×#104—474no—
389C2×#104—944no—
390#419—474no—
391#419—942no—
392C2×#103—942no—
393C2×#100—474no—
394C2×#100—944no—
395#339—474no—
396C47C2×#200411yescharacteristic, complemented, Sylow
397C25—472no—
398C2×#41—21no—
399C2×#41—21no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—22no—
403C2×#41—22no—
404C2×#41C22⋊D4712yes—
405C2×#41C22×D4712yes—
406C22×Q8—22no—
407C22×Q8—21no—
408C22×Q8—21no—
409C22×Q8—21no—
410C22×Q8—22no—
411C22×Q8—21no—
412C22×Q8—21no—
413C22×Q8—22no—
414C22×Q8C22⋊D4711yescharacteristic
415C22×Q8C22⋊D4714yescomplemented
416C22×Q8C22×D4712yes—
417C22×D4—21no—
418C22×D4—472no—
419C22×D4—472no—
420C22×D4—944no—
421C22×D4—941no—
422C22×D4—942no—
423C22×D4—941no—
424C22×D4—942no—
425C22×D4—942no—
426C22×D4—941no—
427C22×D4—941no—
428C22×D4—942no—
429C22×D4C22⋊D4711yescharacteristic
430C23×C4—21no—
431C23×C4—21no—
432C23×C4—941no—
433C23×C4C22×D4711yescharacteristic
434#137—216no—
435#137—28no—
436#137—28no—
437C2×Q16—28no—
438C2×Q16—24no—
439C2×SD16—28no—
440C2×SD16—24no—
441C2×M16—22no—
442C2×M16—21no—
443C2×M16—472no—
444C2×M16—941no—
445C2×M16—942no—
446C2×M16—942no—
447C2×M16C22⋊D4714yescomplemented
448C2×M16C22×D4712yes—
449C22×C8—21no—
450C22×C8—941no—
451#127—944no—
452#127—944no—
453#124—948no—
454#124—9416no—
455#124—944no—
456#124—944no—
457#124—948no—
458#121—948no—
459#121—948no—
460#120—948no—
461#120—948no—
462#117—472no—
463#117—472no—
464#117—941no—
465#117—941no—
466C2×C4⋊C4—472no—
467C2×C22⋊C4—948no—
468C2×C22⋊C4—9416no—
469C2×C22⋊C4—944no—
470C2×C22⋊C4—941no—
471C2×C42—474no—
472C2×C42—472no—
473C2×C42—941no—
474C2×C42—941no—
475C2×C42—942no—
476#104—948no—
477#103—948no—
478#100—948no—
479C24—21no—
480C24—474no—
481C24—942no—
482C24—944no—
483C24—944no—
484C24—942no—
485C24—1888no—
486C24D4×D4711yescharacteristic
487#41—24no—
488#41—24no—
489#41—22no—
490#41—22no—
491#41—22no—
492#41—22no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—42no—
497C2×Q8—24no—
498C2×Q8—21no—
499C2×Q8—22no—
500C2×Q8—24no—
501C2×Q8—22no—
502C2×Q8—24no—
503C2×Q8—24no—
504C2×Q8—24no—
505C2×Q8—22no—
506C2×Q8—22no—
507C2×Q8—22no—
508C2×Q8—22no—
509C2×Q8—22no—
510C2×Q8—41no—
511C2×Q8—41no—
512C2×Q8—42no—
513C2×Q8—42no—
514C2×Q8—42no—
515C2×Q8—42no—
516C2×Q8—42no—
517C2×Q8C2×C22⋊D4712yes—
518C2×Q8C2×C22⋊D4714yescomplemented
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—21no—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—42no—
525C2×D4—42no—
526C2×D4—9416no—
527C2×D4—942no—
528C2×D4—944no—
529C2×D4—942no—
530C2×D4—944no—
531C2×D4—942no—
532C2×D4—942no—
533C2×D4—942no—
534C2×D4—942no—
535C2×D4—942no—
536C2×D4—942no—
537C2×D4—3768no—
538C2×D4—1882no—
539C2×D4—1882no—
540C2×D4—1882no—
541C2×D4—1882no—
542C2×D4—1882no—
543C2×D4—1882no—
544C2×D4—1882no—
545C2×D4—1882no—
546C2×D4C2×C22⋊D4712yes—
547C22×C4—24no—
548C22×C4—21no—
549C22×C4—22no—
550C22×C4—22no—
551C22×C4—22no—
552C22×C4—41no—
553C22×C4—41no—
554C22×C4—42no—
555C22×C4—42no—
556C22×C4—42no—
557C22×C4—944no—
558C22×C4—942no—
559C22×C4—942no—
560C22×C4—1884no—
561C22×C4D4×D4711yescharacteristic
562C22×C4D4×D4712yes—
563C22×C4D4×D4711yescharacteristic
564C22×C4D4×D4711yescharacteristic
565C22×C4#1062535212yes—
566C22×C4C2×C22⋊D4712yes—
567C22×C4C2×C22⋊D4711yescharacteristic
568C22×C4C2×C22⋊D4711yescharacteristic
569C22×C4C2×C22⋊D4712yes—
570C22×C4C23×D4711yescharacteristic
571Q16—216no—
572SD16—216no—
573M16—24no—
574M16—22no—
575M16—942no—
576M16—942no—
577M16—1882no—
578M16C2×C22⋊D4714yescomplemented
579C2×C8—24no—
580C2×C8—22no—
581C2×C8—942no—
582C2×C8—1882no—
583C4⋊C4—474no—
584C4⋊C4—942no—
585C22⋊C4—9432no—
586C22⋊C4—9416no—
587C22⋊C4—942no—
588C22⋊C4—942no—
589C42—474no—
590C42—944no—
591C42—942no—
592C42—942no—
593C42—942no—
594C23—24no—
595C23—22no—
596C23—42no—
597C23—44no—
598C23—944no—
599C23—944no—
600C23—944no—
601C23—3768no—
602C23—3764no—
603C23—18816no—
604C23—1888no—
605C23—1884no—
606C23—1884no—
607C23—1888no—
608C23#1309399211yescharacteristic
609C23#1309399212yes—
610C23#1309399211yescharacteristic
611C23#1309399512yes—
612C23C2×D4×D4711yescharacteristic
613Q8—24no—
614Q8—22no—
615Q8—24no—
616Q8—22no—
617Q8—44no—
618Q8—44no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622D4—22no—
623D4—22no—
624D4—44no—
625D4—42no—
626D4—42no—
627D4—42no—
628D4—37616no—
629D4—1884no—
630D4—1884no—
631D4—1882no—
632D4—1882no—
633D4—1882no—
634D4—1882no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—944no—
646C2×C4—944no—
647C2×C4—1884no—
648C2×C4—1888no—
649C2×C4—1884no—
650C2×C4#1309399211yescharacteristic
651C2×C4#1309399212yes—
652C2×C4#1309399211yescharacteristic
653C2×C4#1309399312yes—
654C2×C4#1309399312yes—
655C2×C4#1309399512yes—
656C2×C4#1309399612yes—
657C2×C4#1309399612yes—
658C2×C4#1309399612yes—
659C2×C4#1309399612yes—
660C2×C4#1309402012yes—
661C2×C4#1309402012yes—
662C2×C4C2×D4×D4711yescharacteristic
663C2×C4C2×D4×D4711yescharacteristic
664C2×C4C22×C22⋊D4711yescharacteristic
665C2×C4C22×C22⋊D4711yescharacteristic
666C2×C4C22×C22⋊D4711yescharacteristic
667C2×C4C2×#1062535211yescharacteristic
668C8—24no—
669C8—1882no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—24no—
674C22—44no—
675C22—42no—
676C22—84no—
677C22—3764no—
678C22—3768no—
679C22—3768no—
680C22—1888no—
681C22—1888no—
682C22—1884no—
683C22#1568903112yes—
684C22#1568903112yes—
685C22#1568929914yes—
686C22C2×#1309399211yescharacteristic
687C22C2×#1309399211yescharacteristic
688C22C2×#1309399511yescharacteristic
689C4—42no—
690C4—44no—
691C4—1884no—
692C4—1884no—
693C4#1568903112yes—
694C4#1568903112yes—
695C4C2×#1309399311yescharacteristic
696C4C2×#1309399611yescharacteristic
697C4C2×#1309399611yescharacteristic
698C4C2×#1309402011yescharacteristic
699C2—21no—
700C2—22no—
701C2—21no—
702C2—82no—
703C2—3764no—
704C2#2024441414yesdirect product factor
705C2C2×#1568903111yescharacteristic
706C2C2×#1568929912yes—
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47#120
prime radicalC2×#15689687C2
hypercenterC22×#137D47
hypocenterC47C2×#2004
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×#13094013C2semidirect
3. C22×#13093951C2semidirect
4. C2×#15689132C2semidirect
5. C22×#13093940C2semidirect
6. #20244414C2direct (N × H)
7. C2×#15688884C2semidirect
8. C22×#137×C47C2semidirect
20. C2×#13094013C22semidirect
27. C2×#13093951C22semidirect
29. #15689132C22semidirect
32. C2×#13093940C22semidirect
33. #15688884C22semidirect
39. C2×#137×C47C22semidirect
84. C2×C4×Dic47D4semidirect
106. C22×Q8×C47D4semidirect
122. C2×M16×C47D4semidirect
164. C2×Q8×C47C2×D4semidirect
221. M16×C47C2×D4semidirect
234. C4×Dic47C2×D4semidirect
346. C22×#137D47semidirect
358. C2×C47#2004semidirect
368. C2×#137C2×D47semidirect
396. C47C2×#2004semidirect
415. C22×Q8C22⋊D47semidirect
447. C2×M16C22⋊D47semidirect
518. C2×Q8C2×C22⋊D47semidirect
578. M16C2×C22⋊D47semidirect
704. C2#20244414direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896872
C22×#130940132
C22×#130939512
C2×#156891322
C22×#130939402
#202444142
C2×#156888842
C22×#137×C472
C2×#200447