Finite Group Databasebeta

← C2×#20243735 #26217601

Subgroups — C2×#20243735 #26217601

Order 12032.

2470 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D376C211yesmaximal, characteristic
2C22×C4×D188C211yesmaximal, characteristic, complemented
3C22×#13093946C211yesmaximal, characteristic, complemented
4C22×#13093946C211yesmaximal, characteristic, complemented
5C2×#15689025C211yesmaximal, characteristic, complemented
6#20243735C218yesmaximal, direct product factor
7C2×#15688886C211yesmaximal, characteristic, complemented
8C2×C47×#336C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C23×D188C2211yescharacteristic
11C23×C4×D47—21no—
12C22×D376—24no—
13C22×D376—24no—
14C22×D376C418yescomplemented
15C22×D376C2214yes—
16C2×C4×D188—24no—
17C2×C4×D188—24no—
18C2×C4×D188—22no—
19C2×C4×D188C2214yescomplemented
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×(C2×D47)⋊C4—21no—
23C22×(C2×D47)⋊C4—21no—
24C22×(C2×D47)⋊C4—21no—
25C2×#13093946—24no—
26C2×#13093946—24no—
27C2×#13093946C2214yescomplemented
28C2×#13093946C2214yescomplemented
29#15689025—22no—
30#15689025C2214yescomplemented
31#15688886C2214yescomplemented
32C23×C8×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C47×#336C2214yescomplemented
36C24×D47—21no—
37C24×D47—21no—
38C22×D188—21no—
39C22×D188—22no—
40C22×D188—22no—
41C22×D188—22no—
42C22×D188—22no—
43C22×D188—22no—
44C22×D188—22no—
45C22×D188—22no—
46C22×D188—21no—
47C22×D188—22no—
48C22×D188C2×C414yes—
49C22×D188C2×C414yescomplemented
50C22×D188D414yes—
51C22×D188D414yes—
52C22×D188C2312yes—
53C22×D188C2312yes—
54C23×Dic47—21no—
55C23×Dic47—21no—
56C22×C4×D47—28no—
57C22×C4×D47—24no—
58C22×C4×D47—44no—
59C22×C4×D47—44no—
60C2×D376—232no—
61C2×D376—28no—
62C2×D376—28no—
63C2×D376C2×C4116yescomplemented
64C4×D188—216no—
65C4×D188—28no—
66C4×D188—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—22no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—22no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47D414yes—
73C2×C4⋊Dic47C2312yes—
74C2×C4⋊Dic47C2312yes—
75C2×(C2×D47)⋊C4—28no—
76C2×(C2×D47)⋊C4—24no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4—44no—
80C2×(C2×D47)⋊C4—44no—
81#13093946—28no—
82#13093946—28no—
83C23×C4×C47D411yescharacteristic
84C23×C4×C47D411yescharacteristic
85C23×C4×C47C2311yescharacteristic
86C22×C8×C47—22no—
87C22×C8×C47—22no—
88C22×C8×C47—22no—
89C22×C8×C47—22no—
90C22×C8×C47—22no—
91C22×C8×C47C2×C414yescomplemented
92C22×C8×C47C2312yes—
93C22×C8×C47C2312yes—
94C2×C42×C47—22no—
95C2×C42×C47—22no—
96C2×C42×C47D412yes—
97C2×C42×C47C2312yes—
98C24×C47C2×D411yescharacteristic
99C23×D47—24no—
100C23×D47—24no—
101C23×D47—24no—
102C23×D47—22no—
103C23×D47—24no—
104C23×D47—22no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C23×D47—42no—
111C23×D47—42no—
112C23×D47—42no—
113C23×D47—42no—
114C23×D47—42no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47D814yescomplemented
121C22×C4×C47C22×C412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47#4112yes—
129C22×C4×C47#4112yes—
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—24no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—42no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C2×D188—28no—
140C2×D188—24no—
141C2×D188—24no—
142C2×D188—28no—
143C2×D188—24no—
144C2×D188—28no—
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—22no—
148C2×D188—28no—
149C2×D188—28no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—24no—
155C2×D188—22no—
156C2×D188—42no—
157C2×D188—44no—
158C2×D188—42no—
159C2×D188—44no—
160C2×D188C22⋊C4116yescomplemented
161C2×D188C22×C414yes—
162C2×D188C22×C414yescomplemented
163C2×D188C2×D414yes—
164C2×D188C2×D414yes—
165C2×C4×D47—216no—
166C2×C4×D47—48no—
167C2×C4×D47—432no—
168C2×C4×D47—48no—
169C2×C8×C47—28no—
170C2×C8×C47—22no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—44no—
176C2×C8×C47C22×C414yescomplemented
177C42×C47—24no—
178C42×C47—22no—
179C42×C47—22no—
180(C2×D47)⋊C4—216no—
181(C2×D47)⋊C4—48no—
182(C2×D47)⋊C4—48no—
183C4⋊Dic47—24no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187C4⋊Dic47—42no—
188C4⋊Dic47C2×D414yes—
189D376—232no—
190C23×C47C4×D412yes—
191C23×C47#12012yes—
192C23×C47#12112yes—
193C23×C47C2×D812yes—
194C23×C47#13512yes—
195C23×C47AGL(1,ℤ/8)12yes—
196C23×C47AGL(1,ℤ/8)12yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—216no—
202C22×D47—24no—
203C22×D47—42no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—48no—
208C22×D47—48no—
209C22×D47—48no—
210C22×D47—48no—
211C22×D47—48no—
212C22×D47—48no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—48no—
220C22×D47—48no—
221C22×D47—48no—
222C22×D47—84no—
223C22×D47—84no—
224C22×D47—84no—
225C22×D47—84no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—28no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—44no—
233C2×C4×C47C2×C22⋊C414yes—
234C2×C4×C47C4×D414yes—
235C2×C4×C47C4×D412yes—
236C2×C4×C47#12012yes—
237C2×C4×C47#12114yes—
238C2×C4×C47#12112yes—
239C2×C4×C47#12314yes—
240C2×C4×C47C2×D814yescomplemented
241C2×C4×C47C23×C411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×Dic47—24no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—48no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—84no—
255D188—24no—
256D188—24no—
257D188—24no—
258D188—24no—
259D188—24no—
260D188—216no—
261D188—42no—
262D188—42no—
263D188—48no—
264D188C2×C22⋊C418yescomplemented
265C4×D47—432no—
266C8×C47—24no—
267C8×C47—42no—
268C2×#1879—471nomaximal, Sylow
269C22×C47#38614yes—
270C22×C47C4×D814yes—
271C22×C47#44214yes—
272C22×C47#44714yes—
273C22×C47#44914yes—
274C22×C47#45914yes—
275C22×C47#46314yes—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×#12111yescharacteristic
279C22×C47C22×D811yescharacteristic
280C22×C47C2×#13511yescharacteristic
281C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
283C2×D47—28no—
284C2×D47—432no—
285C2×D47—44no—
286C2×D47—44no—
287C2×D47—44no—
288C2×D47—44no—
289C2×D47—44no—
290C2×D47—44no—
291C2×D47—44no—
292C2×D47—44no—
293C2×D47—44no—
294C2×D47—816no—
295C2×D47—82no—
296C2×D47—82no—
297C2×D47—82no—
298C2×D47—82no—
299C4×C47—24no—
300C4×C47—42no—
301C4×C47#38614yes—
302C4×C47C22×C22⋊C411yescharacteristic
303C4×C47C2×C4×D411yescharacteristic
304C4×C47C2×#12111yescharacteristic
305C4×C47C2×#12311yescharacteristic
306Dic47—44no—
307Dic47—82no—
308C23×D8—471no—
309C22×C4×D4—471no—
310C22×#102—471no—
311C22×#102—471no—
312C2×#390—471no—
313#1879—478no—
314C2×#340—471no—
315C2×#336D4711yescharacteristic, complemented
316C2×C47#187918yescomplemented
317C2×C47C2×#38611yescharacteristic
318C2×C47C2×C4×D811yescharacteristic
319C2×C47C2×#44211yescharacteristic
320C2×C47C2×#44711yescharacteristic
321C2×C47C2×#44911yescharacteristic
322C2×C47C2×#45911yescharacteristic
323C2×C47C2×#46311yescharacteristic
324D47—48no—
325D47—84no—
326C23×D4—471no—
327C23×D4—471no—
328C24×C4—941no—
329C22×D8—478no—
330C22×D8—474no—
331C22×D8—944no—
332C22×D8—944no—
333C23×C8C2×D4711yescharacteristic
334C23×C8C2×D4711yescharacteristic
335C2×C4×D4—474no—
336C2×C4×D4—944no—
337C2×C4×D4—944no—
338C2×C4×D4—942no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—471no—
341C22×C22⋊C4—941no—
342C22×C22⋊C4—941no—
343C22×C22⋊C4—941no—
344C22×C42C2×D4711yescharacteristic
345C2×#102—474no—
346C2×#102—474no—
347C2×#102—944no—
348C2×#102—944no—
349#390—474no—
350#390—942no—
351#340—474no—
352#336C2×D4714yescomplemented
353C47C2×#187911yescharacteristic, complemented, Sylow
354C25—941no—
355C25—941no—
356C22×D4—474no—
357C22×D4—474no—
358C22×D4—474no—
359C22×D4—474no—
360C22×D4—472no—
361C22×D4—472no—
362C22×D4—941no—
363C22×D4—941no—
364C22×D4—942no—
365C22×D4—942no—
366C22×D4—942no—
367C22×D4—942no—
368C22×D4—942no—
369C22×D4—942no—
370C22×D4—942no—
371C22×D4—942no—
372C23×C4—948no—
373C23×C4—941no—
374C23×C4—944no—
375C23×C4—941no—
376C23×C4—1884no—
377C23×C4—1884no—
378C23×C4D18811yescharacteristic
379C23×C4D18811yescharacteristic
380C23×C4C22×D4711yescharacteristic
381C2×D8—4716no—
382C2×D8—9432no—
383C2×D8—948no—
384C2×D8—948no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—22no—
390C22×C8C4×D4714yescomplemented
391C22×C8C22×D4712yes—
392C22×C8C22×D4712yes—
393C4×D4—948no—
394C4×D4—948no—
395C4×D4—9416no—
396C2×C4⋊C4—474no—
397C2×C4⋊C4—472no—
398C2×C4⋊C4—472no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—942no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4—941no—
404C2×C22⋊C4—948no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—944no—
408C2×C22⋊C4—1884no—
409C2×C22⋊C4—1884no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42D18812yes—
413C2×C42C22×D4712yes—
414#102—948no—
415#102—948no—
416C24—944no—
417C24—944no—
418C24—944no—
419C24—942no—
420C24—944no—
421C24—942no—
422C24—1882no—
423C24—1882no—
424C24—1882no—
425C24—1882no—
426C24—1882no—
427C24—1882no—
428C24—1882no—
429C24—1882no—
430C24—1882no—
431C24—1882no—
432C24C2×D18811yescharacteristic
433C2×D4—4716no—
434C2×D4—474no—
435C2×D4—474no—
436C2×D4—474no—
437C2×D4—474no—
438C2×D4—948no—
439C2×D4—948no—
440C2×D4—948no—
441C2×D4—944no—
442C2×D4—944no—
443C2×D4—944no—
444C2×D4—948no—
445C2×D4—944no—
446C2×D4—948no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—942no—
451C2×D4—942no—
452C2×D4—942no—
453C2×D4—942no—
454C2×D4—942no—
455C2×D4—1882no—
456C2×D4—1884no—
457C2×D4—1882no—
458C2×D4—1884no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—9416no—
465C22×C4—944no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—18832no—
469C22×C4—1888no—
470C22×C4—1882no—
471C22×C4—1882no—
472C22×C4—1882no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4—1888no—
476C22×C4D37614yescomplemented
477C22×C4C2×C4×D4712yes—
478C22×C4C2×D18812yes—
479C22×C4C2×D18812yes—
480C22×C4C2×D18811yescharacteristic
481C22×C4#1062535012yes—
482C22×C4D4×D4712yes—
483C22×C4D4×D4712yes—
484C22×C4D4×D4712yes—
485C22×C4#1062535412yes—
486C22×C4C23×D4711yescharacteristic
487D8—9432no—
488C2×C8—28no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—44no—
495C2×C8C2×C4×D4714yescomplemented
496C4⋊C4—474no—
497C4⋊C4—944no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—942no—
501C4⋊C4—1882no—
502C22⋊C4—9416no—
503C22⋊C4—1888no—
504C22⋊C4—1888no—
505C42—24no—
506C42—22no—
507C42—22no—
508C23—9416no—
509C23—944no—
510C23—944no—
511C23—944no—
512C23—944no—
513C23—3764no—
514C23—3764no—
515C23—3764no—
516C23—3764no—
517C23—1888no—
518C23—1888no—
519C23—1882no—
520C23—1882no—
521C23—1882no—
522C23—1882no—
523C23—1882no—
524C23—1882no—
525C23—1888no—
526C23—1888no—
527C23—1888no—
528C23—1888no—
529C23—1888no—
530C23—1888no—
531C23—1882no—
532C23—1882no—
533C23—1882no—
534C23—1882no—
535C23—1888no—
536C23C4×D18812yes—
537C23#1309399412yes—
538C23#1309399912yes—
539C23C2×D37612yes—
540C23#1309404612yes—
541C23#1309405212yes—
542C23#1309405212yes—
543C23C22×D18811yescharacteristic
544D4—478no—
545D4—9416no—
546D4—944no—
547D4—944no—
548D4—944no—
549D4—944no—
550D4—944no—
551D4—1888no—
552D4—1882no—
553D4—1882no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—944no—
562C2×C4—3764no—
563C2×C4—18832no—
564C2×C4—1882no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4—1888no—
568C2×C4—1882no—
569C2×C4—1882no—
570C2×C4C22⋊C4×D4714yes—
571C2×C4#1309398614yes—
572C2×C4C4×D18812yes—
573C2×C4#1309399412yes—
574C2×C4#1309399714yes—
575C2×C4#1309399912yes—
576C2×C4#1309400914yes—
577C2×C4C2×D37614yescomplemented
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C2×D4×D4711yescharacteristic
580C2×C4C2×D4×D4711yescharacteristic
581C2×C4C2×D4×D4711yescharacteristic
582C2×C4C2×#1062535011yescharacteristic
583C2×C4C2×#1062535411yescharacteristic
584C8—24no—
585C8—42no—
586C22—948no—
587C22—37616no—
588C22—3762no—
589C22—3762no—
590C22—3762no—
591C22—3762no—
592C22—1884no—
593C22—1884no—
594C22—1884no—
595C22—1884no—
596C22—18832no—
597C22—1884no—
598C22—1884no—
599C22—1884no—
600C22—1884no—
601C22—1884no—
602C22#1568900114yes—
603C22C4×D37614yes—
604C22#1568923614yes—
605C22#1568925914yes—
606C22#1568927014yes—
607C22#1568931914yes—
608C22#1568933914yes—
609C22C2×C4×D18811yescharacteristic
610C22C2×#1309399411yescharacteristic
611C22C2×#1309399911yescharacteristic
612C22C22×D37611yescharacteristic
613C22C2×#1309404611yescharacteristic
614C22C2×#1309405211yescharacteristic
615C22C2×#1309405211yescharacteristic
616C4—24no—
617C4—42no—
618C4—3762no—
619C4—1884no—
620C4#1568900114yes—
621C4C2×C22⋊C4×D4711yescharacteristic
622C4C2×#1309398611yescharacteristic
623C4C2×#1309399711yescharacteristic
624C4C2×#1309400911yescharacteristic
625C2—3764no—
626C2—1888no—
627C2#2024373518yesdirect product factor
628C2C2×#1568900111yescharacteristic
629C2C2×C4×D37611yescharacteristic
630C2C2×#1568923611yescharacteristic
631C2C2×#1568925911yescharacteristic
632C2C2×#1568927011yescharacteristic
633C2C2×#1568931911yescharacteristic
634C2C2×#1568933911yescharacteristic
subgroupG/H
centerC24C2×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#336C2
socleC24×C47C2×D4
prime radicalC23×D376C2
cosocleC22×C4×C47C24
hypercenterC2×#336D47
hypocenterC47C2×#1879
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
2. C22×C4×D188C2semidirect
3. C22×#13093946C2semidirect
4. C22×#13093946C2semidirect
5. C2×#15689025C2semidirect
6. #20243735C2direct (N × H)
7. C2×#15688886C2semidirect
8. C2×C47×#336C2semidirect
14. C22×D376C4semidirect
19. C2×C4×D188C22semidirect
27. C2×#13093946C22semidirect
28. C2×#13093946C22semidirect
30. #15689025C22semidirect
31. #15688886C22semidirect
35. C47×#336C22semidirect
49. C22×D188C2×C4semidirect
63. C2×D376C2×C4semidirect
91. C22×C8×C47C2×C4semidirect
120. C22×C4×C47D8semidirect
160. C2×D188C22⋊C4semidirect
162. C2×D188C22×C4semidirect
176. C2×C8×C47C22×C4semidirect
240. C2×C4×C47C2×D8semidirect
264. D188C2×C22⋊C4semidirect
315. C2×#336D47semidirect
316. C2×C47#1879semidirect
352. #336C2×D47semidirect
353. C47C2×#1879semidirect
390. C22×C8C4×D47semidirect
476. C22×C4D376semidirect
495. C2×C8C2×C4×D47semidirect
577. C2×C4C2×D376semidirect
627. C2#20243735direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3762
C22×C4×D1882
C22×#130939462
C2×#156890252
#202437352
C2×#156888862
C2×C47×#3362
C2×#187947