Finite Group Databasebeta

← C2×#24910971 #35572275

Subgroups — C2×#24910971 #35572275

Order 21248.

3830 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581775C212yesmaximal, complemented
2C22×#14581763C211yesmaximal, characteristic
3#24910971C2116yesmaximal, direct product factor
4C2×#19127629C212yesmaximal, complemented
5C2×#19127616C212yesmaximal, complemented
6C2×#19127604C211yesmaximal, characteristic, complemented
7C2×#19127599C211yesmaximal, characteristic, complemented
8C2×#19127574C212yesmaximal, complemented
9C2×#19127570C211yesmaximal, characteristic, complemented
10C2×#19127558C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
12C22×C4×D4×C83C211yesmaximal, characteristic, complemented
13C23×C22⋊D83C2212yes—
14C23×C4×D83C2211yescharacteristic
15C2×#14581775—216no—
16C2×#14581775C22116yescomplemented
17C2×#14581763—28no—
18C2×#14581763C2218yes—
19C22×C4⋊Dic83C2211yescharacteristic
20C22×C4⋊Dic83C2211yescharacteristic
21C22×Dic83⋊C4C2212yes—
22C22×Dic83⋊C4C2212yes—
23C22×Dic83⋊C4C2212yes—
24C22×C22⋊Dic83C2212yes—
25C22×C22⋊Dic83C2212yes—
26C22×C22⋊Dic83C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2211yescharacteristic
29C22×(C2×D83)⋊C4C2211yescharacteristic
30C22×C4×Dic83C2211yescharacteristic
31#19127629C22116yescomplemented
32#19127616C22116yescomplemented
33#19127604C2218yescomplemented
34#19127599C2218yescomplemented
35#19127574C22116yescomplemented
36#19127570C2218yescomplemented
37#19127558C2218yescomplemented
38C4×(C2×D83)⋊C4C2218yescomplemented
39C2×C83⋊#95C2212yes—
40C2×C83⋊#95C2211yescharacteristic
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2211yescharacteristic
43C23×D4×C83C2211yescharacteristic
44C24×C4×C83C2212yes—
45C2×C4×D4×C83—28no—
46C2×C4×D4×C83C2218yescomplemented
47C22×C4⋊C4×C83C2211yescharacteristic
48C22×C22⋊C4×C83C2212yes—
49C22×C42×C83C2211yescharacteristic
50C24×D83C2311yescharacteristic
51C22×C22⋊D83—28no—
52C22×C22⋊D83—28no—
53C22×C22⋊D83—44no—
54C22×C22⋊D83C2318yes—
55C23×Dic83C2312yes—
56C23×Dic83C2311yescharacteristic
57C23×Dic83C2312yes—
58C23×Dic83C2311yescharacteristic
59C23×Dic83C2311yescharacteristic
60C22×C4×D83—24no—
61C22×C4×D83—24no—
62C22×C4×D83—24no—
63C22×C4×D83C2314yes—
64#14581775—264no—
65#14581763—232no—
66C2×C4⋊Dic83—24no—
67C2×C4⋊Dic83—24no—
68C2×C4⋊Dic83C2314yes—
69C2×C4⋊Dic83C2314yes—
70C2×Dic83⋊C4—28no—
71C2×Dic83⋊C4—28no—
72C2×Dic83⋊C4—44no—
73C2×Dic83⋊C4C2318yes—
74C2×Dic83⋊C4C2318yes—
75C2×Dic83⋊C4C2318yes—
76C2×C22⋊Dic83—28no—
77C2×C22⋊Dic83—28no—
78C2×C22⋊Dic83—28no—
79C2×C22⋊Dic83C2318yes—
80C2×C22⋊Dic83C2318yes—
81C2×C22⋊Dic83C2318yes—
82C2×(C2×D83)⋊C4—28no—
83C2×(C2×D83)⋊C4—24no—
84C2×(C2×D83)⋊C4D418yescomplemented
85C2×(C2×D83)⋊C4C2318yes—
86C2×(C2×D83)⋊C4C2314yes—
87C2×(C2×D83)⋊C4C2314yes—
88C2×C4×Dic83—24no—
89C2×C4×Dic83C2314yes—
90C83⋊#95C2318yes—
91C83⋊#95C2314yes—
92C83⋊#95C2318yes—
93C83⋊#95C2314yes—
94C25×C83C2312yes—
95C22×D4×C83—24no—
96C22×D4×C83—24no—
97C22×D4×C83D418yes—
98C22×D4×C83C2314yes—
99C23×C4×C83—28no—
100C23×C4×C83—28no—
101C23×C4×C83—44no—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2312yes—
104C23×C4×C83C2318yes—
105C23×C4×C83C2311yescharacteristic
106C23×C4×C83C2311yescharacteristic
107C4×D4×C83—232no—
108C2×C4⋊C4×C83—24no—
109C2×C4⋊C4×C83C2314yes—
110C2×C22⋊C4×C83—28no—
111C2×C22⋊C4×C83C2318yes—
112C2×C42×C83—24no—
113C2×C42×C83C2314yes—
114C23×D83—22no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—22no—
119C23×D83—22no—
120C23×D83C2×D414yes—
121C23×D83C2412yes—
122C2×C22⋊D83—216no—
123C2×C22⋊D83—216no—
124C2×C22⋊D83—48no—
125C2×C22⋊D83—432no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—24no—
132C22×Dic83—24no—
133C22×Dic83—24no—
134C22×Dic83—22no—
135C22×Dic83—22no—
136C22×Dic83—24no—
137C22×Dic83—24no—
138C22×Dic83—24no—
139C22×Dic83C2×D414yes—
140C22×Dic83#4114yes—
141C22×Dic83C2412yes—
142C22×Dic83C2414yes—
143C22×Dic83C2414yes—
144C22×Dic83C2412yes—
145C22×Dic83C2412yes—
146C2×C4×D83—28no—
147C2×C4×D83—28no—
148C2×C4×D83—28no—
149C2×C4×D83—416no—
150C4⋊Dic83—28no—
151C4⋊Dic83—28no—
152Dic83⋊C4—216no—
153Dic83⋊C4—216no—
154Dic83⋊C4—48no—
155C22⋊Dic83—216no—
156C22⋊Dic83—216no—
157C22⋊Dic83—216no—
158(C2×D83)⋊C4—216no—
159(C2×D83)⋊C4—28no—
160(C2×D83)⋊C4C2×D4116yescomplemented
161C4×Dic83—28no—
162C24×C83—24no—
163C24×C83—24no—
164C24×C83—24no—
165C24×C83—24no—
166C24×C83—24no—
167C24×C83—24no—
168C24×C83C2×D418yes—
169C24×C83C2414yes—
170C24×C83C2411yescharacteristic
171C2×D4×C83—232no—
172C2×D4×C83—28no—
173C2×D4×C83—28no—
174C2×D4×C83C2×D4116yes—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—24no—
179C22×C4×C83—216no—
180C22×C4×C83—24no—
181C22×C4×C83—22no—
182C22×C4×C83—24no—
183C22×C4×C83—22no—
184C22×C4×C83—216no—
185C22×C4×C83—48no—
186C22×C4×C83—432no—
187C22×C4×C83C2×D414yes—
188C22×C4×C83C2×D414yes—
189C22×C4×C83#4114yes—
190C22×C4×C83#4114yes—
191C22×C4×C83C2412yes—
192C22×C4×C83C2414yes—
193C22×C4×C83C2412yes—
194C22×C4×C83C2412yes—
195C4⋊C4×C83—28no—
196C22⋊C4×C83—216no—
197C42×C83—28no—
198C23×C83—216no—
199C23×C83—216no—
200C23×C83—216no—
201C23×C83—24no—
202C23×C83—24no—
203C23×C83—24no—
204C23×C83—24no—
205C23×C83—24no—
206C23×C83—24no—
207C23×C83—48no—
208C23×C83—48no—
209C23×C83—48no—
210C23×C83—48no—
211C23×C83C22×D412yes—
212C23×C83C22×D418yes—
213C23×C83C22×D412yes—
214C23×C83C2×#4112yes—
215C23×C83C2×#4112yes—
216C23×C83C2×#4112yes—
217C23×C83H(5,𝔽2)12yes—
218C23×C832−1+412yes—
219C23×C83C2511yescharacteristic
220C22×D83—28no—
221C22×D83—28no—
222C22×D83—28no—
223C22×D83—22no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—22no—
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—44no—
230C22×D83—44no—
231C22×D83—44no—
232C22×D83—44no—
233C22×D83C22×D414yes—
234D4×C83—232no—
235C2×C4×C83—28no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—22no—
242C2×C4×C83—24no—
243C2×C4×C83—22no—
244C2×C4×C83—432no—
245C2×C4×C83—48no—
246C2×C4×C83—44no—
247C2×C4×C83#123116yescomplemented
248C2×C4×C83C22×D414yes—
249C2×C4×C83C22×D414yes—
250C2×C4×C83C2×#4114yes—
251C2×C4×C83C2×#4114yes—
252C22⋊D83—432no—
253C2×Dic83—28no—
254C2×Dic83—28no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—24no—
260C2×Dic83—22no—
261C2×Dic83—24no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—24no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—24no—
268C2×Dic83—44no—
269C2×Dic83—48no—
270C2×Dic83—48no—
271C2×Dic83C22×D414yes—
272C2×Dic83C2×#4114yes—
273C4×D83—416no—
274C22×C83—28no—
275C22×C83—28no—
276C22×C83—28no—
277C22×C83—44no—
278C22×C83—432no—
279C22×C83—44no—
280C22×C83—44no—
281C22×C83—44no—
282C22×C83C2×#12314yes—
283C22×C83#53214yes—
284C22×C83#54214yes—
285C22×C83D4214yes—
286C22×C83#54714yes—
287C22×C83#54714yes—
288C22×C83#55314yes—
289C22×C83C23×D411yescharacteristic
290C22×C83C23×D411yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C22×#4111yescharacteristic
293C22×C83C22×#4111yescharacteristic
294C22×C83C2×H(5,𝔽2)11yescharacteristic
295C22×C83C2×2−1+411yescharacteristic
296C2×D83—24no—
297C2×D83—24no—
298C2×D83—24no—
299C2×D83—42no—
300C2×D83—416no—
301C2×D83—42no—
302C2×D83—42no—
303C2×D83—42no—
304C4×C83—24no—
305C4×C83—44no—
306C4×C83—42no—
307C4×C83C2×#12318yescomplemented
308Dic83—24no—
309Dic83—24no—
310Dic83—42no—
311Dic83—44no—
312Dic83—44no—
313C2×#2422—831nomaximal, Sylow
314C2×C83—48no—
315C2×C83#242218yescomplemented
316C2×C83C22×#12311yescharacteristic
317C2×C83C2×#53211yescharacteristic
318C2×C83C2×#54211yescharacteristic
319C2×C83C2×D4211yescharacteristic
320C2×C83C2×#54711yescharacteristic
321C2×C83C2×#54711yescharacteristic
322C2×C83C2×#55311yescharacteristic
323D83—44no—
324C22×#123—832no—
325C22×#121—831no—
326C22×C4×D4D8311yescharacteristic, complemented
327#2422—8316no—
328C2×#397—832no—
329C2×#394—832no—
330C2×#390—831no—
331C2×#389—831no—
332C2×#385—832no—
333C2×#384—831no—
334C2×#380—831no—
335C2×C4×C22⋊C4—831no—
336C83C2×#242211yescharacteristic, complemented, Sylow
337C23×D4—832no—
338C23×D4C2×D8311yescharacteristic
339C24×C4—831no—
340C24×C4C2×D8312yes—
341C2×#123—8316no—
342C2×#123—16616no—
343C2×#121—838no—
344C2×#121—1668no—
345C2×C4×D4—28no—
346C2×C4×D4C2×D8318yescomplemented
347C22×C4⋊C4—832no—
348C22×C4⋊C4—832no—
349C22×C4⋊C4—832no—
350C22×C4⋊C4—831no—
351C22×C4⋊C4—831no—
352C22×C4⋊C4C2×D8311yescharacteristic
353C22×C22⋊C4—831no—
354C22×C22⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4—832no—
357C22×C22⋊C4—831no—
358C22×C22⋊C4—832no—
359C22×C22⋊C4C2×D8312yes—
360C22×C42—831no—
361C22×C42C2×D8311yescharacteristic
362#397—8316no—
363#394—8316no—
364#390—838no—
365#389—838no—
366#385—8316no—
367#384—838no—
368#380—838no—
369C4×C22⋊C4—838no—
370C2×#95—832no—
371C2×#95—831no—
372C2×#95—832no—
373C2×#95—831no—
374C25—831no—
375C25C22×D8312yes—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—3324no—
379C22×D4—838no—
380C22×D4—1668no—
381C22×D4—1668no—
382C22×D4C22⋊D8318yes—
383C22×D4C22×D8314yes—
384C23×C4—28no—
385C23×C4—28no—
386C23×C4—44no—
387C23×C4—831no—
388C23×C4—831no—
389C23×C4—832no—
390C23×C4—831no—
391C23×C4—832no—
392C23×C4—834no—
393C23×C4—1664no—
394C23×C4—1664no—
395C23×C4—1664no—
396C23×C4C22×D8311yescharacteristic
397C23×C4C22×D8312yes—
398C23×C4C22×D8318yes—
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401#123—16664no—
402#121—16632no—
403C4×D4—232no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—3324no—
406C2×C4⋊C4—838no—
407C2×C4⋊C4—834no—
408C2×C4⋊C4—838no—
409C2×C4⋊C4—838no—
410C2×C4⋊C4—834no—
411C2×C4⋊C4—1668no—
412C2×C4⋊C4—1668no—
413C2×C4⋊C4—1664no—
414C2×C4⋊C4—1664no—
415C2×C4⋊C4C22×D8314yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—838no—
418C2×C22⋊C4—834no—
419C2×C22⋊C4—838no—
420C2×C22⋊C4—838no—
421C2×C22⋊C4—838no—
422C2×C22⋊C4—838no—
423C2×C22⋊C4—834no—
424C2×C22⋊C4—1664no—
425C2×C22⋊C4—1668no—
426C2×C22⋊C4—1668no—
427C2×C22⋊C4—1668no—
428C2×C22⋊C4—1668no—
429C2×C22⋊C4C22×D8318yes—
430C2×C42—24no—
431C2×C42—834no—
432C2×C42—1664no—
433C2×C42C22×D8314yes—
434#95—834no—
435#95—834no—
436#95—838no—
437#95—838no—
438C24—24no—
439C24—24no—
440C24—24no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—834no—
445C24—832no—
446C24—1662no—
447C24—1662no—
448C24—1662no—
449C24—1662no—
450C24—1662no—
451C24—1662no—
452C24C2×C22⋊D8318yes—
453C24C23×D8314yes—
454C24C23×D8311yescharacteristic
455C2×D4—232no—
456C2×D4—28no—
457C2×D4—28no—
458C2×D4—33232no—
459C2×D4—3328no—
460C2×D4—16616no—
461C2×D4—16616no—
462C2×D4C2×C22⋊D83116yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—216no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—216no—
473C22×C4—48no—
474C22×C4—432no—
475C22×C4—33216no—
476C22×C4—834no—
477C22×C4—834no—
478C22×C4—832no—
479C22×C4—832no—
480C22×C4—832no—
481C22×C4—834no—
482C22×C4—834no—
483C22×C4—1664no—
484C22×C4—1664no—
485C22×C4—1662no—
486C22×C4—1664no—
487C22×C4—1664no—
488C22×C4—1664no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1664no—
492C22×C4—1668no—
493C22×C4—1668no—
494C22×C4—1668no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4D4×D8314yes—
500C22×C4C2×C22⋊D8314yes—
501C22×C4#1305059014yes—
502C22×C4#1305059014yes—
503C22×C4C23×D8312yes—
504C22×C4C23×D8314yes—
505C22×C4C23×D8312yes—
506C22×C4C23×D8312yes—
507C4⋊C4—28no—
508C4⋊C4—3328no—
509C4⋊C4—16616no—
510C4⋊C4—16616no—
511C4⋊C4—1668no—
512C4⋊C4—1668no—
513C22⋊C4—216no—
514C22⋊C4—8316no—
515C22⋊C4—1668no—
516C22⋊C4—16616no—
517C22⋊C4—16616no—
518C22⋊C4—16616no—
519C22⋊C4—16616no—
520C42—28no—
521C42—1668no—
522C23—216no—
523C23—216no—
524C23—216no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—3324no—
536C23—3324no—
537C23—3324no—
538C23—3324no—
539C23—834no—
540C23—1668no—
541C23—1668no—
542C23—1668no—
543C23—1662no—
544C23—1662no—
545C23—1662no—
546C23—1662no—
547C23—1662no—
548C23—1662no—
549C23C2×D4×D8312yes—
550C23C22×C22⋊D8318yes—
551C23C22×C22⋊D8312yes—
552C23#41×D8312yes—
553C23C2×#1305059012yes—
554C23C2×#1305059012yes—
555C23#1458183712yes—
556C23#1458183912yes—
557C23C24×D8311yescharacteristic
558D4—232no—
559D4—33232no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—432no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—3328no—
573C2×C4—3324no—
574C2×C4—3328no—
575C2×C4—33216no—
576C2×C4—834no—
577C2×C4—834no—
578C2×C4—1664no—
579C2×C4—1664no—
580C2×C4—1662no—
581C2×C4—1664no—
582C2×C4—1664no—
583C2×C4—1664no—
584C2×C4—1662no—
585C2×C4—1662no—
586C2×C4—1664no—
587C2×C4—1668no—
588C2×C4—1668no—
589C2×C4—1662no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4#14581775116yescomplemented
594C2×C4C2×D4×D8314yes—
595C2×C4C22×C22⋊D8314yes—
596C2×C4C2×#1305059014yes—
597C2×C4C2×#1305059014yes—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—44no—
602C22—44no—
603C22—432no—
604C22—44no—
605C22—44no—
606C22—3322no—
607C22—3322no—
608C22—3322no—
609C22—33216no—
610C22—3322no—
611C22—1664no—
612C22—1664no—
613C22—1664no—
614C22C2×#1458177514yes—
615C22#1912824814yes—
616C22#1912831614yes—
617C22D4×C22⋊D8314yes—
618C22#1912835014yes—
619C22#1912835614yes—
620C22#1912840114yes—
621C22C22×D4×D8311yescharacteristic
622C22C23×C22⋊D8311yescharacteristic
623C22C2×#41×D8311yescharacteristic
624C22C22×#1305059011yescharacteristic
625C22C22×#1305059011yescharacteristic
626C22C2×#1458183711yescharacteristic
627C22C2×#1458183911yescharacteristic
628C4—24no—
629C4—44no—
630C4—42no—
631C4—3324no—
632C4—3322no—
633C4—3324no—
634C4—1664no—
635C4—1664no—
636C4C2×#1458177518yescomplemented
637C2—48no—
638C2—3324no—
639C2#2491097118yesdirect product factor
640C2C22×#1458177511yescharacteristic
641C2C2×#1912824811yescharacteristic
642C2C2×#1912831611yescharacteristic
643C2C2×D4×C22⋊D8311yescharacteristic
644C2C2×#1912835011yescharacteristic
645C2C2×#1912835611yescharacteristic
646C2C2×#1912840111yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC22×C4×D4×C83C2
socleC24×C83C24
prime radicalC22×#14581763C2
hypercenterC22×C4×D4D83
hypocenterC83C2×#2422
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14581775C2semidirect
3. #24910971C2direct (N × H)
4. C2×#19127629C2semidirect
5. C2×#19127616C2semidirect
6. C2×#19127604C2semidirect
7. C2×#19127599C2semidirect
8. C2×#19127574C2semidirect
9. C2×#19127570C2semidirect
10. C2×#19127558C2semidirect
11. C2×C4×(C2×D83)⋊C4C2semidirect
12. C22×C4×D4×C83C2semidirect
16. C2×#14581775C22semidirect
31. #19127629C22semidirect
32. #19127616C22semidirect
33. #19127604C22semidirect
34. #19127599C22semidirect
35. #19127574C22semidirect
36. #19127570C22semidirect
37. #19127558C22semidirect
38. C4×(C2×D83)⋊C4C22semidirect
46. C2×C4×D4×C83C22semidirect
84. C2×(C2×D83)⋊C4D4semidirect
160. (C2×D83)⋊C4C2×D4semidirect
247. C2×C4×C83#123semidirect
307. C4×C83C2×#123semidirect
315. C2×C83#2422semidirect
326. C22×C4×D4D83semidirect
336. C83C2×#2422semidirect
346. C2×C4×D4C2×D83semidirect
593. C2×C4#14581775semidirect
636. C4C2×#14581775semidirect
639. C2#24910971direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817752
C22×#145817632
#249109712
C2×#191276292
C2×#191276162
C2×#191276042
C2×#191275992
C2×#191275742
C2×#191275702
C2×#191275582
C2×C4×(C2×D83)⋊C42
C22×C4×D4×C832
C2×#242283