Finite Group Databasebeta

← C2×#24910980 #35572282

Subgroups — C2×#24910980 #35572282

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×#14581777C212yesmaximal, complemented
2C22×#14581767C211yesmaximal, characteristic
3C22×#14581754C211yesmaximal, characteristic, complemented
4#24910980C2116yesmaximal, direct product factor
5C2×#19127627C212yesmaximal, complemented
6C2×#19127618C212yesmaximal, complemented
7C2×#19127603C211yesmaximal, characteristic, complemented
8C2×#19127597C211yesmaximal, characteristic, complemented
9C2×#19127572C212yesmaximal, complemented
10C2×#19127558C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
12C2×#384×C83C211yesmaximal, characteristic, complemented
13C23×D332C2212yes—
14C23×D332C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2212yes—
17C2×#14581777—216no—
18C2×#14581777C22116yescomplemented
19C2×#14581767—28no—
20C2×#14581767C2218yes—
21C2×#14581754—28no—
22C2×#14581754C2218yescomplemented
23C22×C4⋊Dic83C2211yescharacteristic
24C22×Dic83⋊C4C2212yes—
25C22×(C2×D83)⋊C4C2212yes—
26C22×(C2×D83)⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2211yescharacteristic
30C22×(C2×D83)⋊C4C2212yes—
31C22×(C2×D83)⋊C4C2211yescharacteristic
32C22×C4×Dic83C2211yescharacteristic
33#19127627C22116yescomplemented
34#19127618C22116yescomplemented
35#19127603C2218yescomplemented
36#19127597C2218yescomplemented
37#19127572C22116yescomplemented
38#19127558C2218yescomplemented
39C4×(C2×D83)⋊C4C2218yescomplemented
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2211yescharacteristic
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2212yes—
47C22×C4⋊C4×C83C2212yes—
48C22×C42×C83C2211yescharacteristic
49#384×C83C2218yescomplemented
50C24×D83C2311yescharacteristic
51C24×D83C2312yes—
52C22×D332—28no—
53C22×D332—28no—
54C22×D332—24no—
55C22×D332—24no—
56C22×D332—44no—
57C22×D332D418yes—
58C22×D332C2318yes—
59C22×D332C2314yes—
60C23×Dic83C2311yescharacteristic
61C23×Dic83C2312yes—
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—28no—
65C22×C4×D83—28no—
66C22×C4×D83—24no—
67C22×C4×D83—24no—
68C22×C4×D83—24no—
69C22×C4×D83—44no—
70C22×C4×D83C2314yes—
71C22×C4×D83C2318yes—
72#14581777—264no—
73#14581767—232no—
74#14581754—232no—
75C2×C4⋊Dic83—24no—
76C2×C4⋊Dic83C2314yes—
77C2×Dic83⋊C4—28no—
78C2×Dic83⋊C4C2318yes—
79C2×(C2×D83)⋊C4—28no—
80C2×(C2×D83)⋊C4—28no—
81C2×(C2×D83)⋊C4—28no—
82C2×(C2×D83)⋊C4—28no—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—24no—
85C2×(C2×D83)⋊C4D418yescomplemented
86C2×(C2×D83)⋊C4C2318yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2314yes—
91C2×(C2×D83)⋊C4C2318yes—
92C2×(C2×D83)⋊C4C2314yes—
93C2×C4×Dic83—24no—
94C2×C4×Dic83C2314yes—
95C83⋊#95C2318yes—
96C83⋊#95C2314yes—
97C83⋊#95C2318yes—
98C83⋊#95C2314yes—
99C23×C4×C83C2312yes—
100C23×C4×C83C2312yes—
101C23×C4×C83C2311yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83—24no—
105C2×C4⋊C4×C83—24no—
106C2×C4⋊C4×C83—28no—
107C2×C4⋊C4×C83—44no—
108C2×C4⋊C4×C83C2314yes—
109C2×C4⋊C4×C83C2314yes—
110C2×C4⋊C4×C83C2318yes—
111C2×C4⋊C4×C83C2318yes—
112C2×C42×C83—24no—
113C2×C42×C83C2314yes—
114C23×D83—22no—
115C23×D83—24no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—24no—
119C23×D83—24no—
120C23×D83—22no—
121C23×D83—24no—
122C23×D83—24no—
123C23×D83—22no—
124C23×D83—24no—
125C23×D83—22no—
126C23×D83C2×D418yes—
127C23×D83C2×D414yes—
128C23×D83C2414yes—
129C23×D83C2412yes—
130C2×D332—232no—
131C2×D332—28no—
132C2×D332—28no—
133C2×D332—216no—
134C2×D332—216no—
135C2×D332—48no—
136C2×D332—432no—
137C2×D332C2×D4116yes—
138C22×Dic83—22no—
139C22×Dic83—22no—
140C22×Dic83—22no—
141C22×Dic83—22no—
142C22×Dic83—24no—
143C22×Dic83—22no—
144C22×Dic83—22no—
145C22×Dic83—24no—
146C22×Dic83—22no—
147C22×Dic83—24no—
148C22×Dic83—22no—
149C22×Dic83C2×D414yes—
150C22×Dic83C2412yes—
151C22×Dic83C2412yes—
152C22×Dic83C2414yes—
153C22×Dic83C2412yes—
154C2×C4×D83—216no—
155C2×C4×D83—28no—
156C2×C4×D83—28no—
157C2×C4×D83—28no—
158C2×C4×D83—216no—
159C2×C4×D83—48no—
160C2×C4×D83—432no—
161C2×C4×D83—416no—
162C4⋊Dic83—28no—
163Dic83⋊C4—216no—
164(C2×D83)⋊C4—216no—
165(C2×D83)⋊C4—216no—
166(C2×D83)⋊C4—216no—
167(C2×D83)⋊C4—216no—
168(C2×D83)⋊C4—216no—
169(C2×D83)⋊C4—28no—
170(C2×D83)⋊C4C2×D4116yescomplemented
171C4×Dic83—28no—
172C24×C83C2411yescharacteristic
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—24no—
180C22×C4×C83—24no—
181C22×C4×C83—24no—
182C22×C4×C83—24no—
183C22×C4×C83C2×D414yes—
184C22×C4×C83C2×D414yes—
185C22×C4×C83#4114yes—
186C22×C4×C83#4114yes—
187C22×C4×C83#4114yes—
188C22×C4×C83C2414yes—
189C22×C4×C83C2414yes—
190C22×C4×C83C2412yes—
191C22×C4×C83C2412yes—
192C22×C4×C83C2412yes—
193C4⋊C4×C83—28no—
194C4⋊C4×C83—216no—
195C4⋊C4×C83—28no—
196C4⋊C4×C83—48no—
197C42×C83—28no—
198C23×C83C22×D412yes—
199C23×C83C22×D412yes—
200C23×C83C2×#4112yes—
201C23×C83C2×#4112yes—
202C23×C83C2×#4112yes—
203C23×C83H(5,𝔽2)12yes—
204C23×C832−1+412yes—
205C23×C83C2511yescharacteristic
206C22×D83—216no—
207C22×D83—28no—
208C22×D83—216no—
209C22×D83—28no—
210C22×D83—28no—
211C22×D83—216no—
212C22×D83—24no—
213C22×D83—22no—
214C22×D83—22no—
215C22×D83—24no—
216C22×D83—22no—
217C22×D83—22no—
218C22×D83—24no—
219C22×D83—24no—
220C22×D83—24no—
221C22×D83—22no—
222C22×D83—24no—
223C22×D83—22no—
224C22×D83—44no—
225C22×D83—48no—
226C22×D83—44no—
227C22×D83—48no—
228C22×D83—48no—
229C22×D83—44no—
230C22×D83—44no—
231C22×D83—48no—
232C22×D83C22×D414yes—
233C22×D83C22×D418yes—
234C2×C4×C83—28no—
235C2×C4×C83—28no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—24no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—48no—
247C2×C4×C83—48no—
248C2×C4×C83#123116yescomplemented
249C2×C4×C83C22×D414yes—
250C2×C4×C83C22×D414yes—
251C2×C4×C83C2×#4114yes—
252C2×C4×C83C2×#4114yes—
253C2×C4×C83C2×#4114yes—
254C2×Dic83—28no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—24no—
259C2×Dic83—22no—
260C2×Dic83—24no—
261C2×Dic83—22no—
262C2×Dic83—24no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—44no—
267C2×Dic83—48no—
268C2×Dic83—44no—
269C2×Dic83C22×D414yes—
270D332—232no—
271D332—432no—
272C4×D83—432no—
273C4×D83—416no—
274C22×C83C2×#12314yes—
275C22×C83#53214yes—
276C22×C83#54214yes—
277C22×C83D4214yes—
278C22×C83#54714yes—
279C22×C83#54714yes—
280C22×C83#55314yes—
281C22×C83C23×D411yescharacteristic
282C22×C83C23×D411yescharacteristic
283C22×C83C22×#4111yescharacteristic
284C22×C83C22×#4111yescharacteristic
285C22×C83C22×#4111yescharacteristic
286C22×C83C2×H(5,𝔽2)11yescharacteristic
287C22×C83C2×2−1+411yescharacteristic
288C2×D83—28no—
289C2×D83—24no—
290C2×D83—28no—
291C2×D83—24no—
292C2×D83—24no—
293C2×D83—28no—
294C2×D83—42no—
295C2×D83—44no—
296C2×D83—432no—
297C2×D83—416no—
298C2×D83—42no—
299C2×D83—44no—
300C2×D83—44no—
301C2×D83—42no—
302C2×D83—42no—
303C2×D83—44no—
304C4×C83—24no—
305C4×C83—24no—
306C4×C83—44no—
307C4×C83—44no—
308C4×C83C2×#12318yescomplemented
309Dic83—24no—
310Dic83—42no—
311Dic83—44no—
312Dic83—42no—
313C2×#2422—831nomaximal, Sylow
314C2×C83#242218yescomplemented
315C2×C83C22×#12311yescharacteristic
316C2×C83C2×#53211yescharacteristic
317C2×C83C2×#54211yescharacteristic
318C2×C83C2×D4211yescharacteristic
319C2×C83C2×#54711yescharacteristic
320C2×C83C2×#54711yescharacteristic
321C2×C83C2×#55311yescharacteristic
322D83—48no—
323D83—44no—
324C22×#123—832no—
325C22×#121—831no—
326C22×C4×D4—831no—
327#2422—8316no—
328C2×#397—832no—
329C2×#394—832no—
330C2×#390—831no—
331C2×#389—831no—
332C2×#385—832no—
333C2×#384D8311yescharacteristic, complemented
334C2×#380—831no—
335C2×C4×C22⋊C4—831no—
336C83C2×#242211yescharacteristic, complemented, Sylow
337C23×D4—832no—
338C23×D4—831no—
339C24×C4—832no—
340C24×C4—831no—
341C2×#123—8316no—
342C2×#123—16616no—
343C2×#121—838no—
344C2×#121—1668no—
345C2×C4×D4—838no—
346C2×C4×D4—1668no—
347C22×C4⋊C4—832no—
348C22×C4⋊C4—831no—
349C22×C4⋊C4C2×D8312yes—
350C22×C4⋊C4C2×D8311yescharacteristic
351C22×C4⋊C4C2×D8311yescharacteristic
352C22×C4⋊C4C2×D8312yes—
353C22×C22⋊C4—831no—
354C22×C22⋊C4—831no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4—832no—
357C22×C22⋊C4—832no—
358C22×C22⋊C4—832no—
359C22×C22⋊C4—832no—
360C22×C42—831no—
361C22×C42C2×D8311yescharacteristic
362#397—8316no—
363#394—8316no—
364#390—838no—
365#389—838no—
366#385—8316no—
367#384C2×D8318yescomplemented
368#380—838no—
369C4×C22⋊C4—838no—
370C2×#95—832no—
371C2×#95—831no—
372C2×#95—832no—
373C2×#95—831no—
374C25—831no—
375C25—832no—
376C22×D4—3324no—
377C22×D4—838no—
378C22×D4—838no—
379C22×D4—834no—
380C22×D4—1668no—
381C22×D4—1668no—
382C22×D4—1664no—
383C22×D4—1664no—
384C23×C4—3324no—
385C23×C4—838no—
386C23×C4—831no—
387C23×C4—832no—
388C23×C4—831no—
389C23×C4—831no—
390C23×C4—834no—
391C23×C4—1668no—
392C23×C4—1668no—
393C23×C4—1664no—
394C23×C4—1664no—
395C23×C4—1664no—
396C23×C4C22×D8312yes—
397C23×C4C22×D8312yes—
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401#123—16664no—
402#121—16632no—
403C4×D4—16632no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—28no—
407C2×C4⋊C4—44no—
408C2×C4⋊C4—838no—
409C2×C4⋊C4—834no—
410C2×C4⋊C4—1668no—
411C2×C4⋊C4—1664no—
412C2×C4⋊C4C22×D8314yes—
413C2×C4⋊C4C22×D8314yes—
414C2×C4⋊C4C22×D8318yes—
415C2×C4⋊C4C22×D8318yes—
416C2×C22⋊C4—838no—
417C2×C22⋊C4—834no—
418C2×C22⋊C4—838no—
419C2×C22⋊C4—838no—
420C2×C22⋊C4—838no—
421C2×C22⋊C4—838no—
422C2×C22⋊C4—834no—
423C2×C22⋊C4—838no—
424C2×C22⋊C4—1668no—
425C2×C22⋊C4—1668no—
426C2×C22⋊C4—1668no—
427C2×C22⋊C4—1668no—
428C2×C22⋊C4—1664no—
429C2×C22⋊C4—1668no—
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—834no—
439C24—838no—
440C24—832no—
441C24—834no—
442C24—1664no—
443C24—1662no—
444C24—1662no—
445C24—1664no—
446C24—1662no—
447C24—1664no—
448C24—1662no—
449C24—1664no—
450C24—1664no—
451C24—1662no—
452C24—1664no—
453C24—1662no—
454C24C23×D8311yescharacteristic
455C2×D4—3328no—
456C2×D4—33232no—
457C2×D4—8316no—
458C2×D4—16616no—
459C2×D4—16616no—
460C2×D4—16632no—
461C2×D4—1668no—
462C2×D4—1668no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—3328no—
474C22×C4—33232no—
475C22×C4—33216no—
476C22×C4—834no—
477C22×C4—832no—
478C22×C4—832no—
479C22×C4—832no—
480C22×C4—834no—
481C22×C4—16616no—
482C22×C4—16616no—
483C22×C4—1662no—
484C22×C4—1664no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1664no—
489C22×C4—1664no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1668no—
493C22×C4—1668no—
494C22×C4—1668no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4D4×D8314yes—
498C22×C4D4×D8314yes—
499C22×C4#1305059014yes—
500C22×C4#1305059114yes—
501C22×C4#1305059114yes—
502C22×C4C23×D8314yes—
503C22×C4C23×D8314yes—
504C22×C4C23×D8312yes—
505C22×C4C23×D8312yes—
506C22×C4C23×D8312yes—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—48no—
511C4⋊C4—16616no—
512C4⋊C4—1668no—
513C22⋊C4—8316no—
514C22⋊C4—16616no—
515C22⋊C4—16616no—
516C22⋊C4—16616no—
517C22⋊C4—16616no—
518C22⋊C4—1668no—
519C22⋊C4—16616no—
520C42—28no—
521C42—1668no—
522C23—3324no—
523C23—3328no—
524C23—3328no—
525C23—3324no—
526C23—3324no—
527C23—3328no—
528C23—3324no—
529C23—3328no—
530C23—834no—
531C23—838no—
532C23—16616no—
533C23—1668no—
534C23—1668no—
535C23—16616no—
536C23—16616no—
537C23—1668no—
538C23—1664no—
539C23—1662no—
540C23—1662no—
541C23—1664no—
542C23—1662no—
543C23—1664no—
544C23—1662no—
545C23—1664no—
546C23—1664no—
547C23—1662no—
548C23—1664no—
549C23—1662no—
550C23C2×D4×D8312yes—
551C23C2×D4×D8312yes—
552C23C2×#1305059012yes—
553C23C2×#1305059112yes—
554C23C2×#1305059112yes—
555C23#1458183712yes—
556C23#1458184012yes—
557C23C24×D8311yescharacteristic
558D4—33232no—
559D4—16632no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—33232no—
575C2×C4—3324no—
576C2×C4—3324no—
577C2×C4—3328no—
578C2×C4—33216no—
579C2×C4—834no—
580C2×C4—1662no—
581C2×C4—1664no—
582C2×C4—1662no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4—1664no—
586C2×C4—1664no—
587C2×C4—1662no—
588C2×C4—1662no—
589C2×C4—1668no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4#14581777116yescomplemented
593C2×C4C2×D4×D8314yes—
594C2×C4C2×D4×D8314yes—
595C2×C4C2×#1305059014yes—
596C2×C4C2×#1305059114yes—
597C2×C4C2×#1305059114yes—
598C22—3322no—
599C22—3324no—
600C22—3324no—
601C22—3322no—
602C22—3322no—
603C22—33232no—
604C22—33216no—
605C22—3322no—
606C22—3324no—
607C22—3324no—
608C22—1668no—
609C22—1664no—
610C22—1664no—
611C22—1668no—
612C22—1668no—
613C22—1664no—
614C22C2×#1458177714yes—
615C22#1912825214yes—
616C22#1912832514yes—
617C22#1912833614yes—
618C22#1912835214yes—
619C22#1912835414yes—
620C22#1912839914yes—
621C22C22×D4×D8311yescharacteristic
622C22C22×D4×D8311yescharacteristic
623C22C22×#1305059011yescharacteristic
624C22C22×#1305059111yescharacteristic
625C22C22×#1305059111yescharacteristic
626C22C2×#1458183711yescharacteristic
627C22C2×#1458184011yescharacteristic
628C4—24no—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—3322no—
633C4—3322no—
634C4—3324no—
635C4—1664no—
636C4C2×#1458177718yescomplemented
637C2—3328no—
638C2—3324no—
639C2#2491098018yesdirect product factor
640C2C22×#1458177711yescharacteristic
641C2C2×#1912825211yescharacteristic
642C2C2×#1912832511yescharacteristic
643C2C2×#1912833611yescharacteristic
644C2C2×#1912835211yescharacteristic
645C2C2×#1912835411yescharacteristic
646C2C2×#1912839911yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#384×C83C2
socleC24×C83C24
prime radicalC22×#14581767C2
hypercenterC2×#384D83
hypocenterC83C2×#2422
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
1. C22×#14581777C2semidirect
3. C22×#14581754C2semidirect
4. #24910980C2direct (N × H)
5. C2×#19127627C2semidirect
6. C2×#19127618C2semidirect
7. C2×#19127603C2semidirect
8. C2×#19127597C2semidirect
9. C2×#19127572C2semidirect
10. C2×#19127558C2semidirect
11. C2×C4×(C2×D83)⋊C4C2semidirect
12. C2×#384×C83C2semidirect
18. C2×#14581777C22semidirect
22. C2×#14581754C22semidirect
33. #19127627C22semidirect
34. #19127618C22semidirect
35. #19127603C22semidirect
36. #19127597C22semidirect
37. #19127572C22semidirect
38. #19127558C22semidirect
39. C4×(C2×D83)⋊C4C22semidirect
49. #384×C83C22semidirect
85. C2×(C2×D83)⋊C4D4semidirect
170. (C2×D83)⋊C4C2×D4semidirect
248. C2×C4×C83#123semidirect
308. C4×C83C2×#123semidirect
314. C2×C83#2422semidirect
333. C2×#384D83semidirect
336. C83C2×#2422semidirect
367. #384C2×D83semidirect
592. C2×C4#14581777semidirect
636. C4C2×#14581777semidirect
639. C2#24910980direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817772
C22×#145817672
C22×#145817542
#249109802
C2×#191276272
C2×#191276182
C2×#191276032
C2×#191275972
C2×#191275722
C2×#191275582
C2×C4×(C2×D83)⋊C42
C2×#384×C832
C2×#242283