Finite Group Databasebeta

← C2×#24915489 #35610531

Subgroups — C2×#24915489 #35610531

Order 21248.

2998 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581768C211yesmaximal, characteristic, complemented
2C22×#14581765C211yesmaximal, characteristic
3#24915489C2116yesmaximal, direct product factor
4C2×#19127638C211yesmaximal, characteristic, complemented
5C2×#19127637C211yesmaximal, characteristic, complemented
6C2×#19127635C212yesmaximal, complemented
7C2×#19127619C212yesmaximal, complemented
8C2×#19127617C212yesmaximal, complemented
9C2×#19127616C212yesmaximal, complemented
10C2×#19127612C211yesmaximal, characteristic, complemented
11C2×#19127563C211yesmaximal, characteristic, complemented
12C2×#393×C83C211yesmaximal, characteristic, complemented
13C23×D332C2211yescharacteristic
14C23×C22⋊D83C2211yescharacteristic
15C23×C22⋊D83C2211yescharacteristic
16C23×C4×D83C2211yescharacteristic
17C2×#14581768—28no—
18C2×#14581768C2218yescomplemented
19C2×#14581765—28no—
20C2×#14581765C2218yes—
21C22×C4⋊Dic83C2211yescharacteristic
22C22×C4⋊Dic83C2212yes—
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2211yescharacteristic
25C22×Dic83⋊C4C2212yes—
26C22×C22⋊Dic83C2212yes—
27C22×(C2×D83)⋊C4C2211yescharacteristic
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×(C2×D83)⋊C4C2212yes—
32#19127638C2218yescomplemented
33#19127637C2218yescomplemented
34#19127635C22116yescomplemented
35#19127619C22116yescomplemented
36#19127617C22116yescomplemented
37#19127616C22116yescomplemented
38#19127612C2218yescomplemented
39#19127563C2218yescomplemented
40C2×C83⋊#95C2211yescharacteristic
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C23×Q8×C83C2211yescharacteristic
44C22×C22⋊C4×C83C2211yescharacteristic
45C22×C22⋊C4×C83C2212yes—
46#393×C83C2218yescomplemented
47C2×#95×C83C2211yescharacteristic
48C2×#95×C83C2212yes—
49C24×D83C2311yescharacteristic
50C24×D83C2311yescharacteristic
51C22×D332—24no—
52C22×D332—44no—
53C22×D332C2314yes—
54C22×C22⋊D83—24no—
55C22×C22⋊D83—24no—
56C22×C22⋊D83—44no—
57C22×C22⋊D83—44no—
58C22×C22⋊D83C2314yes—
59C22×C22⋊D83C2314yes—
60C23×Dic83C2312yes—
61C23×Dic83C2312yes—
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—44no—
65C22×C4×D83D418yes—
66C22×C4×D83C2314yes—
67#14581768—232no—
68#14581765—232no—
69C2×C4⋊Dic83—24no—
70C2×C4⋊Dic83—44no—
71C2×C4⋊Dic83C2314yes—
72C2×C4⋊Dic83C2318yes—
73C2×Dic83⋊C4—24no—
74C2×Dic83⋊C4—28no—
75C2×Dic83⋊C4—44no—
76C2×Dic83⋊C4C2318yes—
77C2×Dic83⋊C4C2314yes—
78C2×Dic83⋊C4C2318yes—
79C2×C22⋊Dic83—44no—
80C2×C22⋊Dic83C2318yes—
81C2×(C2×D83)⋊C4—28no—
82C2×(C2×D83)⋊C4—24no—
83C2×(C2×D83)⋊C4—44no—
84C2×(C2×D83)⋊C4—44no—
85C2×(C2×D83)⋊C4—44no—
86C2×(C2×D83)⋊C4C2314yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2318yes—
91C83⋊#95C2314yes—
92C83⋊#95C2318yes—
93C83⋊#95C2318yes—
94C25×C83C2311yescharacteristic
95C22×Q8×C83—24no—
96C22×Q8×C83—44no—
97C22×Q8×C83C2314yes—
98C23×C4×C83C2312yes—
99C23×C4×C83C2312yes—
100C23×C4×C83C2311yescharacteristic
101C23×C4×C83C2311yescharacteristic
102C2×C22⋊C4×C83—24no—
103C2×C22⋊C4×C83—44no—
104C2×C22⋊C4×C83C2314yes—
105C2×C22⋊C4×C83C2318yes—
106#95×C83C2314yes—
107#95×C83C2318yes—
108C23×D83—24no—
109C23×D83—24no—
110C23×D83—22no—
111C23×D83—22no—
112C23×D83—22no—
113C23×D83—24no—
114C23×D83—24no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83C2×D414yes—
118C23×D83C2412yes—
119C23×D83C2412yes—
120C2×D332—28no—
121C2×D332—48no—
122C2×D332—416no—
123C2×C22⋊D83—28no—
124C2×C22⋊D83—28no—
125C2×C22⋊D83—48no—
126C2×C22⋊D83—48no—
127C2×C22⋊D83—416no—
128C2×C22⋊D83—416no—
129C22×Dic83—24no—
130C22×Dic83—22no—
131C22×Dic83—24no—
132C22×Dic83—24no—
133C22×Dic83—24no—
134C22×Dic83—24no—
135C22×Dic83—24no—
136C22×Dic83—24no—
137C22×Dic83—24no—
138C22×Dic83C2×D414yes—
139C22×Dic83C2412yes—
140C22×Dic83C2414yes—
141C22×Dic83C2414yes—
142C22×Dic83C2412yes—
143C2×C4×D83—416no—
144C2×C4×D83—48no—
145C2×C4×D83C2×D4116yes—
146C4⋊Dic83—28no—
147C4⋊Dic83—48no—
148Dic83⋊C4—28no—
149Dic83⋊C4—216no—
150Dic83⋊C4—48no—
151C22⋊Dic83—48no—
152(C2×D83)⋊C4—216no—
153(C2×D83)⋊C4—28no—
154(C2×D83)⋊C4—48no—
155(C2×D83)⋊C4—48no—
156(C2×D83)⋊C4—48no—
157C24×C83—22no—
158C24×C83—24no—
159C24×C83—24no—
160C24×C83—22no—
161C24×C83—22no—
162C24×C83C2412yes—
163C24×C83C2411yescharacteristic
164C2×Q8×C83—28no—
165C2×Q8×C83—48no—
166C2×Q8×C83—416no—
167C22×C4×C83—24no—
168C22×C4×C83—22no—
169C22×C4×C83—24no—
170C22×C4×C83—24no—
171C22×C4×C83—24no—
172C22×C4×C83—24no—
173C22×C4×C83—24no—
174C22×C4×C83—24no—
175C22×C4×C83—24no—
176C22×C4×C83C2×D414yes—
177C22×C4×C83C2412yes—
178C22×C4×C83C2414yes—
179C22×C4×C83C2414yes—
180C22×C4×C83C2412yes—
181C22⋊C4×C83—28no—
182C22⋊C4×C83—48no—
183C23×C83—22no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—22no—
187C23×C83—22no—
188C23×C83—44no—
189C23×C83—44no—
190C23×C83—44no—
191C23×C83—48no—
192C23×C83—48no—
193C23×C83C22×D412yes—
194C23×C83H(5,𝔽2)14yes—
195C23×C83H(5,𝔽2)14yes—
196C23×C832−1+412yes—
197C23×C832−1+412yes—
198C23×C83C2511yescharacteristic
199C22×D83—28no—
200C22×D83—28no—
201C22×D83—28no—
202C22×D83—24no—
203C22×D83—24no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—22no—
207C22×D83—24no—
208C22×D83—24no—
209C22×D83—22no—
210C22×D83—22no—
211C22×D83—48no—
212C22×D83—48no—
213C22×D83—44no—
214C22×D83—44no—
215C22×D83—44no—
216C22×D83—48no—
217C22×D83—48no—
218C22×D83C22×D414yes—
219Q8×C83—416no—
220C2×C4×C83—28no—
221C2×C4×C83—24no—
222C2×C4×C83—22no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—48no—
231C2×C4×C83—44no—
232C2×C4×C83—48no—
233C2×C4×C83C22×D414yes—
234C22⋊D83—416no—
235C22⋊D83—416no—
236C2×Dic83—28no—
237C2×Dic83—24no—
238C2×Dic83—24no—
239C2×Dic83—22no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—24no—
246C2×Dic83—48no—
247C2×Dic83—44no—
248C2×Dic83—48no—
249C2×Dic83C22×D414yes—
250D332—416no—
251C4×D83—416no—
252C22×C83—42no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—44no—
256C22×C83—44no—
257C22×C83—88no—
258C22×C83#53414yes—
259C22×C83#53514yes—
260C22×C83#53614yes—
261C22×C83#56218yes—
262C22×C83#56218yes—
263C22×C83C23×D411yescharacteristic
264C22×C83C2×H(5,𝔽2)12yes—
265C22×C83C2×H(5,𝔽2)12yes—
266C22×C83C2×2−1+411yescharacteristic
267C22×C83C2×2−1+411yescharacteristic
268C2×D83—24no—
269C2×D83—24no—
270C2×D83—24no—
271C2×D83—416no—
272C2×D83—44no—
273C2×D83—44no—
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—44no—
278C2×D83—44no—
279C2×D83—88no—
280C4×C83—24no—
281C4×C83—44no—
282C4×C83—42no—
283C4×C83—44no—
284Dic83—24no—
285Dic83—44no—
286Dic83—42no—
287Dic83—44no—
288C2×#2816—831nomaximal, Sylow
289C2×C83—82no—
290C2×C83#281618yescomplemented
291C2×C83C2×#53411yescharacteristic
292C2×C83C2×#53511yescharacteristic
293C2×C83C2×#53611yescharacteristic
294C2×C83C2×#56212yes—
295C2×C83C2×#56212yes—
296D83—44no—
297D83—82no—
298C22×C22⋊Q8—831no—
299C22×#121—831no—
300#2816—8316no—
301C2×#399—831no—
302C2×#399—831no—
303C2×#398—832no—
304C2×#394—832no—
305C2×#394—832no—
306C2×#394—832no—
307C2×#393—831no—
308C2×#393D8311yescharacteristic, complemented
309C2×#381—831no—
310C83C2×#281611yescharacteristic, complemented, Sylow
311C23×Q8C2×D8311yescharacteristic
312C23×D4—831no—
313C23×D4—831no—
314C23×D4—831no—
315C24×C4—831no—
316C2×C22⋊Q8—838no—
317C2×C22⋊Q8—1668no—
318C2×#121—838no—
319C2×#121—1668no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—832no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—831no—
324C22×C4⋊C4—831no—
325C22×C22⋊C4—832no—
326C22×C22⋊C4—832no—
327C22×C22⋊C4—832no—
328C22×C22⋊C4—832no—
329C22×C22⋊C4—831no—
330C22×C22⋊C4—832no—
331C22×C22⋊C4C2×D8311yescharacteristic
332C22×C22⋊C4C2×D8312yes—
333#399—838no—
334#399—838no—
335#398—8316no—
336#394—8316no—
337#394—8316no—
338#394—8316no—
339#393—838no—
340#393C2×D8318yescomplemented
341#381—838no—
342C2×#95—832no—
343C2×#95—831no—
344C2×#95—832no—
345C2×#95C2×D8312yes—
346C2×#95C2×D8311yescharacteristic
347C25—831no—
348C25—831no—
349C25C22×D8311yescharacteristic
350C22×Q8—24no—
351C22×Q8—44no—
352C22×Q8C22×D8314yes—
353C22×D4—3324no—
354C22×D4—3324no—
355C22×D4—3324no—
356C22×D4—834no—
357C22×D4—834no—
358C22×D4—834no—
359C22×D4—1664no—
360C22×D4—1664no—
361C22×D4—1664no—
362C23×C4—3324no—
363C23×C4—838no—
364C23×C4—834no—
365C23×C4—831no—
366C23×C4—832no—
367C23×C4—832no—
368C23×C4—831no—
369C23×C4C22×D8312yes—
370C23×C4C22×D8312yes—
371C23×C4C22×D8311yescharacteristic
372C23×C4C22×D8311yescharacteristic
373C22⋊Q8—16632no—
374#121—16632no—
375C2×C4⋊C4—3324no—
376C2×C4⋊C4—3324no—
377C2×C4⋊C4—838no—
378C2×C4⋊C4—838no—
379C2×C4⋊C4—834no—
380C2×C4⋊C4—838no—
381C2×C4⋊C4—834no—
382C2×C4⋊C4—1664no—
383C2×C4⋊C4—1668no—
384C2×C4⋊C4—1664no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—44no—
387C2×C22⋊C4—3324no—
388C2×C22⋊C4—3324no—
389C2×C22⋊C4—3324no—
390C2×C22⋊C4—3324no—
391C2×C22⋊C4—838no—
392C2×C22⋊C4—838no—
393C2×C22⋊C4—838no—
394C2×C22⋊C4—838no—
395C2×C22⋊C4—834no—
396C2×C22⋊C4—838no—
397C2×C22⋊C4—1668no—
398C2×C22⋊C4—1664no—
399C2×C22⋊C4C22×D8314yes—
400C2×C22⋊C4C22×D8318yes—
401#95—834no—
402#95—838no—
403#95—838no—
404#95C22×D8314yes—
405#95C22×D8318yes—
406C24—22no—
407C24—24no—
408C24—24no—
409C24—22no—
410C24—22no—
411C24—834no—
412C24—832no—
413C24—832no—
414C24—1662no—
415C24—1664no—
416C24—1664no—
417C24—1662no—
418C24—1662no—
419C24—1664no—
420C24—1664no—
421C24—1662no—
422C24—1662no—
423C24C23×D8312yes—
424C24C23×D8311yescharacteristic
425C2×Q8—28no—
426C2×Q8—48no—
427C2×Q8—416no—
428C2×D4—3328no—
429C2×D4—33216no—
430C2×D4—33216no—
431C2×D4—3328no—
432C2×D4—3328no—
433C2×D4—33216no—
434C2×D4—1668no—
435C2×D4—1668no—
436C2×D4—1668no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—33216no—
447C22×C4—3328no—
448C22×C4—8316no—
449C22×C4—834no—
450C22×C4—832no—
451C22×C4—834no—
452C22×C4—832no—
453C22×C4—834no—
454C22×C4—1664no—
455C22×C4—1662no—
456C22×C4—1664no—
457C22×C4—1664no—
458C22×C4—1664no—
459C22×C4—1664no—
460C22×C4—1664no—
461C22×C4—1664no—
462C22×C4—1664no—
463C22×C4D4×D8314yes—
464C22×C4C23×D8314yes—
465C22×C4C23×D8314yes—
466C22×C4C23×D8312yes—
467C22×C4C23×D8312yes—
468C4⋊C4—3328no—
469C4⋊C4—3328no—
470C4⋊C4—1668no—
471C4⋊C4—16616no—
472C4⋊C4—1668no—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—3328no—
476C22⋊C4—3328no—
477C22⋊C4—3328no—
478C22⋊C4—3328no—
479C22⋊C4—16616no—
480C22⋊C4—1668no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—3328no—
492C23—3328no—
493C23—3328no—
494C23—3328no—
495C23—3324no—
496C23—3324no—
497C23—3324no—
498C23—834no—
499C23—1662no—
500C23—1664no—
501C23—1664no—
502C23—1662no—
503C23—1662no—
504C23—1668no—
505C23—1668no—
506C23—1668no—
507C23—1664no—
508C23—1664no—
509C23—1662no—
510C23—1662no—
511C23C2×D4×D8312yes—
512C23#1458183714yes—
513C23#1458183814yes—
514C23#1458183912yes—
515C23#1458184012yes—
516C23C24×D8311yescharacteristic
517Q8—416no—
518D4—33216no—
519D4—33216no—
520D4—33216no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—33216no—
535C2×C4—3328no—
536C2×C4—3324no—
537C2×C4—3328no—
538C2×C4—834no—
539C2×C4—1664no—
540C2×C4—1662no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1668no—
548C2×C4—1664no—
549C2×C4C2×D4×D8314yes—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—88no—
556C22—33216no—
557C22—3324no—
558C22—3324no—
559C22—3322no—
560C22—3322no—
561C22—3322no—
562C22—3324no—
563C22—3324no—
564C22—6648no—
565C22—1664no—
566C22—1664no—
567C22—1664no—
568C22#1912826114yes—
569C22#1912826914yes—
570C22#1912827414yes—
571C22#1912844518yes—
572C22#1912844718yes—
573C22C22×D4×D8311yescharacteristic
574C22C2×#1458183712yes—
575C22C2×#1458183812yes—
576C22C2×#1458183911yescharacteristic
577C22C2×#1458184011yescharacteristic
578C4—24no—
579C4—44no—
580C4—42no—
581C4—44no—
582C4—3322no—
583C4—3324no—
584C4—3324no—
585C4—1664no—
586C2—82no—
587C2—3324no—
588C2—6642no—
589C2#2491548918yesdirect product factor
590C2C2×#1912826111yescharacteristic
591C2C2×#1912826911yescharacteristic
592C2C2×#1912827411yescharacteristic
593C2C2×#1912844512yes—
594C2C2×#1912844712yes—
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#393×C83C2
socleC24×C83C24
prime radicalC22×#14581765C2
hypercenterC2×#393D83
hypocenterC83C2×#2816
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×#14581768C2semidirect
3. #24915489C2direct (N × H)
4. C2×#19127638C2semidirect
5. C2×#19127637C2semidirect
6. C2×#19127635C2semidirect
7. C2×#19127619C2semidirect
8. C2×#19127617C2semidirect
9. C2×#19127616C2semidirect
10. C2×#19127612C2semidirect
11. C2×#19127563C2semidirect
12. C2×#393×C83C2semidirect
18. C2×#14581768C22semidirect
32. #19127638C22semidirect
33. #19127637C22semidirect
34. #19127635C22semidirect
35. #19127619C22semidirect
36. #19127617C22semidirect
37. #19127616C22semidirect
38. #19127612C22semidirect
39. #19127563C22semidirect
46. #393×C83C22semidirect
290. C2×C83#2816semidirect
308. C2×#393D83semidirect
310. C83C2×#2816semidirect
340. #393C2×D83semidirect
589. C2#24915489direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817682
C22×#145817652
#249154892
C2×#191276382
C2×#191276372
C2×#191276352
C2×#191276192
C2×#191276172
C2×#191276162
C2×#191276122
C2×#191275632
C2×#393×C832
C2×#281683