Finite Group Databasebeta

← C2×#24911041 #35572549

Subgroups — C2×#24911041 #35572549

Order 21248.

3894 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581781C211yesmaximal, characteristic, complemented
2C22×#14581768C211yesmaximal, characteristic, complemented
3C22×#14581767C211yesmaximal, characteristic
4C22×C4×D332C211yesmaximal, characteristic, complemented
5#24911041C2116yesmaximal, direct product factor
6C2×#19127630C212yesmaximal, complemented
7C2×#19127619C212yesmaximal, complemented
8C2×#19127597C211yesmaximal, characteristic, complemented
9C2×#19127579C212yesmaximal, complemented
10C2×#19127560C212yesmaximal, complemented
11C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
12C2×#391×C83C211yesmaximal, characteristic, complemented
13C23×D332C2212yes—
14C23×D332C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2212yes—
17C2×#14581781—28no—
18C2×#14581781C2218yescomplemented
19C2×#14581768—28no—
20C2×#14581768C2218yescomplemented
21C2×#14581767—28no—
22C2×#14581767C2218yes—
23C2×C4×D332—28no—
24C2×C4×D332C2218yescomplemented
25C22×C4⋊Dic83C2211yescharacteristic
26C22×Dic83⋊C4C2212yes—
27C22×Dic83⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2211yescharacteristic
31C22×(C2×D83)⋊C4C2212yes—
32C22×(C2×D83)⋊C4C2212yes—
33C22×(C2×D83)⋊C4C2212yes—
34C22×(C2×D83)⋊C4C2211yescharacteristic
35C22×C4×Dic83C2211yescharacteristic
36#19127630C22116yescomplemented
37#19127619C22116yescomplemented
38#19127597C2218yescomplemented
39#19127579C22116yescomplemented
40#19127560C22116yescomplemented
41C4×(C2×D83)⋊C4C2218yescomplemented
42C2×C83⋊#95C2211yescharacteristic
43C2×C83⋊#95C2212yes—
44C2×C83⋊#95C2211yescharacteristic
45C23×Q8×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C42×C83C2211yescharacteristic
48#391×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C2×#95×C83C2212yes—
51C24×D83C2312yes—
52C24×D83C2311yescharacteristic
53C22×D332—28no—
54C22×D332—28no—
55C22×D332—24no—
56C22×D332—24no—
57C22×D332—44no—
58C22×D332D418yes—
59C22×D332C2318yes—
60C22×D332C2314yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C23×Dic83C2312yes—
65C22×C4×D83—28no—
66C22×C4×D83—24no—
67C22×C4×D83—24no—
68C22×C4×D83—24no—
69C22×C4×D83—28no—
70C22×C4×D83—44no—
71C22×C4×D83C2314yes—
72C22×C4×D83C2318yes—
73#14581781—232no—
74#14581768—232no—
75#14581767—232no—
76C4×D332—232no—
77C2×C4⋊Dic83D418yescomplemented
78C2×C4⋊Dic83C2314yes—
79C2×Dic83⋊C4—28no—
80C2×Dic83⋊C4—44no—
81C2×Dic83⋊C4C2318yes—
82C2×Dic83⋊C4C2318yes—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—28no—
85C2×(C2×D83)⋊C4—28no—
86C2×(C2×D83)⋊C4—24no—
87C2×(C2×D83)⋊C4—28no—
88C2×(C2×D83)⋊C4—28no—
89C2×(C2×D83)⋊C4—24no—
90C2×(C2×D83)⋊C4C2318yes—
91C2×(C2×D83)⋊C4C2318yes—
92C2×(C2×D83)⋊C4C2314yes—
93C2×(C2×D83)⋊C4C2318yes—
94C2×(C2×D83)⋊C4C2318yes—
95C2×(C2×D83)⋊C4C2314yes—
96C2×(C2×D83)⋊C4C2318yes—
97C2×C4×Dic83—24no—
98C2×C4×Dic83C2314yes—
99C83⋊#95C2314yes—
100C83⋊#95C2318yes—
101C83⋊#95C2314yes—
102C22×Q8×C83—24no—
103C22×Q8×C83—24no—
104C22×Q8×C83—24no—
105C22×Q8×C83C2314yes—
106C23×C4×C83C2311yescharacteristic
107C23×C4×C83C2312yes—
108C23×C4×C83C2312yes—
109C23×C4×C83C2311yescharacteristic
110C23×C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83—24no—
112C2×C4⋊C4×C83C2314yes—
113C2×C42×C83—24no—
114C2×C42×C83C2314yes—
115#95×C83C2318yes—
116#95×C83C2318yes—
117C23×D83—22no—
118C23×D83—24no—
119C23×D83—22no—
120C23×D83—22no—
121C23×D83—24no—
122C23×D83—24no—
123C23×D83—24no—
124C23×D83—22no—
125C23×D83—24no—
126C23×D83—22no—
127C23×D83—24no—
128C23×D83—22no—
129C23×D83C2×D418yes—
130C23×D83#4114yes—
131C23×D83C2414yes—
132C23×D83C2412yes—
133C2×D332—232no—
134C2×D332—28no—
135C2×D332—28no—
136C2×D332—216no—
137C2×D332—216no—
138C2×D332—48no—
139C2×D332—432no—
140C2×D332C2×D4116yes—
141C22×Dic83—24no—
142C22×Dic83—24no—
143C22×Dic83—22no—
144C22×Dic83—22no—
145C22×Dic83—22no—
146C22×Dic83—22no—
147C22×Dic83—22no—
148C22×Dic83—22no—
149C22×Dic83—22no—
150C22×Dic83—22no—
151C22×Dic83—22no—
152C22×Dic83C2×D418yes—
153C22×Dic83C2412yes—
154C22×Dic83C2412yes—
155C22×Dic83C2412yes—
156C22×Dic83C2414yes—
157C2×C4×D83—28no—
158C2×C4×D83—28no—
159C2×C4×D83—28no—
160C2×C4×D83—216no—
161C2×C4×D83—216no—
162C2×C4×D83—48no—
163C2×C4×D83—432no—
164C2×C4×D83—416no—
165C4⋊Dic83C2×D4116yescomplemented
166Dic83⋊C4—216no—
167Dic83⋊C4—48no—
168(C2×D83)⋊C4—216no—
169(C2×D83)⋊C4—216no—
170(C2×D83)⋊C4—216no—
171(C2×D83)⋊C4—28no—
172(C2×D83)⋊C4—28no—
173(C2×D83)⋊C4—216no—
174(C2×D83)⋊C4—216no—
175C4×Dic83—28no—
176C24×C83C2411yescharacteristic
177C2×Q8×C83—28no—
178C2×Q8×C83—28no—
179C2×Q8×C83—28no—
180C2×Q8×C83—416no—
181C22×C4×C83—22no—
182C22×C4×C83—22no—
183C22×C4×C83—24no—
184C22×C4×C83—22no—
185C22×C4×C83—24no—
186C22×C4×C83—24no—
187C22×C4×C83—24no—
188C22×C4×C83—24no—
189C22×C4×C83—22no—
190C22×C4×C83—24no—
191C22×C4×C83—22no—
192C22×C4×C83C2×D414yes—
193C22×C4×C83C2×D414yes—
194C22×C4×C83#4114yes—
195C22×C4×C83#4114yes—
196C22×C4×C83C2414yes—
197C22×C4×C83C2412yes—
198C22×C4×C83C2414yes—
199C22×C4×C83C2412yes—
200C22×C4×C83C2412yes—
201C4⋊C4×C83—28no—
202C42×C83—28no—
203C23×C83C22×D412yes—
204C23×C83C22×D412yes—
205C23×C83C2×#4112yes—
206C23×C83C2×#4112yes—
207C23×C83C2×#4112yes—
208C23×C83H(5,𝔽2)12yes—
209C23×C832−1+412yes—
210C23×C83C2511yescharacteristic
211C22×D83—216no—
212C22×D83—28no—
213C22×D83—28no—
214C22×D83—216no—
215C22×D83—216no—
216C22×D83—28no—
217C22×D83—22no—
218C22×D83—24no—
219C22×D83—22no—
220C22×D83—22no—
221C22×D83—22no—
222C22×D83—24no—
223C22×D83—24no—
224C22×D83—24no—
225C22×D83—24no—
226C22×D83—22no—
227C22×D83—24no—
228C22×D83—22no—
229C22×D83—44no—
230C22×D83—48no—
231C22×D83—44no—
232C22×D83—48no—
233C22×D83—48no—
234C22×D83—44no—
235C22×D83—44no—
236C22×D83—48no—
237C22×D83C22×D418yes—
238C22×D83C2×#4114yes—
239Q8×C83—416no—
240C2×C4×C83—28no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—22no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—24no—
247C2×C4×C83—24no—
248C2×C4×C83—22no—
249C2×C4×C83—24no—
250C2×C4×C83—24no—
251C2×C4×C83—22no—
252C2×C4×C83—48no—
253C2×C4×C83—48no—
254C2×C4×C83—44no—
255C2×C4×C83#121116yes—
256C2×C4×C83C22×D414yes—
257C2×C4×C83C22×D414yes—
258C2×C4×C83C2×#4114yes—
259C2×C4×C83C2×#4114yes—
260C2×Dic83—216no—
261C2×Dic83—24no—
262C2×Dic83—24no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—22no—
272C2×Dic83—44no—
273C2×Dic83—44no—
274C2×Dic83—44no—
275C2×Dic83C22×D418yes—
276D332—232no—
277D332—432no—
278C4×D83—432no—
279C4×D83—416no—
280C22×C83C2×#12114yes—
281C22×C83#52914yes—
282C22×C83#54214yes—
283C22×C83#54614yes—
284C22×C83#54614yes—
285C22×C83#54714yes—
286C22×C83#54814yes—
287C22×C83C23×D411yescharacteristic
288C22×C83C23×D411yescharacteristic
289C22×C83C22×#4111yescharacteristic
290C22×C83C22×#4111yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C2×H(5,𝔽2)11yescharacteristic
293C22×C83C2×2−1+411yescharacteristic
294C2×D83—28no—
295C2×D83—24no—
296C2×D83—24no—
297C2×D83—28no—
298C2×D83—28no—
299C2×D83—24no—
300C2×D83—42no—
301C2×D83—44no—
302C2×D83—432no—
303C2×D83—416no—
304C2×D83—42no—
305C2×D83—44no—
306C2×D83—44no—
307C2×D83—42no—
308C2×D83—42no—
309C2×D83—44no—
310C4×C83—24no—
311C4×C83—44no—
312C4×C83—44no—
313C4×C83—42no—
314C4×C83C2×#12118yes—
315Dic83—28no—
316Dic83—42no—
317Dic83—42no—
318Dic83—42no—
319C2×#2427—831nomaximal, Sylow
320C2×C83#242718yescomplemented
321C2×C83C22×#12111yescharacteristic
322C2×C83C2×#52911yescharacteristic
323C2×C83C2×#54211yescharacteristic
324C2×C83C2×#54611yescharacteristic
325C2×C83C2×#54611yescharacteristic
326C2×C83C2×#54711yescharacteristic
327C2×C83C2×#54811yescharacteristic
328D83—48no—
329D83—44no—
330C22×#124—831no—
331C22×C22⋊Q8—831no—
332C22×#121—831no—
333C22×C4×D4—831no—
334#2427—8316no—
335C2×#397—832no—
336C2×#394—832no—
337C2×#391D8311yescharacteristic, complemented
338C2×#389—831no—
339C2×#386—832no—
340C2×#380—832no—
341C2×C4×C22⋊C4—831no—
342C83C2×#242711yescharacteristic, complemented, Sylow
343C23×Q8C2×D8311yescharacteristic
344C23×D4—832no—
345C23×D4—831no—
346C24×C4—832no—
347C24×C4—831no—
348C2×#124—838no—
349C2×#124—1668no—
350C2×C22⋊Q8—838no—
351C2×C22⋊Q8—1668no—
352C2×#121—838no—
353C2×#121—1668no—
354C2×C4×D4—838no—
355C2×C4×D4—1668no—
356C22×C4⋊C4—832no—
357C22×C4⋊C4—831no—
358C22×C4⋊C4—832no—
359C22×C4⋊C4C2×D8311yescharacteristic
360C22×C22⋊C4—831no—
361C22×C22⋊C4—832no—
362C22×C22⋊C4—832no—
363C22×C22⋊C4—832no—
364C22×C22⋊C4—832no—
365C22×C22⋊C4—832no—
366C22×C22⋊C4—831no—
367C22×C42—831no—
368C22×C42C2×D8311yescharacteristic
369#397—8316no—
370#394—8316no—
371#391C2×D8318yescomplemented
372#389—838no—
373#386—8316no—
374#380—8316no—
375C4×C22⋊C4—838no—
376C2×#95—831no—
377C2×#95—832no—
378C2×#95—831no—
379C2×#95C2×D8312yes—
380C2×#95C2×D8312yes—
381C25—832no—
382C25—831no—
383C22×Q8—24no—
384C22×Q8—24no—
385C22×Q8—24no—
386C22×Q8C22×D8314yes—
387C22×D4—3324no—
388C22×D4—838no—
389C22×D4—838no—
390C22×D4—834no—
391C22×D4—1668no—
392C22×D4—1668no—
393C22×D4—1664no—
394C22×D4—1664no—
395C23×C4—3324no—
396C23×C4—838no—
397C23×C4—832no—
398C23×C4—831no—
399C23×C4—831no—
400C23×C4—831no—
401C23×C4—834no—
402C23×C4—1668no—
403C23×C4—1664no—
404C23×C4—1664no—
405C23×C4—1664no—
406C23×C4—1668no—
407C23×C4C22×D8312yes—
408C23×C4C22×D8311yescharacteristic
409C23×C4C22×D8312yes—
410C23×C4C22×D8311yescharacteristic
411C23×C4C22×D8311yescharacteristic
412#124—16632no—
413C22⋊Q8—16632no—
414#121—16632no—
415C4×D4—16632no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—3324no—
418C2×C4⋊C4—838no—
419C2×C4⋊C4—838no—
420C2×C4⋊C4—834no—
421C2×C4⋊C4—838no—
422C2×C4⋊C4—1668no—
423C2×C4⋊C4C22×D8314yes—
424C2×C22⋊C4—838no—
425C2×C22⋊C4—838no—
426C2×C22⋊C4—838no—
427C2×C22⋊C4—838no—
428C2×C22⋊C4—838no—
429C2×C22⋊C4—834no—
430C2×C22⋊C4—834no—
431C2×C22⋊C4—1668no—
432C2×C22⋊C4—1668no—
433C2×C22⋊C4—1668no—
434C2×C22⋊C4—1668no—
435C2×C22⋊C4—1668no—
436C2×C22⋊C4—1664no—
437C2×C22⋊C4—1664no—
438C2×C42—24no—
439C2×C42—834no—
440C2×C42—1664no—
441C2×C42C22×D8314yes—
442#95—834no—
443#95—834no—
444#95—838no—
445#95C22×D8318yes—
446#95C22×D8318yes—
447C24—838no—
448C24—834no—
449C24—834no—
450C24—832no—
451C24—1662no—
452C24—1664no—
453C24—1662no—
454C24—1664no—
455C24—1662no—
456C24—1664no—
457C24—1662no—
458C24—1664no—
459C24—1664no—
460C24—1662no—
461C24—1662no—
462C24—1664no—
463C24C23×D8311yescharacteristic
464C2×Q8—28no—
465C2×Q8—28no—
466C2×Q8—28no—
467C2×Q8—416no—
468C2×D4—3328no—
469C2×D4—33232no—
470C2×D4—8316no—
471C2×D4—16616no—
472C2×D4—16616no—
473C2×D4—16632no—
474C2×D4—1668no—
475C2×D4—1668no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—33232no—
488C22×C4—3328no—
489C22×C4—33216no—
490C22×C4—838no—
491C22×C4—834no—
492C22×C4—832no—
493C22×C4—832no—
494C22×C4—832no—
495C22×C4—16616no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1668no—
506C22×C4—1668no—
507C22×C4—1668no—
508C22×C4—16616no—
509C22×C4—1664no—
510C22×C4—1664no—
511C22×C4D4×D8314yes—
512C22×C4D4×D8314yes—
513C22×C4#1305059014yes—
514C22×C4#1305059114yes—
515C22×C4C23×D8312yes—
516C22×C4C23×D8314yes—
517C22×C4C23×D8314yes—
518C22×C4C23×D8312yes—
519C22×C4C23×D8312yes—
520C4⋊C4—28no—
521C4⋊C4—3328no—
522C4⋊C4—8316no—
523C4⋊C4—16616no—
524C22⋊C4—16616no—
525C22⋊C4—16616no—
526C22⋊C4—16616no—
527C22⋊C4—16616no—
528C22⋊C4—16616no—
529C22⋊C4—1668no—
530C22⋊C4—1668no—
531C42—28no—
532C42—1668no—
533C23—3324no—
534C23—3328no—
535C23—3328no—
536C23—3324no—
537C23—3324no—
538C23—3328no—
539C23—3324no—
540C23—3328no—
541C23—838no—
542C23—834no—
543C23—1668no—
544C23—16616no—
545C23—16616no—
546C23—1668no—
547C23—16616no—
548C23—1668no—
549C23—1662no—
550C23—1664no—
551C23—1662no—
552C23—1664no—
553C23—1662no—
554C23—1664no—
555C23—1662no—
556C23—1664no—
557C23—1664no—
558C23—1662no—
559C23—1662no—
560C23—1664no—
561C23C2×D4×D8312yes—
562C23C2×D4×D8312yes—
563C23#41×D8312yes—
564C23C2×#1305059012yes—
565C23C2×#1305059112yes—
566C23#1458183812yes—
567C23#1458184012yes—
568C23C24×D8311yescharacteristic
569Q8—416no—
570D4—33232no—
571D4—16632no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—33232no—
588C2×C4—3324no—
589C2×C4—3324no—
590C2×C4—3324no—
591C2×C4—33216no—
592C2×C4—838no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1662no—
597C2×C4—1662no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—16616no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4#14581765116yes—
606C2×C4C2×D4×D8314yes—
607C2×C4C2×D4×D8314yes—
608C2×C4C2×#1305059014yes—
609C2×C4C2×#1305059114yes—
610C22—3322no—
611C22—3324no—
612C22—3324no—
613C22—3322no—
614C22—3322no—
615C22—3324no—
616C22—3322no—
617C22—3324no—
618C22—33232no—
619C22—33216no—
620C22—1664no—
621C22—1668no—
622C22—1668no—
623C22—1664no—
624C22—1668no—
625C22—1664no—
626C22C2×#1458176514yes—
627C22#1912823814yes—
628C22#1912832314yes—
629C22#1912834214yes—
630C22#1912834414yes—
631C22#1912835414yes—
632C22#1912836114yes—
633C22C22×D4×D8311yescharacteristic
634C22C22×D4×D8311yescharacteristic
635C22C2×#41×D8311yescharacteristic
636C22C22×#1305059011yescharacteristic
637C22C22×#1305059111yescharacteristic
638C22C2×#1458183811yescharacteristic
639C22C2×#1458184011yescharacteristic
640C4—24no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—3322no—
645C4—3322no—
646C4—3322no—
647C4—1668no—
648C4C2×#1458176518yes—
649C2—3328no—
650C2—3324no—
651C2#2491104118yesdirect product factor
652C2C22×#1458176511yescharacteristic
653C2C2×#1912823811yescharacteristic
654C2C2×#1912832311yescharacteristic
655C2C2×#1912834211yescharacteristic
656C2C2×#1912834411yescharacteristic
657C2C2×#1912835411yescharacteristic
658C2C2×#1912836111yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#391×C83C2
socleC24×C83C24
prime radicalC22×#14581767C2
hypercenterC2×#391D83
hypocenterC83C2×#2427
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C22×#14581781C2semidirect
2. C22×#14581768C2semidirect
4. C22×C4×D332C2semidirect
5. #24911041C2direct (N × H)
6. C2×#19127630C2semidirect
7. C2×#19127619C2semidirect
8. C2×#19127597C2semidirect
9. C2×#19127579C2semidirect
10. C2×#19127560C2semidirect
11. C2×C4×(C2×D83)⋊C4C2semidirect
12. C2×#391×C83C2semidirect
18. C2×#14581781C22semidirect
20. C2×#14581768C22semidirect
24. C2×C4×D332C22semidirect
36. #19127630C22semidirect
37. #19127619C22semidirect
38. #19127597C22semidirect
39. #19127579C22semidirect
40. #19127560C22semidirect
41. C4×(C2×D83)⋊C4C22semidirect
48. #391×C83C22semidirect
77. C2×C4⋊Dic83D4semidirect
165. C4⋊Dic83C2×D4semidirect
320. C2×C83#2427semidirect
337. C2×#391D83semidirect
342. C83C2×#2427semidirect
371. #391C2×D83semidirect
651. C2#24911041direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817812
C22×#145817682
C22×#145817672
C22×C4×D3322
#249110412
C2×#191276302
C2×#191276192
C2×#191275972
C2×#191275792
C2×#191275602
C2×C4×(C2×D83)⋊C42
C2×#391×C832
C2×#242783