Finite Group Databasebeta

← C2×#24910864 #35573102

Subgroups — C2×#24910864 #35573102

Order 21248.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581776C211yesmaximal, characteristic
2C22×#14581775C211yesmaximal, characteristic
3C23×Dic83⋊C4C211yesmaximal, characteristic, complemented
4#24910864C2116yesmaximal, direct product factor
5C2×#19127631C212yesmaximal, complemented
6C2×#19127623C212yesmaximal, complemented
7C2×#19127599C212yesmaximal, complemented
8C2×#19127592C212yesmaximal, complemented
9C2×#19127578C211yesmaximal, characteristic
10C2×#19127576C211yesmaximal, characteristic, complemented
11C2×#19127562C211yesmaximal, characteristic, complemented
12C22×#117×C83C211yesmaximal, characteristic, complemented
13C23×C22⋊D83C2211yescharacteristic
14C24×Dic83C2211yescharacteristic
15C24×Dic83C2211yescharacteristic
16C2×#14581776—28no—
17C2×#14581776C2218yes—
18C2×#14581775—28no—
19C2×#14581775C2218yes—
20C22×C4⋊Dic83C2212yes—
21C22×C4⋊Dic83C2212yes—
22C22×Dic83⋊C4—28no—
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2211yescharacteristic
26C22×Dic83⋊C4C2211yescharacteristic
27C22×Dic83⋊C4C2218yescomplemented
28C22×C22⋊Dic83C2211yescharacteristic
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2211yescharacteristic
31C22×(C2×D83)⋊C4C2212yes—
32C22×(C2×D83)⋊C4C2211yescharacteristic
33#19127631C22116yescomplemented
34#19127623C22116yescomplemented
35#19127599C22116yescomplemented
36#19127592C22116yescomplemented
37#19127578C2218yes—
38#19127576C2218yescomplemented
39#19127562C2218yescomplemented
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2212yes—
44C24×C4×C83C2211yescharacteristic
45C2×#117×C83—28no—
46C2×#117×C83C2218yescomplemented
47C22×C4⋊C4×C83C2212yes—
48C22×C22⋊C4×C83C2211yescharacteristic
49C22×C22⋊C4×C83C2211yescharacteristic
50C22×C42×C83C2212yes—
51C24×D83C2311yescharacteristic
52C22×C22⋊D83—24no—
53C22×C22⋊D83—24no—
54C22×C22⋊D83—24no—
55C22×C22⋊D83C2314yes—
56C23×Dic83—28no—
57C23×Dic83—24no—
58C23×Dic83—24no—
59C23×Dic83—24no—
60C23×Dic83—24no—
61C23×Dic83C2314yes—
62C23×Dic83C2314yes—
63C23×Dic83C2311yescharacteristic
64C23×Dic83C2311yescharacteristic
65C23×Dic83C2312yes—
66C23×Dic83C2311yescharacteristic
67C23×Dic83C2312yes—
68#14581776—232no—
69#14581775—232no—
70C2×C4⋊Dic83—28no—
71C2×C4⋊Dic83—28no—
72C2×C4⋊Dic83C2318yes—
73C2×C4⋊Dic83C2318yes—
74C2×Dic83⋊C4—28no—
75C2×Dic83⋊C4—28no—
76C2×Dic83⋊C4—232no—
77C2×Dic83⋊C4—24no—
78C2×Dic83⋊C4—24no—
79C2×Dic83⋊C4C2318yes—
80C2×Dic83⋊C4C2314yes—
81C2×Dic83⋊C4C2318yes—
82C2×Dic83⋊C4C2314yes—
83C2×C22⋊Dic83—24no—
84C2×C22⋊Dic83C2314yes—
85C2×(C2×D83)⋊C4—28no—
86C2×(C2×D83)⋊C4—24no—
87C2×(C2×D83)⋊C4—24no—
88C2×(C2×D83)⋊C4—28no—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2314yes—
91C2×(C2×D83)⋊C4C2318yes—
92C2×(C2×D83)⋊C4C2314yes—
93C83⋊#95C2318yes—
94C83⋊#95C2318yes—
95C83⋊#95C2318yes—
96C83⋊#95C2318yes—
97C25×C83C2311yescharacteristic
98C23×C4×C83—24no—
99C23×C4×C83—24no—
100C23×C4×C83D418yes—
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2312yes—
104C23×C4×C83C2311yescharacteristic
105C23×C4×C83C2314yes—
106#117×C83—232no—
107C2×C4⋊C4×C83—28no—
108C2×C4⋊C4×C83C2318yes—
109C2×C22⋊C4×C83—24no—
110C2×C22⋊C4×C83D418yes—
111C2×C22⋊C4×C83C2314yes—
112C2×C22⋊C4×C83C2314yes—
113C2×C42×C83—28no—
114C2×C42×C83C2318yes—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—24no—
119C23×D83—24no—
120C23×D83C2412yes—
121C2×C22⋊D83—28no—
122C2×C22⋊D83—28no—
123C2×C22⋊D83—28no—
124C2×C22⋊D83—416no—
125C22×Dic83—232no—
126C22×Dic83—28no—
127C22×Dic83—24no—
128C22×Dic83—24no—
129C22×Dic83—216no—
130C22×Dic83—28no—
131C22×Dic83—24no—
132C22×Dic83—28no—
133C22×Dic83—24no—
134C22×Dic83—22no—
135C22×Dic83—22no—
136C22×Dic83—22no—
137C22×Dic83—22no—
138C22×Dic83—28no—
139C22×Dic83—24no—
140C22×Dic83—22no—
141C22×Dic83—24no—
142C22×Dic83—24no—
143C22×Dic83—416no—
144C22×Dic83#4118yes—
145C22×Dic83C2414yes—
146C22×Dic83C2412yes—
147C22×Dic83C2412yes—
148C22×Dic83C2412yes—
149C22×Dic83C2414yes—
150C4⋊Dic83—216no—
151C4⋊Dic83—216no—
152Dic83⋊C4—28no—
153Dic83⋊C4—28no—
154Dic83⋊C4—216no—
155Dic83⋊C4—216no—
156C22⋊Dic83—28no—
157(C2×D83)⋊C4—28no—
158(C2×D83)⋊C4—28no—
159(C2×D83)⋊C4—216no—
160(C2×D83)⋊C4—216no—
161C24×C83—24no—
162C24×C83—24no—
163C24×C83C2×D414yes—
164C24×C83C2×D414yes—
165C24×C83#4114yes—
166C24×C83C2412yes—
167C24×C83C2411yescharacteristic
168C22×C4×C83—232no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—28no—
172C22×C4×C83—28no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—24no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—24no—
180C22×C4×C83C2×D418yes—
181C22×C4×C83C2×D4116yes—
182C22×C4×C83C2×D414yes—
183C22×C4×C83C2×D414yes—
184C22×C4×C83C2414yes—
185C22×C4×C83C2412yes—
186C22×C4×C83C2414yes—
187C22×C4×C83C2412yes—
188C4⋊C4×C83—216no—
189C22⋊C4×C83—28no—
190C22⋊C4×C83C2×D4116yes—
191C42×C83—216no—
192C23×C83—28no—
193C23×C83—28no—
194C23×C83—216no—
195C23×C83—216no—
196C23×C83—24no—
197C23×C83—24no—
198C23×C83#121116yes—
199C23×C83C22×D414yes—
200C23×C83C22×D412yes—
201C23×C83C22×D414yes—
202C23×C83C22×D412yes—
203C23×C83C2×#4114yes—
204C23×C83C2×#4114yes—
205C23×C83C2×#4112yes—
206C23×C832−1+414yes—
207C23×C83C2511yescharacteristic
208C22×D83—22no—
209C22×D83—22no—
210C22×D83—24no—
211C22×D83—22no—
212C22×D83—24no—
213C22×D83—44no—
214C22×D83—44no—
215C22×D83—44no—
216C22×D83—48no—
217C22×D83—48no—
218C2×C4×C83—232no—
219C2×C4×C83—28no—
220C2×C4×C83—28no—
221C2×C4×C83—216no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—48no—
232C2×C4×C83C22×D418yes—
233C2×C4×C83C22×D414yes—
234C2×C4×C83C22×D414yes—
235C22⋊D83—416no—
236C2×Dic83—232no—
237C2×Dic83—216no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—22no—
249C2×Dic83—24no—
250C2×Dic83—416no—
251C2×Dic83—48no—
252C2×Dic83—44no—
253C2×Dic83—44no—
254C2×Dic83—44no—
255C2×Dic83C2×#4118yes—
256C22×C83—24no—
257C22×C83—232no—
258C22×C83—24no—
259C22×C83—28no—
260C22×C83—28no—
261C22×C83C2×#12118yes—
262C22×C83C2×#12114yes—
263C22×C83#52514yes—
264C22×C83#53614yes—
265C22×C83#54718yes—
266C22×C83#55218yes—
267C22×C83C23×D411yescharacteristic
268C22×C83C23×D411yescharacteristic
269C22×C83C22×#4112yes—
270C22×C83C22×#4111yescharacteristic
271C22×C83C2×2−1+412yes—
272C2×D83—42no—
273C2×D83—42no—
274C2×D83—42no—
275C2×D83—44no—
276C2×D83—44no—
277C2×D83—88no—
278C4×C83—24no—
279C4×C83—24no—
280C4×C83—28no—
281C4×C83—44no—
282Dic83—28no—
283Dic83—44no—
284Dic83—42no—
285Dic83—42no—
286Dic83—42no—
287C2×#2413—831nomaximal, Sylow
288C2×C83—28no—
289C2×C83#241318yescomplemented
290C2×C83C22×#12111yescharacteristic
291C2×C83C2×#52511yescharacteristic
292C2×C83C2×#53611yescharacteristic
293C2×C83C2×#54712yes—
294C2×C83C2×#55212yes—
295D83—82no—
296C22×#123—831no—
297C22×#123—831no—
298C22×#117D8311yescharacteristic, complemented
299C23×C4⋊C4—831no—
300#2413—8316no—
301C2×#398—832no—
302C2×#396—832no—
303C2×#389—832no—
304C2×#388—832no—
305C2×#386—831no—
306C2×#385—831no—
307C2×#381—831no—
308C83C2×#241311yescharacteristic, complemented, Sylow
309C23×D4—831no—
310C24×C4—831no—
311C24×C4—831no—
312C24×C4C2×D8311yescharacteristic
313C2×#123—838no—
314C2×#123—838no—
315C2×#123—1668no—
316C2×#123—1668no—
317C2×#117—28no—
318C2×#117C2×D8318yescomplemented
319C22×C4⋊C4—832no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—831no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—832no—
324C22×C4⋊C4—831no—
325C22×C4⋊C4—838no—
326C22×C4⋊C4—1668no—
327C22×C4⋊C4C2×D8312yes—
328C22×C22⋊C4—831no—
329C22×C22⋊C4—831no—
330C22×C22⋊C4—832no—
331C22×C22⋊C4—832no—
332C22×C22⋊C4—831no—
333C22×C22⋊C4C2×D8311yescharacteristic
334C22×C22⋊C4C2×D8311yescharacteristic
335C22×C42C2×D8312yes—
336#398—8316no—
337#396—8316no—
338#389—8316no—
339#388—8316no—
340#386—838no—
341#385—838no—
342#381—838no—
343C2×#95—832no—
344C2×#95—832no—
345C2×#95—832no—
346C2×#95—832no—
347C25—831no—
348C25C22×D8311yescharacteristic
349C22×D4—834no—
350C22×D4—1664no—
351C22×D4—1664no—
352C22×D4—1664no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—832no—
356C23×C4—834no—
357C23×C4—834no—
358C23×C4—831no—
359C23×C4—831no—
360C23×C4—832no—
361C23×C4—831no—
362C23×C4—1664no—
363C23×C4—1664no—
364C23×C4—1664no—
365C23×C4—1668no—
366C23×C4—1664no—
367C23×C4D33218yes—
368C23×C4C22×D8312yes—
369C23×C4C22×D8312yes—
370C23×C4C22×D8314yes—
371C23×C4C22×D8311yescharacteristic
372C23×C4C22×D8311yescharacteristic
373#123—16632no—
374#123—16632no—
375#117—232no—
376C2×C4⋊C4—28no—
377C2×C4⋊C4—838no—
378C2×C4⋊C4—838no—
379C2×C4⋊C4—834no—
380C2×C4⋊C4—838no—
381C2×C4⋊C4—834no—
382C2×C4⋊C4—838no—
383C2×C4⋊C4—1668no—
384C2×C4⋊C4—1668no—
385C2×C4⋊C4—1668no—
386C2×C4⋊C4—1668no—
387C2×C4⋊C4—16632no—
388C2×C4⋊C4—1664no—
389C2×C4⋊C4—1664no—
390C2×C4⋊C4C22×D8318yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—834no—
393C2×C22⋊C4—834no—
394C2×C22⋊C4—838no—
395C2×C22⋊C4—834no—
396C2×C22⋊C4—838no—
397C2×C22⋊C4—1668no—
398C2×C22⋊C4—1664no—
399C2×C22⋊C4—1668no—
400C2×C22⋊C4—1664no—
401C2×C22⋊C4—1664no—
402C2×C22⋊C4C22⋊D8318yes—
403C2×C22⋊C4C22×D8314yes—
404C2×C22⋊C4C22×D8314yes—
405C2×C42—28no—
406C2×C42C22×D8318yes—
407#95—838no—
408#95—838no—
409#95—838no—
410#95—838no—
411C24—24no—
412C24—24no—
413C24—832no—
414C24—1662no—
415C24—1662no—
416C24—1664no—
417C24—1662no—
418C24—1664no—
419C24C2×C22⋊D8314yes—
420C24C2×D33214yes—
421C24#1305059014yes—
422C24C23×D8312yes—
423C24C23×D8311yescharacteristic
424C2×D4—33216no—
425C2×D4—1668no—
426C2×D4—1668no—
427C2×D4—1668no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—33216no—
441C22×C4—838no—
442C22×C4—834no—
443C22×C4—832no—
444C22×C4—832no—
445C22×C4—834no—
446C22×C4—832no—
447C22×C4—1664no—
448C22×C4—1662no—
449C22×C4—1668no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1668no—
453C22×C4—1664no—
454C22×C4—1662no—
455C22×C4—1664no—
456C22×C4—1668no—
457C22×C4—1664no—
458C22×C4—1662no—
459C22×C4—1664no—
460C22×C4—16616no—
461C22×C4—16632no—
462C22×C4—1668no—
463C22×C4—1664no—
464C22×C4—1664no—
465C22×C4C2×C22⋊D8318yes—
466C22×C4C2×D33214yes—
467C22×C4C2×D33214yes—
468C22×C4C2×D332116yes—
469C22×C4C23×D8314yes—
470C22×C4C23×D8314yes—
471C22×C4C23×D8312yes—
472C22×C4C23×D8312yes—
473C4⋊C4—216no—
474C4⋊C4—16616no—
475C4⋊C4—16616no—
476C4⋊C4—16616no—
477C4⋊C4—16616no—
478C4⋊C4—1668no—
479C4⋊C4—1668no—
480C22⋊C4—28no—
481C22⋊C4—16616no—
482C22⋊C4—1668no—
483C22⋊C4—16616no—
484C22⋊C4—1668no—
485C22⋊C4—1668no—
486C22⋊C4C2×C22⋊D83116yes—
487C42—216no—
488C23—216no—
489C23—216no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—3328no—
495C23—3328no—
496C23—3324no—
497C23—3324no—
498C23—3324no—
499C23—1662no—
500C23—1662no—
501C23—1664no—
502C23—1662no—
503C23—1664no—
504C23#14581766116yes—
505C23C22×C22⋊D8314yes—
506C23C22×C22⋊D8312yes—
507C23C22×D33214yes—
508C23C22×D33212yes—
509C23#41×D8314yes—
510C23C2×#1305059014yes—
511C23C2×#1305059012yes—
512C23#1458183914yes—
513C23C24×D8311yescharacteristic
514D4—33216no—
515C2×C4—216no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—33216no—
530C2×C4—3328no—
531C2×C4—3324no—
532C2×C4—3324no—
533C2×C4—3324no—
534C2×C4—838no—
535C2×C4—1664no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4—1664no—
540C2×C4—1662no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1662no—
544C2×C4—1664no—
545C2×C4—16632no—
546C2×C4—16616no—
547C2×C4—1664no—
548C2×C4—1664no—
549C2×C4C22×C22⋊D8318yes—
550C2×C4C22×D33214yes—
551C2×C4C22×D33214yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—3324no—
558C22—3322no—
559C22—3322no—
560C22—3324no—
561C22—3322no—
562C22—6648no—
563C22C2×#1458176618yes—
564C22C2×#1458176614yes—
565C22#1912821314yes—
566C22#1912827714yes—
567C22#1912835018yes—
568C22#1912838418yes—
569C22C23×C22⋊D8311yescharacteristic
570C22C23×D33211yescharacteristic
571C22C2×#41×D8312yes—
572C22C22×#1305059011yescharacteristic
573C22C2×#1458183912yes—
574C4—28no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—3324no—
579C4—3322no—
580C4—3322no—
581C4—3322no—
582C4—1668no—
583C2—28no—
584C2—6642no—
585C2#2491086418yesdirect product factor
586C2C22×#1458176611yescharacteristic
587C2C2×#1912821311yescharacteristic
588C2C2×#1912827711yescharacteristic
589C2C2×#1912835012yes—
590C2C2×#1912838412yes—
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC22×#117×C83C2
socleC24×C83C24
prime radicalC23×C22⋊D83C22
hypercenterC22×#117D83
hypocenterC83C2×#2413
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C23×Dic83⋊C4C2semidirect
4. #24910864C2direct (N × H)
5. C2×#19127631C2semidirect
6. C2×#19127623C2semidirect
7. C2×#19127599C2semidirect
8. C2×#19127592C2semidirect
10. C2×#19127576C2semidirect
11. C2×#19127562C2semidirect
12. C22×#117×C83C2semidirect
27. C22×Dic83⋊C4C22semidirect
33. #19127631C22semidirect
34. #19127623C22semidirect
35. #19127599C22semidirect
36. #19127592C22semidirect
38. #19127576C22semidirect
39. #19127562C22semidirect
46. C2×#117×C83C22semidirect
289. C2×C83#2413semidirect
298. C22×#117D83semidirect
308. C83C2×#2413semidirect
318. C2×#117C2×D83semidirect
585. C2#24910864direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817762
C22×#145817752
C23×Dic83⋊C42
#249108642
C2×#191276312
C2×#191276232
C2×#191275992
C2×#191275922
C2×#191275782
C2×#191275762
C2×#191275622
C22×#117×C832
C2×#241383