Finite Group Databasebeta

← C2×#24917379 #36272174

Subgroups — C2×#24917379 #36272174

Order 21248.

2378 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128214C211yesmaximal, characteristic
2C22×#14581750C211yesmaximal, characteristic, complemented
3#24917379C2116yesmaximal, direct product factor
4C2×#19128019C211yesmaximal, characteristic, complemented
5C2×#19128041C212yesmaximal, complemented
6C2×#19128028C212yesmaximal, complemented
7C2×#19128029C211yesmaximal, characteristic, complemented
8C2×#19128015C211yesmaximal, characteristic, complemented
9C2×#19127738C212yesmaximal, complemented
10C2×#19127696C212yesmaximal, complemented
11C2×C22⋊C8×D83C211yesmaximal, characteristic, complemented
12C2×#484×C83C211yesmaximal, characteristic, complemented
13C22×#13050591C2211yescharacteristic
14C23×C4×D83C2211yescharacteristic
15C22×C8×D83C2212yes—
16#19128214—24no—
17#19128214C2218yes—
18C2×#14581777—21no—
19C2×#14581768C2211yescharacteristic
20C2×#14581767C2211yescharacteristic
21C2×#14581766C2211yescharacteristic
22C2×#14581762—21no—
23C2×#14581754—21no—
24C2×(C22⋊D83)⋊C4—21no—
25C2×#14581750—21no—
26C2×#14581750—21no—
27C2×#14581750—24no—
28C2×#14581750C2212yes—
29C2×#14581750C2212yes—
30C2×#14581750C2211yescharacteristic
31C2×#14581750C2218yescomplemented
32C22×C4⋊Dic83C2211yescharacteristic
33C22×(C2×D83)⋊C4—21no—
34C22×C4×Dic83—21no—
35#19128041C22116yescomplemented
36#19128029C2218yescomplemented
37#19128028C22116yescomplemented
38#19128019C2218yescomplemented
39#19128015C2218yescomplemented
40#19127738C22116yescomplemented
41C2×#14581726C2212yescomplemented
42C2×#14581723C2212yescomplemented
43C2×#14581723C2212yescomplemented
44#19127696C22116yescomplemented
45C2×#14581716C2212yescomplemented
46C2×#14581714C2212yescomplemented
47C2×#14581712C2212yescomplemented
48C2×C22⋊(C83⋊C8)C2211yescharacteristic
49C2×(C2×D83)⋊C8C2212yes—
50C22⋊C8×D83C2218yescomplemented
51C2×C22⋊Q8×C83C2211yescharacteristic
52C22×C4⋊C4×C83C2211yescharacteristic
53#484×C83C2218yescomplemented
54C2×#107×C83C2212yescomplemented
55C2×#103×C83C2212yescomplemented
56C2×C22⋊C8×C83C2211yescharacteristic
57C24×D83D411yescharacteristic
58C2×#13050591—22no—
59C2×#13050591—28no—
60C2×#13050591C2314yes—
61C22×D332—21no—
62C22×D332—21no—
63C22×D332C2311yescharacteristic
64C22×C22⋊D83—21no—
65C23×Dic83—21no—
66C23×Dic83D411yescharacteristic
67C22×C4×D83—28no—
68C22×C4×D83—21no—
69C22×C4×D83D411yescharacteristic
70C22×C4×D83D411yescharacteristic
71C22×C4×D83D414yes—
72C22×C4×D83C2311yescharacteristic
73C22×C4×D83C2314yes—
74C22×C4×D83C2311yescharacteristic
75C22×C83⋊C8C2312yes—
76C2×C8×D83—28no—
77C2×C8×D83C2318yes—
78#14581777—28no—
79#14581768—22no—
80#14581768C2314yes—
81#14581767—22no—
82#14581767C2314yes—
83#14581766—22no—
84#14581766C2314yes—
85#14581762—28no—
86#14581754—28no—
87(C22⋊D83)⋊C4—28no—
88#14581750—24no—
89#14581750—24no—
90#14581750—28no—
91#14581750—28no—
92#14581750—22no—
93#14581750C2318yes—
94#14581750C2318yes—
95#14581750C2314yes—
96C2×C4⋊Dic83—22no—
97C2×C4⋊Dic83—21no—
98C2×C4⋊Dic83C2312yes—
99C2×C4⋊Dic83C2314yes—
100C2×C4⋊Dic83C2311yescharacteristic
101C2×Dic83⋊C4—21no—
102C2×(C2×D83)⋊C4—22no—
103C2×(C2×D83)⋊C4—22no—
104C2×(C2×D83)⋊C4—28no—
105C2×(C2×D83)⋊C4—21no—
106C2×(C2×D83)⋊C4—21no—
107C2×(C2×D83)⋊C4—21no—
108C2×(C2×D83)⋊C4—21no—
109C2×C4×Dic83—22no—
110C2×C4×Dic83—22no—
111C2×C4×Dic83—28no—
112C2×C4×Dic83—21no—
113#14581726C2318yescomplemented
114#14581723C2318yescomplemented
115#14581723C2318yescomplemented
116#14581716C2318yescomplemented
117#14581714C2318yescomplemented
118#14581712C2318yescomplemented
119C22⋊(C83⋊C8)C2314yes—
120(C2×D83)⋊C8C2318yes—
121C22×Q8×C83C2311yescharacteristic
122C23×C4×C83—21no—
123C23×C4×C83C2311yescharacteristic
124C22×C8×C83C2312yes—
125C22⋊Q8×C83—22no—
126C22⋊Q8×C83C2314yes—
127C2×C4⋊C4×C83—22no—
128C2×C4⋊C4×C83—21no—
129C2×C4⋊C4×C83—21no—
130C2×C4⋊C4×C83C2314yes—
131C2×C4⋊C4×C83C2312yes—
132C2×C4⋊C4×C83C2311yescharacteristic
133C2×C22⋊C4×C83—21no—
134#107×C83C2318yescomplemented
135#103×C83C2318yescomplemented
136C22⋊C8×C83C2314yes—
137C23×D83—24no—
138C23×D83—24no—
139C23×D83—21no—
140C23×D83—42no—
141C23×D83C2×D411yescharacteristic
142C23×D83C2×D414yes—
143C23×D83C2×D411yescharacteristic
144#13050591—216no—
145#13050591—48no—
146C2×D332—24no—
147C2×D332—24no—
148C2×D332—24no—
149C2×D332—21no—
150C2×D332—44no—
151C2×D332—44no—
152C2×D332C2412yes—
153C2×C22⋊D83—24no—
154C2×C22⋊D83—44no—
155C22×Dic83—24no—
156C22×Dic83—24no—
157C22×Dic83—22no—
158C22×Dic83—21no—
159C22×Dic83—44no—
160C22×Dic83C2×D411yescharacteristic
161C22×Dic83C2×D414yes—
162C22×Dic83C2×D411yescharacteristic
163C2×C4×D83—24no—
164C2×C4×D83—24no—
165C2×C4×D83—24no—
166C2×C4×D83—216no—
167C2×C4×D83—216no—
168C2×C4×D83—24no—
169C2×C4×D83—44no—
170C2×C4×D83C2×D412yes—
171C2×C4×D83C2×D414yes—
172C2×C4×D83C2×D414yes—
173C2×C4×D83C2×D412yes—
174C2×C4×D83#4118yes—
175C2×C4×D83C2412yes—
176C2×C4×D83C2412yes—
177C2×C83⋊C8—24no—
178C2×C83⋊C8C2414yes—
179C8×D83—216no—
180C4⋊Dic83—21no—
181C4⋊Dic83—24no—
182C4⋊Dic83—22no—
183C4⋊Dic83C2412yes—
184C4⋊Dic83C2414yes—
185Dic83⋊C4—24no—
186(C2×D83)⋊C4—24no—
187(C2×D83)⋊C4—24no—
188(C2×D83)⋊C4—28no—
189(C2×D83)⋊C4—28no—
190(C2×D83)⋊C4—24no—
191(C2×D83)⋊C4—24no—
192C4×Dic83—28no—
193C4×Dic83—28no—
194C4×Dic83—24no—
195C24×C83C2×D411yescharacteristic
196C2×Q8×C83—21no—
197C2×Q8×C83—24no—
198C2×Q8×C83C2412yes—
199C22×C4×C83—24no—
200C22×C4×C83—21no—
201C22×C4×C83—21no—
202C22×C4×C83—24no—
203C22×C4×C83—22no—
204C22×C4×C83—44no—
205C22×C4×C83C2×D412yes—
206C22×C4×C83C2×D411yescharacteristic
207C22×C4×C83C2412yes—
208C22×C4×C83C2411yescharacteristic
209C2×C8×C83—24no—
210C2×C8×C83C2414yes—
211C4⋊C4×C83—21no—
212C4⋊C4×C83—24no—
213C4⋊C4×C83—24no—
214C4⋊C4×C83—22no—
215C4⋊C4×C83C2412yes—
216C4⋊C4×C83C2414yes—
217C22⋊C4×C83—24no—
218C23×C83—22no—
219C23×C83—22no—
220C23×C83#13514yes—
221C23×C83C22×D412yes—
222C23×C83C22×D411yescharacteristic
223C22×D83—22no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—28no—
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—28no—
230C22×D83—22no—
231C22×D83—44no—
232C22×D83—416no—
233C22×D83—42no—
234C22×D83—42no—
235C22×D83—42no—
236C22×D83#12314yes—
237C22×D83C22×D412yes—
238C22×D83C22×D412yes—
239Q8×C83—24no—
240Q8×C83—42no—
241C2×C4×C83—22no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—24no—
247C2×C4×C83—48no—
248C2×C4×C83—42no—
249C2×C4×C83—44no—
250C2×C4×C83—42no—
251C2×C4×C83C22×D412yes—
252C2×C4×C83C22×D411yescharacteristic
253C2×C4×C83C2×#4114yes—
254C2×C4×C83C2511yescharacteristic
255C22⋊D83—48no—
256C2×Dic83—22no—
257C2×Dic83—28no—
258C2×Dic83—24no—
259C2×Dic83—22no—
260C2×Dic83—48no—
261C2×Dic83—44no—
262C2×Dic83—42no—
263C2×Dic83#12314yes—
264C2×Dic83C22×D412yes—
265C2×Dic83C22×D412yes—
266D332—24no—
267D332—48no—
268D332—48no—
269D332—42no—
270C4×D83—28no—
271C4×D83—24no—
272C4×D83—24no—
273C4×D83—28no—
274C4×D83—48no—
275C4×D83#12318yes—
276C4×D83C2×#4118yes—
277C8×C83—24no—
278C83⋊C8—24no—
279C22×C83—28no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83C2×#12312yes—
283C22×C83C2×#13514yes—
284C22×C83C2×#13512yes—
285C22×C83#57512yes—
286C22×C83C23×D411yescharacteristic
287C2×D83—28no—
288C2×D83—22no—
289C2×D83—22no—
290C2×D83—22no—
291C2×D83—22no—
292C2×D83—22no—
293C2×D83—416no—
294C2×D83—44no—
295C2×D83—42no—
296C2×D83—42no—
297C2×D83—42no—
298C2×D83—84no—
299C2×D83C2×#12314yes—
300C4×C83—22no—
301C4×C83—42no—
302C4×C83—42no—
303C4×C83—44no—
304C4×C83C2×#12312yes—
305C4×C83C22×#4112yes—
306Dic83—22no—
307Dic83—44no—
308Dic83—42no—
309Dic83C2×#12314yes—
310C2×#3091—831nomaximal, Sylow
311C2×C83—24no—
312C2×C83#309114yescomplemented
313C2×C83C22×#12311yescharacteristic
314C2×C83C22×#13511yescharacteristic
315C2×C83C2×#57511yescharacteristic
316D83—24no—
317D83—42no—
318D83—82no—
319C2×#525—831no—
320C22×#117—831no—
321#3091—8316no—
322C2×#484D8311yescharacteristic, complemented
323C2×#481—831no—
324C2×#485—832no—
325C2×#482—832no—
326C2×#483—831no—
327C2×#480—831no—
328C2×#427—832no—
329C2×#416—832no—
330C22×C22⋊C8—831no—
331C83C2×#309111yescharacteristic, complemented, Sylow
332C22×#41—831no—
333C24×C4—831no—
334C23×C8—832no—
335#525—838no—
336#525—1664no—
337C2×#123—1661no—
338C2×C22⋊Q8—831no—
339C2×C22⋊Q8C2×D8311yescharacteristic
340C2×#121—831no—
341C2×#121—831no—
342C2×#120—1661no—
343C2×C4×D4—1661no—
344C2×C4×D4—1661no—
345C2×#117—832no—
346C2×#117—838no—
347C2×#117—832no—
348C2×#117—831no—
349C2×#117—1661no—
350C2×#117—1661no—
351C2×#117—1664no—
352C22×C4⋊C4—831no—
353C22×C4⋊C4C2×D8311yescharacteristic
354C22×C22⋊C4—1661no—
355C22×C42—1661no—
356#485—8316no—
357#484C2×D8318yescomplemented
358#483—838no—
359#482—8316no—
360#481—838no—
361#480—838no—
362#427—8316no—
363C2×#107—832no—
364C2×#107C2×D8312yescomplemented
365C2×#106—832no—
366C2×#106—832no—
367#416—8316no—
368C2×#103—832no—
369C2×#103C2×D8312yescomplemented
370C2×#102—832no—
371C2×#102—832no—
372C2×C22⋊C8—831no—
373C2×C22⋊C8—832no—
374C2×C22⋊C8—838no—
375C2×C22⋊C8C2×D8311yescharacteristic
376C25—831no—
377C2×#41—834no—
378C2×#41—1668no—
379C2×#41—1662no—
380C22×Q8C22×D8311yescharacteristic
381C22×D4—831no—
382C22×D4—1661no—
383C22×D4—1661no—
384C22×D4—1661no—
385C23×C4—21no—
386C23×C4—831no—
387C23×C4—831no—
388C23×C4—831no—
389C23×C4—831no—
390C23×C4—834no—
391C23×C4—831no—
392C23×C4—834no—
393C23×C4—1661no—
394C23×C4—1668no—
395C23×C4—1661no—
396C23×C4C22×D8311yescharacteristic
397C22×C8—832no—
398C22×C8—838no—
399C22×C8—1668no—
400C22×C8C22×D8312yes—
401#123—1668no—
402C22⋊Q8—22no—
403C22⋊Q8—834no—
404C22⋊Q8—1662no—
405C22⋊Q8C22×D8314yes—
406#121—834no—
407#121—834no—
408#121—1662no—
409#121—1662no—
410#120—1668no—
411C4×D4—1668no—
412C4×D4—1668no—
413#117—838no—
414#117—838no—
415#117—834no—
416#117—1664no—
417#117—1668no—
418#117—1664no—
419#117—1668no—
420#117—1662no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—832no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—831no—
427C2×C4⋊C4—1661no—
428C2×C4⋊C4—1662no—
429C2×C4⋊C4—1661no—
430C2×C4⋊C4C22×D8312yes—
431C2×C4⋊C4C22×D8314yes—
432C2×C4⋊C4C22×D8311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1661no—
435C2×C22⋊C4—1668no—
436C2×C22⋊C4—1662no—
437C2×C22⋊C4—1662no—
438C2×C22⋊C4—1661no—
439C2×C22⋊C4—1661no—
440C2×C22⋊C4—1661no—
441C2×C42—1668no—
442C2×C42—1662no—
443C2×C42—1662no—
444C2×C42—1661no—
445#107—838no—
446#107C22×D8318yescomplemented
447#106—838no—
448#106—838no—
449#103—838no—
450#103C22×D8318yescomplemented
451#102—838no—
452#102—838no—
453C22⋊C8—834no—
454C22⋊C8—838no—
455C22⋊C8C22×D8314yes—
456C24—3322no—
457C24—834no—
458C24—831no—
459C24—831no—
460C24—1664no—
461C24—1661no—
462C24—1664no—
463C24D4×D8311yescharacteristic
464#41—3328no—
465#41—16616no—
466C2×Q8—21no—
467C2×Q8—24no—
468C2×Q8C23×D8312yes—
469C2×D4—3324no—
470C2×D4—3324no—
471C2×D4—3324no—
472C2×D4—832no—
473C2×D4—1664no—
474C2×D4—1664no—
475C2×D4—1664no—
476C2×D4—1661no—
477C2×D4—1664no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—44no—
484C22×C4—3324no—
485C22×C4—3324no—
486C22×C4—831no—
487C22×C4—832no—
488C22×C4—838no—
489C22×C4—832no—
490C22×C4—834no—
491C22×C4—831no—
492C22×C4—834no—
493C22×C4—832no—
494C22×C4—834no—
495C22×C4—832no—
496C22×C4—1664no—
497C22×C4—1664no—
498C22×C4—16616no—
499C22×C4—16616no—
500C22×C4—1664no—
501C22×C4—1664no—
502C22×C4—1662no—
503C22×C4—1661no—
504C22×C4—1664no—
505C22×C4—1664no—
506C22×C4D4×D8312yes—
507C22×C4D4×D8311yescharacteristic
508C22×C4C23×D8312yes—
509C22×C4C23×D8311yescharacteristic
510C2×C8—24no—
511C2×C8—834no—
512C2×C8—16616no—
513C2×C8—1664no—
514C2×C8C23×D8314yes—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—834no—
520C4⋊C4—832no—
521C4⋊C4—1664no—
522C4⋊C4—1664no—
523C4⋊C4—1662no—
524C4⋊C4—1661no—
525C4⋊C4C23×D8314yes—
526C4⋊C4C23×D8312yes—
527C22⋊C4—24no—
528C22⋊C4—1664no—
529C22⋊C4—1668no—
530C22⋊C4—1668no—
531C22⋊C4—1664no—
532C22⋊C4—1664no—
533C22⋊C4—1664no—
534C42—1668no—
535C42—1668no—
536C42—1664no—
537C23—22no—
538C23—22no—
539C23—33216no—
540C23—3324no—
541C23—3322no—
542C23—3322no—
543C23—3322no—
544C23—834no—
545C23—832no—
546C23—832no—
547C23—1668no—
548C23—1662no—
549C23—1662no—
550C23—1668no—
551C23—1662no—
552C23—1662no—
553C23—1662no—
554C23—1662no—
555C23#1458181514yes—
556C23C2×D4×D8312yes—
557C23C2×D4×D8311yescharacteristic
558Q8—24no—
559Q8—42no—
560D4—3328no—
561D4—3328no—
562D4—3328no—
563D4—3322no—
564D4—1664no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—3328no—
576C2×C4—3328no—
577C2×C4—3324no—
578C2×C4—3322no—
579C2×C4—834no—
580C2×C4—832no—
581C2×C4—838no—
582C2×C4—838no—
583C2×C4—832no—
584C2×C4—1668no—
585C2×C4—1668no—
586C2×C4—1664no—
587C2×C4—1662no—
588C2×C4—1668no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4C2×D4×D8312yes—
593C2×C4C2×D4×D8311yescharacteristic
594C2×C4#41×D8314yes—
595C2×C4C24×D8311yescharacteristic
596C8—24no—
597C8—1664no—
598C22—22no—
599C22—22no—
600C22—28no—
601C22—33216no—
602C22—3324no—
603C22—3322no—
604C22—3322no—
605C22—3322no—
606C22—834no—
607C22—6644no—
608C22—1662no—
609C22—1662no—
610C22—1662no—
611C22—1662no—
612C22—1662no—
613C22—1668no—
614C22#123×D8312yes—
615C22C2×#1458181514yes—
616C22C2×#1458181512yes—
617C22#1912849712yes—
618C22C22×D4×D8311yescharacteristic
619C4—22no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—3324no—
624C4—3322no—
625C4—834no—
626C4—1662no—
627C4#123×D8312yes—
628C4C2×#41×D8312yes—
629C2—24no—
630C2—3322no—
631C2—6642no—
632C2—1664no—
633C2#2491737914yesdirect product factor
634C2C2×#123×D8311yescharacteristic
635C2C22×#1458181511yescharacteristic
636C2C2×#1912849711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#484×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128214C2
hypercenterC2×#484D83
hypocenterC83C2×#3091
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#14581750C2semidirect
3. #24917379C2direct (N × H)
4. C2×#19128019C2semidirect
5. C2×#19128041C2semidirect
6. C2×#19128028C2semidirect
7. C2×#19128029C2semidirect
8. C2×#19128015C2semidirect
9. C2×#19127738C2semidirect
10. C2×#19127696C2semidirect
11. C2×C22⋊C8×D83C2semidirect
12. C2×#484×C83C2semidirect
31. C2×#14581750C22semidirect
35. #19128041C22semidirect
36. #19128029C22semidirect
37. #19128028C22semidirect
38. #19128019C22semidirect
39. #19128015C22semidirect
40. #19127738C22semidirect
41. C2×#14581726C22semidirect
42. C2×#14581723C22semidirect
43. C2×#14581723C22semidirect
44. #19127696C22semidirect
45. C2×#14581716C22semidirect
46. C2×#14581714C22semidirect
47. C2×#14581712C22semidirect
50. C22⋊C8×D83C22semidirect
53. #484×C83C22semidirect
54. C2×#107×C83C22semidirect
55. C2×#103×C83C22semidirect
113. #14581726C23semidirect
114. #14581723C23semidirect
115. #14581723C23semidirect
116. #14581716C23semidirect
117. #14581714C23semidirect
118. #14581712C23semidirect
134. #107×C83C23semidirect
135. #103×C83C23semidirect
312. C2×C83#3091semidirect
322. C2×#484D83semidirect
331. C83C2×#3091semidirect
357. #484C2×D83semidirect
364. C2×#107C2×D83semidirect
369. C2×#103C2×D83semidirect
446. #107C22×D83semidirect
450. #103C22×D83semidirect
633. C2#24917379direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282142
C22×#145817502
#249173792
C2×#191280192
C2×#191280412
C2×#191280282
C2×#191280292
C2×#191280152
C2×#191277382
C2×#191276962
C2×C22⋊C8×D832
C2×#484×C832
C2×#309183