Finite Group Databasebeta

← C2×#24917386 #36272182

Subgroups — C2×#24917386 #36272182

Order 21248.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128212C211yesmaximal, characteristic
2#24917386C2116yesmaximal, direct product factor
3C2×#19128036C212yesmaximal, complemented
4C2×#19128019C212yesmaximal, complemented
5C2×#19128031C212yesmaximal, complemented
6C2×#19128014C212yesmaximal, complemented
7C2×#19127741C212yesmaximal, complemented
8C2×#19127700C212yesmaximal, complemented
9C2×#19127665C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C83C211yesmaximal, characteristic, complemented
11C22×#13050590C2211yescharacteristic
12C22×#13050574C2212yes—
13#19128212—24no—
14#19128212C2218yes—
15C2×#14581775—21no—
16C2×#14581771C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×#14581766C2211yescharacteristic
19C2×#14581766C2211yescharacteristic
20C2×#14581761—21no—
21C2×C4×C22⋊D83—21no—
22C2×C4×C22⋊D83—21no—
23C2×#14581751C2211yescharacteristic
24#19128036C22116yescomplemented
25#19128031C22116yescomplemented
26#19128019C22116yescomplemented
27#19128014C22116yescomplemented
28#19127741C22116yescomplemented
29C2×#14581725C2214yescomplemented
30C2×#14581723C2214yescomplemented
31#19127700C22116yescomplemented
32C2×#14581715C2214yescomplemented
33C2×#14581712C2214yescomplemented
34#19127665C2218yescomplemented
35C2×C22⋊(C83⋊C8)C2212yes—
36C2×C22⋊(C83⋊C8)C2212yes—
37C24×C4×C83—21no—
38C24×C4×C83C2211yescharacteristic
39C22×C4⋊C4×C83—24no—
40C22×C4⋊C4×C83—21no—
41C22×C4⋊C4×C83—21no—
42C22×C4⋊C4×C83—21no—
43C22×C4⋊C4×C83C2212yes—
44C22×C4⋊C4×C83C2212yes—
45C22×C4⋊C4×C83C2212yes—
46C22×C4⋊C4×C83C2218yescomplemented
47C2×#13050590—22no—
48C2×#13050590—28no—
49C2×#13050590C2314yes—
50C22×Dic166C2311yescharacteristic
51C22×D332C2311yescharacteristic
52C22×C22⋊D83—21no—
53C22×C22⋊D83—21no—
54C22×C22⋊D83—21no—
55C22×C4×D83—21no—
56C2×#13050574—28no—
57C2×#13050574C2318yes—
58C22×C83⋊C8C2314yes—
59#14581775—28no—
60#14581771—22no—
61#14581771—22no—
62#14581771C2314yes—
63#14581771C2314yes—
64#14581766—22no—
65#14581766—22no—
66#14581766C2314yes—
67#14581766C2314yes—
68#14581761—28no—
69C4×C22⋊D83—28no—
70C4×C22⋊D83—28no—
71#14581751—22no—
72#14581751C2314yes—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×C4×Dic83—21no—
83#14581725C23116yescomplemented
84#14581723C23116yescomplemented
85#14581715C23116yescomplemented
86#14581712C23116yescomplemented
87C22⋊(C83⋊C8)C2318yes—
88C22⋊(C83⋊C8)C2318yes—
89C25×C83D411yescharacteristic
90C23×C4×C83—28no—
91C23×C4×C83—22no—
92C23×C4×C83—22no—
93C23×C4×C83—22no—
94C23×C4×C83—28no—
95C23×C4×C83—48no—
96C23×C4×C83D411yescharacteristic
97C23×C4×C83D411yescharacteristic
98C23×C4×C83D411yescharacteristic
99C23×C4×C83D414yes—
100C23×C4×C83C2311yescharacteristic
101C23×C4×C83C2311yescharacteristic
102C23×C4×C83C2314yes—
103C23×C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83—24no—
105C2×C4⋊C4×C83—24no—
106C2×C4⋊C4×C83—22no—
107C2×C4⋊C4×C83—22no—
108C2×C4⋊C4×C83—22no—
109C2×C4⋊C4×C83—24no—
110C2×C4⋊C4×C83—28no—
111C2×C4⋊C4×C83—28no—
112C2×C4⋊C4×C83—28no—
113C2×C4⋊C4×C83C2314yes—
114C2×C4⋊C4×C83C2318yes—
115C2×C4⋊C4×C83C2318yes—
116C2×C4⋊C4×C83C2318yes—
117C23×D83—21no—
118#13050590—216no—
119#13050590—48no—
120C2×Dic166—24no—
121C2×Dic166—21no—
122C2×Dic166C2412yes—
123C2×D332—24no—
124C2×D332—21no—
125C2×D332C2412yes—
126C2×C22⋊D83—24no—
127C2×C22⋊D83—24no—
128C2×C22⋊D83—24no—
129C2×C22⋊D83—44no—
130C2×C22⋊D83—44no—
131C2×C22⋊D83—44no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—24no—
136C2×C4×D83—44no—
137#13050574—216no—
138C2×C83⋊C8—28no—
139C2×C83⋊C8C2418yes—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83C2412yes—
143C4⋊Dic83C2412yes—
144Dic83⋊C4—24no—
145Dic83⋊C4—24no—
146C22⋊Dic83—24no—
147C22⋊Dic83—24no—
148C22⋊Dic83—24no—
149(C2×D83)⋊C4—24no—
150(C2×D83)⋊C4—24no—
151C4×Dic83—24no—
152C24×C83—24no—
153C24×C83—24no—
154C24×C83—24no—
155C24×C83C2×D414yes—
156C24×C83C2×D411yescharacteristic
157C24×C83C2×D411yescharacteristic
158C24×C83C2×D411yescharacteristic
159C22×C4×C83—24no—
160C22×C4×C83—24no—
161C22×C4×C83—24no—
162C22×C4×C83—216no—
163C22×C4×C83—24no—
164C22×C4×C83—24no—
165C22×C4×C83—216no—
166C22×C4×C83—28no—
167C22×C4×C83—28no—
168C22×C4×C83—28no—
169C22×C4×C83—24no—
170C22×C4×C83—48no—
171C22×C4×C83—48no—
172C22×C4×C83—432no—
173C22×C4×C83—48no—
174C22×C4×C83C2×D412yes—
175C22×C4×C83C2×D412yes—
176C22×C4×C83C2×D414yes—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D414yes—
180C22×C4×C83C2×D414yes—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D412yes—
183C22×C4×C83#4118yes—
184C22×C4×C83C2412yes—
185C22×C4×C83C2412yes—
186C22×C4×C83C2411yescharacteristic
187C22×C4×C83C2412yes—
188C4⋊C4×C83—28no—
189C4⋊C4×C83—28no—
190C4⋊C4×C83—28no—
191C4⋊C4×C83—24no—
192C4⋊C4×C83C2418yes—
193C23×C83—232no—
194C23×C83—28no—
195C23×C83—22no—
196C23×C83—22no—
197C23×C83—28no—
198C23×C83—22no—
199C23×C83—22no—
200C23×C83—28no—
201C23×C83—22no—
202C23×C83—22no—
203C23×C83#12314yes—
204C23×C83AGL(1,ℤ/8)14yes—
205C23×C83#13714yes—
206C23×C83C22×D412yes—
207C23×C83C22×D412yes—
208C23×C83C22×D412yes—
209C23×C83C22×D411yescharacteristic
210C22×D83—22no—
211C22×D83—42no—
212C22×D83—42no—
213C22×D83—42no—
214C2×C4×C83—28no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—28no—
220C2×C4×C83—28no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
223C2×C4×C83—28no—
224C2×C4×C83—416no—
225C2×C4×C83—416no—
226C2×C4×C83—416no—
227C2×C4×C83—48no—
228C2×C4×C83#12314yes—
229C2×C4×C83#12318yes—
230C2×C4×C83C22×D412yes—
231C2×C4×C83C22×D412yes—
232C2×C4×C83C22×D412yes—
233C2×C4×C83C22×D411yescharacteristic
234C2×C4×C83C2×#4114yes—
235C2×C4×C83C2×#4118yes—
236C2×C4×C83C2511yescharacteristic
237C22⋊D83—48no—
238C22⋊D83—48no—
239C22⋊D83—48no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246D332—24no—
247D332—42no—
248C4×D83—48no—
249Dic166—24no—
250Dic166—42no—
251C83⋊C8—28no—
252C22×C83—232no—
253C22×C83—28no—
254C22×C83—28no—
255C22×C83—28no—
256C22×C83—22no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—22no—
261C22×C83—22no—
262C22×C83C2×#12314yes—
263C22×C83C2×#12312yes—
264C22×C83C2×AGL(1,ℤ/8)14yes—
265C22×C83C2×AGL(1,ℤ/8)12yes—
266C22×C83C2×#13714yes—
267C22×C83C2×#13712yes—
268C22×C83C23×D411yescharacteristic
269C2×D83—42no—
270C2×D83—42no—
271C2×D83—42no—
272C2×D83—84no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—48no—
276C4×C83C2×#12312yes—
277C4×C83C2×#12314yes—
278C4×C83C22×#4112yes—
279Dic83—42no—
280Dic83—42no—
281Dic83—42no—
282C2×#3092—831nomaximal, Sylow
283C2×C83—24no—
284C2×C83—24no—
285C2×C83—24no—
286C2×C83#309214yescomplemented
287C2×C83C22×#12311yescharacteristic
288C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
289C2×C83C22×#13711yescharacteristic
290D83—82no—
291C2×#525—831no—
292C23×C4⋊C4D8311yescharacteristic, complemented
293#3092—8316no—
294C2×#484—832no—
295C2×#481—832no—
296C2×#483—832no—
297C2×#480—832no—
298C2×#428—832no—
299C2×#417—832no—
300C2×#407—831no—
301C83C2×#309211yescharacteristic, complemented, Sylow
302C22×#41—831no—
303C24×C4—21no—
304C24×C4C2×D8311yescharacteristic
305C22×M16—832no—
306#525—838no—
307#525—1664no—
308C2×#123—1661no—
309C2×C22⋊Q8—831no—
310C2×C22⋊Q8—831no—
311C2×#121—831no—
312C2×#121—831no—
313C2×#120—1661no—
314C2×C4×D4—1661no—
315C2×C4×D4—1661no—
316C2×#117—831no—
317C22×C4⋊C4—24no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D8312yes—
322C22×C4⋊C4C2×D8312yes—
323C22×C4⋊C4C2×D8312yes—
324C22×C4⋊C4C2×D8318yescomplemented
325#484—8316no—
326#483—8316no—
327#481—8316no—
328#480—8316no—
329#428—8316no—
330C2×#107—834no—
331C2×#106—834no—
332#417—8316no—
333C2×#103—834no—
334C2×#102—834no—
335#407—838no—
336C2×C22⋊C8—832no—
337C2×C22⋊C8—832no—
338C25C22⋊D8311yescharacteristic
339C2×#41—834no—
340C2×#41—1668no—
341C2×#41—1662no—
342C22×Q8—831no—
343C22×D4—831no—
344C22×D4—1661no—
345C22×D4—1661no—
346C22×D4—1661no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—1661no—
354C23×C4C22⋊D8311yescharacteristic
355C23×C4C22⋊D8311yescharacteristic
356C23×C4C22⋊D8311yescharacteristic
357C23×C4C22⋊D8314yes—
358C23×C4C22×D8311yescharacteristic
359C23×C4C22×D8311yescharacteristic
360C23×C4C22×D8314yes—
361C23×C4C22×D8311yescharacteristic
362C2×M16—838no—
363C2×M16—1668no—
364C22×C8—834no—
365#123—1668no—
366C22⋊Q8—834no—
367C22⋊Q8—834no—
368C22⋊Q8—1662no—
369C22⋊Q8—1662no—
370#121—834no—
371#121—834no—
372#121—1662no—
373#121—1662no—
374#120—1668no—
375C4×D4—1668no—
376C4×D4—1668no—
377#117—834no—
378#117—1662no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4C22×D8314yes—
393C2×C4⋊C4C22×D8318yes—
394C2×C4⋊C4C22×D8318yes—
395C2×C4⋊C4C22×D8318yes—
396C2×C22⋊C4—1661no—
397C2×C22⋊C4—1661no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C42—1661no—
402#107—8316no—
403#106—8316no—
404#103—8316no—
405#102—8316no—
406C22⋊C8—838no—
407C22⋊C8—838no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—1661no—
412C24C2×C22⋊D8314yes—
413C24C2×C22⋊D8311yescharacteristic
414C24C2×C22⋊D8311yescharacteristic
415C24C2×C22⋊D8311yescharacteristic
416#41—3328no—
417#41—16616no—
418C2×Q8—832no—
419C2×Q8—1664no—
420C2×Q8—1661no—
421C2×D4—3324no—
422C2×D4—3324no—
423C2×D4—3324no—
424C2×D4—832no—
425C2×D4—1664no—
426C2×D4—1664no—
427C2×D4—1664no—
428C2×D4—1661no—
429C2×D4—1664no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—3324no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1664no—
450C22×C4C2×C22⋊D8314yes—
451C22×C4C2×C22⋊D8314yes—
452C22×C4C2×C22⋊D8312yes—
453C22×C4C2×C22⋊D8312yes—
454C22×C4C2×C22⋊D8314yes—
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8312yes—
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4#1305059018yes—
460C22×C4C23×D8312yes—
461C22×C4C23×D8312yes—
462C22×C4C23×D8312yes—
463C22×C4C23×D8311yescharacteristic
464M16—16616no—
465C2×C8—838no—
466C2×C8—1668no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—832no—
472C4⋊C4—832no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1664no—
476C4⋊C4—1664no—
477C4⋊C4C23×D8318yes—
478C22⋊C4—1664no—
479C22⋊C4—1664no—
480C22⋊C4—1664no—
481C22⋊C4—1664no—
482C22⋊C4—1664no—
483C42—1664no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—3322no—
495C23—3322no—
496C23—3322no—
497C23—1662no—
498C23#1458177514yes—
499C23#1458181814yes—
500C23#1458182314yes—
501C23C22×C22⋊D8312yes—
502C23C22×C22⋊D8312yes—
503C23C22×C22⋊D8312yes—
504C23C22×C22⋊D8311yescharacteristic
505Q8—3322no—
506Q8—1664no—
507D4—3328no—
508D4—3328no—
509D4—3328no—
510D4—3322no—
511D4—1664no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—3322no—
527C2×C4—3322no—
528C2×C4—3322no—
529C2×C4—3328no—
530C2×C4—1662no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4#1458177514yes—
534C2×C4#1458177518yes—
535C2×C4C22×C22⋊D8312yes—
536C2×C4C22×C22⋊D8312yes—
537C2×C4C22×C22⋊D8312yes—
538C2×C4C22×C22⋊D8311yescharacteristic
539C2×C4C2×#1305059014yes—
540C2×C4C2×#1305059018yes—
541C2×C4C24×D8311yescharacteristic
542C8—1668no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—3322no—
554C22—3322no—
555C22—3322no—
556C22—6644no—
557C22C2×#1458177514yes—
558C22C2×#1458177512yes—
559C22C2×#1458181814yes—
560C22C2×#1458181812yes—
561C22C2×#1458182314yes—
562C22C2×#1458182312yes—
563C22C23×C22⋊D8311yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—3322no—
568C4—3322no—
569C4—3322no—
570C4C2×#1458177514yes—
571C4C2×#1458177512yes—
572C4C22×#1305059012yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—6642no—
577C2#2491738614yesdirect product factor
578C2C22×#1458177511yescharacteristic
579C2C22×#1458181811yescharacteristic
580C2C22×#1458182311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC23×C4⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128212C2
hypercenterC23×C4⋊C4D83
hypocenterC83C2×#3092
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #24917386C2direct (N × H)
3. C2×#19128036C2semidirect
4. C2×#19128019C2semidirect
5. C2×#19128031C2semidirect
6. C2×#19128014C2semidirect
7. C2×#19127741C2semidirect
8. C2×#19127700C2semidirect
9. C2×#19127665C2semidirect
10. C23×C4⋊C4×C83C2semidirect
24. #19128036C22semidirect
25. #19128031C22semidirect
26. #19128019C22semidirect
27. #19128014C22semidirect
28. #19127741C22semidirect
29. C2×#14581725C22semidirect
30. C2×#14581723C22semidirect
31. #19127700C22semidirect
32. C2×#14581715C22semidirect
33. C2×#14581712C22semidirect
34. #19127665C22semidirect
46. C22×C4⋊C4×C83C22semidirect
83. #14581725C23semidirect
84. #14581723C23semidirect
85. #14581715C23semidirect
86. #14581712C23semidirect
286. C2×C83#3092semidirect
292. C23×C4⋊C4D83semidirect
301. C83C2×#3092semidirect
324. C22×C4⋊C4C2×D83semidirect
577. C2#24917386direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282122
#249173862
C2×#191280362
C2×#191280192
C2×#191280312
C2×#191280142
C2×#191277412
C2×#191277002
C2×#191276652
C23×C4⋊C4×C832
C2×#309283