Finite Group Databasebeta

← #35405379

Subgroups — #35405379

Order 21248.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#13050590C211yesmaximal, characteristic
2C2×#19127741C211yesmaximal, characteristic, complemented
3C2×#19127700C211yesmaximal, characteristic
4C2×#19127700C211yesmaximal, characteristic
5#24907730C212yesmaximal, complemented
6#24907723C212yesmaximal, complemented
7#24907716C212yesmaximal, complemented
8#24907709C212yesmaximal, complemented
9#24907182C211yesmaximal, characteristic, complemented
10#24907157C211yesmaximal, characteristic, complemented
11C2×#389×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C22×#13050574C2211yescharacteristic
15C4×#13050590—24no—
16C4×#13050590—24no—
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×C4×D332C2211yescharacteristic
20C2×C4×D332C2211yescharacteristic
21C2×C4×C22⋊D83—21no—
22C2×C4×C22⋊D83—21no—
23C2×#14581751C2211yescharacteristic
24C2×#14581749—21no—
25C2×C42×D83—21no—
26#19127741—24no—
27C2×#14581725C2212yes—
28C2×#14581723C2212yes—
29#19127700—24no—
30#19127700C418yescomplemented
31C2×#14581715C2212yes—
32C2×#14581715C2212yes—
33C2×#14581712C2212yes—
34C2×#14581712C2212yes—
35#19127463C2212yes—
36#19127463C2212yes—
37#19127456C2212yes—
38#19127456C2212yes—
39C22×C4⋊C4×C83—21no—
40C22×C4⋊C4×C83C2211yescharacteristic
41C22×C4⋊C4×C83C2211yescharacteristic
42C22×C42×C83C2211yescharacteristic
43#389×C83—21no—
44#389×C83—21no—
45#389×C83C2212yes—
46#389×C83C2212yes—
47C2×#95×C83—21no—
48C2×#13050590—22no—
49C2×#13050590—22no—
50C2×#13050590—22no—
51C2×#13050590C2×C414yes—
52C2×#13050590D414yes—
53C2×#13050590Q814yes—
54C22×Dic166C2311yescharacteristic
55C22×D332C2311yescharacteristic
56C22×C22⋊D83—21no—
57C22×C4×D83—21no—
58C22×C4×D83—21no—
59C22×C4×D83—21no—
60C2×#13050574—22no—
61C2×#13050574—22no—
62C2×#13050574—22no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574C2×C414yescomplemented
66C22×C83⋊C8C2312yes—
67C22×C83⋊C8C2312yes—
68C4×Dic166—22no—
69C4×Dic166—22no—
70C4×Dic166—22no—
71C4×Dic166—22no—
72C4×D332—22no—
73C4×D332—22no—
74C4×D332—22no—
75C4×D332—22no—
76C4×C22⋊D83—28no—
77C4×C22⋊D83—28no—
78#14581751—22no—
79#14581751D414yes—
80#14581749—28no—
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86C2×(C2×D83)⋊C4—21no—
87C2×(C2×D83)⋊C4—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83—21no—
91C42×D83—28no—
92#14581725—24no—
93#14581723—24no—
94#14581715—24no—
95#14581715C2×C418yescomplemented
96#14581712—24no—
97#14581712C2×C418yescomplemented
98C23×C4×C83—21no—
99C23×C4×C83—21no—
100C23×C4×C83D411yescharacteristic
101C23×C4×C83D411yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—22no—
105C2×C4⋊C4×C83—22no—
106C2×C4⋊C4×C83—22no—
107C2×C4⋊C4×C83—22no—
108C2×C4⋊C4×C83—22no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—44no—
111C2×C4⋊C4×C83C2×C414yescomplemented
112C2×C4⋊C4×C83C2312yes—
113C2×C4⋊C4×C83C2312yes—
114C2×C42×C83—22no—
115C2×C42×C83—22no—
116C2×C42×C83D411yescharacteristic
117C2×C42×C83D411yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—22no—
121#95×C83—22no—
122C23×D83—21no—
123#13050590—28no—
124#13050590—28no—
125#13050590—28no—
126#13050590C4⋊C4116yescomplemented
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166C22×C412yes—
131C2×Dic166C2×D412yes—
132C2×Dic166C2×Q812yes—
133C2×D332—21no—
134C2×D332—21no—
135C2×D332—21no—
136C2×D332C22×C412yes—
137C2×D332C2×D412yes—
138C2×D332C2×Q812yes—
139C2×C22⋊D83—24no—
140C2×C22⋊D83—24no—
141C2×C22⋊D83—24no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C2×C4×D83—24no—
146C2×C4×D83—24no—
147C2×C4×D83—24no—
148C2×C4×D83—24no—
149C2×C4×D83—24no—
150C2×C4×D83—42no—
151C2×C4×D83—42no—
152C2×C4×D83—42no—
153C2×C4×D83—42no—
154#13050574—28no—
155#13050574—44no—
156C2×C83⋊C8—22no—
157C2×C83⋊C8—22no—
158C2×C83⋊C8—22no—
159C2×C83⋊C8—22no—
160C2×C83⋊C8—22no—
161C2×C83⋊C8C22×C414yes—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83C2×D412yes—
165C4⋊Dic83C2×D412yes—
166Dic83⋊C4—24no—
167Dic83⋊C4—24no—
168C22⋊Dic83—24no—
169(C2×D83)⋊C4—24no—
170(C2×D83)⋊C4—24no—
171C4×Dic83—24no—
172C4×Dic83—24no—
173C4×Dic83—24no—
174C24×C83C2×D411yescharacteristic
175C22×C4×C83—24no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83—22no—
180C22×C4×C83—22no—
181C22×C4×C83—22no—
182C22×C4×C83—41no—
183C22×C4×C83—41no—
184C22×C4×C83—42no—
185C22×C4×C83—42no—
186C22×C4×C83—42no—
187C22×C4×C83—42no—
188C22×C4×C83—42no—
189C22×C4×C83C22×C412yes—
190C22×C4×C83C2×D412yes—
191C22×C4×C83C2×D412yes—
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×Q812yes—
198C22×C4×C83#4112yes—
199C22×C4×C83#4112yes—
200C22×C4×C83C2411yescharacteristic
201C4⋊C4×C83—24no—
202C4⋊C4×C83—22no—
203C4⋊C4×C83—22no—
204C4⋊C4×C83—22no—
205C4⋊C4×C83—44no—
206C4⋊C4×C83—42no—
207C4⋊C4×C83—44no—
208C4⋊C4×C83C22×C414yes—
209C42×C83—22no—
210C42×C83—22no—
211C42×C83—21no—
212C42×C83—21no—
213C42×C83—21no—
214C42×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83—21no—
219C23×C83C4×D412yes—
220C23×C83#12012yes—
221C23×C83C22⋊Q812yes—
222C23×C83C22×D411yescharacteristic
223C22×D83—22no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—41no—
227C22×D83—41no—
228C22×D83—41no—
229C22×D83—41no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—24no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—44no—
242C2×C4×C83—44no—
243C2×C4×C83—48no—
244C2×C4×C83—42no—
245C2×C4×C83—42no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83—84no—
250C2×C4×C83C2×C4⋊C414yes—
251C2×C4×C83C4×D412yes—
252C2×C4×C83C4×D414yes—
253C2×C4×C83#12012yes—
254C2×C4×C83C22⋊Q812yes—
255C2×C4×C83C22⋊Q814yes—
256C2×C4×C83#12314yes—
257C2×C4×C83AGL(1,ℤ/8)12yes—
258C2×C4×C83#13712yes—
259C2×C4×C83C23×C411yescharacteristic
260C2×C4×C83C22×D411yescharacteristic
261C2×C4×C83C22×D411yescharacteristic
262C2×C4×C83C22×Q811yescharacteristic
263C2×C4×C83C2×#4111yescharacteristic
264C2×C4×C83C2×#4111yescharacteristic
265C22⋊D83—216no—
266C2×Dic83—22no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275D332—22no—
276D332—22no—
277D332—22no—
278D332C2×C4⋊C414yescomplemented
279C4×D83—216no—
280C4×D83—48no—
281C4×D83—48no—
282Dic166—22no—
283Dic166—22no—
284Dic166—22no—
285Dic166C2×C4⋊C414yescomplemented
286C83⋊C8—24no—
287C83⋊C8—42no—
288C22×C83—22no—
289C22×C83—22no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83#38514yes—
295C22×C83C2×C4×D411yescharacteristic
296C22×C83C2×#12011yescharacteristic
297C22×C83C2×C22⋊Q811yescharacteristic
298C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C83C2×#13711yescharacteristic
300C22×C83#57511yescharacteristic
301C22×C83#57511yescharacteristic
302C2×D83—24no—
303C2×D83—42no—
304C2×D83—42no—
305C2×D83—42no—
306C2×D83—42no—
307C2×D83—42no—
308C2×D83—42no—
309C4×C83—22no—
310C4×C83—22no—
311C4×C83—44no—
312C4×C83—41no—
313C4×C83—41no—
314C4×C83—82no—
315C4×C83#38514yes—
316C4×C83C22×C4⋊C411yescharacteristic
317C4×C83C2×C4×D411yescharacteristic
318C4×C83C2×C22⋊Q811yescharacteristic
319C4×C83C2×#12311yescharacteristic
320Dic83—24no—
321Dic83—42no—
322Dic83—42no—
323#20875—831nomaximal, Sylow
324C2×C83—24no—
325C2×C83C2×#38511yescharacteristic
326C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
327C2×C83C4×#13711yescharacteristic
328C2×C83#300611yescharacteristic
329C2×C83#300711yescharacteristic
330C2×C83#307911yescharacteristic
331C2×C83#308011yescharacteristic
332D83—44no—
333C2×C4×#41—831no—
334C2×#428—831no—
335C2×#417—831no—
336C2×#417—831no—
337C2×#389D8311yescharacteristic, complemented
338#1865—832no—
339#1864—832no—
340#1863—832no—
341#1862—832no—
342#1736—831no—
343#1728—831no—
344C83#2087511yescharacteristic, complemented, Sylow
345C22×#41—831no—
346C22×M16—831no—
347C22×M16—831no—
348C4×#41—1664no—
349C4×#41—1664no—
350C2×C4×Q8—831no—
351C2×C4×Q8—831no—
352C2×C4×D4—831no—
353C2×C4×D4—831no—
354C2×C4×D4—1661no—
355C2×C4×D4—1661no—
356C2×#117—831no—
357C2×#117—1661no—
358C22×C4⋊C4—21no—
359C22×C4⋊C4C2×D8311yescharacteristic
360C22×C4⋊C4C2×D8311yescharacteristic
361C22×C42—1661no—
362C22×C42C2×D8311yescharacteristic
363#428—1664no—
364C2×#107—832no—
365C2×#106—832no—
366#417—838no—
367#417—1664no—
368C2×#103—832no—
369C2×#103—832no—
370C2×#102—832no—
371C2×#102—832no—
372#389—21no—
373#389—21no—
374#389C2×D8312yes—
375#389C2×D8312yes—
376C2×#95—21no—
377#340—832no—
378#340—832no—
379#336—832no—
380#336—832no—
381C2×#41—834no—
382C2×#41—834no—
383C2×#41—834no—
384C2×#41—1662no—
385C2×#41—1662no—
386C2×#41—1662no—
387C22×Q8—831no—
388C22×D4—831no—
389C22×D4—1661no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—1661no—
393C23×C4—1661no—
394C23×C4—1661no—
395C23×C4C22⋊D8311yescharacteristic
396C23×C4C22⋊D8311yescharacteristic
397C23×C4C22×D8311yescharacteristic
398C2×M16—834no—
399C2×M16—1662no—
400C2×M16—1662no—
401C2×M16—1662no—
402C2×M16—1662no—
403C2×M16—1662no—
404C22×C8—832no—
405C22×C8—832no—
406C4×Q8—1662no—
407C4×Q8—1662no—
408C4×Q8—1662no—
409C4×Q8—1662no—
410C4×D4—1662no—
411C4×D4—1662no—
412C4×D4—1662no—
413C4×D4—1662no—
414C4×D4—1668no—
415C4×D4—1668no—
416#117—834no—
417#117—1668no—
418#117—1662no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—44no—
427C2×C4⋊C4—831no—
428C2×C4⋊C4—831no—
429C2×C4⋊C4—1661no—
430C2×C4⋊C4—1661no—
431C2×C4⋊C4C4×D8314yescomplemented
432C2×C4⋊C4C22×D8312yes—
433C2×C4⋊C4C22×D8312yes—
434C2×C22⋊C4—1661no—
435C2×C22⋊C4—1661no—
436C2×C22⋊C4—1661no—
437C2×C42—22no—
438C2×C42—22no—
439C2×C42—1668no—
440C2×C42—1661no—
441C2×C42—1661no—
442C2×C42—1661no—
443C2×C42C22⋊D8311yescharacteristic
444C2×C42C22⋊D8311yescharacteristic
445C2×C42C22×D8311yescharacteristic
446C2×C42C22×D8311yescharacteristic
447#107—1664no—
448#106—1664no—
449#103—838no—
450#103—1664no—
451#102—838no—
452#102—1664no—
453#95—22no—
454#95—22no—
455C24—1661no—
456C24C2×C22⋊D8311yescharacteristic
457#41—8316no—
458#41—1668no—
459#41—1668no—
460#41—1668no—
461C2×Q8—832no—
462C2×Q8—832no—
463C2×Q8—832no—
464C2×Q8—1661no—
465C2×Q8—1661no—
466C2×Q8—1661no—
467C2×D4—832no—
468C2×D4—832no—
469C2×D4—832no—
470C2×D4—1664no—
471C2×D4—1664no—
472C2×D4—1664no—
473C2×D4—1661no—
474C2×D4—1661no—
475C2×D4—1661no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—3322no—
491C22×C4—3322no—
492C22×C4—3322no—
493C22×C4—3322no—
494C22×C4—1664no—
495C22×C4—1664no—
496C22×C4—1661no—
497C22×C4—1661no—
498C22×C4—1661no—
499C22×C4—1664no—
500C22×C4—1664no—
501C22×C4—1664no—
502C22×C4C2×C4×D8312yes—
503C22×C4D4×D8312yes—
504C22×C4D4×D8312yes—
505C22×C4C2×C22⋊D8311yescharacteristic
506C22×C4C2×C22⋊D8311yescharacteristic
507C22×C4C2×C22⋊D8311yescharacteristic
508C22×C4C2×C22⋊D8311yescharacteristic
509C22×C4C2×C22⋊D8311yescharacteristic
510C22×C4Q8×D8312yes—
511C22×C4#1305059012yes—
512C22×C4#1305059112yes—
513C22×C4C23×D8311yescharacteristic
514M16—3324no—
515M16—1668no—
516C2×C8—834no—
517C2×C8—1662no—
518C2×C8—1662no—
519C2×C8—1662no—
520C2×C8—1662no—
521C2×C8—1662no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—832no—
530C4⋊C4—832no—
531C4⋊C4—1664no—
532C4⋊C4—1664no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4C2×C4×D8314yes—
536C22⋊C4—1664no—
537C22⋊C4—1664no—
538C22⋊C4—1664no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1664no—
546C42—1664no—
547C42—1664no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—3321no—
553C23—3321no—
554C23—3321no—
555C23—3321no—
556C23—1662no—
557C23—1662no—
558C23—1662no—
559C23C4×C22⋊D8312yes—
560C23#1458176012yes—
561C23#1458176812yes—
562C23C22×C22⋊D8311yescharacteristic
563Q8—834no—
564Q8—1662no—
565Q8—1662no—
566Q8—1662no—
567D4—834no—
568D4—16616no—
569D4—1662no—
570D4—1662no—
571D4—1662no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—3328no—
593C2×C4—3328no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4—16616no—
604C2×C4C4⋊C4×D8314yes—
605C2×C4C4×C22⋊D8312yes—
606C2×C4#1458175414yes—
607C2×C4#1458176012yes—
608C2×C4#1458176812yes—
609C2×C4#1458177014yes—
610C2×C4#1458177714yes—
611C2×C4#1458181812yes—
612C2×C4#1458182312yes—
613C2×C4C22×C4×D8311yescharacteristic
614C2×C4C2×D4×D8311yescharacteristic
615C2×C4C2×D4×D8311yescharacteristic
616C2×C4C2×Q8×D8311yescharacteristic
617C2×C4C2×#1305059011yescharacteristic
618C2×C4C2×#1305059111yescharacteristic
619C8—3322no—
620C8—1664no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—3322no—
628C22—3322no—
629C22—3322no—
630C22—3322no—
631C22—3322no—
632C22—3322no—
633C22—1664no—
634C22#1912757214yes—
635C22C2×C4×C22⋊D8311yescharacteristic
636C22C2×#1458176011yescharacteristic
637C22C2×#1458176811yescharacteristic
638C22C2×#1458181811yescharacteristic
639C22C2×#1458182311yescharacteristic
640C22#1912849911yescharacteristic
641C22#1912849911yescharacteristic
642C4—22no—
643C4—22no—
644C4—44no—
645C4—41no—
646C4—41no—
647C4—82no—
648C4—3322no—
649C4—3322no—
650C4—1664no—
651C4#1912757214yes—
652C4C2×C4⋊C4×D8311yescharacteristic
653C4C2×#1458175411yescharacteristic
654C4C2×#1458177011yescharacteristic
655C4C2×#1458177711yescharacteristic
656C2—24no—
657C2—3324no—
658C2C2×#1912757211yescharacteristic
659C2C4×#1458181811yescharacteristic
660C2C4×#1458182311yescharacteristic
661C2#2491671311yescharacteristic
662C2#2491672111yescharacteristic
663C2#2491727211yescharacteristic
664C2#2491728111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#389×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC2×#389D83
hypocenterC83#20875
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×#19127741C2semidirect
5. #24907730C2semidirect
6. #24907723C2semidirect
7. #24907716C2semidirect
8. #24907709C2semidirect
9. #24907182C2semidirect
10. #24907157C2semidirect
11. C2×#389×C83C2semidirect
30. #19127700C4semidirect
65. C2×#13050574C2×C4semidirect
95. #14581715C2×C4semidirect
97. #14581712C2×C4semidirect
111. C2×C4⋊C4×C83C2×C4semidirect
126. #13050590C4⋊C4semidirect
278. D332C2×C4⋊C4semidirect
285. Dic166C2×C4⋊C4semidirect
337. C2×#389D83semidirect
344. C83#20875semidirect
431. C2×C4⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#130505902
C2×#191277412
C2×#191277002
#249077302
#249077232
#249077162
#249077092
#249071822
#249071572
C2×#389×C832
#2087583