Finite Group Databasebeta

← #35422027

Subgroups — #35422027

Order 21248.

850 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D83C211yesmaximal, characteristic
2#24910336C211yesmaximal, characteristic
3#24910331C211yesmaximal, characteristic
4#24908190C211yesmaximal, characteristic, complemented
5#24908188C211yesmaximal, characteristic, complemented
6#24908049C211yesmaximal, characteristic, complemented
7#24908047C211yesmaximal, characteristic, complemented
8#24908043C211yesmaximal, characteristic, complemented
9#24907552C211yesmaximal, characteristic
10#24907548C211yesmaximal, characteristic
11#24907487C211yesmaximal, characteristic
12C4×#14581723C211yesmaximal, characteristic, complemented
13#24907351C211yesmaximal, characteristic, complemented
14#24907233C211yesmaximal, characteristic
15C83×#1919C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17C2×#14581750—21no—
18C2×C4⋊C4×D83C2211yescharacteristic
19C2×C4⋊C4×D83C2211yescharacteristic
20C2×#14581724C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25#106×D83C418yescomplemented
26C2×(C2×D83)⋊C8C2211yescharacteristic
27C2×C4×C83⋊C8C2211yescharacteristic
28C2×C8×Dic83C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127597C2211yescharacteristic
32#19127588—21no—
33#19127585—21no—
34#19127583—21no—
35#19127582—21no—
36#19127563—21no—
37#19127560—21no—
38#19127558C2211yescharacteristic
39#19127558C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4×C4⋊Dic83C2211yescharacteristic
43C4×(C2×D83)⋊C4—21no—
44C4×(C2×D83)⋊C4C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83—21no—
62C22×C4×D83C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C2×C8×D83—22no—
66C2×C8×D83—22no—
67C2×C8×D83C2×C414yes—
68#14581750—44no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C4⋊C4×D83—22no—
79C4⋊C4×D83—22no—
80C4⋊C4×D83C2×C414yes—
81C4⋊C4×D83C2×C414yes—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4D411yescharacteristic
85C2×(C2×D83)⋊C4D411yescharacteristic
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83C2311yescharacteristic
90#14581724C2×C414yes—
91#14581723—22no—
92#14581723—22no—
93#14581723C2×C414yes—
94#14581723C2×C414yes—
95(C2×D83)⋊C8—22no—
96C4×C83⋊C8—22no—
97C8×Dic83—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95D411yescharacteristic
107C83⋊#95D411yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C42×C83—21no—
116C2×C42×C83C2311yescharacteristic
117#106×C83C2×C414yes—
118#95×C83—21no—
119#95×C83—21no—
120C23×D83C2×D411yescharacteristic
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83—21no—
132C2×C4×D83—24no—
133C2×C4×D83—24no—
134C2×C4×D83—42no—
135C2×C4×D83—42no—
136C2×C4×D83—81no—
137C2×C4×D83—81no—
138C2×C4×D83C22×C412yes—
139C2×C4×D83#4112yes—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8C22×C412yes—
144C2×C83⋊C8#4112yes—
145C2×C83⋊C8#4112yes—
146C8×D83—28no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—42no—
154C4⋊Dic83C22×C412yes—
155C4⋊Dic83C22×C412yes—
156Dic83⋊C4—24no—
157Dic83⋊C4—24no—
158Dic83⋊C4—81no—
159Dic83⋊C4—81no—
160(C2×D83)⋊C4—22no—
161(C2×D83)⋊C4—42no—
162(C2×D83)⋊C4—42no—
163(C2×D83)⋊C4SD1614yes—
164C4×Dic83—21no—
165C4×Dic83—21no—
166C4×Dic83—42no—
167C4×Dic83—42no—
168C4×Dic83—81no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2411yescharacteristic
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83C22×C412yes—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83C22×C412yes—
190C4⋊C4×C83C22×C412yes—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—81no—
194C23×C83C22×D411yescharacteristic
195C22×D83—21no—
196C22×D83—21no—
197C22×D83—21no—
198C22×D83—21no—
199C22×D83C4×D412yes—
200C22×D83C2×SD1612yes—
201C22×D83#13512yes—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83C2×SD1612yes—
220C2×C4×C83C23×C411yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×C4×C832−1+411yescharacteristic
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—22no—
231C2×Dic83—22no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83C4×D412yes—
246C2×Dic83C2×SD1612yes—
247C4×D83—22no—
248C4×D83—22no—
249C4×D83—22no—
250C4×D83—22no—
251C4×D83—48no—
252C4×D83—84no—
253C8×C83—22no—
254C8×C83—41no—
255C83⋊C8—41no—
256C83⋊C8#11714yescomplemented
257C22×C83C2×C4×D411yescharacteristic
258C22×C83#54711yescharacteristic
259C22×C83#54711yescharacteristic
260C22×C83C22×SD1611yescharacteristic
261C22×C83C2×#13511yescharacteristic
262C22×C83#57711yescharacteristic
263C22×C83#57711yescharacteristic
264C2×D83—22no—
265C2×D83—22no—
266C2×D83—22no—
267C2×D83—22no—
268C2×D83—22no—
269C2×D83—22no—
270C2×D83C4×SD1614yes—
271C4×C83—22no—
272C4×C83—41no—
273C4×C83—42no—
274C4×C83—42no—
275C4×C83—81no—
276C4×C83—81no—
277C4×C83C2×#11711yescharacteristic
278C4×C83#51911yescharacteristic
279C4×C83#55211yescharacteristic
280C4×C83#55211yescharacteristic
281Dic83—22no—
282Dic83—22no—
283Dic83—41no—
284Dic83—42no—
285Dic83—42no—
286Dic83—81no—
287Dic83—81no—
288#22609—831nomaximal, Sylow
289C2×C83#234411yescharacteristic
290C2×C83C2×C4×SD1611yescharacteristic
291C2×C83#294911yescharacteristic
292C2×C83#333511yescharacteristic
293C2×C83#333511yescharacteristic
294C2×C83#333711yescharacteristic
295C2×C83#333711yescharacteristic
296D83—24no—
297C22×#106—831no—
298#2345—831no—
299#2344—831no—
300#1941—831no—
301#1941—831no—
302#1920—831no—
303#1920—831no—
304#1919—831no—
305#1919D8311yescharacteristic, complemented
306#1824—831no—
307#1823—831no—
308#1807—831no—
309C4×#106—831no—
310C4×#106—831no—
311C8×C22⋊C4—831no—
312C83#2260911yescharacteristic, complemented, Sylow
313C23×C8—831no—
314C2×#117—1661no—
315C22×C4⋊C4—831no—
316C22×C4⋊C4—831no—
317C2×#106—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×#106—838no—
321C2×#106—831no—
322C2×#106—831no—
323C2×#106C2×D8311yescharacteristic
324C2×C22⋊C8—831no—
325C2×C4×C8—831no—
326C2×C4×C8—831no—
327#389—831no—
328#389—831no—
329#389—831no—
330#389C2×D8311yescharacteristic
331#387—1661no—
332#387—1661no—
333#387—1661no—
334#387—1661no—
335#381—1661no—
336#380—831no—
337#380—831no—
338#380—1661no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4C2×D8311yescharacteristic
343C4×C22⋊C4—831no—
344C4×C22⋊C4—1661no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340C2×D8311yescharacteristic
353#340C2×D8311yescharacteristic
354#340C2×D8311yescharacteristic
355#336—831no—
356#336C2×D8311yescharacteristic
357C23×C4—831no—
358C23×C4—1661no—
359C23×C4—1661no—
360C22×C8—831no—
361C22×C8—834no—
362C22×C8—831no—
363C22×C8—1662no—
364C22×C8—1662no—
365C22×C8C22×D8311yescharacteristic
366C22×C8C22×D8311yescharacteristic
367#117—3324no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—834no—
372C2×C4⋊C4—834no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—1662no—
376C2×C4⋊C4—1662no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4C22×D8311yescharacteristic
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C22⋊C4—831no—
387C2×C22⋊C4—831no—
388C2×C22⋊C4—1661no—
389C2×C22⋊C4—1661no—
390C2×C42—21no—
391C2×C42—831no—
392C2×C42—1661no—
393C2×C42—1661no—
394C2×C42—1661no—
395C2×C42C22×D8311yescharacteristic
396#106—834no—
397#106—834no—
398#106—834no—
399#106—1662no—
400#106—1662no—
401#106C4×D8314yes—
402C22⋊C8—1662no—
403C4×C8—1662no—
404C4×C8—1662no—
405#95—21no—
406#95—21no—
407#95—831no—
408#95—831no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417C24—831no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—3322no—
423C22×C4—3322no—
424C22×C4—831no—
425C22×C4—831no—
426C22×C4—832no—
427C22×C4—832no—
428C22×C4—831no—
429C22×C4—6641no—
430C22×C4—6641no—
431C22×C4—1664no—
432C22×C4—1664no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4D4×D8311yescharacteristic
442C22×C4D4×D8311yescharacteristic
443C22×C4C23×D8311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—832no—
450C2×C8—832no—
451C2×C8—832no—
452C2×C8—1668no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8C2×C4×D8312yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—3322no—
465C4⋊C4—832no—
466C4⋊C4—832no—
467C4⋊C4—6641no—
468C4⋊C4—6641no—
469C4⋊C4—1664no—
470C4⋊C4—1664no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4C2×C4×D8312yes—
478C4⋊C4C2×C4×D8312yes—
479C22⋊C4—3322no—
480C22⋊C4—3322no—
481C22⋊C4—834no—
482C22⋊C4—1662no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—3322no—
487C42—3322no—
488C42—6641no—
489C42—1661no—
490C42—1661no—
491C23—832no—
492C23—832no—
493C23—832no—
494C23—1661no—
495C23—1661no—
496C23—1661no—
497C23—1661no—
498C23C2×D4×D8311yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—3328no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—832no—
528C2×C4—832no—
529C2×C4—6644no—
530C2×C4—1661no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4—1661no—
536C2×C4—1661no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4SD16×D8312yes—
544C2×C4C22×C4×D8311yescharacteristic
545C2×C4#41×D8311yescharacteristic
546C2×C4#41×D8311yescharacteristic
547C2×C4#41×D8311yescharacteristic
548C2×C4#1458183911yescharacteristic
549C2×C4#1458184011yescharacteristic
550C8—22no—
551C8—41no—
552C8—3321no—
553C8—834no—
554C22—834no—
555C22—1662no—
556C22—1662no—
557C22—1662no—
558C22—1662no—
559C22—1662no—
560C22—1662no—
561C22C4×D4×D8311yescharacteristic
562C22#1912834911yescharacteristic
563C22#1912835311yescharacteristic
564C22C2×SD16×D8311yescharacteristic
565C22#135×D8311yescharacteristic
566C22#1912851011yescharacteristic
567C22#1912851111yescharacteristic
568C4—22no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—3321no—
575C4—3322no—
576C4—3322no—
577C4—6641no—
578C4—6641no—
579C4—1662no—
580C4—1662no—
581C4#117×D8311yescharacteristic
582C4#1912817711yescharacteristic
583C4#1912838311yescharacteristic
584C4#1912838711yescharacteristic
585C2—1664no—
586C2#2491032911yescharacteristic
587C2C4×SD16×D8311yescharacteristic
588C2#2491631911yescharacteristic
589C2#2491981811yescharacteristic
590C2#2491982311yescharacteristic
591C2#2491984511yescharacteristic
592C2#2491984911yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1919C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1919D83
hypocenterC83#22609
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #24908190C2semidirect
5. #24908188C2semidirect
6. #24908049C2semidirect
7. #24908047C2semidirect
8. #24908043C2semidirect
12. C4×#14581723C2semidirect
13. #24907351C2semidirect
15. C83×#1919C2semidirect
25. #106×D83C4semidirect
256. C83⋊C8#117semidirect
305. #1919D83semidirect
312. C83#22609semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D832
#249103362
#249103312
#249081902
#249081882
#249080492
#249080472
#249080432
#249075522
#249075482
#249074872
C4×#145817232
#249073512
#249072332
C83×#19192
#2260983