Finite Group Databasebeta

← #35504438

Subgroups — #35504438

Order 21248.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14581766C211yesmaximal, characteristic
2#24909030C211yesmaximal, characteristic, complemented
3#24908987C211yesmaximal, characteristic, complemented
4#24908798C211yesmaximal, characteristic, complemented
5#24908606C211yesmaximal, characteristic, complemented
6#24908052C211yesmaximal, characteristic, complemented
7#24908038C211yesmaximal, characteristic, complemented
8#24907889C211yesmaximal, characteristic
9#24907889C211yesmaximal, characteristic
10#24907723C211yesmaximal, characteristic, complemented
11#24907709C211yesmaximal, characteristic, complemented
12#24907490C211yesmaximal, characteristic, complemented
13#24907486C211yesmaximal, characteristic, complemented
14C2×#19127456C211yesmaximal, characteristic, complemented
15C83×#2851C211yesmaximal, characteristic, complemented
16C23×C83⋊C8C2211yescharacteristic
17C23×C83⋊C8C2211yescharacteristic
18C2×#14581766C2211yescharacteristic
19C2×C4×D332C2211yescharacteristic
20C2×C4×C22⋊D83—21no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29#19127603C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127592—21no—
32#19127580—21no—
33#19127561C2211yescharacteristic
34C4×C4⋊Dic83C2211yescharacteristic
35C4×(C2×D83)⋊C4—21no—
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#125×C83C2211yescharacteristic
49C22×C42×C83C2211yescharacteristic
50#389×C83C2211yescharacteristic
51#389×C83C2211yescharacteristic
52#387×C83—21no—
53#387×C83—21no—
54#385×C83—21no—
55#385×C83—21no—
56#382×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59C22×D332C2311yescharacteristic
60C22×C22⋊D83—21no—
61C22×C4×D83—21no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8—22no—
66C22×C83⋊C8—41no—
67C22×C83⋊C8—41no—
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71C22×C83⋊C8C2311yescharacteristic
72#14581766—22no—
73#14581766Q814yes—
74C4×D332—41no—
75C4×D332—41no—
76C4×C22⋊D83—44no—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×C4×Dic83—21no—
87#14581725—22no—
88#14581725—41no—
89#14581723—22no—
90#14581723—41no—
91#14581712—22no—
92#14581712—22no—
93#14581712—41no—
94#14581712—41no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C23×C4×C83D411yescharacteristic
99C23×C4×C83D411yescharacteristic
100C23×C4×C83C2311yescharacteristic
101#125×C83—41no—
102#125×C83—41no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C42×C83—41no—
114C2×C42×C83—41no—
115C2×C42×C83D411yescharacteristic
116C2×C42×C83D411yescharacteristic
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125C23×D83—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C2×Q812yes—
132C2×C22⋊D83—24no—
133C2×C22⋊D83—42no—
134C2×C22⋊D83—42no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C2×C4×D83—42no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—44no—
152C2×C83⋊C8—44no—
153C2×C83⋊C8#4112yes—
154C2×C83⋊C8#4112yes—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83C2×Q812yes—
161Dic83⋊C4—42no—
162C22⋊Dic83—42no—
163(C2×D83)⋊C4—24no—
164(C2×D83)⋊C4—42no—
165(C2×D83)⋊C4—42no—
166(C2×D83)⋊C4—42no—
167C4×Dic83—42no—
168C24×C83C2×D411yescharacteristic
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—41no—
181C22×C4×C83—41no—
182C22×C4×C83—41no—
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×Q812yes—
189C22×C4×C83#4112yes—
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—42no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C22⋊C4×C83—42no—
204C22⋊C4×C83—42no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—21no—
209C42×C83—41no—
210C42×C83—41no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83C22⋊Q812yes—
216C23×C83#13512yes—
217C23×C83AGL(1,ℤ/8)12yes—
218C23×C83C22×D411yescharacteristic
219C22×D83—22no—
220C22×D83—41no—
221C22×D83—41no—
222C22×D83—41no—
223C22×D83—41no—
224C22×D83—41no—
225C22×D83—41no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—21no—
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—21no—
242C2×C4×C83—21no—
243C2×C4×C83—44no—
244C2×C4×C83—44no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83C22⋊Q812yes—
258C2×C4×C83#13512yes—
259C2×C4×C83C22×Q811yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83C2×#4111yescharacteristic
263C2×C4×C83H(5,𝔽2)11yescharacteristic
264C2×C4×C83H(5,𝔽2)11yescharacteristic
265C22⋊D83—48no—
266C2×Dic83—22no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275D332—22no—
276D332—22no—
277D332—41no—
278D332—41no—
279C4×D83—84no—
280C83⋊C8—22no—
281C83⋊C8—22no—
282C83⋊C8—41no—
283C83⋊C8—41no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—22no—
290C22×C83#47814yes—
291C22×C83C2×C22⋊Q811yescharacteristic
292C22×C83#53211yescharacteristic
293C22×C83#53411yescharacteristic
294C22×C83C2×#13511yescharacteristic
295C22×C83C2×#13511yescharacteristic
296C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C83#57511yescharacteristic
298C2×D83—42no—
299C2×D83—42no—
300C2×D83—42no—
301C2×D83—81no—
302C2×D83—81no—
303C2×D83—81no—
304C2×D83—81no—
305C4×C83—22no—
306C4×C83—22no—
307C4×C83—41no—
308C4×C83—41no—
309C4×C83—41no—
310C4×C83—81no—
311C4×C83—81no—
312C4×C83—81no—
313C4×C83—81no—
314C4×C83#55111yescharacteristic
315C4×C83#55111yescharacteristic
316C4×C83#55411yescharacteristic
317C4×C83#55411yescharacteristic
318Dic83—42no—
319Dic83—81no—
320Dic83—81no—
321#29161—831nomaximal, Sylow
322C2×C83—24no—
323C2×C83#256411yescharacteristic
324C2×C83C2×#47811yescharacteristic
325C2×C83#307611yescharacteristic
326C2×C83#316811yescharacteristic
327C2×C83#318011yescharacteristic
328C2×C83#318811yescharacteristic
329C2×C83#319111yescharacteristic
330D83—82no—
331#2851D8311yescharacteristic, complemented
332C4×#121—831no—
333#2101—831no—
334#2092—831no—
335#2059—831no—
336#2025—831no—
337#1921—831no—
338#1919—831no—
339#1893—831no—
340#1893—831no—
341#1864—831no—
342#1862—831no—
343#1808—831no—
344#1807—831no—
345C2×#336—831no—
346C83#2916111yescharacteristic, complemented, Sylow
347C23×C8—831no—
348C23×C8—831no—
349C2×#125C2×D8311yescharacteristic
350C2×#121—831no—
351C2×C4×D4—831no—
352C2×C4×D4—1661no—
353C22×C42C2×D8311yescharacteristic
354C2×#107—831no—
355C2×#107—831no—
356C2×#106—831no—
357C2×#106—831no—
358C2×#102—831no—
359C2×#102—831no—
360C2×#102—831no—
361C2×#102—831no—
362#390—831no—
363#389—831no—
364#389C2×D8311yescharacteristic
365#389C2×D8311yescharacteristic
366#388—1661no—
367#387—21no—
368#387—21no—
369#386—1661no—
370#385—21no—
371#385—21no—
372#382C2×D8311yescharacteristic
373#380—831no—
374#380C2×D8311yescharacteristic
375#380C2×D8311yescharacteristic
376C4×C4⋊C4—831no—
377C4×C22⋊C4—1661no—
378#340—831no—
379#340—831no—
380#340—831no—
381#340—831no—
382#340—831no—
383#340—831no—
384#340—831no—
385#340—831no—
386#336—831no—
387#336—831no—
388#336—831no—
389#336—831no—
390C22×D4—831no—
391C22×D4—1661no—
392C23×C4—1661no—
393C23×C4C22⋊D8311yescharacteristic
394C23×C4C22⋊D8311yescharacteristic
395C23×C4C22×D8311yescharacteristic
396C22×C8—3321no—
397C22×C8—3321no—
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8—831no—
402C22×C8—1662no—
403C22×C8—1662no—
404C22×C8—1662no—
405C22×C8—1662no—
406#125—41no—
407#125—41no—
408#121—834no—
409#121—1662no—
410C4×D4—3321no—
411C4×D4—3321no—
412C4×D4—3324no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—831no—
419C2×C4⋊C4—831no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4C22×D8311yescharacteristic
422C2×C4⋊C4C22×D8311yescharacteristic
423C2×C4⋊C4C22×D8311yescharacteristic
424C2×C4⋊C4C22×D8311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1661no—
428C2×C22⋊C4—1661no—
429C2×C22⋊C4—1661no—
430C2×C22⋊C4—1661no—
431C2×C22⋊C4—1661no—
432C2×C42—41no—
433C2×C42—41no—
434C2×C42—1661no—
435C2×C42C22⋊D8311yescharacteristic
436C2×C42C22⋊D8311yescharacteristic
437C2×C42C22×D8311yescharacteristic
438C2×C42C22×D8311yescharacteristic
439#107—3321no—
440#107—1662no—
441#106—3321no—
442#106—1662no—
443#102—3321no—
444#102—3321no—
445#102—1662no—
446#102—1662no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1661no—
454#95—1661no—
455#95—1661no—
456C24—1661no—
457C24C2×C22⋊D8311yescharacteristic
458C2×D4—3322no—
459C2×D4—3322no—
460C2×D4—832no—
461C2×D4—1664no—
462C2×D4—1661no—
463C2×D4—1661no—
464C2×D4—1661no—
465C2×D4—1661no—
466C2×D4—1661no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—3322no—
482C22×C4—3322no—
483C22×C4—3322no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1661no—
487C22×C4C2×C22⋊D8311yescharacteristic
488C22×C4C2×C22⋊D8311yescharacteristic
489C22×C4C2×C22⋊D8311yescharacteristic
490C22×C4C2×C22⋊D8311yescharacteristic
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4Q8×D8312yes—
493C22×C4#1305059112yes—
494C22×C4C23×D8311yescharacteristic
495C2×C8—3324no—
496C2×C8—3324no—
497C2×C8—832no—
498C2×C8—832no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C2×C8—1661no—
505C2×C8—1661no—
506C2×C8—1661no—
507C2×C8—1661no—
508C2×C8—1661no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—3322no—
522C4⋊C4—832no—
523C4⋊C4—1661no—
524C4⋊C4—1661no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—3322no—
531C22⋊C4—3322no—
532C22⋊C4—3322no—
533C22⋊C4—3322no—
534C22⋊C4—1664no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42—41no—
541C42—3322no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—3321no—
547C23—3321no—
548C23—3321no—
549C23—3321no—
550C23—3321no—
551C23—3321no—
552C23—1662no—
553C23#1458176812yes—
554C23#1458181212yes—
555C23#1458181712yes—
556C23C22×C22⋊D8311yescharacteristic
557D4—3328no—
558D4—3321no—
559D4—3321no—
560D4—1662no—
561D4—1662no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—6644no—
602C2×C4—1662no—
603C2×C4#1458176812yes—
604C2×C4#1458181212yes—
605C2×C4C2×Q8×D8311yescharacteristic
606C2×C4#41×D8311yescharacteristic
607C2×C4#41×D8311yescharacteristic
608C2×C4C2×#1305059111yescharacteristic
609C2×C4#1458183711yescharacteristic
610C2×C4#1458183711yescharacteristic
611C8—3321no—
612C8—3321no—
613C8—1662no—
614C8—1662no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—3322no—
622C22—3322no—
623C22—3322no—
624C22—6641no—
625C22—6641no—
626C22—6641no—
627C22—6641no—
628C22#1912799814yes—
629C22C2×#1458176811yescharacteristic
630C22#1912824711yescharacteristic
631C22#1912826011yescharacteristic
632C22C2×#1458181211yescharacteristic
633C22C2×#1458181211yescharacteristic
634C22C2×#1458181711yescharacteristic
635C22#1912849911yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—3322no—
646C4—6641no—
647C4—6641no—
648C4#1912837811yescharacteristic
649C4#1912837811yescharacteristic
650C4#1912840511yescharacteristic
651C4#1912840511yescharacteristic
652C2—24no—
653C2—6642no—
654C2#2491264511yescharacteristic
655C2C2×#1912799811yescharacteristic
656C2#2491724511yescharacteristic
657C2#2491802711yescharacteristic
658C2#2491816211yescharacteristic
659C2#2491822411yescharacteristic
660C2#2491825111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2851C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
hypercenter#2851D83
hypocenterC83#29161
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
2. #24909030C2semidirect
3. #24908987C2semidirect
4. #24908798C2semidirect
5. #24908606C2semidirect
6. #24908052C2semidirect
7. #24908038C2semidirect
10. #24907723C2semidirect
11. #24907709C2semidirect
12. #24907490C2semidirect
13. #24907486C2semidirect
14. C2×#19127456C2semidirect
15. C83×#2851C2semidirect
331. #2851D83semidirect
346. C83#29161semidirect

Maximal subgroups

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

subgroupindex
C4×#145817662
#249090302
#249089872
#249087982
#249086062
#249080522
#249080382
#249078892
#249077232
#249077092
#249074902
#249074862
C2×#191274562
C83×#28512
#2916183