Finite Group Databasebeta

← #35423826

Subgroups — #35423826

Order 21248.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127811C211yesmaximal, characteristic
2#24910374C211yesmaximal, characteristic, complemented
3#24908163C211yesmaximal, characteristic, complemented
4#24908149C211yesmaximal, characteristic, complemented
5#24908048C211yesmaximal, characteristic, complemented
6#24908041C211yesmaximal, characteristic, complemented
7#24907814C211yesmaximal, characteristic
8#24907814C211yesmaximal, characteristic
9#24907598C211yesmaximal, characteristic, complemented
10#24907473C211yesmaximal, characteristic, complemented
11#24907460C211yesmaximal, characteristic, complemented
12#24907338C211yesmaximal, characteristic, complemented
13#24907291C211yesmaximal, characteristic, complemented
14C4×#14581712C211yesmaximal, characteristic, complemented
15C83×#2354C211yesmaximal, characteristic, complemented
16C22×#13050575C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19C2×C4×D332C2211yescharacteristic
20#19127811C418yescomplemented
21C2×#14581723C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×C4×C83⋊C8C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127572—21no—
35#19127572—21no—
36#19127567C2211yescharacteristic
37#19127560—21no—
38#95×D83—21no—
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50C2×#117×C83—21no—
51#390×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#388×C83—21no—
54#388×C83—21no—
55#384×C83C2211yescharacteristic
56#380×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C2×#13050575—22no—
65C2×#13050575—22no—
66C2×#13050575C2×C414yes—
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71C4×D332—22no—
72C4×D332—22no—
73C4×D332—22no—
74C4×D332—41no—
75C4×D332—41no—
76C4×D332C2×C414yes—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85#14581723C2×C414yescomplemented
86C4⋊(C83⋊C8)—22no—
87#14581713—41no—
88#14581713—41no—
89#14581713C2×C414yescomplemented
90#14581712—22no—
91#14581712—22no—
92#14581712C2×C414yes—
93#14581703—41no—
94#14581703C2×C414yescomplemented
95C4×C83⋊C8—41no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C22×D4×C83C2311yescharacteristic
101C23×C4×C83—21no—
102C4×D4×C83—22no—
103C4×D4×C83C2×C414yescomplemented
104#117×C83—44no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83D411yescharacteristic
108C2×C4⋊C4×C83D411yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C42×C83—21no—
118C2×C42×C83—21no—
119C2×C42×C83C2311yescharacteristic
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122#95×C83—21no—
123#95×C83—21no—
124#95×C83D411yescharacteristic
125#95×C83D411yescharacteristic
126C23×D83—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332C22×C412yes—
131C2×D332#4112yes—
132C2×D332#4112yes—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C2×C4×D83—24no—
137C2×C4×D83—42no—
138C2×C4×D83—42no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C4×D83—81no—
142C2×C4×D83—81no—
143C2×C4×D83—81no—
144C2×C4×D83—81no—
145#13050575—28no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8C22×C412yes—
157C2×C83⋊C8C22×C412yes—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83C22×C412yes—
164Dic83⋊C4—81no—
165Dic83⋊C4—81no—
166(C2×D83)⋊C4—24no—
167(C2×D83)⋊C4—42no—
168(C2×D83)⋊C4—42no—
169C24×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83—21no—
174C2×D4×C83—21no—
175C2×D4×C83C22×C412yes—
176C22×C4×C83—21no—
177C22×C4×C83—24no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—42no—
181C22×C4×C83—42no—
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2411yescharacteristic
189C4⋊C4×C83—22no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—41no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—81no—
198C4⋊C4×C83C22×C412yes—
199C22⋊C4×C83—24no—
200C22⋊C4×C83—42no—
201C22⋊C4×C83—42no—
202C22⋊C4×C83—81no—
203C22⋊C4×C83—81no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—42no—
209C42×C83—42no—
210C42×C83C22×C412yes—
211C42×C83#4112yes—
212C23×C83—22no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83—41no—
217C23×C83—41no—
218C23×C83—41no—
219C23×C83C22×D411yescharacteristic
220C22×D83—22no—
221C22×D83—22no—
222C22×D83—22no—
223C22×D83—41no—
224C22×D83—41no—
225C22×D83—41no—
226C22×D83—41no—
227D4×C83—22no—
228D4×C83—22no—
229D4×C83—41no—
230D4×C83—41no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
247C2×C4×C83—48no—
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×C83C4×D412yes—
257C2×C4×C83C4×D412yes—
258C2×C4×C83AGL(1,ℤ/8)12yes—
259C2×C4×C83C23×C411yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83C2×#4111yescharacteristic
263C2×C4×C83H(5,𝔽2)11yescharacteristic
264C2×C4×C832−1+411yescharacteristic
265C2×Dic83—22no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274D332—22no—
275D332—22no—
276D332—22no—
277D332#11714yescomplemented
278C4×D83—84no—
279C4×D83—84no—
280C4×D83—84no—
281C83⋊C8—22no—
282C83⋊C8—22no—
283C83⋊C8—41no—
284C83⋊C8—41no—
285C22×C83—42no—
286C22×C83—42no—
287C22×C83—42no—
288C22×C83—81no—
289C22×C83—81no—
290C22×C83—81no—
291C22×C83—81no—
292C22×C83C2×C4×D411yescharacteristic
293C22×C83#54611yescharacteristic
294C22×C83#54811yescharacteristic
295C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C83#57511yescharacteristic
297C22×C83#57611yescharacteristic
298C22×C83#57711yescharacteristic
299C2×D83—24no—
300C2×D83—42no—
301C2×D83—42no—
302C2×D83—42no—
303C2×D83—42no—
304C2×D83—42no—
305C2×D83—42no—
306C4×C83—22no—
307C4×C83—22no—
308C4×C83—22no—
309C4×C83—42no—
310C4×C83—41no—
311C4×C83—41no—
312C4×C83—41no—
313C4×C83—81no—
314C4×C83—81no—
315C4×C83C2×#11711yescharacteristic
316C4×C83#52011yescharacteristic
317C4×C83#55211yescharacteristic
318C4×C83#55311yescharacteristic
319Dic83—42no—
320Dic83—81no—
321Dic83—81no—
322#22780—831nomaximal, Sylow
323C2×C83—82no—
324C2×C83#235211yescharacteristic
325C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C83#294611yescharacteristic
327C2×C83#330611yescharacteristic
328C2×C83#331911yescharacteristic
329C2×C83#336611yescharacteristic
330C2×C83#337611yescharacteristic
331D83—44no—
332C2×#442—831no—
333#2354D8311yescharacteristic, complemented
334#2349—831no—
335#1936—831no—
336#1934—831no—
337#1920—831no—
338#1919—831no—
339#1879—831no—
340#1879—831no—
341#1836—831no—
342#1804—831no—
343#1802—831no—
344#1771—831no—
345#1762—831no—
346C4×#102—831no—
347C83#2278011yescharacteristic, complemented, Sylow
348C22×D8—831no—
349C2×C4×D4—831no—
350C2×C4×D4—831no—
351C2×C4×D4—831no—
352C2×C4×D4C2×D8311yescharacteristic
353C2×#117—21no—
354#442—838no—
355C2×#106—831no—
356C2×C4⋊C8—831no—
357C2×#102—831no—
358C2×#102—831no—
359C2×#102—831no—
360C2×#102—831no—
361C2×#102—831no—
362C2×#102—831no—
363C2×#97—831no—
364C2×#97—831no—
365C2×C4×C8—831no—
366#390C2×D8311yescharacteristic
367#390C2×D8311yescharacteristic
368#389—831no—
369#389—831no—
370#388—21no—
371#388—21no—
372#385—1661no—
373#385—1661no—
374#384C2×D8311yescharacteristic
375#382—831no—
376#380—21no—
377#380—1661no—
378C4×C4⋊C4C2×D8311yescharacteristic
379C4×C4⋊C4C2×D8311yescharacteristic
380C4×C4⋊C4C2×D8311yescharacteristic
381C2×#95—1661no—
382#340—831no—
383#340—831no—
384#340—831no—
385#340—831no—
386#340—831no—
387#340—831no—
388#340—831no—
389#340—831no—
390#336—831no—
391#336—831no—
392C22×D4—831no—
393C22×D4C22×D8311yescharacteristic
394C23×C4—21no—
395C23×C4—1661no—
396C23×C4—1661no—
397C23×C4—1661no—
398C2×D8—834no—
399C2×D8—1662no—
400C2×D8—1662no—
401C22×C8—831no—
402C22×C8—831no—
403C22×C8—831no—
404C22×C8—831no—
405C4×D4—22no—
406C4×D4—3321no—
407C4×D4—3321no—
408C4×D4—834no—
409C4×D4—1662no—
410C4×D4—1662no—
411C4×D4—1662no—
412C4×D4C4×D8314yescomplemented
413#117—44no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—831no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4C22⋊D8311yescharacteristic
422C2×C4⋊C4C22⋊D8311yescharacteristic
423C2×C4⋊C4C22×D8311yescharacteristic
424C2×C4⋊C4C22×D8311yescharacteristic
425C2×C4⋊C4C22×D8311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1661no—
432C2×C22⋊C4—1661no—
433C2×C22⋊C4—1661no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D8311yescharacteristic
437C2×C42C22×D8311yescharacteristic
438C2×C42C22×D8311yescharacteristic
439#106—834no—
440C4⋊C8—1662no—
441#102—3321no—
442#102—3321no—
443#102—834no—
444#102—834no—
445#102—1662no—
446#102—1662no—
447#97—3321no—
448#97—834no—
449C4×C8—3321no—
450#95—21no—
451#95—21no—
452#95—1661no—
453#95—1661no—
454#95—1661no—
455#95—1661no—
456#95C22⋊D8311yescharacteristic
457#95C22⋊D8311yescharacteristic
458C24—21no—
459C24—1661no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—832no—
466C2×D4—832no—
467C2×D4—832no—
468C2×D4—1661no—
469C2×D4—1661no—
470C2×D4—1661no—
471C2×D4C2×C4×D8312yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—3322no—
479C22×C4—3322no—
480C22×C4—3322no—
481C22×C4—3322no—
482C22×C4—6641no—
483C22×C4—6641no—
484C22×C4—6641no—
485C22×C4—6641no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4—1664no—
490C22×C4C2×C22⋊D8311yescharacteristic
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4C2×C22⋊D8311yescharacteristic
493C22×C4C2×C22⋊D8311yescharacteristic
494C22×C4C2×C22⋊D8311yescharacteristic
495C22×C4C2×C22⋊D8311yescharacteristic
496C22×C4C23×D8311yescharacteristic
497D8—1668no—
498C2×C8—832no—
499C2×C8—832no—
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—
509C2×C8—1661no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—832no—
520C4⋊C4—6641no—
521C4⋊C4—6641no—
522C4⋊C4—1661no—
523C4⋊C4—1661no—
524C4⋊C4—1661no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4C2×C4×D8312yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C22⋊C4—3322no—
534C22⋊C4—3322no—
535C22⋊C4—1664no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C42C2×C4×D8312yes—
543C42#1305059012yes—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—3321no—
552C23—3321no—
553C23—3321no—
554C23—3321no—
555C23—1662no—
556C23—1662no—
557C23—1662no—
558C23C22×C22⋊D8311yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—834no—
564D4—1662no—
565D4—1662no—
566D4—1662no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—48no—
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—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—6644no—
601C2×C4—6644no—
602C2×C4—6644no—
603C2×C4—1662no—
604C2×C4C4×C22⋊D8312yes—
605C2×C4C4×C22⋊D8312yes—
606C2×C4#1458181812yes—
607C2×C4C22×C4×D8311yescharacteristic
608C2×C4#41×D8311yescharacteristic
609C2×C4#41×D8311yescharacteristic
610C2×C4C2×#1305059011yescharacteristic
611C2×C4#1458183811yescharacteristic
612C2×C4#1458183911yescharacteristic
613C8—3321no—
614C8—3321no—
615C8—1662no—
616C8—1662no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—3322no—
625C22—3322no—
626C22—3322no—
627C22—3322no—
628C22—3322no—
629C22—3322no—
630C22—1664no—
631C22C2×C4×C22⋊D8311yescharacteristic
632C22#1912833911yescharacteristic
633C22#1912835811yescharacteristic
634C22C2×#1458181811yescharacteristic
635C22#1912849911yescharacteristic
636C22#1912850411yescharacteristic
637C22#1912850911yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—3322no—
648C4—6641no—
649C4—6641no—
650C4#117×D8311yescharacteristic
651C4#1912818111yescharacteristic
652C4#1912839011yescharacteristic
653C4#1912839511yescharacteristic
654C2—82no—
655C2—3324no—
656C2#2491039311yescharacteristic
657C2C4×#1458181811yescharacteristic
658C2#2491629111yescharacteristic
659C2#2491941811yescharacteristic
660C2#2491961311yescharacteristic
661C2#2492016611yescharacteristic
662C2#2492028411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2354C2
socleC23×C83C22×D4
prime radicalC22×#13050575C22
cosocleC22×C4×C83C24
hypercenter#2354D83
hypocenterC83#22780
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #24910374C2semidirect
3. #24908163C2semidirect
4. #24908149C2semidirect
5. #24908048C2semidirect
6. #24908041C2semidirect
9. #24907598C2semidirect
10. #24907473C2semidirect
11. #24907460C2semidirect
12. #24907338C2semidirect
13. #24907291C2semidirect
14. C4×#14581712C2semidirect
15. C83×#2354C2semidirect
20. #19127811C4semidirect
85. #14581723C2×C4semidirect
89. #14581713C2×C4semidirect
94. #14581703C2×C4semidirect
103. C4×D4×C83C2×C4semidirect
277. D332#117semidirect
333. #2354D83semidirect
347. C83#22780semidirect
412. C4×D4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191278112
#249103742
#249081632
#249081492
#249080482
#249080412
#249078142
#249075982
#249074732
#249074602
#249073382
#249072912
C4×#145817122
C83×#23542
#2278083