Finite Group Databasebeta

← #35421069

Subgroups — #35421069

Order 21248.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128411C211yesmaximal, characteristic
2C2×#19127832C211yesmaximal, characteristic, complemented
3#24908435C211yesmaximal, characteristic, complemented
4#24908431C211yesmaximal, characteristic, complemented
5#24908427C211yesmaximal, characteristic, complemented
6#24908420C211yesmaximal, characteristic, complemented
7#24908420C211yesmaximal, characteristic, complemented
8#24908415C211yesmaximal, characteristic, complemented
9#24907470C211yesmaximal, characteristic
10#24907470C211yesmaximal, characteristic
11#24907460C211yesmaximal, characteristic
12#24907291C211yesmaximal, characteristic
13#24907291C211yesmaximal, characteristic
14C4×#14581712C211yesmaximal, characteristic
15C83×#2355C211yesmaximal, characteristic, complemented
16#19128411—24no—
17#19128411C418yescomplemented
18C2×#14581792C2211yescharacteristic
19C2×#14581784C2211yescharacteristic
20C2×#14581784C2211yescharacteristic
21C2×#14581770—21no—
22C2×#14581770—21no—
23C2×#14581768—21no—
24C2×C4×D332C2211yescharacteristic
25C2×C4×D332C2211yescharacteristic
26C2×C4×D332C2211yescharacteristic
27C2×C4⋊C4×D83—21no—
28#19127832—24no—
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×C4⋊(C83⋊C8)C2211yescharacteristic
31C2×C4⋊(C83⋊C8)C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×#14581712C2211yescharacteristic
35C2×#14581712C2211yescharacteristic
36C2×#14581703C2211yescharacteristic
37C2×#14581703C2211yescharacteristic
38C2×C4×C83⋊C8C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52#391×C83—21no—
53#387×C83—21no—
54#387×C83—21no—
55#384×C83C2211yescharacteristic
56#382×C83C2211yescharacteristic
57#382×C83C2211yescharacteristic
58C4×C4⋊C4×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C4×C4⋊C4×C83C2211yescharacteristic
62C22×Dic166—21no—
63C22×D332C2311yescharacteristic
64C22×C4×D83—21no—
65C22×C4×D83—21no—
66C22×C4×D83—21no—
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71#14581792—22no—
72#14581792C2×C414yes—
73#14581784—22no—
74#14581784—22no—
75#14581784C2×C414yes—
76#14581784C2×C414yes—
77#14581770—28no—
78#14581770—28no—
79#14581768—28no—
80C4×D332—22no—
81C4×D332—22no—
82C4×D332—22no—
83C4×D332C2×C414yes—
84C4×D332C2×C414yes—
85C4×D332C2×C414yes—
86C2×C4⋊Dic83—21no—
87C2×C4⋊Dic83C2311yescharacteristic
88C2×C4⋊Dic83C2311yescharacteristic
89C2×C4⋊Dic83C2311yescharacteristic
90C2×Dic83⋊C4—21no—
91C2×Dic83⋊C4—21no—
92C2×Dic83⋊C4—21no—
93C2×Dic83⋊C4—21no—
94C4⋊C4×D83—28no—
95C2×(C2×D83)⋊C4—21no—
96C2×(C2×D83)⋊C4—21no—
97C2×(C2×D83)⋊C4—21no—
98C4⋊(C83⋊C8)—22no—
99C4⋊(C83⋊C8)—22no—
100C4⋊(C83⋊C8)—22no—
101#14581712—22no—
102#14581712—22no—
103#14581712—22no—
104#14581712Q814yescomplemented
105#14581703—22no—
106#14581703—22no—
107C4×C83⋊C8—22no—
108C22×Q8×C83D411yescharacteristic
109C4×Q8×C83C2×C414yes—
110C4×Q8×C83D414yes—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83D411yescharacteristic
114C2×C4⋊C4×C83D411yescharacteristic
115C2×C4⋊C4×C83D411yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83—21no—
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83—21no—
130#95×C83—21no—
131#95×C83—21no—
132#95×C83—21no—
133C23×D83—21no—
134C2×Dic166—24no—
135C2×Dic166—81no—
136C2×Dic166—81no—
137C2×D332—21no—
138C2×D332—21no—
139C2×D332—21no—
140C2×D332C22×C412yes—
141C2×D332C2×Q812yes—
142C2×D332#4112yes—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C2×C4×D83—24no—
147C2×C4×D83—24no—
148C2×C4×D83—24no—
149C2×C4×D83—42no—
150C2×C4×D83—42no—
151C2×C4×D83—81no—
152C2×C4×D83—81no—
153C2×C4×D83—81no—
154C2×C4×D83—81no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8C2×Q812yes—
167C4⋊Dic83—24no—
168C4⋊Dic83—21no—
169C4⋊Dic83—21no—
170C4⋊Dic83—21no—
171C4⋊Dic83C22×C412yes—
172C4⋊Dic83C22×C412yes—
173C4⋊Dic83C22×C412yes—
174Dic83⋊C4—24no—
175Dic83⋊C4—24no—
176Dic83⋊C4—24no—
177Dic83⋊C4—24no—
178(C2×D83)⋊C4—24no—
179(C2×D83)⋊C4—24no—
180(C2×D83)⋊C4—24no—
181C2×Q8×C83—41no—
182C2×Q8×C83—41no—
183C2×Q8×C83C22⋊C414yes—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2411yescharacteristic
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83—81no—
204C4⋊C4×C83C22⋊C414yes—
205C4⋊C4×C83C22⋊C414yes—
206C4⋊C4×C83C22⋊C414yes—
207C4⋊C4×C83C2×Q812yes—
208C42×C83—42no—
209C42×C83—81no—
210C42×C83—81no—
211C42×C83C22×C412yes—
212C42×C83C22×C412yes—
213C42×C83C22×C412yes—
214C42×C83C2×D412yes—
215C42×C83C2×D412yes—
216C42×C83C2×D412yes—
217C23×C83C22×D411yescharacteristic
218C22×D83—22no—
219C22×D83—22no—
220C22×D83—22no—
221C22×D83—41no—
222C22×D83—41no—
223C22×D83—41no—
224C22×D83—41no—
225Q8×C83—44no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
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—21no—
238C2×C4×C83—22no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83C2×C22⋊C412yes—
251C2×C4×C83C2×C22⋊C412yes—
252C2×C4×C83C2×C22⋊C412yes—
253C2×C4×C83C2×C22⋊C412yes—
254C2×C4×C83C2×C22⋊C412yes—
255C2×C4×C83C2×C22⋊C412yes—
256C2×C4×C83C23×C411yescharacteristic
257C2×C4×C83C22×D411yescharacteristic
258C2×C4×C83C22×Q811yescharacteristic
259C2×C4×C83C2×#4111yescharacteristic
260C2×C4×C832−1+411yescharacteristic
261C2×C4×C832−1+411yescharacteristic
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271D332—22no—
272D332—22no—
273D332—22no—
274D332C4×Q814yes—
275C4×D83—48no—
276C4×D83—84no—
277C4×D83—84no—
278Dic166—84no—
279C83⋊C8—22no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C83⋊C8—41no—
283C22×C83C22×C22⋊C411yescharacteristic
284C22×C83#54811yescharacteristic
285C22×C83D4×Q811yescharacteristic
286C22×C83#57511yescharacteristic
287C22×C83#57511yescharacteristic
288C22×C83#57611yescharacteristic
289C22×C83#57711yescharacteristic
290C2×D83—24no—
291C2×D83—42no—
292C2×D83—42no—
293C2×D83—42no—
294C2×D83—42no—
295C2×D83—42no—
296C2×D83—42no—
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—22no—
300C4×C83—22no—
301C4×C83—22no—
302C4×C83—22no—
303C4×C83—42no—
304C4×C83—81no—
305C4×C83—81no—
306C4×C83—81no—
307C4×C83C2×C4×Q811yescharacteristic
308C4×C83#51911yescharacteristic
309C4×C83#54811yescharacteristic
310C4×C83D4×Q811yescharacteristic
311Dic83—42no—
312Dic83—42no—
313Dic83—42no—
314#22526—831nomaximal, Sylow
315C2×C83Q8×C22⋊C411yescharacteristic
316C2×C83#289611yescharacteristic
317C2×C83#289711yescharacteristic
318C2×C83#337011yescharacteristic
319C2×C83#337611yescharacteristic
320C2×C83#338811yescharacteristic
321C2×C83#338911yescharacteristic
322D83—44no—
323C2×#554—831no—
324C2×#446—831no—
325#2355D8311yescharacteristic, complemented
326#1993—831no—
327#1992—831no—
328#1991—831no—
329#1989—831no—
330#1989—831no—
331#1988—831no—
332#1804—831no—
333#1804—831no—
334#1802—831no—
335#1762—831no—
336#1762—831no—
337C4×#102—831no—
338C83#2252611yescharacteristic, complemented, Sylow
339#554—838no—
340#554—1664no—
341C2×C4⋊Q8—831no—
342C2×#125—831no—
343C2×#125—831no—
344C2×C22⋊Q8—1661no—
345C2×C22⋊Q8—1661no—
346C2×C22⋊Q8—1661no—
347C2×C4×Q8C2×D8311yescharacteristic
348C2×C4×D4—831no—
349C2×C4×D4—831no—
350C2×C4×D4—831no—
351C22×C4⋊C4—1661no—
352#446—1664no—
353C2×C4⋊C8—831no—
354C2×C4⋊C8—831no—
355C2×C4⋊C8—831no—
356C2×#102—831no—
357C2×#102—831no—
358C2×#102—831no—
359C2×#102—831no—
360C2×#97—831no—
361C2×#97—831no—
362C2×C4×C8—831no—
363#391—21no—
364#387—21no—
365#387—21no—
366#384C2×D8311yescharacteristic
367#382C2×D8311yescharacteristic
368#382C2×D8311yescharacteristic
369C4×C4⋊C4—21no—
370C4×C4⋊C4C2×D8311yescharacteristic
371C4×C4⋊C4C2×D8311yescharacteristic
372C4×C4⋊C4C2×D8311yescharacteristic
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379#340—831no—
380#340—831no—
381#340—831no—
382#340—831no—
383#340—831no—
384#340—831no—
385C22×Q8—1661no—
386C22×Q8C22⋊D8311yescharacteristic
387C22×D4—831no—
388C23×C4—1661no—
389C23×C4—1661no—
390C23×C4—1661no—
391C22×C8—831no—
392C22×C8—831no—
393C22×C8—831no—
394C22×C8—831no—
395C4⋊Q8—834no—
396C4⋊Q8—1662no—
397#125—834no—
398#125—834no—
399#125—1662no—
400#125—1662no—
401C22⋊Q8—1668no—
402C22⋊Q8—1668no—
403C22⋊Q8—1668no—
404C4×Q8C4×D8314yes—
405C4×Q8D33214yes—
406C4×D4—834no—
407C4×D4—834no—
408C4×D4—834no—
409C4×D4—1662no—
410C4×D4—1662no—
411C4×D4—1662no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1668no—
423C2×C4⋊C4C22⋊D8311yescharacteristic
424C2×C4⋊C4C22⋊D8311yescharacteristic
425C2×C4⋊C4C22⋊D8311yescharacteristic
426C2×C4⋊C4C22×D8311yescharacteristic
427C2×C4⋊C4C22×D8311yescharacteristic
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8311yescharacteristic
430C2×C22⋊C4—1661no—
431C2×C22⋊C4—1661no—
432C2×C22⋊C4—1661no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D8311yescharacteristic
437C2×C42C22×D8311yescharacteristic
438C2×C42C22×D8311yescharacteristic
439C4⋊C8—1662no—
440C4⋊C8—1662no—
441C4⋊C8—1662no—
442#102—834no—
443#102—1662no—
444#102—1662no—
445#102—1662no—
446#97—1662no—
447#97—1662no—
448C4×C8—1662no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456C24—1661no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—6641no—
460C2×Q8—6641no—
461C2×Q8—1664no—
462C2×Q8(C2×D83)⋊C414yes—
463C2×D4—832no—
464C2×D4—832no—
465C2×D4—832no—
466C2×D4—1661no—
467C2×D4—1661no—
468C2×D4—1661no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—3322no—
474C22×C4—3322no—
475C22×C4—6641no—
476C22×C4—6641no—
477C22×C4—6641no—
478C22×C4—6641no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4—1661no—
482C22×C4—1664no—
483C22×C4—1664no—
484C22×C4—1664no—
485C22×C4C2×C22⋊D8311yescharacteristic
486C22×C4C2×C22⋊D8311yescharacteristic
487C22×C4C2×C22⋊D8311yescharacteristic
488C22×C4C2×C22⋊D8311yescharacteristic
489C22×C4C2×C22⋊D8311yescharacteristic
490C22×C4C2×C22⋊D8311yescharacteristic
491C22×C4C23×D8311yescharacteristic
492C2×C8—832no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C2×C8—1661no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—832no—
514C4⋊C4—832no—
515C4⋊C4—832no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1664no—
520C4⋊C4—1664no—
521C4⋊C4—1664no—
522C4⋊C4—1664no—
523C4⋊C4—1664no—
524C4⋊C4(C2×D83)⋊C414yes—
525C4⋊C4(C2×D83)⋊C414yes—
526C4⋊C4(C2×D83)⋊C414yes—
527C4⋊C4Q8×D8312yes—
528C22⋊C4—1664no—
529C22⋊C4—1664no—
530C22⋊C4—1664no—
531C42—42no—
532C42—81no—
533C42—81no—
534C42C2×C4×D8312yes—
535C42C2×C4×D8312yes—
536C42C2×C4×D8312yes—
537C42C2×D33212yes—
538C42C2×D33212yes—
539C42C2×D33212yes—
540C23—3321no—
541C23—3321no—
542C23—3321no—
543C23—3321no—
544C23—1662no—
545C23—1662no—
546C23—1662no—
547C23C22×C22⋊D8311yescharacteristic
548Q8—44no—
549Q8—6644no—
550D4—834no—
551D4—1662no—
552D4—1662no—
553D4—1662no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—3328no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—6644no—
586C2×C4—6644no—
587C2×C4—1662no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4C2×(C2×D83)⋊C412yes—
591C2×C4C2×(C2×D83)⋊C412yes—
592C2×C4C2×(C2×D83)⋊C412yes—
593C2×C4C2×(C2×D83)⋊C412yes—
594C2×C4C2×(C2×D83)⋊C412yes—
595C2×C4C2×(C2×D83)⋊C412yes—
596C2×C4C22×C4×D8311yescharacteristic
597C2×C4C22×D33211yescharacteristic
598C2×C4C2×Q8×D8311yescharacteristic
599C2×C4#41×D8311yescharacteristic
600C2×C4#1458183911yescharacteristic
601C2×C4#1458184011yescharacteristic
602C8—3321no—
603C8—3321no—
604C8—3321no—
605C8—1662no—
606C22—3322no—
607C22—3322no—
608C22—3322no—
609C22—3322no—
610C22—3322no—
611C22—3322no—
612C22—1664no—
613C22C22×(C2×D83)⋊C411yescharacteristic
614C22#1912835811yescharacteristic
615C22Q8×C22⋊D8311yescharacteristic
616C22#1912849911yescharacteristic
617C22#1912849911yescharacteristic
618C22#1912850411yescharacteristic
619C22#1912850911yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—3322no—
631C4—3322no—
632C4—3322no—
633C4C4×Q8×D8311yescharacteristic
634C4#1912817711yescharacteristic
635C4#1912836311yescharacteristic
636C4Q8×D33211yescharacteristic
637C2—3324no—
638C2Q8×(C2×D83)⋊C411yescharacteristic
639C2#2491598011yescharacteristic
640C2#2491598611yescharacteristic
641C2#2492021011yescharacteristic
642C2#2492028411yescharacteristic
643C2#2492040211yescharacteristic
644C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2355C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2355D83
hypocenterC83#22526
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×#19127832C2semidirect
3. #24908435C2semidirect
4. #24908431C2semidirect
5. #24908427C2semidirect
6. #24908420C2semidirect
7. #24908420C2semidirect
8. #24908415C2semidirect
15. C83×#2355C2semidirect
17. #19128411C4semidirect
104. #14581712Q8semidirect
325. #2355D83semidirect
338. C83#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#191284112
C2×#191278322
#249084352
#249084312
#249084272
#249084202
#249084152
#249074702
#249074602
#249072912
C4×#145817122
C83×#23552
#2252683