Finite Group Databasebeta

← #35327136

Subgroups — #35327136

Order 21248.

2222 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581749C211yesmaximal, characteristic
2C2×#19127572C212yesmaximal
3C2×#19127563C211yesmaximal, characteristic
4C2×#95×D83C211yesmaximal, characteristic
5#24905714C212yesmaximal, complemented
6#24905713C212yesmaximal, complemented
7Dic83⋊#95C211yesmaximal, characteristic, complemented
8#24905690C211yesmaximal, characteristic, complemented
9#24905689C211yesmaximal, characteristic, complemented
10(C2×D83)⋊#95C212yesmaximal
11C83×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D83C2212yes—
13C23×C4×D83C2211yescharacteristic
14C2×#14581749—28no—
15C2×#14581749C418yescomplemented
16C22×Dic83⋊C4C2212yes—
17C22×Dic83⋊C4C2211yescharacteristic
18C22×Dic83⋊C4C2211yescharacteristic
19C22×(C2×D83)⋊C4C2212yes—
20C22×(C2×D83)⋊C4C2211yescharacteristic
21C22×(C2×D83)⋊C4C2211yescharacteristic
22C22×C4×Dic83C2211yescharacteristic
23#19127572—28no—
24#19127563—24no—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34#95×D83—24no—
35C22×C4⋊C4×C83C2212yes—
36C22×C42×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2211yescharacteristic
39C2×#95×C83C2212yes—
40C24×D83C2311yescharacteristic
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46C22×C4×D83—24no—
47C22×C4×D83—24no—
48C22×C4×D83—24no—
49C22×C4×D83—24no—
50C22×C4×D83—24no—
51C22×C4×D83—24no—
52C22×C4×D83—24no—
53C22×C4×D83—42no—
54C22×C4×D83—42no—
55C22×C4×D83C2×C414yes—
56C22×C4×D83D414yes—
57C22×C4×D83Q814yes—
58#14581749—232no—
59C2×Dic83⋊C4—24no—
60C2×Dic83⋊C4—24no—
61C2×Dic83⋊C4—24no—
62C2×Dic83⋊C4—24no—
63C2×Dic83⋊C4—42no—
64C2×Dic83⋊C4C2×C414yescomplemented
65C2×Dic83⋊C4C2×C414yescomplemented
66C2×(C2×D83)⋊C4—24no—
67C2×(C2×D83)⋊C4—24no—
68C2×(C2×D83)⋊C4—24no—
69C2×(C2×D83)⋊C4—24no—
70C2×(C2×D83)⋊C4C2×C414yes—
71C2×(C2×D83)⋊C4C2×C414yes—
72C2×(C2×D83)⋊C4D418yes—
73C2×C4×Dic83—24no—
74C2×C4×Dic83C2×C414yescomplemented
75C83⋊#95—24no—
76C83⋊#95—24no—
77C83⋊#95—24no—
78C83⋊#95—24no—
79C83⋊#95—22no—
80C83⋊#95—24no—
81C83⋊#95—22no—
82C83⋊#95—24no—
83C83⋊#95—42no—
84C23×C4×C83C2312yes—
85C23×C4×C83C2311yescharacteristic
86C23×C4×C83C2312yes—
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83—24no—
90C2×C4⋊C4×C83—24no—
91C2×C4⋊C4×C83—42no—
92C2×C42×C83—24no—
93C2×C42×C83C2×C414yescomplemented
94#95×C83—22no—
95#95×C83—22no—
96#95×C83—42no—
97C23×D83—22no—
98C23×D83—22no—
99C23×D83—21no—
100C23×D83—21no—
101C23×D83—21no—
102C23×D83—21no—
103C23×D83C22×C412yes—
104C23×D83C2×D414yes—
105C23×D83C2×D412yes—
106C23×D83C2×Q812yes—
107C23×D83#4114yes—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—21no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83C22×C412yes—
131C22×Dic83C22×C412yes—
132C22×Dic83C22×C412yes—
133C22×Dic83C2×D414yes—
134C22×Dic83C2×D412yes—
135C22×Dic83C2×Q812yes—
136C22×Dic83#4114yes—
137C2×C4×D83—28no—
138C2×C4×D83—28no—
139C2×C4×D83—216no—
140C2×C4×D83—216no—
141C2×C4×D83—216no—
142C2×C4×D83—48no—
143C2×C4×D83—48no—
144C2×C4×D83—48no—
145C2×C4×D83—48no—
146C2×C4×D83—48no—
147C2×C4×D83—48no—
148C2×C4×D83C4⋊C4116yescomplemented
149Dic83⋊C4—28no—
150Dic83⋊C4—28no—
151Dic83⋊C4—48no—
152(C2×D83)⋊C4—28no—
153(C2×D83)⋊C4—48no—
154(C2×D83)⋊C4C22⋊C4116yes—
155C4×Dic83—28no—
156C24×C83C2411yescharacteristic
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83C22×C412yes—
180C22×C4×C83C22×C412yes—
181C22×C4×C83C22×C412yes—
182C22×C4×C83C2×D414yes—
183C22×C4×C83C2×D412yes—
184C22×C4×C83C2×Q812yes—
185C22×C4×C83#4114yes—
186C4⋊C4×C83—48no—
187C42×C83—28no—
188C23×C83C23×C411yescharacteristic
189C23×C83C22×D412yes—
190C23×C83C22×D411yescharacteristic
191C23×C83C22×Q811yescharacteristic
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C83C2×#4112yes—
195C23×C83H(5,𝔽2)11yescharacteristic
196C23×C83H(5,𝔽2)11yescharacteristic
197C23×C832−1+411yescharacteristic
198C23×C832−1+411yescharacteristic
199C22×D83—22no—
200C22×D83—22no—
201C22×D83—22no—
202C22×D83—22no—
203C22×D83—24no—
204C22×D83—24no—
205C22×D83—22no—
206C22×D83—22no—
207C22×D83—22no—
208C22×D83—22no—
209C22×D83—24no—
210C22×D83—24no—
211C22×D83—24no—
212C22×D83—24no—
213C22×D83—24no—
214C22×D83—24no—
215C22×D83—24no—
216C22×D83—24no—
217C22×D83C2×C22⋊C414yes—
218C22×D83C2×C4⋊C414yes—
219C22×D83#11714yes—
220C22×D83#12118yes—
221C22×D83C22⋊Q818yes—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—42no—
239C2×C4×C83—42no—
240C2×C4×C83—42no—
241C2×C4×C83—42no—
242C2×C4×C83—42no—
243C2×C4×C83—42no—
244C2×C4×C83—42no—
245C2×C4×C83—42no—
246C2×C4×C83—42no—
247C2×C4×C83—42no—
248C2×C4×C83—42no—
249C2×C4×C83C2×C22⋊C414yes—
250C2×C4×C83C2×C4⋊C414yes—
251C2×Dic83—22no—
252C2×Dic83—22no—
253C2×Dic83—22no—
254C2×Dic83—22no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—22no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—24no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83—42no—
269C2×Dic83—42no—
270C2×Dic83—42no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—42no—
275C2×Dic83—42no—
276C2×Dic83—42no—
277C2×Dic83—42no—
278C2×Dic83C2×C22⋊C414yes—
279C2×Dic83C2×C4⋊C414yes—
280C4×D83—232no—
281C4×D83—432no—
282C22×C83C22×C22⋊C411yescharacteristic
283C22×C83C22×C4⋊C411yescharacteristic
284C22×C83C2×#11711yescharacteristic
285C22×C83C4×#4111yescharacteristic
286C22×C83C4×#4111yescharacteristic
287C22×C83#52011yescharacteristic
288C22×C83#52011yescharacteristic
289C22×C83C2×#12112yes—
290C22×C83C2×C22⋊Q812yes—
291C22×C83#52512yes—
292C22×C83#53312yes—
293C22×C83#53511yescharacteristic
294C22×C83#53611yescharacteristic
295C22×C83#54411yescharacteristic
296C22×C83#54411yescharacteristic
297C22×C83#54612yes—
298C22×C83#54612yes—
299C22×C83#54712yes—
300C22×C83#54712yes—
301C22×C83#55112yes—
302C22×C83#55112yes—
303C22×C83#55212yes—
304C22×C83#55212yes—
305C2×D83—24no—
306C2×D83—24no—
307C2×D83—24no—
308C2×D83—24no—
309C2×D83—24no—
310C2×D83—28no—
311C2×D83—28no—
312C2×D83—28no—
313C2×D83—28no—
314C2×D83—24no—
315C2×D83#38018yes—
316C4×C83—24no—
317C4×C83—24no—
318C4×C83—42no—
319C4×C83—44no—
320C4×C83—82no—
321Dic83—24no—
322Dic83—24no—
323Dic83—42no—
324Dic83—44no—
325Dic83—82no—
326#13962—831nomaximal, Sylow
327C2×C83C2×#38011yescharacteristic
328C2×C83#229611yescharacteristic
329C2×C83#231711yescharacteristic
330C2×C83#233911yescharacteristic
331C2×C83#233911yescharacteristic
332C2×C83#234311yescharacteristic
333C2×C83#234311yescharacteristic
334C2×C83#241112yes—
335C2×C83#241312yes—
336C2×C83#243812yes—
337C2×C83#243812yes—
338D83—28no—
339C22×#117—831no—
340C2×#385—832no—
341C2×#381—831no—
342C22×#95—831no—
343#1441—832no—
344#1441—832no—
345C4⋊#95—831no—
346C4⋊#95D8311yescharacteristic, complemented
347#1434—831no—
348#1434—831no—
349C22⋊#95—832no—
350C83#1396211yescharacteristic, complemented, Sylow
351C24×C4—832no—
352C24×C4—831no—
353C2×#117—838no—
354C2×#117—1668no—
355C22×C4⋊C4—832no—
356C22×C4⋊C4—831no—
357C22×C4⋊C4—831no—
358C22×C4⋊C4C2×D8312yes—
359C22×C22⋊C4—831no—
360C22×C22⋊C4—831no—
361C22×C22⋊C4—832no—
362C22×C42—831no—
363C22×C42C2×D8311yescharacteristic
364#385—1668no—
365#381—1664no—
366C2×#95—832no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—831no—
370C2×#95—831no—
371C2×#95—832no—
372C2×#95—832no—
373C2×#95—832no—
374C2×#95—832no—
375C2×#95—1664no—
376C2×#95C2×D8311yescharacteristic
377C2×#95C2×D8311yescharacteristic
378C2×#95C2×D8312yes—
379C25—831no—
380C23×C4—3322no—
381C23×C4—3322no—
382C23×C4—831no—
383C23×C4—831no—
384C23×C4—834no—
385C23×C4—834no—
386C23×C4—834no—
387C23×C4—831no—
388C23×C4—832no—
389C23×C4—832no—
390C23×C4—1664no—
391C23×C4—1664no—
392C23×C4—1664no—
393C23×C4—1664no—
394C23×C4—1664no—
395C23×C4—1664no—
396C23×C4—1664no—
397C23×C4C22×D8312yes—
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8312yes—
400C23×C4C22×D8311yescharacteristic
401C23×C4C22×D8311yescharacteristic
402#117—16632no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—42no—
406C2×C4⋊C4—3322no—
407C2×C4⋊C4—834no—
408C2×C4⋊C4—834no—
409C2×C4⋊C4—1664no—
410C2×C4⋊C4—1664no—
411C2×C4⋊C4—1664no—
412C2×C4⋊C4—1664no—
413C2×C22⋊C4—834no—
414C2×C22⋊C4—838no—
415C2×C22⋊C4—834no—
416C2×C22⋊C4—1664no—
417C2×C22⋊C4—1664no—
418C2×C22⋊C4—1664no—
419C2×C22⋊C4—1664no—
420C2×C42—24no—
421C2×C42—834no—
422C2×C42—1664no—
423C2×C42C4×D8314yescomplemented
424#95—22no—
425#95—22no—
426#95—42no—
427#95—3322no—
428#95—1664no—
429#95—1664no—
430#95—1664no—
431#95—1664no—
432#95—1662no—
433#95—1662no—
434#95—1664no—
435#95—1664no—
436C24—832no—
437C24—834no—
438C24—832no—
439C24—832no—
440C24—834no—
441C24—1662no—
442C24—1662no—
443C24—1661no—
444C24—1661no—
445C24—1661no—
446C24—1661no—
447C24C23×D8311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—3328no—
471C22×C4—3328no—
472C22×C4—3328no—
473C22×C4—3328no—
474C22×C4—3328no—
475C22×C4—3328no—
476C22×C4—832no—
477C22×C4—832no—
478C22×C4—832no—
479C22×C4—834no—
480C22×C4—832no—
481C22×C4—8316no—
482C22×C4—832no—
483C22×C4—834no—
484C22×C4—16616no—
485C22×C4—1668no—
486C22×C4—1668no—
487C22×C4—16616no—
488C22×C4—16616no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1661no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1661no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4—1661no—
510C22×C4—1661no—
511C22×C4C2×C4×D8312yes—
512C22×C4C2×C4×D8312yes—
513C22×C4C2×C4×D8312yes—
514C22×C4D4×D8314yes—
515C22×C4D4×D8312yes—
516C22×C4Q8×D8312yes—
517C22×C4#1305059014yes—
518C4⋊C4—48no—
519C4⋊C4—3328no—
520C4⋊C4—1668no—
521C4⋊C4—1668no—
522C22⋊C4—3328no—
523C22⋊C4—8316no—
524C22⋊C4—1668no—
525C42—28no—
526C42—1668no—
527C23—834no—
528C23—834no—
529C23—834no—
530C23—838no—
531C23—838no—
532C23—1662no—
533C23—1662no—
534C23—1664no—
535C23—1664no—
536C23—1664no—
537C23—1664no—
538C23—1664no—
539C23—1664no—
540C23—1664no—
541C23—1664no—
542C23—1662no—
543C23—1662no—
544C23—1662no—
545C23—1662no—
546C23—1664no—
547C23—1664no—
548C23—1662no—
549C23—1662no—
550C23C22×C4×D8311yescharacteristic
551C23C2×D4×D8312yes—
552C23C2×D4×D8311yescharacteristic
553C23C2×Q8×D8311yescharacteristic
554C23#41×D8312yes—
555C23#41×D8312yes—
556C23C2×#1305059012yes—
557C23#1458183711yescharacteristic
558C23#1458183711yescharacteristic
559C23#1458184011yescharacteristic
560C23#1458184011yescharacteristic
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—33232no—
589C2×C4—3322no—
590C2×C4—3322no—
591C2×C4—3322no—
592C2×C4—3322no—
593C2×C4—3322no—
594C2×C4—3322no—
595C2×C4—3322no—
596C2×C4—3322no—
597C2×C4—3322no—
598C2×C4—3322no—
599C2×C4—3322no—
600C2×C4—3322no—
601C2×C4—3322no—
602C2×C4—834no—
603C2×C4—834no—
604C2×C4—16632no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1662no—
608C2×C4—1662no—
609C2×C4—1662no—
610C2×C4—1664no—
611C2×C4—1664no—
612C2×C4—1664no—
613C2×C4—1662no—
614C2×C4—1662no—
615C2×C4—1662no—
616C2×C4—1662no—
617C2×C4—1664no—
618C2×C4—1664no—
619C2×C4C22⋊C4×D8314yes—
620C2×C4C4⋊C4×D8314yes—
621C22—838no—
622C22—1664no—
623C22—1668no—
624C22—1668no—
625C22—1668no—
626C22—1664no—
627C22—1664no—
628C22—1664no—
629C22—1664no—
630C22—1664no—
631C22—1668no—
632C22C2×C22⋊C4×D8311yescharacteristic
633C22C2×C4⋊C4×D8311yescharacteristic
634C22#117×D8311yescharacteristic
635C22#1912816911yescharacteristic
636C22C4×#1305059011yescharacteristic
637C22#1912818211yescharacteristic
638C22#1912818211yescharacteristic
639C22#121×D8312yes—
640C22C22⋊Q8×D8312yes—
641C22#1912821012yes—
642C22#1912825512yes—
643C22#1912826411yescharacteristic
644C22#1912827611yescharacteristic
645C22#1912832811yescharacteristic
646C22#1912832811yescharacteristic
647C22#1912834112yes—
648C22#1912834812yes—
649C22#1912835212yes—
650C22#1912835312yes—
651C22#1912838012yes—
652C22#1912838112yes—
653C22#1912838312yes—
654C22#1912839212yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—3322no—
661C4—3324no—
662C4—6642no—
663C4—1664no—
664C4—1664no—
665C2—1668no—
666C2#380×D8311yescharacteristic
667C2#2490994311yescharacteristic
668C2#2491008411yescharacteristic
669C2#2491029011yescharacteristic
670C2#2491029111yescharacteristic
671C2#2491031611yescharacteristic
672C2#2491032611yescharacteristic
673C2#2491084512yes—
674C2#2491086712yes—
675C2#2491113612yes—
676C2#2491114612yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×C4⋊#95C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC4⋊#95D83
hypocenterC83#13962
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #24905714C2semidirect
6. #24905713C2semidirect
7. Dic83⋊#95C2semidirect
8. #24905690C2semidirect
9. #24905689C2semidirect
11. C83×C4⋊#95C2semidirect
15. C2×#14581749C4semidirect
64. C2×Dic83⋊C4C2×C4semidirect
65. C2×Dic83⋊C4C2×C4semidirect
74. C2×C4×Dic83C2×C4semidirect
93. C2×C42×C83C2×C4semidirect
148. C2×C4×D83C4⋊C4semidirect
346. C4⋊#95D83semidirect
350. C83#13962semidirect
423. C2×C42C4×D83semidirect

Maximal subgroups

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

subgroupindex
C22×#145817492
C2×#191275722
C2×#191275632
C2×#95×D832
#249057142
#249057132
Dic83⋊#952
#249056902
#249056892
(C2×D83)⋊#952
C83×C4⋊#952
#1396283