Finite Group Databasebeta

← #35330471

Subgroups — #35330471

Order 21248.

1822 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127638C211yesmaximal, characteristic
2C2×#19127599C211yesmaximal, characteristic, complemented
3C2×#19127586C212yesmaximal, complemented
4C2×#19127575C212yesmaximal
5C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
6#24905730C211yesmaximal, characteristic, complemented
7#24905728C211yesmaximal, characteristic, complemented
8#24905709C211yesmaximal, characteristic
9#24905701C212yesmaximal
10#24905693C212yesmaximal, complemented
11C4×#95×C83C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×C4⋊Dic83C2212yes—
14C22×C4⋊Dic83C2211yescharacteristic
15C22×C4⋊Dic83C2212yes—
16C22×C4⋊Dic83C2211yescharacteristic
17C22×Dic83⋊C4C2212yes—
18C22×(C2×D83)⋊C4C2212yes—
19C22×(C2×D83)⋊C4C2212yes—
20C22×(C2×D83)⋊C4C2211yescharacteristic
21C22×(C2×D83)⋊C4C2211yescharacteristic
22C22×C4×Dic83C2211yescharacteristic
23#19127638C418yescomplemented
24#19127599—24no—
25#19127586—28no—
26#19127575—28no—
27C4×(C2×D83)⋊C4—24no—
28C2×C83⋊#95C2211yescharacteristic
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2211yescharacteristic
35C22×C42×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—22no—
47C22×C4×D83—22no—
48C22×C4×D83—22no—
49C22×C4×D83—24no—
50C22×C4×D83—42no—
51C2×C4⋊Dic83—24no—
52C2×C4⋊Dic83—24no—
53C2×C4⋊Dic83—22no—
54C2×C4⋊Dic83—24no—
55C2×C4⋊Dic83—22no—
56C2×C4⋊Dic83—22no—
57C2×C4⋊Dic83—24no—
58C2×C4⋊Dic83—42no—
59C2×C4⋊Dic83—42no—
60C2×C4⋊Dic83C2×C414yescomplemented
61C2×C4⋊Dic83C2×C418yescomplemented
62C2×Dic83⋊C4—24no—
63C2×Dic83⋊C4—24no—
64C2×Dic83⋊C4—24no—
65C2×(C2×D83)⋊C4—24no—
66C2×(C2×D83)⋊C4—24no—
67C2×(C2×D83)⋊C4—22no—
68C2×(C2×D83)⋊C4—22no—
69C2×(C2×D83)⋊C4—24no—
70C2×(C2×D83)⋊C4—22no—
71C2×(C2×D83)⋊C4—22no—
72C2×(C2×D83)⋊C4—24no—
73C2×(C2×D83)⋊C4—42no—
74C2×(C2×D83)⋊C4C2×C418yes—
75C2×(C2×D83)⋊C4C2×C414yes—
76C2×(C2×D83)⋊C4Q814yes—
77C2×C4×Dic83—24no—
78C2×C4×Dic83—22no—
79C83⋊#95—24no—
80C83⋊#95—24no—
81C83⋊#95—24no—
82C83⋊#95—24no—
83C83⋊#95—24no—
84C83⋊#95—22no—
85C83⋊#95—22no—
86C23×C4×C83C2312yes—
87C23×C4×C83C2312yes—
88C23×C4×C83C2311yescharacteristic
89C23×C4×C83C2311yescharacteristic
90C23×C4×C83C2311yescharacteristic
91C2×C42×C83—24no—
92C2×C42×C83—24no—
93C2×C42×C83—42no—
94C2×C42×C83—42no—
95C2×C42×C83D414yes—
96#95×C83—22no—
97#95×C83C2×C414yescomplemented
98#95×C83D418yes—
99C23×D83—21no—
100C23×D83—22no—
101C23×D83—21no—
102C23×D83—21no—
103C23×D83—21no—
104C23×D83—22no—
105C23×D83—22no—
106C23×D83—22no—
107C23×D83C22×C412yes—
108C23×D83C2×Q812yes—
109C23×D83#4112yes—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—21no—
114C22×Dic83—21no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—22no—
133C22×Dic83—22no—
134C22×Dic83—22no—
135C22×Dic83C22×C412yes—
136C22×Dic83C22×C414yes—
137C22×Dic83C2×Q812yes—
138C22×Dic83#4112yes—
139C2×C4×D83—28no—
140C2×C4×D83—48no—
141C2×C4×D83—48no—
142C2×C4×D83—44no—
143C2×C4×D83—44no—
144C4⋊Dic83—216no—
145C4⋊Dic83—28no—
146C4⋊Dic83—48no—
147C4⋊Dic83—44no—
148Dic83⋊C4—48no—
149(C2×D83)⋊C4—216no—
150(C2×D83)⋊C4—28no—
151(C2×D83)⋊C4—28no—
152(C2×D83)⋊C4—48no—
153C4×Dic83—44no—
154C24×C83C2411yescharacteristic
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
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—22no—
171C22×C4×C83—22no—
172C22×C4×C83C22×C414yes—
173C22×C4×C83C22×C412yes—
174C22×C4×C83C2×D414yes—
175C22×C4×C83C2×D414yes—
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×D412yes—
178C22×C4×C83C2×D414yes—
179C22×C4×C83C2×D412yes—
180C22×C4×C83C2×Q812yes—
181C22×C4×C83#4112yes—
182C22×C4×C83#4112yes—
183C22×C4×C83#4114yes—
184C42×C83—44no—
185C42×C83—48no—
186C23×C83C23×C411yescharacteristic
187C23×C83C22×D411yescharacteristic
188C23×C83C22×D412yes—
189C23×C83C22×Q811yescharacteristic
190C23×C83C2×#4111yescharacteristic
191C23×C83C2×#4111yescharacteristic
192C23×C83C2×#4111yescharacteristic
193C23×C83C2×#4111yescharacteristic
194C23×C83C2×#4112yes—
195C23×C83H(5,𝔽2)12yes—
196C23×C832−1+412yes—
197C22×D83—24no—
198C22×D83—22no—
199C22×D83—22no—
200C22×D83—24no—
201C22×D83—22no—
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—24no—
206C22×D83—24no—
207C22×D83—24no—
208C22×D83—24no—
209C22×D83—41no—
210C22×D83—41no—
211C22×D83—42no—
212C22×D83—41no—
213C22×D83—41no—
214C22×D83—42no—
215C22×D83—42no—
216C22×D83—42no—
217C22×D83—42no—
218C22×D83—42no—
219C22×D83C4×Q814yes—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—24no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—42no—
242C2×C4×C83—42no—
243C2×C4×C83—42no—
244C2×C4×C83—42no—
245C2×C4×C83C4×D414yes—
246C2×C4×C83C4×D418yes—
247C2×C4×C83C22⋊Q814yes—
248C2×C4×C83#12714yes—
249C2×Dic83—22no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—22no—
255C2×Dic83—22no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—42no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—41no—
268C2×Dic83—41no—
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—
278C4×D83—416no—
279C22×C83C2×C4×D411yescharacteristic
280C22×C83C2×C4×D412yes—
281C22×C83C2×C4×Q811yescharacteristic
282C22×C83C4×#4111yescharacteristic
283C22×C83#52012yes—
284C22×C83C2×C22⋊Q811yescharacteristic
285C22×C83#52511yescharacteristic
286C22×C83#52911yescharacteristic
287C22×C83C2×#12711yescharacteristic
288C22×C83#53211yescharacteristic
289C22×C83#53212yes—
290C22×C83#53311yescharacteristic
291C22×C83#53411yescharacteristic
292C22×C83#53611yescharacteristic
293C22×C83#53912yes—
294C22×C83#54612yes—
295C22×C83#54612yes—
296C22×C83#54712yes—
297C22×C83D4×Q812yes—
298C22×C83#55112yes—
299C22×C83#55212yes—
300C22×C83#55212yes—
301C22×C83#55412yes—
302C2×D83—28no—
303C2×D83—24no—
304C2×D83—42no—
305C2×D83—42no—
306C2×D83—42no—
307C2×D83—42no—
308C2×D83—44no—
309C2×D83—42no—
310C2×D83—42no—
311C2×D83—44no—
312C2×D83—44no—
313C4×C83—24no—
314C4×C83—24no—
315C4×C83—28no—
316C4×C83—24no—
317C4×C83—44no—
318Dic83—42no—
319Dic83—44no—
320Dic83—42no—
321Dic83—42no—
322Dic83—82no—
323#14230—831nomaximal, Sylow
324C2×C83C4×C22⋊Q811yescharacteristic
325C2×C83C4×#12711yescharacteristic
326C2×C83#232211yescharacteristic
327C2×C83#235912yes—
328C2×C83#236012yes—
329C2×C83#253911yescharacteristic
330C2×C83#254211yescharacteristic
331C2×C83#255312yes—
332C2×C83#255612yes—
333C2×C83#256411yescharacteristic
334C2×C83#261111yescharacteristic
335D83—44no—
336C2×#399—831no—
337C2×#389—831no—
338C2×#387—832no—
339C2×#385—832no—
340C2×C4×C22⋊C4—831no—
341#1445—831no—
342#1444—831no—
343#1439—831no—
344#1437—832no—
345C4⋊#95—832no—
346C4×#95D8311yescharacteristic, complemented
347C83#1423011yescharacteristic, complemented, Sylow
348C24×C4—831no—
349C22×C4⋊C4—831no—
350C22×C4⋊C4—832no—
351C22×C4⋊C4—832no—
352C22×C4⋊C4—831no—
353C22×C4⋊C4—832no—
354C22×C22⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4—831no—
357C22×C22⋊C4—831no—
358C22×C42—831no—
359C22×C42C2×D8312yes—
360C22×C42C2×D8311yescharacteristic
361#399—838no—
362#389—1664no—
363#387—1668no—
364#385—1668no—
365C4×C22⋊C4—1664no—
366C2×#95—831no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95—831no—
373C2×#95C2×D8311yescharacteristic
374C2×#95C2×D8311yescharacteristic
375C2×#95C2×D8312yes—
376C25—831no—
377C23×C4—3322no—
378C23×C4—832no—
379C23×C4—831no—
380C23×C4—831no—
381C23×C4—831no—
382C23×C4—832no—
383C23×C4—1662no—
384C23×C4—1662no—
385C23×C4—1664no—
386C23×C4—1662no—
387C23×C4C22×D8312yes—
388C23×C4C22×D8312yes—
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8311yescharacteristic
391C23×C4C22×D8311yescharacteristic
392C2×C4⋊C4—3322no—
393C2×C4⋊C4—3322no—
394C2×C4⋊C4—838no—
395C2×C4⋊C4—834no—
396C2×C4⋊C4—1664no—
397C2×C4⋊C4—1664no—
398C2×C4⋊C4—1664no—
399C2×C4⋊C4—1664no—
400C2×C4⋊C4—1664no—
401C2×C4⋊C4—1662no—
402C2×C4⋊C4—1664no—
403C2×C4⋊C4—1662no—
404C2×C4⋊C4—1662no—
405C2×C4⋊C4—1664no—
406C2×C22⋊C4—3322no—
407C2×C22⋊C4—838no—
408C2×C22⋊C4—834no—
409C2×C22⋊C4—834no—
410C2×C22⋊C4—1664no—
411C2×C22⋊C4—1664no—
412C2×C22⋊C4—1662no—
413C2×C22⋊C4—1662no—
414C2×C22⋊C4—1664no—
415C2×C22⋊C4—1662no—
416C2×C22⋊C4—1662no—
417C2×C22⋊C4—1664no—
418C2×C42—24no—
419C2×C42—24no—
420C2×C42—42no—
421C2×C42—42no—
422C2×C42—1662no—
423C2×C42—1664no—
424C2×C42D33214yes—
425#95—22no—
426#95—1662no—
427#95—1662no—
428#95—1664no—
429#95—1664no—
430#95—1664no—
431#95—1664no—
432#95—1664no—
433#95C4×D8314yescomplemented
434#95D33218yes—
435C24—832no—
436C24—832no—
437C24—832no—
438C24—1662no—
439C24—1661no—
440C24—1661no—
441C24—1661no—
442C24—1662no—
443C24—1661no—
444C24—1662no—
445C24—1662no—
446C24C23×D8311yescharacteristic
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—3324no—
465C22×C4—3328no—
466C22×C4—3328no—
467C22×C4—3324no—
468C22×C4—834no—
469C22×C4—832no—
470C22×C4—832no—
471C22×C4—832no—
472C22×C4—1668no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1661no—
482C22×C4—1661no—
483C22×C4—1661no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1661no—
490C22×C4—1661no—
491C22×C4—1662no—
492C22×C4—1661no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4C2×C4×D8314yes—
499C22×C4C2×C4×D8312yes—
500C22×C4C2×D33214yes—
501C22×C4C2×D33212yes—
502C22×C4C2×D33214yes—
503C22×C4C2×D33212yes—
504C22×C4C2×D33214yes—
505C22×C4C2×D33212yes—
506C22×C4Q8×D8312yes—
507C22×C4#1305058912yes—
508C22×C4#1305059012yes—
509C22×C4#1305059014yes—
510C4⋊C4—3324no—
511C4⋊C4—3328no—
512C4⋊C4—3328no—
513C4⋊C4—16616no—
514C4⋊C4—1668no—
515C22⋊C4—3328no—
516C22⋊C4—1668no—
517C22⋊C4—1668no—
518C22⋊C4—16616no—
519C42—44no—
520C42—48no—
521C42—3324no—
522C23—3322no—
523C23—3322no—
524C23—3322no—
525C23—3322no—
526C23—3321no—
527C23—3321no—
528C23—3322no—
529C23—3321no—
530C23—3321no—
531C23—3322no—
532C23—834no—
533C23—1664no—
534C23—1664no—
535C23—1664no—
536C23—1664no—
537C23—1662no—
538C23—1664no—
539C23—1662no—
540C23—1664no—
541C23—1662no—
542C23—1662no—
543C23—1662no—
544C23—1662no—
545C23C22×C4×D8311yescharacteristic
546C23C22×D33212yes—
547C23C22×D33211yescharacteristic
548C23C2×Q8×D8311yescharacteristic
549C23#41×D8311yescharacteristic
550C23#41×D8311yescharacteristic
551C23C2×#1305058911yescharacteristic
552C23C2×#1305059012yes—
553C23C2×#1305059011yescharacteristic
554C23#1458183712yes—
555C23#1458183912yes—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—33216no—
582C2×C4—3322no—
583C2×C4—3322no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3322no—
587C2×C4—3321no—
588C2×C4—3321no—
589C2×C4—3322no—
590C2×C4—3322no—
591C2×C4—3322no—
592C2×C4—3322no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3322no—
596C2×C4—3322no—
597C2×C4—3322no—
598C2×C4—3322no—
599C2×C4—3322no—
600C2×C4—3322no—
601C2×C4—3322no—
602C2×C4—1664no—
603C2×C4—1664no—
604C2×C4—1662no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1664no—
608C2×C4—1664no—
609C2×C4—1662no—
610C2×C4—1662no—
611C2×C4C4×D33214yes—
612C2×C4C4×D33218yes—
613C2×C4#1458177214yes—
614C2×C4#1458178914yes—
615C22—3322no—
616C22—3322no—
617C22—3322no—
618C22—3322no—
619C22—3322no—
620C22—3324no—
621C22—3324no—
622C22—3324no—
623C22—3322no—
624C22—1668no—
625C22—1664no—
626C22C2×C4×D33211yescharacteristic
627C22C2×C4×D33212yes—
628C22C4×Q8×D8311yescharacteristic
629C22C4×#1305058911yescharacteristic
630C22#1912818012yes—
631C22C2×#1458177211yescharacteristic
632C22#1912821311yescharacteristic
633C22#1912823311yescharacteristic
634C22C2×#1458178911yescharacteristic
635C22#1912825111yescharacteristic
636C22#1912825112yes—
637C22#1912825511yescharacteristic
638C22#1912826311yescharacteristic
639C22#1912827711yescharacteristic
640C22#1912829312yes—
641C22#1912834612yes—
642C22#1912834612yes—
643C22#1912835512yes—
644C22Q8×D33212yes—
645C22#1912837712yes—
646C22#1912838412yes—
647C22#1912839012yes—
648C22#1912840612yes—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—28no—
653C4—44no—
654C4—3322no—
655C4—3322no—
656C4—3322no—
657C4—3324no—
658C4—6642no—
659C2—3324no—
660C2C4×#1458177211yescharacteristic
661C2C4×#1458178911yescharacteristic
662C2#2491013411yescharacteristic
663C2#2491045712yes—
664C2#2491047012yes—
665C2#2491238911yescharacteristic
666C2#2491241211yescharacteristic
667C2#2491250212yes—
668C2#2491254112yes—
669C2#2491264311yescharacteristic
670C2#2491315511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC4×#95×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC4×#95D83
hypocenterC83#14230
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#19127599C2semidirect
3. C2×#19127586C2semidirect
6. #24905730C2semidirect
7. #24905728C2semidirect
10. #24905693C2semidirect
11. C4×#95×C83C2semidirect
23. #19127638C4semidirect
60. C2×C4⋊Dic83C2×C4semidirect
61. C2×C4⋊Dic83C2×C4semidirect
97. #95×C83C2×C4semidirect
346. C4×#95D83semidirect
347. C83#14230semidirect
433. #95C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276382
C2×#191275992
C2×#191275862
C2×#191275752
C2×C4×(C2×D83)⋊C42
#249057302
#249057282
#249057092
#249057012
#249056932
C4×#95×C832
#1423083