Finite Group Databasebeta

← #35325988

Subgroups — #35325988

Order 21248.

1906 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127591C212yesmaximal
2C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
3C2×#19127548C211yesmaximal, characteristic
4#24905701C212yesmaximal
5#24905695C212yesmaximal, complemented
6#24905690C211yesmaximal, characteristic, complemented
7#24905689C211yesmaximal, characteristic, complemented
8(C2×D83)⋊#95C211yesmaximal, characteristic
9C4×C83⋊#95C212yesmaximal, complemented
10#95×Dic83C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C83C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×(C2×D83)⋊C4C2212yes—
14C22×(C2×D83)⋊C4C2211yescharacteristic
15C22×(C2×D83)⋊C4C2212yes—
16C22×(C2×D83)⋊C4C2211yescharacteristic
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2211yescharacteristic
19C22×C4×Dic83C2212yes—
20#19127591C4116yescomplemented
21C4×(C2×D83)⋊C4C418yescomplemented
22#19127548—24no—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C22×C4⋊C4×C83C2212yes—
34C22×C42×C83C2212yes—
35C22×C42×C83C2211yescharacteristic
36C4×C4⋊C4×C83—24no—
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2211yescharacteristic
39C24×D83C2311yescharacteristic
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C23×Dic83C2311yescharacteristic
45C22×C4×D83—24no—
46C22×C4×D83—22no—
47C22×C4×D83—22no—
48C22×C4×D83—42no—
49C22×C4×D83C2×C414yes—
50C2×(C2×D83)⋊C4—24no—
51C2×(C2×D83)⋊C4—22no—
52C2×(C2×D83)⋊C4—24no—
53C2×(C2×D83)⋊C4—22no—
54C2×(C2×D83)⋊C4—42no—
55C2×(C2×D83)⋊C4—42no—
56C2×(C2×D83)⋊C4C2×C418yes—
57C2×(C2×D83)⋊C4C2×C414yes—
58C2×(C2×D83)⋊C4C2×C418yes—
59C2×(C2×D83)⋊C4C2×C418yes—
60C2×(C2×D83)⋊C4C2×C414yes—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—24no—
63C2×C4×Dic83—42no—
64C2×C4×Dic83—42no—
65C2×C4×Dic83—42no—
66C2×C4×Dic83—42no—
67C2×C4×Dic83C2×C414yes—
68C2×C4×Dic83C2×C418yescomplemented
69C83⋊#95—24no—
70C83⋊#95—22no—
71C83⋊#95—22no—
72C83⋊#95—24no—
73C83⋊#95—24no—
74C83⋊#95—42no—
75C83⋊#95C2×C414yes—
76C83⋊#95C2×C414yes—
77C83⋊#95C2×C418yescomplemented
78C83⋊#95C2×C418yescomplemented
79C23×C4×C83C2312yes—
80C23×C4×C83C2311yescharacteristic
81C23×C4×C83C2312yes—
82C23×C4×C83C2311yescharacteristic
83C23×C4×C83C2311yescharacteristic
84C2×C4⋊C4×C83—24no—
85C2×C4⋊C4×C83—24no—
86C2×C4⋊C4×C83C2×C418yescomplemented
87C2×C42×C83—22no—
88C2×C42×C83—22no—
89C2×C42×C83—24no—
90C2×C42×C83—24no—
91C2×C42×C83—42no—
92C2×C42×C83C2×C414yes—
93#95×C83—22no—
94#95×C83D414yes—
95C23×D83—21no—
96C23×D83—22no—
97C23×D83—21no—
98C23×D83—21no—
99C23×D83—22no—
100C23×D83—21no—
101C23×D83—22no—
102C23×D83—21no—
103C23×D83—21no—
104C23×D83C22×C412yes—
105C23×D83C22×C414yes—
106C22×Dic83—21no—
107C22×Dic83—21no—
108C22×Dic83—22no—
109C22×Dic83—21no—
110C22×Dic83—22no—
111C22×Dic83—21no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—21no—
121C22×Dic83—22no—
122C22×Dic83—21no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83C22×C412yes—
129C22×Dic83C22×C414yes—
130C22×Dic83C22×C414yes—
131C22×Dic83C22×C412yes—
132C22×Dic83C22×C414yes—
133C22×Dic83C22×C412yes—
134C22×Dic83#4114yes—
135C2×C4×D83—28no—
136C2×C4×D83—28no—
137C2×C4×D83—44no—
138C2×C4×D83—48no—
139C2×C4×D83—48no—
140(C2×D83)⋊C4—28no—
141(C2×D83)⋊C4—216no—
142(C2×D83)⋊C4—48no—
143(C2×D83)⋊C4C42116yescomplemented
144C4×Dic83—44no—
145C4×Dic83—48no—
146C4×Dic83—48no—
147C24×C83C2411yescharacteristic
148C22×C4×C83—21no—
149C22×C4×C83—21no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—21no—
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×C83C22×C414yes—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C22×C414yes—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C22×C414yes—
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2×D414yes—
175C22×C4×C83#4112yes—
176C22×C4×C83#4112yes—
177C22×C4×C83#4114yes—
178C22×C4×C83#4112yes—
179C4⋊C4×C83—216no—
180C42×C83—28no—
181C42×C83—48no—
182C23×C83C23×C411yescharacteristic
183C23×C83C23×C412yes—
184C23×C83C22×D411yescharacteristic
185C23×C83C2×#4112yes—
186C23×C83C2×#4111yescharacteristic
187C23×C83C2×#4111yescharacteristic
188C23×C83C2×#4112yes—
189C23×C83C2×#4111yescharacteristic
190C23×C83H(5,𝔽2)11yescharacteristic
191C23×C83H(5,𝔽2)12yes—
192C23×C832−1+411yescharacteristic
193C22×D83—22no—
194C22×D83—22no—
195C22×D83—24no—
196C22×D83—22no—
197C22×D83—24no—
198C22×D83—22no—
199C22×D83—24no—
200C22×D83—24no—
201C22×D83—24no—
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—24no—
205C22×D83—42no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—42no—
209C22×D83—41no—
210C22×D83—42no—
211C22×D83—41no—
212C22×D83—42no—
213C22×D83—42no—
214C22×D83—42no—
215C22×D83C2×C4214yes—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—22no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—22no—
231C2×C4×C83—24no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—22no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—24no—
239C2×C4×C83—42no—
240C2×C4×C83—42no—
241C2×C4×C83—42no—
242C2×C4×C83—42no—
243C2×C4×C83C2×C4214yes—
244C2×C4×C83#11714yes—
245C2×C4×C83C4×D414yes—
246C2×C4×C83C4×D418yes—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—22no—
250C2×Dic83—22no—
251C2×Dic83—22no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—22no—
256C2×Dic83—24no—
257C2×Dic83—22no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—22no—
263C2×Dic83—24no—
264C2×Dic83—42no—
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×Dic83C2×C4214yes—
275C2×Dic83#11714yes—
276C4×D83—416no—
277C22×C83C22×C4211yescharacteristic
278C22×C83C2×#11711yescharacteristic
279C22×C83C2×#11711yescharacteristic
280C22×C83C2×C4×D411yescharacteristic
281C22×C83C2×C4×D412yes—
282C22×C83C4×#4112yes—
283C22×C83C4×#4111yescharacteristic
284C22×C83C4×#4112yes—
285C22×C83C4×#4112yes—
286C22×C83#51812yes—
287C22×C83#51811yescharacteristic
288C22×C83#52011yescharacteristic
289C22×C83#52012yes—
290C22×C83#54612yes—
291C22×C83#54611yescharacteristic
292C22×C83#54811yescharacteristic
293C22×C83#55112yes—
294C22×C83#55211yescharacteristic
295C22×C83#55312yes—
296C22×C83#55311yescharacteristic
297C22×C83#55312yes—
298C22×C83#55612yes—
299C22×C83#55912yes—
300C2×D83—24no—
301C2×D83—24no—
302C2×D83—24no—
303C2×D83—42no—
304C2×D83—42no—
305C2×D83—42no—
306C2×D83—42no—
307C2×D83—44no—
308C2×D83—44no—
309C2×D83—44no—
310C2×D83—44no—
311C4×C83—24no—
312C4×C83—24no—
313C4×C83—28no—
314C4×C83—24no—
315C4×C83—44no—
316Dic83—24no—
317Dic83—24no—
318Dic83—42no—
319Dic83—44no—
320Dic83—44no—
321#13822—831nomaximal, Sylow
322C2×C83C4×#11711yescharacteristic
323C2×C83C42×D411yescharacteristic
324C2×C83#227511yescharacteristic
325C2×C83#234211yescharacteristic
326C2×C83#234711yescharacteristic
327C2×C83#235211yescharacteristic
328C2×C83#235711yescharacteristic
329C2×C83#236412yes—
330C2×C83#236712yes—
331C2×C83#237312yes—
332C2×C83#237412yes—
333D83—44no—
334C2×#388—832no—
335C2×C4×C4⋊C4D8311yescharacteristic, complemented
336C2×C4×C22⋊C4—831no—
337C2×C4×C22⋊C4—831no—
338#1437—832no—
339#1436—832no—
340#1434—831no—
341#1434—831no—
342C22⋊#95—831no—
343C4×#95—832no—
344C4×#95—831no—
345C83#1382211yescharacteristic, complemented, Sylow
346C24×C4—831no—
347C22×C4⋊C4C2×D8312yes—
348C22×C22⋊C4—832no—
349C22×C22⋊C4—831no—
350C22×C22⋊C4—831no—
351C22×C22⋊C4—832no—
352C22×C42—832no—
353C22×C42—831no—
354C22×C42—832no—
355C22×C42C2×D8312yes—
356C22×C42C2×D8311yescharacteristic
357#388—8316no—
358C4×C4⋊C4—24no—
359C4×C22⋊C4—838no—
360C4×C22⋊C4—1664no—
361C2×#95—832no—
362C2×#95—831no—
363C2×#95—832no—
364C2×#95—832no—
365C2×#95—831no—
366C2×#95—831no—
367C2×#95—832no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—831no—
371C2×#95C2×D8311yescharacteristic
372C2×#95C2×D8311yescharacteristic
373C25—831no—
374C23×C4—3322no—
375C23×C4—834no—
376C23×C4—831no—
377C23×C4—831no—
378C23×C4—832no—
379C23×C4—832no—
380C23×C4—831no—
381C23×C4—1664no—
382C23×C4—1662no—
383C23×C4—1662no—
384C23×C4C22×D8312yes—
385C23×C4C22×D8311yescharacteristic
386C23×C4C22×D8312yes—
387C23×C4C22×D8311yescharacteristic
388C23×C4C22×D8311yescharacteristic
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4C4×D8318yescomplemented
392C2×C22⋊C4—3322no—
393C2×C22⋊C4—3322no—
394C2×C22⋊C4—838no—
395C2×C22⋊C4—834no—
396C2×C22⋊C4—838no—
397C2×C22⋊C4—834no—
398C2×C22⋊C4—838no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4—1662no—
401C2×C22⋊C4—1664no—
402C2×C22⋊C4—1662no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—24no—
406C2×C42—24no—
407C2×C42—42no—
408C2×C42—3322no—
409C2×C42—3322no—
410C2×C42—3322no—
411C2×C42—3322no—
412C2×C42—834no—
413C2×C42—838no—
414C2×C42—1664no—
415C2×C42—1664no—
416C2×C42C4×D8314yes—
417#95—22no—
418#95—3322no—
419#95—834no—
420#95—834no—
421#95—838no—
422#95—838no—
423#95—1664no—
424#95—1662no—
425#95—1662no—
426#95—1664no—
427#95—1664no—
428#95C22⋊D8314yes—
429C24—834no—
430C24—832no—
431C24—1661no—
432C24—1662no—
433C24—1661no—
434C24—1662no—
435C24—1661no—
436C24—1661no—
437C24—1661no—
438C24—1662no—
439C24—1661no—
440C24C23×D8311yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—3328no—
461C22×C4—3328no—
462C22×C4—3324no—
463C22×C4—832no—
464C22×C4—834no—
465C22×C4—832no—
466C22×C4—834no—
467C22×C4—834no—
468C22×C4—832no—
469C22×C4—834no—
470C22×C4—1668no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1661no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1661no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1668no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4—1661no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1661no—
494C22×C4C2×C4×D8314yes—
495C22×C4C2×C4×D8312yes—
496C22×C4C2×C4×D8314yes—
497C22×C4C2×C4×D8312yes—
498C22×C4C2×C4×D8312yes—
499C22×C4C2×C4×D8314yes—
500C22×C4C2×C22⋊D8312yes—
501C22×C4C2×C22⋊D8314yes—
502C22×C4#1305058912yes—
503C22×C4#1305059012yes—
504C22×C4#1305059014yes—
505C22×C4#1305059112yes—
506C4⋊C4—216no—
507C22⋊C4—3328no—
508C22⋊C4—8316no—
509C22⋊C4—1668no—
510C22⋊C4—16616no—
511C42—28no—
512C42—48no—
513C42—3328no—
514C42—3328no—
515C42—3324no—
516C23—3322no—
517C23—3321no—
518C23—3321no—
519C23—3322no—
520C23—3321no—
521C23—3322no—
522C23—3321no—
523C23—3322no—
524C23—3322no—
525C23—3322no—
526C23—834no—
527C23—1662no—
528C23—1662no—
529C23—1664no—
530C23—1662no—
531C23—1664no—
532C23—1664no—
533C23—1662no—
534C23—1664no—
535C23—1664no—
536C23—1662no—
537C23—1662no—
538C23—1664no—
539C23C22×C4×D8311yescharacteristic
540C23C22×C4×D8312yes—
541C23C22×C22⋊D8311yescharacteristic
542C23#41×D8312yes—
543C23C2×#1305058911yescharacteristic
544C23C2×#1305059011yescharacteristic
545C23C2×#1305059012yes—
546C23C2×#1305059111yescharacteristic
547C23#1458183711yescharacteristic
548C23#1458183812yes—
549C23#1458184011yescharacteristic
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—33216no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3322no—
582C2×C4—3322no—
583C2×C4—3322no—
584C2×C4—3322no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—834no—
589C2×C4—834no—
590C2×C4—1662no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1662no—
594C2×C4—1664no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1662no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1662no—
602C2×C4—1664no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4—1664no—
607C2×C4C42×D8314yes—
608C2×C4#1458175014yes—
609C2×C4C4×C22⋊D8314yes—
610C2×C4C4×C22⋊D8318yes—
611C22—3324no—
612C22—3322no—
613C22—3322no—
614C22—3322no—
615C22—3324no—
616C22—3324no—
617C22—3322no—
618C22—3324no—
619C22—1664no—
620C22—1664no—
621C22—1664no—
622C22C2×C42×D8311yescharacteristic
623C22#117×D8311yescharacteristic
624C22C2×#1458175011yescharacteristic
625C22C2×C4×C22⋊D8311yescharacteristic
626C22C2×C4×C22⋊D8312yes—
627C22C4×#1305058912yes—
628C22C4×#1305059012yes—
629C22C4×#1305059011yescharacteristic
630C22C4×#1305059112yes—
631C22#1912817312yes—
632C22#1912817511yescharacteristic
633C22#1912818211yescharacteristic
634C22#1912818312yes—
635C22#1912833912yes—
636C22#1912834511yescharacteristic
637C22#1912836211yescharacteristic
638C22#1912837512yes—
639C22#1912838611yescharacteristic
640C22#1912839712yes—
641C22#1912839911yescharacteristic
642C22#1912840012yes—
643C22#1912842312yes—
644C22#1912843612yes—
645C4—24no—
646C4—28no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—3324no—
651C4—3322no—
652C4—3324no—
653C4—1664no—
654C4—1664no—
655C2—3324no—
656C2C4×#1458175011yescharacteristic
657C2C42×C22⋊D8311yescharacteristic
658C2#2490982311yescharacteristic
659C2#2491031311yescharacteristic
660C2#2491035711yescharacteristic
661C2#2491038811yescharacteristic
662C2#2491043411yescharacteristic
663C2#2491050812yes—
664C2#2491053012yes—
665C2#2491058812yes—
666C2#2491059212yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC22×C83C22×C42
FittingC2×C4×C4⋊C4×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×C4×C4⋊C4D83
hypocenterC83#13822
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #24905695C2semidirect
6. #24905690C2semidirect
7. #24905689C2semidirect
9. C4×C83⋊#95C2semidirect
10. #95×Dic83C2semidirect
11. C2×C4×C4⋊C4×C83C2semidirect
20. #19127591C4semidirect
21. C4×(C2×D83)⋊C4C4semidirect
68. C2×C4×Dic83C2×C4semidirect
77. C83⋊#95C2×C4semidirect
78. C83⋊#95C2×C4semidirect
86. C2×C4⋊C4×C83C2×C4semidirect
143. (C2×D83)⋊C4C42semidirect
335. C2×C4×C4⋊C4D83semidirect
345. C83#13822semidirect
391. C2×C4⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275912
C2×C4×(C2×D83)⋊C42
C2×#191275482
#249057012
#249056952
#249056902
#249056892
(C2×D83)⋊#952
C4×C83⋊#952
#95×Dic832
C2×C4×C4⋊C4×C832
#1382283