Finite Group Databasebeta

← #35327481

Subgroups — #35327481

Order 21248.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127574C211yesmaximal, characteristic
2C2×#19127564C211yesmaximal, characteristic
3C2×#19127564C211yesmaximal, characteristic
4C2×C4×C22⋊Dic83C211yesmaximal, characteristic
5#24905724C212yesmaximal, complemented
6#24905718C212yesmaximal, complemented
7#24905694C212yesmaximal, complemented
8#24905687C212yesmaximal, complemented
9#24905686C211yesmaximal, characteristic
10#24905686C211yesmaximal, characteristic
11C22×#95×C83C211yesmaximal, characteristic
12C22×Dic83⋊C4C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C22⋊Dic83C2211yescharacteristic
17C22×C4×Dic83C2211yescharacteristic
18C22×C4×Dic83C2211yescharacteristic
19#19127574—24no—
20#19127564—24no—
21#19127564C418yescomplemented
22C4×C22⋊Dic83—24no—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2211yescharacteristic
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C24×C4×C83C2211yescharacteristic
35C24×C4×C83C2211yescharacteristic
36C24×C4×C83C2211yescharacteristic
37C2×#95×C83—24no—
38C2×#95×C83C2212yes—
39C2×#95×C83C2212yes—
40C23×Dic83C2312yes—
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—24no—
48C2×Dic83⋊C4—42no—
49C2×C22⋊Dic83—22no—
50C2×C22⋊Dic83—22no—
51C2×C22⋊Dic83—22no—
52C2×C22⋊Dic83—22no—
53C2×C22⋊Dic83—22no—
54C2×C22⋊Dic83—22no—
55C2×C22⋊Dic83—22no—
56C2×C22⋊Dic83—22no—
57C2×C22⋊Dic83—22no—
58C2×C22⋊Dic83C2×C414yes—
59C2×C22⋊Dic83C2×C414yes—
60C2×C22⋊Dic83D414yes—
61C2×C4×Dic83—22no—
62C2×C4×Dic83—22no—
63C2×C4×Dic83—24no—
64C2×C4×Dic83—42no—
65C83⋊#95—24no—
66C83⋊#95—24no—
67C83⋊#95—24no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—24no—
71C83⋊#95—24no—
72C83⋊#95—42no—
73C83⋊#95—42no—
74C83⋊#95C2×C418yescomplemented
75C83⋊#95C2×C418yescomplemented
76C25×C83C2311yescharacteristic
77C23×C4×C83—22no—
78C23×C4×C83—22no—
79C23×C4×C83—22no—
80C23×C4×C83—22no—
81C23×C4×C83—22no—
82C23×C4×C83—22no—
83C23×C4×C83—22no—
84C23×C4×C83—22no—
85C23×C4×C83—22no—
86C23×C4×C83—24no—
87C23×C4×C83—24no—
88C23×C4×C83—22no—
89C23×C4×C83—22no—
90C23×C4×C83—42no—
91C23×C4×C83—42no—
92C23×C4×C83C2×C414yes—
93C23×C4×C83D414yes—
94C23×C4×C83C2312yes—
95C23×C4×C83C2311yescharacteristic
96C23×C4×C83C2312yes—
97C23×C4×C83C2311yescharacteristic
98#95×C83—24no—
99#95×C83—24no—
100C22×Dic83—21no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—21no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—21no—
111C22×Dic83—21no—
112C22×Dic83—21no—
113C22×Dic83—21no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83C22×C412yes—
130C22×Dic83C22×C414yes—
131C22×Dic83C22×C412yes—
132C22×Dic83C2×D412yes—
133C22×Dic83C2×D412yes—
134Dic83⋊C4—48no—
135C22⋊Dic83—28no—
136C22⋊Dic83—28no—
137C22⋊Dic83—28no—
138C22⋊Dic83—44no—
139C4×Dic83—44no—
140C4×Dic83—44no—
141C24×C83—22no—
142C24×C83—22no—
143C24×C83—22no—
144C24×C83—22no—
145C24×C83C22×C412yes—
146C24×C83C2×D412yes—
147C24×C83C2×D412yes—
148C24×C83#4112yes—
149C24×C83#4112yes—
150C24×C83#4112yes—
151C24×C83#4112yes—
152C24×C83C2411yescharacteristic
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—22no—
156C22×C4×C83—28no—
157C22×C4×C83—28no—
158C22×C4×C83—28no—
159C22×C4×C83—28no—
160C22×C4×C83—28no—
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—21no—
169C22×C4×C83—21no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—28no—
178C22×C4×C83—44no—
179C22×C4×C83—48no—
180C22×C4×C83—48no—
181C22×C4×C83—48no—
182C22×C4×C83—48no—
183C22×C4×C83—44no—
184C22×C4×C83—44no—
185C22×C4×C83—44no—
186C22×C4×C83—44no—
187C22×C4×C83—44no—
188C22×C4×C83—44no—
189C22×C4×C83C22×C414yes—
190C22×C4×C83C2×D412yes—
191C22×C4×C83C2×D412yes—
192C22×C4×C83#4114yes—
193C22×C4×C83#4114yes—
194C23×C83—24no—
195C23×C83—24no—
196C23×C83—22no—
197C23×C83—22no—
198C23×C83—24no—
199C23×C83—22no—
200C23×C83—22no—
201C23×C83—24no—
202C23×C83—24no—
203C23×C83—24no—
204C23×C83—24no—
205C23×C83—24no—
206C23×C83—24no—
207C23×C83—24no—
208C23×C83—24no—
209C23×C83—24no—
210C23×C83#11714yes—
211C23×C83C4×D414yes—
212C23×C83C4×D414yes—
213C23×C83#12114yes—
214C23×C83#12314yes—
215C23×C83#12414yes—
216C23×C83#12614yes—
217C23×C83C23×C411yescharacteristic
218C23×C83C22×D411yescharacteristic
219C23×C83C22×D411yescharacteristic
220C23×C83C2×#4111yescharacteristic
221C23×C83C2×#4111yescharacteristic
222C23×C83C2×#4111yescharacteristic
223C23×C83C2×#4111yescharacteristic
224C23×C83C2×#4112yes—
225C23×C83C2×#4112yes—
226C23×C83H(5,𝔽2)12yes—
227C23×C832−1+412yes—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—24no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—416no—
242C2×C4×C83—416no—
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×C83—42no—
250C2×C4×C83—42no—
251C2×C4×C83—42no—
252C2×C4×C83—42no—
253C2×C4×C83—416no—
254C2×C4×C83#12314yes—
255C2×Dic83—24no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—24no—
265C2×Dic83—24no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—42no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—42no—
276C2×Dic83—42no—
277C2×Dic83—42no—
278C2×Dic83—42no—
279C2×Dic83—42no—
280C2×Dic83—42no—
281C2×Dic83—42no—
282C2×Dic83—42no—
283C2×Dic83—42no—
284C2×Dic83—42no—
285C2×Dic83—42no—
286C2×Dic83—42no—
287C2×Dic83—42no—
288C2×Dic83—42no—
289C22×C83—24no—
290C22×C83—28no—
291C22×C83—28no—
292C22×C83—24no—
293C22×C83—24no—
294C22×C83—24no—
295C22×C83—28no—
296C22×C83—28no—
297C22×C83—24no—
298C22×C83—24no—
299C22×C83#38818yes—
300C22×C83C2×#11711yescharacteristic
301C22×C83C2×C4×D411yescharacteristic
302C22×C83C2×C4×D411yescharacteristic
303C22×C83C4×#4112yes—
304C22×C83#52012yes—
305C22×C83C2×#12111yescharacteristic
306C22×C83C2×#12311yescharacteristic
307C22×C83C2×#12311yescharacteristic
308C22×C83#52511yescharacteristic
309C22×C83C2×#12411yescharacteristic
310C22×C83C2×#12611yescharacteristic
311C22×C83#52912yes—
312C22×C83#53411yescharacteristic
313C22×C83#53611yescharacteristic
314C22×C83#53912yes—
315C22×C83#54612yes—
316C22×C83#54712yes—
317C22×C83#55112yes—
318C22×C83#55112yes—
319C22×C83#55112yes—
320C22×C83#55212yes—
321C22×C83#55212yes—
322C22×C83#55212yes—
323C4×C83—24no—
324C4×C83—42no—
325C4×C83—44no—
326C4×C83—44no—
327Dic83—42no—
328Dic83—44no—
329Dic83—42no—
330Dic83—42no—
331Dic83—42no—
332Dic83—82no—
333#13992—831nomaximal, Sylow
334C2×C83—28no—
335C2×C83C2×#38811yescharacteristic
336C2×C83C4×#12311yescharacteristic
337C2×C83#232211yescharacteristic
338C2×C83#234312yes—
339C2×C83#235912yes—
340C2×C83#248311yescharacteristic
341C2×C83#248411yescharacteristic
342C2×C83#248611yescharacteristic
343C2×C83#248812yes—
344C2×C83#249312yes—
345C2×C83#253411yescharacteristic
346C2×#385—831no—
347C2×#381—831no—
348C2×#381—831no—
349C2×C4×C22⋊C4—831no—
350C22×#95D8311yescharacteristic
351#1443—832no—
352#1442—832no—
353#1436—832no—
354#1434—832no—
355C22⋊#95—831no—
356C22⋊#95—831no—
357C83#1399211yescharacteristic, complemented, Sylow
358C24×C4C2×D8311yescharacteristic
359C24×C4C2×D8311yescharacteristic
360C24×C4C2×D8311yescharacteristic
361C22×C4⋊C4—832no—
362C22×C22⋊C4—831no—
363C22×C22⋊C4—831no—
364C22×C22⋊C4—831no—
365C22×C22⋊C4—831no—
366C22×C42—831no—
367C22×C42—831no—
368#385—1664no—
369#381—838no—
370#381—1664no—
371C4×C22⋊C4—1664no—
372C2×#95—24no—
373C2×#95—832no—
374C2×#95—832no—
375C2×#95—832no—
376C2×#95—832no—
377C2×#95—832no—
378C2×#95—832no—
379C2×#95—832no—
380C2×#95—832no—
381C2×#95—832no—
382C2×#95—831no—
383C2×#95—831no—
384C2×#95C2×D8312yes—
385C2×#95C2×D8312yes—
386C25C22×D8311yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—22no—
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—24no—
397C23×C4—24no—
398C23×C4—22no—
399C23×C4—22no—
400C23×C4—42no—
401C23×C4—42no—
402C23×C4—831no—
403C23×C4—832no—
404C23×C4—831no—
405C23×C4—831no—
406C23×C4—831no—
407C23×C4—832no—
408C23×C4C4×D8314yes—
409C23×C4C22⋊D8314yes—
410C23×C4C22×D8312yes—
411C23×C4C22×D8311yescharacteristic
412C23×C4C22×D8312yes—
413C23×C4C22×D8311yescharacteristic
414C2×C4⋊C4—3322no—
415C2×C4⋊C4—1664no—
416C2×C4⋊C4—1664no—
417C2×C22⋊C4—834no—
418C2×C22⋊C4—834no—
419C2×C22⋊C4—834no—
420C2×C22⋊C4—1662no—
421C2×C22⋊C4—1662no—
422C2×C22⋊C4—1662no—
423C2×C22⋊C4—1662no—
424C2×C22⋊C4—1662no—
425C2×C22⋊C4—1662no—
426C2×C22⋊C4—1662no—
427C2×C22⋊C4—1662no—
428C2×C22⋊C4—1662no—
429C2×C42—3322no—
430C2×C42—1662no—
431C2×C42—1662no—
432C2×C42—1664no—
433#95—24no—
434#95—24no—
435#95—3322no—
436#95—3322no—
437#95—838no—
438#95—838no—
439#95—1664no—
440#95—1664no—
441#95—1664no—
442#95—1662no—
443#95—1662no—
444#95—1664no—
445#95—1664no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D8312yes—
451C24D4×D8312yes—
452C24C2×C22⋊D8312yes—
453C24#1305058912yes—
454C24#1305058912yes—
455C24#1305058912yes—
456C24#1305059012yes—
457C24C23×D8311yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—834no—
495C22×C4—832no—
496C22×C4—832no—
497C22×C4—832no—
498C22×C4—832no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—1662no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1661no—
508C22×C4—1661no—
509C22×C4—1661no—
510C22×C4—1661no—
511C22×C4—1661no—
512C22×C4—1662no—
513C22×C4—1661no—
514C22×C4—1662no—
515C22×C4—1662no—
516C22×C4—1662no—
517C22×C4—1662no—
518C22×C4—1661no—
519C22×C4—1661no—
520C22×C4—1662no—
521C22×C4—1662no—
522C22×C4—1662no—
523C22×C4—1662no—
524C22×C4—1662no—
525C22×C4—1662no—
526C22×C4—1662no—
527C22×C4—1662no—
528C22×C4C2×C4×D8314yes—
529C22×C4C2×C22⋊D8312yes—
530C22×C4C2×C22⋊D8312yes—
531C22×C4#1305059014yes—
532C22×C4#1305059014yes—
533C4⋊C4—3328no—
534C22⋊C4—3324no—
535C22⋊C4—1668no—
536C22⋊C4—1668no—
537C22⋊C4—1668no—
538C42—3324no—
539C42—3324no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23#1458174814yes—
557C23(C22⋊D83)⋊C414yes—
558C23C4×C22⋊D8314yes—
559C23#1458176314yes—
560C23#1458177314yes—
561C23#1458177814yes—
562C23#1458178514yes—
563C23C22×C4×D8311yescharacteristic
564C23C2×D4×D8311yescharacteristic
565C23C22×C22⋊D8311yescharacteristic
566C23C2×#1305058911yescharacteristic
567C23C2×#1305058911yescharacteristic
568C23C2×#1305058911yescharacteristic
569C23C2×#1305059012yes—
570C23C2×#1305059012yes—
571C23C2×#1305059011yescharacteristic
572C23#1458183712yes—
573C23#1458184012yes—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—3322no—
601C2×C4—3322no—
602C2×C4—3322no—
603C2×C4—3322no—
604C2×C4—3322no—
605C2×C4—3322no—
606C2×C4—3321no—
607C2×C4—3321no—
608C2×C4—3321no—
609C2×C4—3321no—
610C2×C4—3322no—
611C2×C4—3321no—
612C2×C4—3321no—
613C2×C4—3321no—
614C2×C4—3321no—
615C2×C4—3322no—
616C2×C4—3322no—
617C2×C4—3322no—
618C2×C4—3322no—
619C2×C4—3322no—
620C2×C4—3322no—
621C2×C4—3322no—
622C2×C4—3322no—
623C2×C4—1664no—
624C2×C4—1664no—
625C2×C4—1664no—
626C2×C4—1662no—
627C2×C4—1662no—
628C2×C4—1664no—
629C2×C4—1664no—
630C2×C4—1664no—
631C2×C4—1664no—
632C2×C4—1662no—
633C2×C4—1662no—
634C2×C4#1458177514yes—
635C22—24no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22#1912758918yes—
646C22C2×#1458174811yescharacteristic
647C22C2×(C22⋊D83)⋊C411yescharacteristic
648C22C2×C4×C22⋊D8311yescharacteristic
649C22C4×#1305059012yes—
650C22#1912818212yes—
651C22C2×#1458176311yescharacteristic
652C22C2×#1458177311yescharacteristic
653C22C2×#1458177511yescharacteristic
654C22#1912821211yescharacteristic
655C22C2×#1458177811yescharacteristic
656C22C2×#1458178511yescharacteristic
657C22#1912823512yes—
658C22#1912826011yescharacteristic
659C22#1912827511yescharacteristic
660C22#1912829812yes—
661C22#1912834112yes—
662C22#1912835212yes—
663C22#1912837712yes—
664C22#1912837712yes—
665C22#1912837712yes—
666C22#1912838612yes—
667C22#1912838612yes—
668C22#1912838612yes—
669C4—24no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—3322no—
674C4—3324no—
675C4—3322no—
676C4—3322no—
677C4—3322no—
678C4—6642no—
679C2—28no—
680C2C2×#1912758911yescharacteristic
681C2C4×#1458177511yescharacteristic
682C2#2491013211yescharacteristic
683C2#2491031912yes—
684C2#2491045112yes—
685C2#2491174111yescharacteristic
686C2#2491175011yescharacteristic
687C2#2491176511yescharacteristic
688C2#2491178912yes—
689C2#2491184812yes—
690C2#2491235311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×#95×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22×#95D83
hypocenterC83#13992
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. #24905724C2semidirect
6. #24905718C2semidirect
7. #24905694C2semidirect
8. #24905687C2semidirect
21. #19127564C4semidirect
74. C83⋊#95C2×C4semidirect
75. C83⋊#95C2×C4semidirect
357. C83#13992semidirect

Maximal subgroups

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

subgroupindex
C2×#191275742
C2×#191275642
C2×C4×C22⋊Dic832
#249057242
#249057182
#249056942
#249056872
#249056862
C22×#95×C832
#1399283