Finite Group Databasebeta

← #35329913

Subgroups — #35329913

Order 21248.

1894 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127612C211yesmaximal, characteristic
2C2×#19127592C211yesmaximal, characteristic
3C2×#19127591C212yesmaximal
4#24905734C211yesmaximal, characteristic, complemented
5#24905722C211yesmaximal, characteristic, complemented
6#24905705C211yesmaximal, characteristic
7#24905701C212yesmaximal
8#24905696C212yesmaximal, complemented
9#24905695C212yesmaximal, complemented
10C4×C83⋊#95C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C83C211yesmaximal, characteristic, complemented
12C22×Dic83⋊C4C2211yescharacteristic
13C22×(C2×D83)⋊C4C2211yescharacteristic
14C22×(C2×D83)⋊C4C2211yescharacteristic
15C22×(C2×D83)⋊C4C2212yes—
16C22×(C2×D83)⋊C4C2212yes—
17C22×(C2×D83)⋊C4C2211yescharacteristic
18C22×C4×Dic83C2212yes—
19#19127612C418yescomplemented
20#19127592—24no—
21#19127591—28no—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2211yescharacteristic
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2211yescharacteristic
32C2×C83⋊#95C2212yes—
33C23×Q8×C83C2211yescharacteristic
34C2×C4×Q8×C83—28no—
35C2×C4×Q8×C83—24no—
36C22×C4⋊C4×C83C2211yescharacteristic
37C22×C4⋊C4×C83C2212yes—
38C22×C42×C83C2212yes—
39C22×C42×C83C2211yescharacteristic
40C24×D83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2312yes—
46C2×Dic83⋊C4—22no—
47C2×Dic83⋊C4—24no—
48C2×(C2×D83)⋊C4—24no—
49C2×(C2×D83)⋊C4—24no—
50C2×(C2×D83)⋊C4—22no—
51C2×(C2×D83)⋊C4—24no—
52C2×(C2×D83)⋊C4—24no—
53C2×(C2×D83)⋊C4—22no—
54C2×(C2×D83)⋊C4—22no—
55C2×(C2×D83)⋊C4—24no—
56C2×(C2×D83)⋊C4—22no—
57C2×(C2×D83)⋊C4—24no—
58C2×(C2×D83)⋊C4—42no—
59C2×(C2×D83)⋊C4C2×C418yes—
60C2×(C2×D83)⋊C4C2×C414yes—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—24no—
63C2×C4×Dic83—42no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—24no—
67C83⋊#95—22no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—24no—
71C83⋊#95—24no—
72C83⋊#95—42no—
73C83⋊#95C2×C414yes—
74C83⋊#95C2×C418yes—
75C22×Q8×C83—24no—
76C22×Q8×C83—22no—
77C22×Q8×C83—22no—
78C22×Q8×C83—24no—
79C22×Q8×C83C2×C414yes—
80C23×C4×C83C2312yes—
81C23×C4×C83C2312yes—
82C23×C4×C83C2311yescharacteristic
83C23×C4×C83C2311yescharacteristic
84C23×C4×C83C2311yescharacteristic
85C4×Q8×C83—232no—
86C2×C4⋊C4×C83—24no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—24no—
89C2×C4⋊C4×C83—24no—
90C2×C4⋊C4×C83D418yes—
91C2×C42×C83—24no—
92C2×C42×C83—24no—
93C2×C42×C83—24no—
94C2×C42×C83—24no—
95C2×C42×C83D414yes—
96C23×D83—21no—
97C23×D83—22no—
98C23×D83—22no—
99C23×D83—21no—
100C23×D83—22no—
101C23×D83—22no—
102C23×D83—21no—
103C23×D83—21no—
104C23×D83—21no—
105C23×D83—21no—
106C23×D83C22×C412yes—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—21no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—21no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83C22×C412yes—
134C22×Dic83C22×C414yes—
135C22×Dic83#4114yes—
136Dic83⋊C4—44no—
137(C2×D83)⋊C4—216no—
138(C2×D83)⋊C4—28no—
139(C2×D83)⋊C4—48no—
140(C2×D83)⋊C4—44no—
141(C2×D83)⋊C4—44no—
142C4×Dic83—48no—
143C24×C83C2411yescharacteristic
144C2×Q8×C83—28no—
145C2×Q8×C83—28no—
146C2×Q8×C83—216no—
147C2×Q8×C83—28no—
148C2×Q8×C83—216no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
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×C83C2×D414yes—
170C22×C4×C83C2×D414yes—
171C22×C4×C83C2×D412yes—
172C22×C4×C83C2×D414yes—
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2×D412yes—
175C22×C4×C83#4112yes—
176C22×C4×C83#4114yes—
177C22×C4×C83#4114yes—
178C4⋊C4×C83—28no—
179C4⋊C4×C83—216no—
180C42×C83—216no—
181C42×C83—28no—
182C23×C83C23×C411yescharacteristic
183C23×C83C22×D411yescharacteristic
184C23×C83C22×D412yes—
185C23×C83C2×#4112yes—
186C23×C83C2×#4111yescharacteristic
187C23×C83C2×#4112yes—
188C23×C83C2×#4112yes—
189C23×C83H(5,𝔽2)11yescharacteristic
190C23×C832−1+411yescharacteristic
191C23×C832−1+411yescharacteristic
192C23×C832−1+411yescharacteristic
193C22×D83—22no—
194C22×D83—24no—
195C22×D83—24no—
196C22×D83—22no—
197C22×D83—22no—
198C22×D83—41no—
199C22×D83—41no—
200C22×D83—42no—
201C22×D83—41no—
202C22×D83—42no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—42no—
209C22×D83—42no—
210C22×D83—42no—
211C22×D83—42no—
212C22×D83—42no—
213C22×D83—42no—
214C22×D83—42no—
215C22×D83—42no—
216Q8×C83—232no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—24no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
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—42no—
239C2×C4×C83—42no—
240C2×C4×C83—42no—
241C2×C4×C83—42no—
242C2×C4×C83C4×D414yes—
243C2×C4×C83C4×D418yes—
244C2×C4×C83#12014yes—
245C2×C4×C83#12414yes—
246C2×Dic83—24no—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—42no—
255C2×Dic83—41no—
256C2×Dic83—42no—
257C2×Dic83—41no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83—42no—
274C2×Dic83—42no—
275C22×C83C2×C4×D411yescharacteristic
276C22×C83C2×C4×D412yes—
277C22×C83C4×#4112yes—
278C22×C83#51911yescharacteristic
279C22×C83#52011yescharacteristic
280C22×C83C2×#12011yescharacteristic
281C22×C83#52511yescharacteristic
282C22×C83#52512yes—
283C22×C83C2×#12411yescharacteristic
284C22×C83#52912yes—
285C22×C83#53211yescharacteristic
286C22×C83#53611yescharacteristic
287C22×C83#53711yescharacteristic
288C22×C83#54111yescharacteristic
289C22×C83#54211yescharacteristic
290C22×C83#54612yes—
291C22×C83#54712yes—
292C22×C83#54812yes—
293C22×C83#54812yes—
294C22×C83#55212yes—
295C22×C83#55612yes—
296C22×C83#55612yes—
297C22×C83#55912yes—
298C2×D83—42no—
299C2×D83—44no—
300C2×D83—44no—
301C2×D83—42no—
302C2×D83—42no—
303C2×D83—81no—
304C2×D83—81no—
305C2×D83—81no—
306C2×D83—81no—
307C2×D83—82no—
308C2×D83—82no—
309C4×C83—24no—
310C4×C83—24no—
311C4×C83—28no—
312C4×C83—24no—
313C4×C83—44no—
314Dic83—44no—
315Dic83—44no—
316Dic83—42no—
317Dic83—81no—
318Dic83—81no—
319#14188—831nomaximal, Sylow
320C2×C83C4×#12011yescharacteristic
321C2×C83C4×#12411yescharacteristic
322C2×C83#232311yescharacteristic
323C2×C83#236212yes—
324C2×C83#236712yes—
325C2×C83#243011yescharacteristic
326C2×C83#243611yescharacteristic
327C2×C83#245812yes—
328C2×C83#246912yes—
329C2×C83#259211yescharacteristic
330C2×C83#260411yescharacteristic
331D83—82no—
332C22×C4×Q8D8311yescharacteristic, complemented
333C2×#393—831no—
334C2×#388—831no—
335C2×#388—832no—
336#1446—831no—
337#1442—831no—
338#1439—831no—
339#1437—832no—
340#1436—832no—
341#1436—832no—
342C4×#95—831no—
343C83#1418811yescharacteristic, complemented, Sylow
344C23×Q8C2×D8311yescharacteristic
345C2×C4×Q8—28no—
346C2×C4×Q8—24no—
347C22×C4⋊C4—831no—
348C22×C4⋊C4C2×D8311yescharacteristic
349C22×C4⋊C4C2×D8312yes—
350C22×C22⋊C4—832no—
351C22×C22⋊C4—832no—
352C22×C22⋊C4—831no—
353C22×C22⋊C4—831no—
354C22×C22⋊C4—831no—
355C22×C42—832no—
356C22×C42C2×D8312yes—
357C22×C42C2×D8311yescharacteristic
358#393—838no—
359#388—1668no—
360#388—1664no—
361C2×#95—832no—
362C2×#95—831no—
363C2×#95—832no—
364C2×#95—832no—
365C2×#95—831no—
366C2×#95—832no—
367C2×#95—831no—
368C2×#95—832no—
369C2×#95—832no—
370C2×#95—832no—
371C2×#95—831no—
372C25—831no—
373C22×Q8—24no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8—24no—
377C22×Q8C4×D8314yes—
378C23×C4—831no—
379C23×C4—832no—
380C23×C4—832no—
381C23×C4—831no—
382C23×C4—831no—
383C23×C4C22×D8312yes—
384C23×C4C22×D8312yes—
385C23×C4C22×D8311yescharacteristic
386C23×C4C22×D8311yescharacteristic
387C23×C4C22×D8311yescharacteristic
388C4×Q8—232no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—1662no—
394C2×C4⋊C4—1664no—
395C2×C4⋊C4C22⋊D8318yes—
396C2×C22⋊C4—3322no—
397C2×C22⋊C4—838no—
398C2×C22⋊C4—834no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4—1664no—
401C2×C22⋊C4—1664no—
402C2×C22⋊C4—1662no—
403C2×C22⋊C4—1662no—
404C2×C22⋊C4—1664no—
405C2×C22⋊C4—1664no—
406C2×C22⋊C4—1662no—
407C2×C22⋊C4—1664no—
408C2×C22⋊C4—1662no—
409C2×C42—24no—
410C2×C42—24no—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—3322no—
414C2×C42—1664no—
415C2×C42—1664no—
416C2×C42C22⋊D8314yes—
417#95—3322no—
418#95—834no—
419#95—838no—
420#95—1664no—
421#95—1662no—
422#95—1664no—
423#95—1664no—
424#95—1662no—
425#95—1664no—
426#95—1662no—
427#95—1664no—
428C24—832no—
429C24—1662no—
430C24—1661no—
431C24—1662no—
432C24—1661no—
433C24—1662no—
434C24—1662no—
435C24—1661no—
436C24—1661no—
437C24—1661no—
438C24—1661no—
439C24C23×D8311yescharacteristic
440C2×Q8—28no—
441C2×Q8—216no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—216no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—832no—
464C22×C4—834no—
465C22×C4—834no—
466C22×C4—1662no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1661no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1661no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4C2×C4×D8314yes—
493C22×C4C2×C4×D8312yes—
494C22×C4C2×C22⋊D8314yes—
495C22×C4C2×C22⋊D8312yes—
496C22×C4C2×C22⋊D8314yes—
497C22×C4C2×C22⋊D8312yes—
498C22×C4C2×C22⋊D8314yes—
499C22×C4C2×C22⋊D8312yes—
500C22×C4#1305059012yes—
501C22×C4#1305059014yes—
502C22×C4#1305059114yes—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—3324no—
506C22⋊C4—3328no—
507C22⋊C4—3324no—
508C22⋊C4—3324no—
509C22⋊C4—1668no—
510C22⋊C4—16616no—
511C42—216no—
512C42—28no—
513C42—3328no—
514C23—3321no—
515C23—3322no—
516C23—3322no—
517C23—3322no—
518C23—3322no—
519C23—3322no—
520C23—3322no—
521C23—3322no—
522C23—3322no—
523C23—3321no—
524C23—3322no—
525C23—3321no—
526C23—3321no—
527C23—3322no—
528C23—3321no—
529C23—3321no—
530C23—3321no—
531C23—3321no—
532C23—1662no—
533C23—1664no—
534C23—1664no—
535C23—1662no—
536C23—1662no—
537C23C22×C4×D8311yescharacteristic
538C23C22×C22⋊D8312yes—
539C23C22×C22⋊D8311yescharacteristic
540C23#41×D8312yes—
541C23C2×#1305059011yescharacteristic
542C23C2×#1305059012yes—
543C23C2×#1305059112yes—
544C23#1458183811yescharacteristic
545C23#1458183911yescharacteristic
546C23#1458184011yescharacteristic
547C23#1458184011yescharacteristic
548Q8—232no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3322no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3322no—
582C2×C4—3321no—
583C2×C4—3322no—
584C2×C4—3321no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—3322no—
589C2×C4—3321no—
590C2×C4—3321no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1662no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—1664no—
603C2×C4C4×C22⋊D8314yes—
604C2×C4C4×C22⋊D8318yes—
605C2×C4#1458176114yes—
606C2×C4#1458178114yes—
607C22—3322no—
608C22—3322no—
609C22—3324no—
610C22—3322no—
611C22—3324no—
612C22—6641no—
613C22—6642no—
614C22—6641no—
615C22—6642no—
616C22—6641no—
617C22—6641no—
618C22C2×C4×C22⋊D8311yescharacteristic
619C22C2×C4×C22⋊D8312yes—
620C22C4×#1305059112yes—
621C22#1912817711yescharacteristic
622C22#1912818311yescharacteristic
623C22C2×#1458176111yescharacteristic
624C22#1912821212yes—
625C22#1912821211yescharacteristic
626C22C2×#1458178111yescharacteristic
627C22#1912823812yes—
628C22#1912824711yescharacteristic
629C22#1912827511yescharacteristic
630C22#1912827911yescharacteristic
631C22#1912830911yescharacteristic
632C22#1912832311yescharacteristic
633C22#1912833912yes—
634C22#1912835012yes—
635C22#1912836212yes—
636C22#1912836212yes—
637C22#1912839212yes—
638C22#1912842312yes—
639C22#1912842412yes—
640C22#1912843612yes—
641C4—24no—
642C4—28no—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—3324no—
647C4—3322no—
648C4—3324no—
649C4—6641no—
650C4—6641no—
651C2—6642no—
652C2C4×#1458176111yescharacteristic
653C2C4×#1458178111yescharacteristic
654C2#2491013911yescharacteristic
655C2#2491048612yes—
656C2#2491053012yes—
657C2#2491106711yescharacteristic
658C2#2491111711yescharacteristic
659C2#2491137912yes—
660C2#2491153812yes—
661C2#2491295111yescharacteristic
662C2#2491309511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×C4×Q8×C83C2
socle, cosocleC24×C83C24
prime radicalC24×D83C23
hypercenterC22×C4×Q8D83
hypocenterC83#14188
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #24905734C2semidirect
5. #24905722C2semidirect
8. #24905696C2semidirect
9. #24905695C2semidirect
10. C4×C83⋊#95C2semidirect
11. C22×C4×Q8×C83C2semidirect
19. #19127612C4semidirect
332. C22×C4×Q8D83semidirect
343. C83#14188semidirect

Maximal subgroups

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

subgroupindex
C2×#191276122
C2×#191275922
C2×#191275912
#249057342
#249057222
#249057052
#249057012
#249056962
#249056952
C4×C83⋊#952
C22×C4×Q8×C832
#1418883