Finite Group Databasebeta

← #35332804

Subgroups — #35332804

Order 21248.

2018 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×#19127581C212yesmaximal
2C2×#19127576C212yesmaximal, complemented
3#24905716C212yesmaximal, complemented
4#24905708C212yesmaximal, complemented
5#24905708C212yesmaximal, complemented
6#24905693C211yesmaximal, characteristic, complemented
7#24905690C211yesmaximal, characteristic, complemented
8#24905685C212yesmaximal, complemented
9C2×#390×C83C211yesmaximal, characteristic
10C24×Dic83C2212yes—
11C22×C4⋊Dic83C2212yes—
12C22×C4⋊Dic83C2212yes—
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2212yes—
15C22×C22⋊Dic83C2212yes—
16#19127581—28no—
17#19127576—28no—
18C2×C83⋊#95C2212yes—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2211yescharacteristic
22C2×C83⋊#95C2211yescharacteristic
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C23×D4×C83C2211yescharacteristic
29C22×C4⋊C4×C83C2211yescharacteristic
30C22×C22⋊C4×C83C2212yes—
31C22×C22⋊C4×C83C2212yes—
32C22×C42×C83C2211yescharacteristic
33#390×C83C418yescomplemented
34C23×Dic83—24no—
35C23×Dic83—24no—
36C23×Dic83—24no—
37C23×Dic83—42no—
38C23×Dic83—42no—
39C23×Dic83—42no—
40C23×Dic83—42no—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C23×Dic83C2312yes—
45C2×C4⋊Dic83—24no—
46C2×C4⋊Dic83—24no—
47C2×C4⋊Dic83—24no—
48C2×C4⋊Dic83—42no—
49C2×C4⋊Dic83—42no—
50C2×C4⋊Dic83—42no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83—24no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—24no—
55C2×C22⋊Dic83—24no—
56C2×C22⋊Dic83—42no—
57C2×C22⋊Dic83—42no—
58C2×C22⋊Dic83—42no—
59C2×C22⋊Dic83—42no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—22no—
64C83⋊#95—42no—
65C83⋊#95—42no—
66C83⋊#95—42no—
67C83⋊#95—41no—
68C83⋊#95—41no—
69C83⋊#95D414yescomplemented
70C25×C83C2312yes—
71C22×D4×C83—42no—
72C22×D4×C83—42no—
73C22×D4×C83C2×C414yes—
74C22×D4×C83D414yes—
75C22×D4×C83Q814yes—
76C23×C4×C83C2311yescharacteristic
77C23×C4×C83C2312yes—
78C23×C4×C83C2311yescharacteristic
79C23×C4×C83C2311yescharacteristic
80C2×C4⋊C4×C83—42no—
81C2×C4⋊C4×C83C2×C414yescomplemented
82C2×C22⋊C4×C83—24no—
83C2×C22⋊C4×C83—24no—
84C2×C22⋊C4×C83—42no—
85C2×C22⋊C4×C83—42no—
86C2×C22⋊C4×C83C2×C418yescomplemented
87C2×C22⋊C4×C83C2×C418yescomplemented
88C2×C42×C83—42no—
89C2×C42×C83C2×C414yescomplemented
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—48no—
117C22×Dic83—48no—
118C22×Dic83—48no—
119C22×Dic83—48no—
120C22×Dic83—48no—
121C22×Dic83—48no—
122C22×Dic83—48no—
123C22×Dic83C2×D414yes—
124C22×Dic83#4114yes—
125C4⋊Dic83—48no—
126C4⋊Dic83—48no—
127C22⋊Dic83—48no—
128C22⋊Dic83—48no—
129C22⋊Dic83—48no—
130C24×C83—22no—
131C24×C83—22no—
132C24×C83—22no—
133C24×C83—22no—
134C24×C83—22no—
135C24×C83—22no—
136C24×C83—22no—
137C24×C83—22no—
138C24×C83—22no—
139C24×C83—22no—
140C24×C83—22no—
141C24×C83—22no—
142C24×C83C22×C414yes—
143C24×C83C2×D414yes—
144C24×C83C2×Q814yes—
145C24×C83C2411yescharacteristic
146C2×D4×C83—44no—
147C2×D4×C83—44no—
148C2×D4×C83—48no—
149C2×D4×C83—48no—
150C2×D4×C83C4⋊C4116yescomplemented
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—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83C22×C412yes—
166C22×C4×C83C22×C414yes—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2×D412yes—
171C22×C4×C83C2×Q812yes—
172C22×C4×C83#4112yes—
173C4⋊C4×C83—44no—
174C22⋊C4×C83—216no—
175C22⋊C4×C83—48no—
176C42×C83—44no—
177C23×C83—24no—
178C23×C83—24no—
179C23×C83—24no—
180C23×C83—24no—
181C23×C83—24no—
182C23×C83—24no—
183C23×C83—24no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—24no—
187C23×C83—24no—
188C23×C83—24no—
189C23×C83—24no—
190C23×C83—24no—
191C23×C83—24no—
192C23×C83—24no—
193C23×C83—24no—
194C23×C83—24no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83—42no—
198C23×C83—42no—
199C23×C83—42no—
200C23×C83—42no—
201C23×C83—42no—
202C23×C83—42no—
203C23×C83—42no—
204C23×C83—42no—
205C23×C83—42no—
206C23×C83—42no—
207C23×C83—42no—
208C23×C83—42no—
209C23×C83—42no—
210C23×C83—42no—
211C23×C83C2×C4⋊C418yes—
212C23×C83C23×C411yescharacteristic
213C23×C83C22×D411yescharacteristic
214C23×C83C22×D411yescharacteristic
215C23×C83C22×Q811yescharacteristic
216C23×C83C2×#4112yes—
217C23×C83C2×#4111yescharacteristic
218C23×C83H(5,𝔽2)12yes—
219C23×C83H(5,𝔽2)12yes—
220C23×C83H(5,𝔽2)12yes—
221C23×C83H(5,𝔽2)11yescharacteristic
222C23×C832−1+411yescharacteristic
223D4×C83—416no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
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×C83C2×C4⋊C414yes—
245C2×C4×C83C4×D414yes—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—432no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83—42no—
256C2×Dic83—42no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
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×Dic83—42no—
275C22×C83—28no—
276C22×C83—28no—
277C22×C83—28no—
278C22×C83—44no—
279C22×C83—44no—
280C22×C83—44no—
281C22×C83—44no—
282C22×C83—44no—
283C22×C83—44no—
284C22×C83—44no—
285C22×C83—44no—
286C22×C83—44no—
287C22×C83—44no—
288C22×C83—44no—
289C22×C83—44no—
290C22×C83C22×C4⋊C411yescharacteristic
291C22×C83C2×C4×D411yescharacteristic
292C22×C83C4×#4111yescharacteristic
293C22×C83#51811yescharacteristic
294C22×C83#51811yescharacteristic
295C22×C83#51811yescharacteristic
296C22×C83#52011yescharacteristic
297C22×C83#53412yes—
298C22×C83#53512yes—
299C22×C83#53812yes—
300C22×C83#54012yes—
301C22×C83#54212yes—
302C22×C83#54312yes—
303C22×C83D4211yescharacteristic
304C22×C83#54612yes—
305C22×C83#54612yes—
306C22×C83#54711yescharacteristic
307C22×C83D4×Q811yescharacteristic
308C22×C83#55312yes—
309C22×C83#55411yescharacteristic
310C22×C83#55412yes—
311C22×C83#56012yes—
312C22×C83#56212yes—
313C4×C83—24no—
314C4×C83—24no—
315C4×C83—44no—
316C4×C83—42no—
317Dic83—44no—
318Dic83—44no—
319Dic83—82no—
320Dic83—82no—
321#14438—831nomaximal, Sylow
322C2×C83—48no—
323C2×C83#231511yescharacteristic
324C2×C83#232811yescharacteristic
325C2×C83#233711yescharacteristic
326C2×C83D4×C4⋊C411yescharacteristic
327C2×C83#234911yescharacteristic
328C2×C83#236411yescharacteristic
329C2×C83#237711yescharacteristic
330C2×C83#267812yes—
331C2×C83#269212yes—
332C2×C83#270112yes—
333C2×C83#278012yes—
334C2×#390D8311yescharacteristic
335C2×#386—832no—
336C2×#385—832no—
337#1441—832no—
338#1439—832no—
339#1439—832no—
340C4⋊#95—831no—
341#1434—831no—
342C22⋊#95—832no—
343C83#1443811yescharacteristic, complemented, Sylow
344C23×D4C2×D8311yescharacteristic
345C24×C4—832no—
346C22×C4⋊C4—832no—
347C22×C4⋊C4—832no—
348C22×C4⋊C4C2×D8311yescharacteristic
349C22×C22⋊C4—832no—
350C22×C22⋊C4—832no—
351C22×C22⋊C4—832no—
352C22×C22⋊C4C2×D8312yes—
353C22×C22⋊C4C2×D8312yes—
354C22×C42C2×D8311yescharacteristic
355#390Dic8318yescomplemented
356#386—1668no—
357#385—1668no—
358C2×#95—831no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—831no—
362C2×#95—832no—
363C2×#95—831no—
364C2×#95—832no—
365C2×#95—831no—
366C2×#95—832no—
367C2×#95—832no—
368C25C22×D8312yes—
369C22×D4—42no—
370C22×D4—42no—
371C22×D4Dic16614yes—
372C22×D4D33214yes—
373C22×D4C2×Dic8314yes—
374C23×C4—3322no—
375C23×C4—3322no—
376C23×C4—3322no—
377C23×C4—3322no—
378C23×C4—832no—
379C23×C4—832no—
380C23×C4—832no—
381C23×C4—832no—
382C23×C4—1664no—
383C23×C4—1664no—
384C23×C4—1664no—
385C23×C4C22×D8312yes—
386C23×C4C22×D8311yescharacteristic
387C23×C4C22×D8311yescharacteristic
388C23×C4C22×D8311yescharacteristic
389C2×C4⋊C4—42no—
390C2×C4⋊C4—3322no—
391C2×C4⋊C4—3322no—
392C2×C4⋊C4—3322no—
393C2×C4⋊C4—1664no—
394C2×C4⋊C4—1664no—
395C2×C4⋊C4—1664no—
396C2×C4⋊C4C2×Dic8314yescomplemented
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—3322no—
402C2×C22⋊C4—3322no—
403C2×C22⋊C4—3322no—
404C2×C22⋊C4—3322no—
405C2×C22⋊C4—1664no—
406C2×C22⋊C4—1664no—
407C2×C22⋊C4—1664no—
408C2×C22⋊C4—1664no—
409C2×C22⋊C4—1664no—
410C2×C22⋊C4C2×Dic8318yescomplemented
411C2×C22⋊C4C2×Dic8318yescomplemented
412C2×C42—42no—
413C2×C42C2×Dic8314yescomplemented
414#95—3322no—
415#95—3322no—
416#95—3322no—
417#95—3321no—
418#95—3321no—
419#95—834no—
420#95—1664no—
421#95—1664no—
422#95—1664no—
423#95—1662no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24C22×Dic8314yes—
437C24C2×D33214yes—
438C24C2×Dic16614yes—
439C24C23×D8311yescharacteristic
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C4⋊Dic83116yescomplemented
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—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—3328no—
460C22×C4—3328no—
461C22×C4—3328no—
462C22×C4—3328no—
463C22×C4—3328no—
464C22×C4—3328no—
465C22×C4—3328no—
466C22×C4—834no—
467C22×C4—834no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
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—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1662no—
494C22×C4C22×Dic8312yes—
495C22×C4C22×Dic8314yes—
496C22×C4C22×Dic8312yes—
497C22×C4C22×Dic8312yes—
498C22×C4D4×D8312yes—
499C22×C4C2×D33212yes—
500C22×C4C2×Dic16612yes—
501C22×C4#1305058912yes—
502C4⋊C4—44no—
503C4⋊C4—3328no—
504C4⋊C4—3328no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—3328no—
508C22⋊C4—3328no—
509C22⋊C4—3328no—
510C42—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Dic8318yes—
546C23C23×Dic8311yescharacteristic
547C23C2×D4×D8311yescharacteristic
548C23C22×D33211yescharacteristic
549C23C22×Dic16611yescharacteristic
550C23#41×D8312yes—
551C23C2×#1305058911yescharacteristic
552C23#1458183712yes—
553C23#1458183712yes—
554C23#1458183712yes—
555C23#1458183811yescharacteristic
556C23#1458183911yescharacteristic
557D4—416no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—33232no—
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—3322no—
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—1664no—
602C2×C4—1664no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4C2×C4⋊Dic8314yes—
608C2×C4D4×Dic8314yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×C4⋊Dic8311yescharacteristic
625C22C2×D4×Dic8311yescharacteristic
626C22#41×Dic8311yescharacteristic
627C22#1912817611yescharacteristic
628C22#1912817611yescharacteristic
629C22#1912817611yescharacteristic
630C22#1912818411yescharacteristic
631C22#1912826312yes—
632C22#1912826412yes—
633C22#1912828512yes—
634C22#1912830512yes—
635C22#1912831512yes—
636C22#1912832712yes—
637C22D4×D33211yescharacteristic
638C22#1912834012yes—
639C22#1912834612yes—
640C22#1912835511yescharacteristic
641C22D4×Dic16611yescharacteristic
642C22#1912839412yes—
643C22#1912840811yescharacteristic
644C22#1912841012yes—
645C22#1912843912yes—
646C22#1912844612yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—3324no—
652C4—3324no—
653C4—6642no—
654C4—6642no—
655C2—48no—
656C2#2491007711yescharacteristic
657C2#2491018911yescharacteristic
658C2#2491027811yescharacteristic
659C2D4×C4⋊Dic8311yescharacteristic
660C2#2491037311yescharacteristic
661C2#2491051511yescharacteristic
662C2#2491062011yescharacteristic
663C2#2491378312yes—
664C2#2491395612yes—
665C2#2491407412yes—
666C2#2491511212yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#390×C83C2
socle, cosocleC24×C83C24
prime radicalC23×D4×C83C22
hypercenterC2×#390D83
hypocenterC83#14438
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#19127576C2semidirect
3. #24905716C2semidirect
4. #24905708C2semidirect
5. #24905708C2semidirect
6. #24905693C2semidirect
7. #24905690C2semidirect
8. #24905685C2semidirect
33. #390×C83C4semidirect
69. C83⋊#95D4semidirect
81. C2×C4⋊C4×C83C2×C4semidirect
86. C2×C22⋊C4×C83C2×C4semidirect
87. C2×C22⋊C4×C83C2×C4semidirect
89. C2×C42×C83C2×C4semidirect
150. C2×D4×C83C4⋊C4semidirect
343. C83#14438semidirect
355. #390Dic83semidirect
396. C2×C4⋊C4C2×Dic83semidirect
410. C2×C22⋊C4C2×Dic83semidirect
411. C2×C22⋊C4C2×Dic83semidirect
413. C2×C42C2×Dic83semidirect
444. C2×D4C4⋊Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275812
C2×#191275762
#249057162
#249057082
#249056932
#249056902
#249056852
C2×#390×C832
#1443883