Finite Group Databasebeta

← #35467490

Subgroups — #35467490

Order 21248.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127996C211yesmaximal, characteristic, complemented
2C2×C4×#13050577C211yesmaximal, characteristic
3#24911313C211yesmaximal, characteristic
4#24908987C211yesmaximal, characteristic
5#24908945C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908796C211yesmaximal, characteristic, complemented
8#24908193C211yesmaximal, characteristic, complemented
9#24908022C211yesmaximal, characteristic, complemented
10#24907818C211yesmaximal, characteristic
11#24907818C211yesmaximal, characteristic
12#24907731C211yesmaximal, characteristic, complemented
13#24907717C211yesmaximal, characteristic, complemented
14#24907709C211yesmaximal, characteristic
15C83×#2351C211yesmaximal, characteristic, complemented
16C22×#13050577C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581770—21no—
19C2×C4×D332C2211yescharacteristic
20#19127996—24no—
21C4×#13050577—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×C4×C83⋊C8C2211yescharacteristic
35#19127623—21no—
36#19127614—21no—
37#19127603C2211yescharacteristic
38#19127603C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127599C2211yescharacteristic
41#19127575—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#389×C83—21no—
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#382×C83—21no—
58#382×C83C2211yescharacteristic
59C4×C4⋊C4×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×D332C2311yescharacteristic
63C22×C4×D83—21no—
64C2×#13050577—22no—
65C2×#13050577—22no—
66C2×#13050577Q814yescomplemented
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71#14581792—22no—
72#14581792—41no—
73#14581770—44no—
74C4×D332—22no—
75C4×D332—41no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87#14581725—22no—
88#14581725—22no—
89#14581725—22no—
90#14581723—22no—
91#14581723—41no—
92C4⋊(C83⋊C8)—22no—
93#14581716—22no—
94#14581716—22no—
95#14581716D414yes—
96#14581712—22no—
97#14581712—22no—
98#14581712—41no—
99C4×C83⋊C8—22no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×Q8×C83C2311yescharacteristic
103C4×Q8×C83—22no—
104C4×Q8×C83—22no—
105C4×Q8×C83—22no—
106C4×Q8×C83—22no—
107C4×Q8×C83—41no—
108C4×Q8×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83D411yescharacteristic
115C2×C4⋊C4×C83D411yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C42×C83—21no—
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C23×D83—21no—
132C2×Dic166—42no—
133C2×Dic166—42no—
134C2×Dic166—81no—
135C2×D332—21no—
136C2×D332—21no—
137C2×D332—21no—
138C2×D332—21no—
139C2×D332—21no—
140C2×D332C2×Q812yes—
141C22×Dic83—21no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C2×C4×D83—42no—
145C2×C4×D83—42no—
146C2×C4×D83—81no—
147#13050577—28no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8C2×D412yes—
159C2×C83⋊C8C2×Q812yes—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—42no—
166C4⋊Dic83C2×D412yes—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169(C2×D83)⋊C4—42no—
170(C2×D83)⋊C4—42no—
171(C2×D83)⋊C4—42no—
172(C2×D83)⋊C4—42no—
173(C2×D83)⋊C4—81no—
174C2×Q8×C83—21no—
175C2×Q8×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83C2×D412yes—
178C2×Q8×C83C2×Q812yes—
179C2×Q8×C83#4112yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—24no—
192C4⋊C4×C83—22no—
193C4⋊C4×C83—22no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—42no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83#4112yes—
204C42×C83—24no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83—42no—
212C42×C83—42no—
213C23×C83C22×D411yescharacteristic
214C22×D83—22no—
215C22×D83—41no—
216C22×D83—41no—
217C22×D83—41no—
218C22×D83—41no—
219C22×D83—41no—
220C22×D83—41no—
221Q8×C83—22no—
222Q8×C83—22no—
223Q8×C83—22no—
224Q8×C83C22⋊Q814yescomplemented
225C2×C4×C83—22no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—22no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83#13512yes—
254C2×C4×C83AGL(1,ℤ/8)12yes—
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C22×Q811yescharacteristic
257C2×C4×C83C2×#4111yescharacteristic
258C2×C4×C83C2×#4111yescharacteristic
259C2×C4×C83H(5,𝔽2)11yescharacteristic
260C2×C4×C832−1+411yescharacteristic
261C2×Dic83—22no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270D332—22no—
271D332—22no—
272D332—41no—
273D332—41no—
274C4×D83—84no—
275Dic166—84no—
276C83⋊C8—22no—
277C83⋊C8—22no—
278C83⋊C8—41no—
279C83⋊C8—41no—
280C22×C83#52511yescharacteristic
281C22×C83D4211yescharacteristic
282C22×C83D4×Q811yescharacteristic
283C22×C83C2×#13511yescharacteristic
284C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C83#57711yescharacteristic
286C22×C83#57811yescharacteristic
287C2×D83—42no—
288C2×D83—42no—
289C2×D83—42no—
290C2×D83—81no—
291C2×D83—81no—
292C2×D83—81no—
293C2×D83—81no—
294C4×C83—24no—
295C4×C83—22no—
296C4×C83—22no—
297C4×C83—41no—
298C4×C83—42no—
299C4×C83—41no—
300C4×C83—41no—
301C4×C83—41no—
302C4×C83—81no—
303C4×C83—81no—
304C4×C83C2×C22⋊Q811yescharacteristic
305C4×C83#53911yescharacteristic
306C4×C83#54711yescharacteristic
307C4×C83#55411yescharacteristic
308Dic83—42no—
309Dic83—81no—
310Dic83—81no—
311#26377—831nomaximal, Sylow
312C2×C83#244911yescharacteristic
313C2×C83#310611yescharacteristic
314C2×C83#312111yescharacteristic
315C2×C83#326511yescharacteristic
316C2×C83#328511yescharacteristic
317C2×C83#338111yescharacteristic
318C2×C83#338611yescharacteristic
319D83—82no—
320C2×#477—831no—
321C2×C4×SD16—831no—
322#2452—831no—
323#2351D8311yescharacteristic, complemented
324#2092—831no—
325#2085—831no—
326#2067—831no—
327#2058—831no—
328#1942—831no—
329#1916—831no—
330#1880—831no—
331#1880—831no—
332#1865—831no—
333#1863—831no—
334#1862—831no—
335C83#2637711yescharacteristic, complemented, Sylow
336C22×SD16—831no—
337C2×C4⋊Q8—831no—
338C2×C22⋊Q8—1661no—
339C2×C4×Q8C2×D8311yescharacteristic
340C2×C4×Q8C2×D8311yescharacteristic
341C2×C4×Q8C2×D8311yescharacteristic
342C2×C4×D4—831no—
343#477—1664no—
344C4×SD16—1664no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#107—831no—
348C2×#106—831no—
349C2×#106—831no—
350C2×C4⋊C8—831no—
351C2×#103—831no—
352C2×#103—831no—
353C2×#103—831no—
354C2×#102—831no—
355C2×#102—831no—
356C2×#102—831no—
357C2×C4×C8—831no—
358#396—1661no—
359#393—1661no—
360#390—831no—
361#390—831no—
362#389—21no—
363#389—21no—
364#389—831no—
365#389—831no—
366#389C2×D8311yescharacteristic
367#389C2×D8311yescharacteristic
368#385—1661no—
369#382—21no—
370#382C2×D8311yescharacteristic
371C4×C4⋊C4—21no—
372C4×C4⋊C4C2×D8311yescharacteristic
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#336—831no—
378#336—831no—
379#336—831no—
380#336—831no—
381C22×Q8—1661no—
382C22×Q8C22×D8311yescharacteristic
383C22×D4—831no—
384C23×C4—1661no—
385C2×SD16—834no—
386C2×SD16—1662no—
387C2×SD16—1662no—
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391C22×C8—831no—
392C4⋊Q8—3321no—
393C4⋊Q8—1662no—
394C22⋊Q8—3324no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—41no—
401C4×D4—3321no—
402C4×D4—1662no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—831no—
409C2×C4⋊C4—831no—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4C22⋊D8311yescharacteristic
415C2×C4⋊C4C22⋊D8311yescharacteristic
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C22⋊C4—1661no—
420C2×C22⋊C4—1661no—
421C2×C22⋊C4—1661no—
422C2×C22⋊C4—1661no—
423C2×C22⋊C4—1661no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42C22×D8311yescharacteristic
428C2×C42C22×D8311yescharacteristic
429C2×C42C22×D8311yescharacteristic
430#107—1662no—
431#107—1662no—
432#107—1662no—
433#106—3321no—
434#106—1662no—
435C4⋊C8—1662no—
436#103—834no—
437#103—1662no—
438#103—1662no—
439#102—3321no—
440#102—1662no—
441#102—1662no—
442C4×C8—1662no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1661no—
448#95—1661no—
449#95C22⋊D8311yescharacteristic
450#95C22⋊D8311yescharacteristic
451C24—1661no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—3322no—
456C2×Q8—3322no—
457C2×Q8—6641no—
458C2×Q8D4×D8312yes—
459C2×Q8Q8×D8312yes—
460C2×Q8#1305059012yes—
461C2×D4—832no—
462C2×D4—1661no—
463C2×D4—1661no—
464C2×D4—1661no—
465C2×D4—1661no—
466C2×D4—1661no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—3322no—
472C22×C4—3322no—
473C22×C4—6641no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4C2×C22⋊D8311yescharacteristic
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C2×C22⋊D8311yescharacteristic
481C22×C4C2×C22⋊D8311yescharacteristic
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C23×D8311yescharacteristic
484SD16—1668no—
485C2×C8—832no—
486C2×C8—832no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—3322no—
510C4⋊C4—3322no—
511C4⋊C4—3322no—
512C4⋊C4—832no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4#1305059012yes—
519C22⋊C4—3322no—
520C22⋊C4—3322no—
521C22⋊C4—3322no—
522C22⋊C4—3322no—
523C22⋊C4—6641no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—42no—
533C23—3321no—
534C23—3321no—
535C23—3321no—
536C23—3321no—
537C23—3321no—
538C23—3321no—
539C23—1662no—
540C23C22×C22⋊D8311yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—6644no—
545Q8#1458177014yescomplemented
546D4—3321no—
547D4—3321no—
548D4—1662no—
549D4—1662no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—6644no—
587C2×C4—1662no—
588C2×C4#1458181212yes—
589C2×C4#1458181712yes—
590C2×C4C2×D4×D8311yescharacteristic
591C2×C4C2×Q8×D8311yescharacteristic
592C2×C4C2×#1305059011yescharacteristic
593C2×C4C2×#1305059011yescharacteristic
594C2×C4#1458183711yescharacteristic
595C2×C4#1458184011yescharacteristic
596C8—3321no—
597C8—3321no—
598C8—1662no—
599C8—1662no—
600C22—3322no—
601C22—3322no—
602C22—3322no—
603C22—6641no—
604C22—6641no—
605C22—6641no—
606C22—6641no—
607C22#1912821211yescharacteristic
608C22D4×C22⋊D8311yescharacteristic
609C22Q8×C22⋊D8311yescharacteristic
610C22C2×#1458181211yescharacteristic
611C22C2×#1458181711yescharacteristic
612C22#1912850911yescharacteristic
613C22#1912851611yescharacteristic
614C4—22no—
615C4—24no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—3322no—
625C4—6641no—
626C4—6641no—
627C4C2×#1458177011yescharacteristic
628C4#1912829811yescharacteristic
629C4#1912835211yescharacteristic
630C4#1912840611yescharacteristic
631C2—6642no—
632C2#2491127211yescharacteristic
633C2#2491752311yescharacteristic
634C2#2491769211yescharacteristic
635C2#2491895911yescharacteristic
636C2#2491920011yescharacteristic
637C2#2492033411yescharacteristic
638C2#2492038011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2351C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2351D83
hypocenterC83#26377
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#19127996C2semidirect
5. #24908945C2semidirect
7. #24908796C2semidirect
8. #24908193C2semidirect
9. #24908022C2semidirect
12. #24907731C2semidirect
13. #24907717C2semidirect
15. C83×#2351C2semidirect
66. C2×#13050577Q8semidirect
224. Q8×C83C22⋊Q8semidirect
323. #2351D83semidirect
335. C83#26377semidirect
545. Q8#14581770semidirect

Maximal subgroups

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

subgroupindex
C2×#191279962
C2×C4×#130505772
#249113132
#249089872
#249089452
#249088482
#249087962
#249081932
#249080222
#249078182
#249077312
#249077172
#249077092
C83×#23512
#2637783