Finite Group Databasebeta

← #35567276

Subgroups — #35567276

Order 21248.

2378 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581769C211yesmaximal, characteristic
2#24910623C214yesmaximal, complemented
3#24910487C214yesmaximal, complemented
4#24910341C212yesmaximal, complemented
5#24910338C212yesmaximal, complemented
6#24910335C212yesmaximal, complemented
7#24910295C212yesmaximal, complemented
8#24910220C214yesmaximal, complemented
9#24910138C214yesmaximal, complemented
10#24910061C211yesmaximal, characteristic
11#24910059C211yesmaximal, characteristic
12#24910059C211yesmaximal, characteristic
13#24909841C211yesmaximal, characteristic
14#24909841C211yesmaximal, characteristic
15C83×#2312C211yesmaximal, characteristic
16C23×Dic166C2211yescharacteristic
17C24×Dic83C2211yescharacteristic
18C2×#14581769C4116yescomplemented
19C2×#14581769C2212yes—
20C2×#14581769C2212yes—
21C2×#14581769C2212yes—
22C2×#14581769C2212yes—
23C2×#14581757C2214yescomplemented
24C2×#14581748C2212yes—
25C2×#14581748C2212yes—
26C22×C4⋊Dic83C2211yescharacteristic
27C22×Dic83⋊C4C2211yescharacteristic
28C22×Dic83⋊C4C2211yescharacteristic
29C22×C22⋊Dic83C2211yescharacteristic
30#19127606C2214yescomplemented
31#19127605C2214yescomplemented
32#19127605C2214yescomplemented
33#19127599C2214yescomplemented
34#19127598C2214yescomplemented
35#19127596C2214yescomplemented
36#19127595C2214yes—
37#19127595C2214yes—
38#19127594C2214yes—
39#19127594C2214yes—
40#19127589C2214yes—
41#19127589C2214yes—
42#19127588C2214yescomplemented
43#19127585C2214yescomplemented
44#19127584C2214yescomplemented
45#19127584C2214yescomplemented
46#19127583C2214yescomplemented
47#19127582C2214yescomplemented
48#19127562C2212yes—
49#19127562C2212yes—
50#19127561C2212yes—
51#19127559C2212yes—
52#19127559C2212yes—
53#19127559C2212yes—
54#19127559C2212yes—
55#19127559C2212yes—
56#19127559C2212yes—
57#19127557C2211yescharacteristic
58#19127557C2211yescharacteristic
59C4×Dic83⋊C4C2214yescomplemented
60C4×C4⋊Dic83C2214yescomplemented
61#19127552C2214yescomplemented
62C22⋊C4×Dic83C2212yes—
63C22⋊C4×Dic83C2212yes—
64C22⋊C4×Dic83C2212yes—
65C22⋊C4×Dic83C2212yes—
66C22⋊C4×Dic83C2212yes—
67C22⋊C4×Dic83C2212yes—
68C4×C22⋊Dic83C2212yes—
69C22×C22⋊C4×C83C2211yescharacteristic
70#388×C83C2214yes—
71#388×C83C2214yes—
72#380×C83C2212yes—
73#379×C83C2211yescharacteristic
74#379×C83C2211yescharacteristic
75C4×C22⋊C4×C83C2212yes—
76C22×Dic166—44no—
77C22×Dic166C2×C418yes—
78C22×Dic166C2312yes—
79C22×Dic166C2312yes—
80C23×Dic83—44no—
81C23×Dic83C2×C418yes—
82C23×Dic83C2311yescharacteristic
83C23×Dic83C2311yescharacteristic
84C23×Dic83C2311yescharacteristic
85C23×Dic83C2311yescharacteristic
86C23×Dic83C2311yescharacteristic
87C23×Dic83C2311yescharacteristic
88C23×Dic83C2311yescharacteristic
89C23×Dic83C2311yescharacteristic
90#14581769C2×C4116yes—
91#14581769C2×C4116yes—
92#14581769C2×C4116yes—
93#14581769C2×C4116yes—
94#14581757—48no—
95#14581748—44no—
96#14581748—44no—
97C2×C4⋊Dic83C2×C418yes—
98C2×C4⋊Dic83C2314yes—
99C2×C4⋊Dic83C2312yes—
100C2×C4⋊Dic83C2312yes—
101C2×Dic83⋊C4C2×C418yes—
102C2×Dic83⋊C4C2×C418yes—
103C2×Dic83⋊C4C2314yes—
104C2×Dic83⋊C4C2314yes—
105C2×Dic83⋊C4C2314yes—
106C2×Dic83⋊C4C2314yes—
107C2×Dic83⋊C4C2312yes—
108C2×Dic83⋊C4C2312yes—
109C2×Dic83⋊C4C2312yes—
110C2×Dic83⋊C4C2312yes—
111C2×C22⋊Dic83C2×C418yes—
112C2×C22⋊Dic83C2312yes—
113C2×C22⋊Dic83C2312yes—
114C2×C22⋊Dic83C2312yes—
115C2×C22⋊Dic83C2312yes—
116C2×C22⋊Dic83C2312yes—
117C2×C22⋊Dic83C2312yes—
118C2×C22⋊Dic83C2312yes—
119C2×C22⋊Dic83C2312yes—
120C2×C4×Dic83C2314yes—
121C2×C4×Dic83C2314yes—
122C2×C4×Dic83C2314yes—
123C2×C4×Dic83C2314yes—
124C2×C4×Dic83C2314yes—
125C83⋊#95C2314yes—
126C83⋊#95C2314yes—
127C83⋊#95C2314yes—
128C83⋊#95C2314yes—
129C83⋊#95C2314yes—
130C83⋊#95C2314yes—
131C83⋊#95C2314yes—
132C83⋊#95C2314yes—
133C83⋊#95C2314yes—
134C83⋊#95C2314yes—
135C25×C83C2311yescharacteristic
136C23×C4×C83C2311yescharacteristic
137C23×C4×C83C2311yescharacteristic
138C2×C4⋊C4×C83C2314yes—
139C2×C22⋊C4×C83C2×C418yes—
140C2×C22⋊C4×C83C2312yes—
141C2×C22⋊C4×C83C2312yes—
142C2×C22⋊C4×C83C2312yes—
143C2×C22⋊C4×C83C2312yes—
144C2×C22⋊C4×C83C2312yes—
145C2×C22⋊C4×C83C2312yes—
146C2×C22⋊C4×C83C2312yes—
147C2×C22⋊C4×C83C2312yes—
148C2×C42×C83C2314yes—
149#95×C83C2314yes—
150#95×C83C2314yes—
151C2×Dic166—28no—
152C2×Dic166—416no—
153C2×Dic166—44no—
154C2×Dic166C22×C418yes—
155C2×Dic166C22×C418yes—
156C22×Dic83—24no—
157C22×Dic83—24no—
158C22×Dic83—24no—
159C22×Dic83—416no—
160C22×Dic83—42no—
161C22×Dic83—42no—
162C22×Dic83—42no—
163C22×Dic83—42no—
164C22×Dic83—42no—
165C22×Dic83C22×C414yes—
166C22×Dic83C22×C414yes—
167C22×Dic83C22×C414yes—
168C22×Dic83C22×C414yes—
169C22×Dic83C22×C414yes—
170C22×Dic83C22×C414yes—
171C22×Dic83C22×C414yes—
172C22×Dic83C22×C414yes—
173C22×Dic83C2414yes—
174C22×Dic83C2414yes—
175C22×Dic83C2411yescharacteristic
176C22×Dic83C2411yescharacteristic
177C22×Dic83C2411yescharacteristic
178C22×Dic83C2412yes—
179C22×Dic83C2412yes—
180C22×Dic83C2411yescharacteristic
181C4⋊Dic83—44no—
182C4⋊Dic83C22×C418yes—
183C4⋊Dic83C22×C418yes—
184Dic83⋊C4—28no—
185Dic83⋊C4—44no—
186Dic83⋊C4—44no—
187Dic83⋊C4—44no—
188Dic83⋊C4C22×C418yes—
189Dic83⋊C4C22×C418yes—
190Dic83⋊C4C22×C418yes—
191Dic83⋊C4C22×C418yes—
192C22⋊Dic83—24no—
193C22⋊Dic83—24no—
194C22⋊Dic83—42no—
195C22⋊Dic83—42no—
196C22⋊Dic83—42no—
197C22⋊Dic83—42no—
198C22⋊Dic83C22×C418yes—
199C22⋊Dic83C22×C418yes—
200C4×Dic83—28no—
201C4×Dic83—44no—
202C4×Dic83—44no—
203C4×Dic83—44no—
204C4×Dic83—44no—
205C24×C83—24no—
206C24×C83—42no—
207C24×C83—42no—
208C24×C83C22×C414yes—
209C24×C83C2411yescharacteristic
210C24×C83C2411yescharacteristic
211C24×C83C2411yescharacteristic
212C22×C4×C83—42no—
213C22×C4×C83—42no—
214C22×C4×C83C22×C414yes—
215C22×C4×C83C22×C414yes—
216C22×C4×C83C2414yes—
217C22×C4×C83C2414yes—
218C22×C4×C83C2412yes—
219C22×C4×C83C2412yes—
220C4⋊C4×C83—44no—
221C22⋊C4×C83—24no—
222C22⋊C4×C83—24no—
223C22⋊C4×C83—42no—
224C22⋊C4×C83—42no—
225C22⋊C4×C83—42no—
226C22⋊C4×C83—42no—
227C22⋊C4×C83C22×C418yes—
228C22⋊C4×C83C22×C418yes—
229C42×C83—44no—
230C23×C83—22no—
231C23×C83—22no—
232C23×C83—28no—
233C23×C83—22no—
234C23×C83—22no—
235C23×C83—22no—
236C23×C83—22no—
237C23×C83—22no—
238C23×C83—22no—
239C23×C83—44no—
240C23×C83—44no—
241C23×C83—416no—
242C23×C83C23×C412yes—
243C23×C83C23×C412yes—
244C23×C83C23×C412yes—
245C23×C832−1+414yes—
246C23×C83C2511yescharacteristic
247C2×C4×C83—24no—
248C2×C4×C83—24no—
249C2×C4×C83—24no—
250C2×C4×C83—24no—
251C2×C4×C83—24no—
252C2×C4×C83—24no—
253C2×C4×C83—24no—
254C2×C4×C83—24no—
255C2×C4×C83—44no—
256C2×C4×C83—44no—
257C2×C4×C83C23×C414yes—
258C2×C4×C83C23×C414yes—
259C2×Dic83—22no—
260C2×Dic83—24no—
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—24no—
264C2×Dic83—28no—
265C2×Dic83—28no—
266C2×Dic83—28no—
267C2×Dic83—24no—
268C2×Dic83—24no—
269C2×Dic83—24no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—44no—
274C2×Dic83—44no—
275C2×Dic83—44no—
276C2×Dic83—44no—
277C2×Dic83—44no—
278C2×Dic83C23×C414yes—
279C2×Dic83C23×C414yes—
280C2×Dic83C23×C412yes—
281C2×Dic83C23×C412yes—
282C2×Dic83C23×C412yes—
283C2×Dic83C23×C412yes—
284C2×Dic83H(5,𝔽2)14yes—
285Dic166—216no—
286Dic166—48no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—28no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83—22no—
295C22×C83—22no—
296C22×C83—416no—
297C22×C83—44no—
298C22×C83—44no—
299C22×C83#51914yes—
300C22×C83C24×C411yescharacteristic
301C22×C83C2×H(5,𝔽2)12yes—
302C22×C83C2×2−1+412yes—
303C22×C83#58512yes—
304C4×C83—24no—
305C4×C83—24no—
306C4×C83—44no—
307C4×C83—44no—
308Dic83—24no—
309Dic83—22no—
310Dic83—22no—
311Dic83—22no—
312Dic83—24no—
313Dic83—44no—
314Dic83—44no—
315Dic83#51814yes—
316#34370—831nomaximal, Sylow
317C2×C83—24no—
318C2×C83—42no—
319C2×C83—42no—
320C2×C83C2×#51811yescharacteristic
321C2×C83C2×#51911yescharacteristic
322C2×C83#342811yescharacteristic
323C2×C83#354114yes—
324C22×C22⋊Q8—831no—
325#2378—834no—
326#2362—834no—
327#2345—832no—
328#2345—832no—
329#2345—832no—
330Q8×C22⋊C4—832no—
331#2332—834no—
332#2323—834no—
333#2312—831no—
334#2312—831no—
335#2312—831no—
336#2312D8311yescharacteristic
337#2280—831no—
338#2280—831no—
339C83#3437011yescharacteristic, complemented, Sylow
340C23×Q8—831no—
341C24×C4—831no—
342C2×C22⋊Q8—8316no—
343C2×C22⋊Q8—832no—
344C2×C22⋊Q8—832no—
345C2×C22⋊Q8—832no—
346C2×C22⋊Q8—832no—
347C2×C4×Q8—834no—
348C2×#117—832no—
349C2×#117—832no—
350C22×C4⋊C4—831no—
351C22×C4⋊C4—831no—
352C22×C4⋊C4—831no—
353C22×C22⋊C4—831no—
354C22×C22⋊C4C2×D8311yescharacteristic
355#391—834no—
356#391—834no—
357#391—834no—
358#389—834no—
359#389—834no—
360#389—834no—
361#388—834no—
362#388—834no—
363#388—834no—
364#388—834no—
365#388—834no—
366#388—834no—
367#388C2×D8314yes—
368#388C2×D8314yes—
369#387—834no—
370#387—834no—
371#387—834no—
372#387—834no—
373#387—834no—
374#387—834no—
375#381—832no—
376#381—832no—
377#380—832no—
378#380—832no—
379#380—832no—
380#380—832no—
381#380—832no—
382#380—832no—
383#380—832no—
384#380C2×D8312yes—
385#379—831no—
386#379—831no—
387#379C2×D8311yescharacteristic
388#379C2×D8311yescharacteristic
389C4×C4⋊C4—834no—
390C4×C4⋊C4—834no—
391C4×C4⋊C4—834no—
392C4×C22⋊C4—832no—
393C4×C22⋊C4—832no—
394C4×C22⋊C4—832no—
395C4×C22⋊C4—832no—
396C4×C22⋊C4—832no—
397C4×C22⋊C4—832no—
398C4×C22⋊C4—832no—
399C4×C22⋊C4C2×D8312yes—
400C25C22×D8311yescharacteristic
401C22×Q8—3324no—
402C22×Q8—838no—
403C22×Q8—832no—
404C22×Q8—832no—
405C23×C4—3324no—
406C23×C4—838no—
407C23×C4—831no—
408C23×C4—831no—
409C23×C4—831no—
410C23×C4—831no—
411C23×C4—831no—
412C23×C4—831no—
413C23×C4—831no—
414C23×C4—831no—
415C23×C4C22×D8311yescharacteristic
416C23×C4C22×D8311yescharacteristic
417C22⋊Q8—8316no—
418C22⋊Q8—8316no—
419C22⋊Q8—8316no—
420C22⋊Q8—8316no—
421C4×Q8—3328no—
422#117—3324no—
423#117—3324no—
424C2×C4⋊C4—834no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—834no—
427C2×C4⋊C4—834no—
428C2×C4⋊C4—838no—
429C2×C4⋊C4—838no—
430C2×C4⋊C4—838no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—832no—
433C2×C4⋊C4—832no—
434C2×C4⋊C4—832no—
435C2×C4⋊C4—832no—
436C2×C4⋊C4—832no—
437C2×C4⋊C4—834no—
438C2×C4⋊C4C22×D8314yes—
439C2×C22⋊C4—832no—
440C2×C22⋊C4—832no—
441C2×C22⋊C4—832no—
442C2×C22⋊C4—832no—
443C2×C22⋊C4—838no—
444C2×C22⋊C4—832no—
445C2×C22⋊C4—832no—
446C2×C22⋊C4—832no—
447C2×C22⋊C4—832no—
448C2×C22⋊C4C4×D8318yes—
449C2×C22⋊C4C22×D8312yes—
450C2×C22⋊C4C22×D8312yes—
451C2×C22⋊C4C22×D8312yes—
452C2×C22⋊C4C22×D8312yes—
453C2×C22⋊C4C22×D8312yes—
454C2×C22⋊C4C22×D8312yes—
455C2×C22⋊C4C22×D8312yes—
456C2×C22⋊C4C22×D8312yes—
457C2×C42—834no—
458C2×C42—834no—
459C2×C42—834no—
460C2×C42—834no—
461C2×C42—834no—
462C2×C42C22×D8314yes—
463#95—834no—
464#95—834no—
465#95—834no—
466#95—834no—
467#95—834no—
468#95—834no—
469#95—834no—
470#95—834no—
471#95—834no—
472#95—834no—
473#95C22×D8314yes—
474#95C22×D8314yes—
475C24—24no—
476C24—42no—
477C24—42no—
478C24C2×C4×D8314yes—
479C24C23×D8311yescharacteristic
480C24C23×D8311yescharacteristic
481C24C23×D8311yescharacteristic
482C2×Q8—3324no—
483C2×Q8—33216no—
484C2×Q8—838no—
485C2×Q8—838no—
486C2×Q8—1668no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—3322no—
490C22×C4—3322no—
491C22×C4—3322no—
492C22×C4—3322no—
493C22×C4—3322no—
494C22×C4—33216no—
495C22×C4—834no—
496C22×C4—834no—
497C22×C4—834no—
498C22×C4—834no—
499C22×C4—834no—
500C22×C4—831no—
501C22×C4—834no—
502C22×C4—834no—
503C22×C4—834no—
504C22×C4—831no—
505C22×C4—831no—
506C22×C4—832no—
507C22×C4—832no—
508C22×C4—831no—
509C22×C4—834no—
510C22×C4—834no—
511C22×C4—1664no—
512C22×C4—1664no—
513C22×C4—1664no—
514C22×C4C2×C4×D8314yes—
515C22×C4C2×C4×D8314yes—
516C22×C4C23×D8314yes—
517C22×C4C23×D8314yes—
518C22×C4C23×D8312yes—
519C22×C4C23×D8312yes—
520C4⋊C4—44no—
521C4⋊C4—3324no—
522C4⋊C4—3324no—
523C4⋊C4—3324no—
524C4⋊C4—3324no—
525C4⋊C4—838no—
526C4⋊C4—838no—
527C4⋊C4—838no—
528C4⋊C4—838no—
529C4⋊C4—838no—
530C4⋊C4—838no—
531C4⋊C4—1668no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—3322no—
539C22⋊C4—3322no—
540C22⋊C4—3322no—
541C22⋊C4—3322no—
542C22⋊C4—838no—
543C22⋊C4—838no—
544C22⋊C4—1664no—
545C22⋊C4—1664no—
546C22⋊C4C2×C4×D8318yes—
547C22⋊C4C2×C4×D8318yes—
548C42—44no—
549C42—3324no—
550C42—3324no—
551C42—3324no—
552C42—3324no—
553C42—1668no—
554C23—22no—
555C23—22no—
556C23—28no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—44no—
564C23—44no—
565C23—416no—
566C23C22×C4×D8312yes—
567C23C22×C4×D8312yes—
568C23C22×C4×D8312yes—
569C23#1458183914yes—
570C23C24×D8311yescharacteristic
571Q8—3328no—
572Q8—16616no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—3324no—
584C2×C4—3324no—
585C2×C4—3324no—
586C2×C4—3324no—
587C2×C4—3324no—
588C2×C4—832no—
589C2×C4—834no—
590C2×C4—834no—
591C2×C4—834no—
592C2×C4—832no—
593C2×C4—832no—
594C2×C4—832no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1664no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1668no—
602C2×C4—1668no—
603C2×C4—1668no—
604C2×C4—1662no—
605C2×C4—1664no—
606C2×C4—1662no—
607C2×C4—1662no—
608C2×C4—1664no—
609C2×C4C22×C4×D8314yes—
610C2×C4C22×C4×D8314yes—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—416no—
621C22—44no—
622C22—44no—
623C22#1912817814yes—
624C22C23×C4×D8311yescharacteristic
625C22H(5,𝔽2)×D8312yes—
626C22C2×#1458183912yes—
627C22#1912853612yes—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—3324no—
633C4—3324no—
634C4—834no—
635C4—1662no—
636C4—1662no—
637C4—1662no—
638C4—1664no—
639C4—1664no—
640C2—24no—
641C2—42no—
642C2—42no—
643C2#518×D8311yescharacteristic
644C2C2×#1912817811yescharacteristic
645C2#2492066111yescharacteristic
646C2#2492174314yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2312C2
socle, cosocleC23×C83C25
prime radicalC25×C83C23
hypercenter#2312D83
hypocenterC83#34370
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #24910623C2semidirect
3. #24910487C2semidirect
4. #24910341C2semidirect
5. #24910338C2semidirect
6. #24910335C2semidirect
7. #24910295C2semidirect
8. #24910220C2semidirect
9. #24910138C2semidirect
18. C2×#14581769C4semidirect
23. C2×#14581757C22semidirect
30. #19127606C22semidirect
31. #19127605C22semidirect
32. #19127605C22semidirect
33. #19127599C22semidirect
34. #19127598C22semidirect
35. #19127596C22semidirect
42. #19127588C22semidirect
43. #19127585C22semidirect
44. #19127584C22semidirect
45. #19127584C22semidirect
46. #19127583C22semidirect
47. #19127582C22semidirect
59. C4×Dic83⋊C4C22semidirect
60. C4×C4⋊Dic83C22semidirect
61. #19127552C22semidirect
339. C83#34370semidirect

Maximal subgroups

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

subgroupindex
C22×#145817692
#249106232
#249104872
#249103412
#249103382
#249103352
#249102952
#249102202
#249101382
#249100612
#249100592
#249098412
C83×#23122
#3437083