Finite Group Databasebeta

← #35577160

Subgroups — #35577160

Order 21248.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915780C214yesmaximal, complemented
2#24913419C214yesmaximal, complemented
3#24912029C214yesmaximal, complemented
4#24911854C214yesmaximal, complemented
5#24911721C214yesmaximal, complemented
6#24911714C214yesmaximal, complemented
7#24910075C211yesmaximal, characteristic, complemented
8#24910038C211yesmaximal, characteristic
9#24910038C211yesmaximal, characteristic
10C2×#19127578C211yesmaximal, characteristic
11C2×#19127576C211yesmaximal, characteristic, complemented
12C2×C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
13C83×#2315C211yesmaximal, characteristic, complemented
14C23×C22⋊D83C2211yescharacteristic
15C24×Dic83C2211yescharacteristic
16C2×#14581787C2214yescomplemented
17C22×C4⋊Dic83C2212yescomplemented
18C22×C22⋊Dic83C2211yescharacteristic
19C22×C22⋊Dic83C2211yescharacteristic
20C22×(C2×D83)⋊C4C2212yes—
21C22×C4×Dic83C2212yescomplemented
22C83⋊#401C2214yescomplemented
23#19127643C2214yescomplemented
24#19127640C2214yescomplemented
25#19127639C2214yescomplemented
26#19127630C2214yescomplemented
27#19127628C2214yescomplemented
28#19127627C2214yescomplemented
29#19127623C2214yescomplemented
30#19127593C2214yescomplemented
31#19127591C2214yescomplemented
32#19127590C2214yescomplemented
33#19127587C2214yescomplemented
34#19127584C2214yescomplemented
35#19127582C2214yescomplemented
36#19127580C2214yes—
37#19127580C2214yes—
38#19127578C2212yes—
39#19127578C2214yes—
40#19127578C2212yes—
41#19127578C2214yes—
42#19127578C2212yes—
43#19127578C2212yes—
44#19127576C2214yes—
45#19127576C2212yescomplemented
46#19127576C2212yescomplemented
47#19127574C2214yes—
48#19127573C2214yescomplemented
49#19127568C2214yescomplemented
50#19127562C2212yes—
51#19127562C2212yes—
52#19127559C2212yescomplemented
53#19127559C2212yescomplemented
54#19127556C2212yes—
55#19127556C2212yes—
56C22⋊C4×Dic83C2214yes—
57C22⋊C4×Dic83C2214yescomplemented
58C2×C83⋊#95C2212yes—
59C2×C83⋊#95C2212yescomplemented
60C2×C83⋊#95C2212yescomplemented
61C22×C22⋊C4×C83C2211yescharacteristic
62C22×C22⋊C4×C83C2211yescharacteristic
63C22×C22⋊C4×C83C2211yescharacteristic
64#385×C83C2214yescomplemented
65#385×C83C2214yescomplemented
66#380×C83C2214yes—
67C24×D83C2311yescharacteristic
68C22×C22⋊D83—24no—
69C22×C22⋊D83—24no—
70C22×C22⋊D83—24no—
71C22×C22⋊D83C2312yes—
72C22×C22⋊D83C2312yes—
73C23×Dic83—24no—
74C23×Dic83—44no—
75C23×Dic83C2311yescharacteristic
76C23×Dic83C2311yescharacteristic
77C23×Dic83C2312yes—
78C23×Dic83C2312yes—
79C23×Dic83C2311yescharacteristic
80C23×Dic83C2311yescharacteristic
81C23×Dic83C2312yes—
82C23×Dic83C2311yescharacteristic
83#14581787—48no—
84C2×C4⋊Dic83—44no—
85C2×C4⋊Dic83C2314yes—
86C2×C4⋊Dic83C2314yes—
87C2×Dic83⋊C4C2314yes—
88C2×Dic83⋊C4C2314yes—
89C2×C22⋊Dic83—24no—
90C2×C22⋊Dic83—24no—
91C2×C22⋊Dic83C2312yes—
92C2×C22⋊Dic83C2312yes—
93C2×C22⋊Dic83C2312yes—
94C2×C22⋊Dic83C2312yes—
95C2×(C2×D83)⋊C4—44no—
96C2×(C2×D83)⋊C4C2314yes—
97C2×(C2×D83)⋊C4C2314yes—
98C2×(C2×D83)⋊C4C2314yes—
99C2×(C2×D83)⋊C4C2314yes—
100C2×(C2×D83)⋊C4C2314yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×C4×Dic83—44no—
103C2×C4×Dic83C2314yes—
104C2×C4×Dic83C2314yes—
105C83⋊#95C2314yes—
106C83⋊#95C2314yes—
107C83⋊#95C2314yes—
108C83⋊#95C2314yes—
109C83⋊#95C2314yes—
110C83⋊#95C2314yes—
111C83⋊#95C2314yes—
112C83⋊#95C2314yes—
113C83⋊#95C2314yes—
114C83⋊#95C2314yes—
115C83⋊#95C2314yes—
116C83⋊#95C2314yes—
117C25×C83C2311yescharacteristic
118C23×C4×C83C2312yes—
119C23×C4×C83C2312yes—
120C23×C4×C83C2312yes—
121C2×C4⋊C4×C83C2314yes—
122C2×C4⋊C4×C83C2314yes—
123C2×C22⋊C4×C83—24no—
124C2×C22⋊C4×C83—24no—
125C2×C22⋊C4×C83—24no—
126C2×C22⋊C4×C83C2312yes—
127C2×C22⋊C4×C83C2312yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C22⋊C4×C83C2312yes—
130C2×C22⋊C4×C83C2312yes—
131C2×C22⋊C4×C83C2312yes—
132#95×C83C2314yes—
133#95×C83C2314yes—
134C23×D83—22no—
135C23×D83—22no—
136C23×D83—22no—
137C23×D83—42no—
138C23×D83—42no—
139C23×D83C2412yes—
140C2×C22⋊D83—24no—
141C2×C22⋊D83—24no—
142C2×C22⋊D83—24no—
143C2×C22⋊D83—24no—
144C2×C22⋊D83—24no—
145C2×C22⋊D83—24no—
146C2×C22⋊D83—416no—
147C22×Dic83—22no—
148C22×Dic83—24no—
149C22×Dic83—24no—
150C22×Dic83—24no—
151C22×Dic83—22no—
152C22×Dic83—22no—
153C22×Dic83—22no—
154C22×Dic83—22no—
155C22×Dic83—24no—
156C22×Dic83—22no—
157C22×Dic83—22no—
158C22×Dic83—24no—
159C22×Dic83—24no—
160C22×Dic83—42no—
161C22×Dic83—42no—
162C22×Dic83—42no—
163C22×Dic83—42no—
164C22×Dic83—42no—
165C22×Dic83—42no—
166C22×Dic83—42no—
167C22×Dic83—416no—
168C22×Dic83C2412yes—
169C22×Dic83C2412yes—
170C22×Dic83C2412yes—
171C22×Dic83C2414yes—
172C22×Dic83C2412yes—
173C22×Dic83C2412yes—
174C4⋊Dic83—28no—
175C4⋊Dic83—44no—
176Dic83⋊C4—44no—
177Dic83⋊C4—44no—
178C22⋊Dic83—24no—
179C22⋊Dic83—24no—
180C22⋊Dic83—24no—
181C22⋊Dic83—24no—
182(C2×D83)⋊C4—28no—
183(C2×D83)⋊C4—44no—
184(C2×D83)⋊C4—44no—
185(C2×D83)⋊C4—44no—
186(C2×D83)⋊C4—44no—
187(C2×D83)⋊C4—44no—
188C4×Dic83—28no—
189C4×Dic83—44no—
190C24×C83—42no—
191C24×C83—42no—
192C24×C83#4114yes—
193C24×C83#4114yes—
194C24×C83#4114yes—
195C24×C83C2411yescharacteristic
196C24×C83C2411yescharacteristic
197C24×C83C2411yescharacteristic
198C22×C4×C83—24no—
199C22×C4×C83—24no—
200C22×C4×C83—24no—
201C22×C4×C83—42no—
202C22×C4×C83—42no—
203C22×C4×C83—42no—
204C22×C4×C83—42no—
205C22×C4×C83—42no—
206C22×C4×C83—42no—
207C22×C4×C83C2414yes—
208C22×C4×C83C2414yes—
209C22×C4×C83C2414yes—
210C4⋊C4×C83—44no—
211C4⋊C4×C83—44no—
212C22⋊C4×C83—24no—
213C22⋊C4×C83—24no—
214C22⋊C4×C83—24no—
215C22⋊C4×C83—24no—
216C22⋊C4×C83—24no—
217C22⋊C4×C83—24no—
218C23×C83—22no—
219C23×C83—22no—
220C23×C83—22no—
221C23×C83—22no—
222C23×C83—22no—
223C23×C83—22no—
224C23×C83—48no—
225C23×C83—48no—
226C23×C83—44no—
227C23×C83—44no—
228C23×C83#126116yescomplemented
229C23×C83C2×#4112yes—
230C23×C83C2×#4112yes—
231C23×C83C2×#4112yes—
232C23×C83C2×#4112yes—
233C23×C83C2×#4112yes—
234C23×C83C2×#4112yes—
235C23×C83C2×#4112yes—
236C23×C83C2×#4112yes—
237C23×C83C2×#4112yes—
238C23×C83C2511yescharacteristic
239C22×D83—22no—
240C22×D83—22no—
241C22×D83—22no—
242C22×D83—24no—
243C22×D83—24no—
244C22×D83—48no—
245C22×D83—44no—
246C22×D83—82no—
247C22×D83—82no—
248C22×D83—84no—
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—24no—
256C2×C4×C83—24no—
257C2×C4×C83—24no—
258C2×C4×C83—48no—
259C2×C4×C83—48no—
260C2×C4×C83—84no—
261C22⋊D83—48no—
262C22⋊D83—48no—
263C2×Dic83—28no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—24no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—22no—
272C2×Dic83—22no—
273C2×Dic83—24no—
274C2×Dic83—22no—
275C2×Dic83—24no—
276C2×Dic83—22no—
277C2×Dic83—24no—
278C2×Dic83—24no—
279C2×Dic83—48no—
280C2×Dic83—48no—
281C2×Dic83—48no—
282C2×Dic83—44no—
283C2×Dic83—44no—
284C2×Dic83—82no—
285C2×Dic83—82no—
286C2×Dic83H(5,𝔽2)14yes—
287C22×C83—24no—
288C22×C83—24no—
289C22×C83—24no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83—22no—
295C22×C83—22no—
296C22×C83—24no—
297C22×C83—24no—
298C22×C83—24no—
299C22×C83—416no—
300C22×C83C2×#12614yes—
301C22×C83C2×#12614yes—
302C22×C83C2×#12614yes—
303C22×C83C22×#4111yescharacteristic
304C22×C83C22×#4111yescharacteristic
305C22×C83C22×#4111yescharacteristic
306C22×C83C2×H(5,𝔽2)12yes—
307C22×C83#58512yes—
308C2×D83—42no—
309C2×D83—44no—
310C2×D83—42no—
311C2×D83—42no—
312C2×D83—44no—
313C2×D83—88no—
314C4×C83—44no—
315C4×C83—44no—
316C4×C83—44no—
317Dic83—24no—
318Dic83—42no—
319Dic83—42no—
320Dic83—42no—
321Dic83—42no—
322Dic83—44no—
323#34995—831nomaximal, Sylow
324C2×C83—24no—
325C2×C83—24no—
326C2×C83—24no—
327C2×C83C22×#12611yescharacteristic
328C2×C83#345912yes—
329C2×C83#349011yescharacteristic
330C2×C83#349111yescharacteristic
331C2×C83#350512yes—
332D83—82no—
333#2852—834no—
334#2643—834no—
335#2510—834no—
336#2493—834no—
337#2481—834no—
338#2480—834no—
339#2315—831no—
340#2315D8311yescharacteristic, complemented
341#2308—831no—
342#2308—831no—
343C2×#386—831no—
344C2×#385—831no—
345C2×C4×C22⋊C4—831no—
346C83#3499511yescharacteristic, complemented, Sylow
347C23×D4—831no—
348C24×C4—831no—
349C2×#126—834no—
350C22×C4⋊C4—832no—
351C22×C22⋊C4—832no—
352C22×C22⋊C4—831no—
353C22×C22⋊C4—831no—
354C22×C22⋊C4C2×D8311yescharacteristic
355C22×C22⋊C4C2×D8311yescharacteristic
356C22×C22⋊C4C2×D8311yescharacteristic
357C22×C42—832no—
358#401—834no—
359#400—834no—
360#400—834no—
361#400—834no—
362#397—834no—
363#397—834no—
364#397—834no—
365#396—834no—
366#388—834no—
367#388—834no—
368#388—834no—
369#387—834no—
370#387—834no—
371#387—834no—
372#386—834no—
373#386—834no—
374#386—834no—
375#386—832no—
376#386—832no—
377#386—832no—
378#386—832no—
379#386—834no—
380#385—834no—
381#385—834no—
382#385—834no—
383#385—832no—
384#385—832no—
385#385C2×D8314yescomplemented
386#385C2×D8314yescomplemented
387#383—834no—
388#381—832no—
389#381—832no—
390#380—832no—
391#380—832no—
392#380C2×D8314yes—
393#379—832no—
394#379—832no—
395C4×C22⋊C4—834no—
396C4×C22⋊C4—834no—
397C2×#95—832no—
398C2×#95—832no—
399C2×#95—832no—
400C25—831no—
401C25C22×D8311yescharacteristic
402C22×D4—832no—
403C22×D4—832no—
404C22×D4—1664no—
405C22×D4—1664no—
406C22×D4—1664no—
407C23×C4—3324no—
408C23×C4—831no—
409C23×C4—831no—
410C23×C4—832no—
411C23×C4—831no—
412C23×C4—832no—
413C23×C4—832no—
414C23×C4—831no—
415C23×C4—831no—
416C23×C4—1664no—
417C23×C4C22×D8312yes—
418C23×C4C22×D8312yes—
419C23×C4C22×D8312yes—
420#126—3328no—
421C2×C4⋊C4—3324no—
422C2×C4⋊C4—834no—
423C2×C4⋊C4—834no—
424C2×C4⋊C4—834no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4C22×D8314yes—
427C2×C4⋊C4C22×D8314yes—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—24no—
431C2×C22⋊C4—3324no—
432C2×C22⋊C4—834no—
433C2×C22⋊C4—834no—
434C2×C22⋊C4—834no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4—834no—
437C2×C22⋊C4—832no—
438C2×C22⋊C4—832no—
439C2×C22⋊C4—832no—
440C2×C22⋊C4—832no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4—1664no—
443C2×C22⋊C4—1664no—
444C2×C22⋊C4C22×D8312yes—
445C2×C22⋊C4C22×D8312yes—
446C2×C22⋊C4C22×D8312yes—
447C2×C22⋊C4C22×D8312yes—
448C2×C22⋊C4C22×D8312yes—
449C2×C22⋊C4C22×D8312yes—
450C2×C42—3324no—
451C2×C42—834no—
452C2×C42—834no—
453#95—834no—
454#95—834no—
455#95—834no—
456#95—834no—
457#95—834no—
458#95—834no—
459#95—834no—
460#95—834no—
461#95—834no—
462#95—834no—
463#95—834no—
464#95—834no—
465#95C22×D8314yes—
466#95C22×D8314yes—
467C24—42no—
468C24—42no—
469C24—3322no—
470C24—3322no—
471C24—832no—
472C24—1662no—
473C24—1662no—
474C24—1662no—
475C24#1305058914yes—
476C24#1305059014yes—
477C24#1305059114yes—
478C24C23×D8311yescharacteristic
479C24C23×D8311yescharacteristic
480C24C23×D8311yescharacteristic
481C2×D4—33216no—
482C2×D4—1664no—
483C2×D4—1664no—
484C2×D4—1664no—
485C2×D4—1664no—
486C2×D4—1664no—
487C2×D4—1664no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—33216no—
498C22×C4—3322no—
499C22×C4—3322no—
500C22×C4—3322no—
501C22×C4—3322no—
502C22×C4—3322no—
503C22×C4—3322no—
504C22×C4—3322no—
505C22×C4—832no—
506C22×C4—832no—
507C22×C4—832no—
508C22×C4—832no—
509C22×C4—834no—
510C22×C4—832no—
511C22×C4—1664no—
512C22×C4—1664no—
513C22×C4—1662no—
514C22×C4—1664no—
515C22×C4—1662no—
516C22×C4—1662no—
517C22×C4—1664no—
518C22×C4—1664no—
519C22×C4—1662no—
520C22×C4—1664no—
521C22×C4—1662no—
522C22×C4—1662no—
523C22×C4—1662no—
524C22×C4C23×D8314yes—
525C22×C4C23×D8314yes—
526C22×C4C23×D8314yes—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—3324no—
530C4⋊C4—3324no—
531C4⋊C4—3324no—
532C4⋊C4—1668no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—3324no—
540C22⋊C4—3324no—
541C22⋊C4—3324no—
542C22⋊C4—3324no—
543C22⋊C4—3324no—
544C22⋊C4—1668no—
545C22⋊C4—1664no—
546C22⋊C4—1664no—
547C22⋊C4—1664no—
548C22⋊C4—1664no—
549C42—3324no—
550C42—1668no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—3328no—
562C23—3324no—
563C23—6642no—
564C23—6642no—
565C23—6644no—
566C23—1662no—
567C23—1662no—
568C23—1662no—
569C23—1664no—
570C23—1664no—
571C23#14581787116yescomplemented
572C23C2×#1305058912yes—
573C23C2×#1305058912yes—
574C23C2×#1305058912yes—
575C23C2×#1305059012yes—
576C23C2×#1305059012yes—
577C23C2×#1305059012yes—
578C23C2×#1305059112yes—
579C23C2×#1305059112yes—
580C23C2×#1305059112yes—
581C23C24×D8311yescharacteristic
582D4—3328no—
583D4—3328no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—84no—
596C2×C4—3324no—
597C2×C4—3328no—
598C2×C4—3324no—
599C2×C4—3328no—
600C2×C4—3328no—
601C2×C4—834no—
602C2×C4—6642no—
603C2×C4—6642no—
604C2×C4—1662no—
605C2×C4—1664no—
606C2×C4—1662no—
607C2×C4—1662no—
608C2×C4—1662no—
609C2×C4—1662no—
610C2×C4—1662no—
611C2×C4—1664no—
612C2×C4—1662no—
613C2×C4—1664no—
614C2×C4—1662no—
615C2×C4—1664no—
616C2×C4—1664no—
617C2×C4—1668no—
618C2×C4—1662no—
619C2×C4—1662no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—416no—
633C22—3324no—
634C22—3322no—
635C22—3322no—
636C22—3322no—
637C22—3324no—
638C22—6648no—
639C22C2×#1458178714yes—
640C22C2×#1458178714yes—
641C22C2×#1458178714yes—
642C22C22×#1305058911yescharacteristic
643C22C22×#1305059011yescharacteristic
644C22C22×#1305059111yescharacteristic
645C22H(5,𝔽2)×D8312yes—
646C22#1912853612yes—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—3322no—
651C4—3322no—
652C4—3322no—
653C4—3322no—
654C4—3324no—
655C4—1664no—
656C2—24no—
657C2—24no—
658C2—24no—
659C2—6642no—
660C2C22×#1458178711yescharacteristic
661C2#2492086912yes—
662C2#2492115311yescharacteristic
663C2#2492116211yescharacteristic
664C2#2492129312yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2315C2
prime radicalC23×C22⋊D83C22
hypercenter#2315D83
hypocenterC83#34995
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #24915780C2semidirect
2. #24913419C2semidirect
3. #24912029C2semidirect
4. #24911854C2semidirect
5. #24911721C2semidirect
6. #24911714C2semidirect
7. #24910075C2semidirect
11. C2×#19127576C2semidirect
12. C2×C22⋊C4×Dic83C2semidirect
13. C83×#2315C2semidirect
16. C2×#14581787C22semidirect
17. C22×C4⋊Dic83C22semidirect
21. C22×C4×Dic83C22semidirect
22. C83⋊#401C22semidirect
23. #19127643C22semidirect
24. #19127640C22semidirect
25. #19127639C22semidirect
26. #19127630C22semidirect
27. #19127628C22semidirect
28. #19127627C22semidirect
29. #19127623C22semidirect
30. #19127593C22semidirect
31. #19127591C22semidirect
32. #19127590C22semidirect
33. #19127587C22semidirect
34. #19127584C22semidirect
35. #19127582C22semidirect
45. #19127576C22semidirect
46. #19127576C22semidirect
48. #19127573C22semidirect
49. #19127568C22semidirect
52. #19127559C22semidirect
53. #19127559C22semidirect
57. C22⋊C4×Dic83C22semidirect
59. C2×C83⋊#95C22semidirect
60. C2×C83⋊#95C22semidirect
64. #385×C83C22semidirect
65. #385×C83C22semidirect
228. C23×C83#126semidirect
340. #2315D83semidirect
346. C83#34995semidirect
385. #385C2×D83semidirect
386. #385C2×D83semidirect
571. C23#14581787semidirect

Maximal subgroups

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

subgroupindex
#249157802
#249134192
#249120292
#249118542
#249117212
#249117142
#249100752
#249100382
C2×#191275782
C2×#191275762
C2×C22⋊C4×Dic832
C83×#23152
#3499583