Finite Group Databasebeta

← #35563003

Subgroups — #35563003

Order 21248.

2390 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D83C211yesmaximal, characteristic
2#24910073C211yesmaximal, characteristic, complemented
3#24910072C211yesmaximal, characteristic, complemented
4#24910071C211yesmaximal, characteristic, complemented
5#24910070C212yesmaximal, complemented
6#24910068C212yesmaximal, complemented
7#24910062C214yesmaximal, complemented
8#24910057C214yesmaximal
9#24909957C214yesmaximal, complemented
10#24909955C214yesmaximal, complemented
11#24909869C211yesmaximal, characteristic, complemented
12#24909868C211yesmaximal, characteristic, complemented
13#24909845C212yesmaximal
14C4×#14581750C212yesmaximal, complemented
15C83×#2314C211yesmaximal, characteristic, complemented
16C23×C4×D83C2211yescharacteristic
17C2×#14581751C2211yescharacteristic
18C2×#14581750C2212yes—
19C2×#14581750C2212yes—
20C2×#14581750C2212yes—
21C2×#14581749C2211yescharacteristic
22C2×#14581749C2211yescharacteristic
23C2×#14581748C2212yes—
24#117×D83C4116yescomplemented
25C2×C4⋊C4×D83C2212yes—
26C2×C22⋊C4×D83C2212yes—
27C2×C42×D83C2211yescharacteristic
28C2×C42×D83C2211yescharacteristic
29#19127595C2214yes—
30#19127594C2214yes—
31#19127593C2214yes—
32#19127592C2214yes—
33#19127591C2214yes—
34#19127590C2214yes—
35#19127589C2214yes—
36#19127588C2214yescomplemented
37#19127587C2214yescomplemented
38#19127586C2214yescomplemented
39#19127585C2214yescomplemented
40#19127584C2214yescomplemented
41#19127583C2214yescomplemented
42#19127582C2214yescomplemented
43#19127571C2211yescharacteristic, complemented
44#19127570C2212yescomplemented
45#19127567C2211yescharacteristic, complemented
46#19127566C2212yescomplemented
47#19127563C2214yes—
48#19127561C2212yes—
49#19127560C2214yes—
50#19127558C2212yes—
51#19127558C2212yes—
52#19127558C2214yes—
53#19127556C2214yes—
54C4⋊C4×Dic83C2214yescomplemented
55C4⋊C4×Dic83C2212yescomplemented
56C4⋊C4×Dic83C2212yescomplemented
57C4×C4⋊Dic83C2212yescomplemented
58#19127552C2214yescomplemented
59C22⋊C4×Dic83C2214yes—
60C4×(C2×D83)⋊C4C2214yes—
61#19127548C2214yes—
62#19127548C2214yes—
63#19127546C2214yescomplemented
64C42×Dic83C2214yescomplemented
65C2×#117×C83C2211yescharacteristic
66#388×C83C2214yes—
67#387×C83C2214yescomplemented
68#384×C83C2211yescharacteristic, complemented
69#382×C83C2211yescharacteristic, complemented
70#380×C83C2212yes—
71C4×C4⋊C4×C83C2212yescomplemented
72C24×D83C2311yescharacteristic
73C23×Dic83C2311yescharacteristic
74C22×C4×D83—24no—
75C22×C4×D83—24no—
76C22×C4×D83C2×C418yes—
77C22×C4×D83C2311yescharacteristic
78C22×C4×D83C2311yescharacteristic
79C22×C4×D83C2311yescharacteristic
80C22×C4×D83C2312yes—
81C22×C4×D83C2312yes—
82C22×C4×D83C2311yescharacteristic
83#14581751C2×C418yes—
84#14581750—28no—
85#14581750—28no—
86#14581750C2×C4116yes—
87#14581749C2×C418yes—
88#14581749C2×C418yes—
89#14581748C2×C4116yes—
90C2×C4⋊Dic83C2312yes—
91C2×C4⋊Dic83C2312yes—
92C2×C4⋊Dic83C2314yes—
93C2×C4⋊Dic83C2312yes—
94C2×Dic83⋊C4C2314yes—
95C2×Dic83⋊C4C2314yes—
96C2×Dic83⋊C4C2312yes—
97C2×Dic83⋊C4C2312yes—
98C4⋊C4×D83C2×C4116yes—
99C2×C22⋊Dic83C2314yes—
100C2×C22⋊Dic83C2312yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2314yes—
104C2×(C2×D83)⋊C4C2314yes—
105C2×(C2×D83)⋊C4C2312yes—
106C2×(C2×D83)⋊C4C2312yes—
107C22⋊C4×D83C2×C4116yes—
108C2×C4×Dic83C2314yes—
109C2×C4×Dic83C2314yes—
110C2×C4×Dic83C2314yes—
111C2×C4×Dic83C2314yes—
112C2×C4×Dic83C2314yes—
113C2×C4×Dic83C2312yes—
114C2×C4×Dic83C2311yescharacteristic
115C2×C4×Dic83C2311yescharacteristic
116C2×C4×Dic83C2312yes—
117C42×D83C2×C418yes—
118C42×D83C2×C418yes—
119C83⋊#95C2314yes—
120C83⋊#95C2314yes—
121C83⋊#95C2314yes—
122C83⋊#95C2314yes—
123C83⋊#95C2314yes—
124C83⋊#95C2314yes—
125C83⋊#95C2314yes—
126C83⋊#95C2314yes—
127C83⋊#95C2314yes—
128C23×C4×C83C2311yescharacteristic
129#117×C83C2×C418yes—
130C2×C4⋊C4×C83C2312yes—
131C2×C4⋊C4×C83C2312yes—
132C2×C4⋊C4×C83C2314yes—
133C2×C4⋊C4×C83C2312yes—
134C2×C22⋊C4×C83C2314yes—
135C2×C22⋊C4×C83C2312yes—
136C2×C42×C83C2314yes—
137C2×C42×C83C2311yescharacteristic
138C2×C42×C83C2311yescharacteristic
139#95×C83C2314yes—
140#95×C83C2314yes—
141#95×C83C2314yes—
142C23×D83—22no—
143C23×D83—22no—
144C23×D83—42no—
145C23×D83—42no—
146C23×D83C22×C414yes—
147C23×D83C2411yescharacteristic
148C23×D83C2411yescharacteristic
149C22×Dic83—22no—
150C22×Dic83—22no—
151C22×Dic83C22×C414yes—
152C22×Dic83C2414yes—
153C22×Dic83C2414yes—
154C22×Dic83C2411yescharacteristic
155C22×Dic83C2412yes—
156C22×Dic83C2412yes—
157C22×Dic83C2411yescharacteristic
158C2×C4×D83—22no—
159C2×C4×D83—22no—
160C2×C4×D83—22no—
161C2×C4×D83—22no—
162C2×C4×D83—22no—
163C2×C4×D83—22no—
164C2×C4×D83—232no—
165C2×C4×D83—42no—
166C2×C4×D83—42no—
167C2×C4×D83—42no—
168C2×C4×D83—42no—
169C2×C4×D83C22×C414yes—
170C2×C4×D83C22×C414yes—
171C2×C4×D83C22×C414yes—
172C2×C4×D83C22×C418yes—
173C2×C4×D83C22×C418yes—
174C2×C4×D83C22×C414yes—
175C2×C4×D83#4114yes—
176C2×C4×D83#4114yes—
177C4⋊Dic83—24no—
178C4⋊Dic83—24no—
179C4⋊Dic83—44no—
180C4⋊Dic83C22×C418yes—
181Dic83⋊C4—44no—
182Dic83⋊C4—44no—
183Dic83⋊C4C22×C418yes—
184Dic83⋊C4C22×C418yes—
185C22⋊Dic83—44no—
186C22⋊Dic83C22×C418yes—
187(C2×D83)⋊C4—28no—
188(C2×D83)⋊C4—28no—
189(C2×D83)⋊C4—44no—
190(C2×D83)⋊C4—44no—
191(C2×D83)⋊C4C22×C418yes—
192(C2×D83)⋊C4C22×C418yes—
193C4×Dic83—28no—
194C4×Dic83—28no—
195C4×Dic83—44no—
196C4×Dic83—44no—
197C4×Dic83—44no—
198C4×Dic83C22×C418yes—
199C4×Dic83C22×C418yes—
200C4×Dic83C22×C414yes—
201C4×Dic83C22×C414yes—
202C24×C83C2411yescharacteristic
203C22×C4×C83—22no—
204C22×C4×C83—22no—
205C22×C4×C83C22×C414yes—
206C22×C4×C83C2414yes—
207C22×C4×C83C2414yes—
208C22×C4×C83C2411yescharacteristic
209C22×C4×C83C2412yes—
210C22×C4×C83C2412yes—
211C22×C4×C83C2411yescharacteristic
212C4⋊C4×C83—24no—
213C4⋊C4×C83—24no—
214C4⋊C4×C83—44no—
215C4⋊C4×C83C22×C418yes—
216C22⋊C4×C83—44no—
217C22⋊C4×C83C22×C418yes—
218C42×C83—44no—
219C42×C83C22×C414yes—
220C42×C83C22×C414yes—
221C23×C83—21no—
222C23×C83—21no—
223C23×C83—22no—
224C23×C83—22no—
225C23×C83C23×C412yes—
226C23×C83C2511yescharacteristic
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—21no—
230C22×D83—21no—
231C22×D83—22no—
232C22×D83—22no—
233C22×D83—28no—
234C22×D83—48no—
235C22×D83—44no—
236C22×D83—48no—
237C22×D83—44no—
238C22×D83C23×C412yes—
239C22×D83C23×C412yes—
240C22×D83C2×#4112yes—
241C22×D83C2×#4112yes—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—28no—
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×C83C23×C414yes—
256C2×C4×C83C23×C414yes—
257C2×C4×C83C23×C412yes—
258C2×C4×C83C23×C412yes—
259C2×C4×C83C2×#4112yes—
260C2×C4×C83C2×#4112yes—
261C2×Dic83—21no—
262C2×Dic83—21no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—28no—
268C2×Dic83—24no—
269C2×Dic83—24no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—24no—
274C2×Dic83C23×C414yes—
275C2×Dic83C23×C414yes—
276C2×Dic83C23×C412yes—
277C2×Dic83C23×C412yes—
278C2×Dic83C2×#4112yes—
279C2×Dic83C2×#4112yes—
280C4×D83—28no—
281C4×D83—28no—
282C4×D83—28no—
283C4×D83—48no—
284C4×D83—48no—
285C4×D83#117116yes—
286C22×C83—22no—
287C22×C83—24no—
288C22×C83—41no—
289C22×C83—42no—
290C22×C83—41no—
291C22×C83C24×C411yescharacteristic
292C22×C83C22×#4111yescharacteristic
293C22×C83C22×#4111yescharacteristic
294C22×C83#58512yes—
295C22×C83#58512yes—
296C2×D83—24no—
297C2×D83—22no—
298C2×D83—24no—
299C2×D83—22no—
300C2×D83—22no—
301C2×D83—24no—
302C2×D83—416no—
303C2×D83—41no—
304C2×D83—42no—
305C2×D83—41no—
306C2×D83C2×#11714yes—
307C4×C83—22no—
308C4×C83—24no—
309C4×C83—24no—
310C4×C83—44no—
311C4×C83—44no—
312C4×C83C2×#11714yes—
313Dic83—22no—
314Dic83—24no—
315Dic83—24no—
316Dic83—44no—
317Dic83—44no—
318Dic83C2×#11714yes—
319#34048—831nomaximal, Sylow
320C2×C83—42no—
321C2×C83C22×#11711yescharacteristic
322C2×C83#342812yes—
323C2×C83#345511yescharacteristic
324C2×C83#345512yes—
325C2×C83#345511yescharacteristic
326D83—24no—
327D83—42no—
328C22×#117—831no—
329#2314—831no—
330#2314—831no—
331#2314—832no—
332#2314—832no—
333#2314—831no—
334#2314D8311yescharacteristic, complemented
335#2313—834no—
336#2312—834no—
337#2298—834no—
338#2298—834no—
339#2285—831no—
340#2285—831no—
341#2280—832no—
342C4×#117—832no—
343C83#3404811yescharacteristic, complemented, Sylow
344C24×C4—831no—
345C2×#117—831no—
346C2×#117—8316no—
347C2×#117—832no—
348C2×#117—832no—
349C2×#117—832no—
350C2×#117—832no—
351C2×#117—831no—
352C2×#117—831no—
353C2×#117C2×D8311yescharacteristic
354C22×C4⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C42—831no—
357C22×C42—831no—
358#388—834no—
359#388—834no—
360#388—834no—
361#388—834no—
362#388—834no—
363#388—834no—
364#388—834no—
365#388C2×D8314yes—
366#387—834no—
367#387—834no—
368#387—834no—
369#387—834no—
370#387—834no—
371#387—834no—
372#387—834no—
373#387C2×D8314yescomplemented
374#384—832no—
375#384—831no—
376#384C2×D8311yescharacteristic, complemented
377#382—832no—
378#382—831no—
379#382C2×D8311yescharacteristic, complemented
380#381—834no—
381#380—832no—
382#380—834no—
383#380—832no—
384#380—832no—
385#380—834no—
386#380C2×D8312yes—
387#379—834no—
388C4×C4⋊C4—832no—
389C4×C4⋊C4—832no—
390C4×C4⋊C4—834no—
391C4×C4⋊C4—832no—
392C4×C4⋊C4—834no—
393C4×C4⋊C4C2×D8312yescomplemented
394C4×C22⋊C4—834no—
395C4×C22⋊C4—834no—
396C4×C22⋊C4—834no—
397C4×C22⋊C4—834no—
398#376—834no—
399C43—834no—
400C25—831no—
401C23×C4—832no—
402C23×C4—832no—
403C23×C4—831no—
404C23×C4—831no—
405C23×C4—838no—
406C23×C4—831no—
407C23×C4—831no—
408C23×C4—831no—
409C23×C4—1664no—
410C23×C4—1664no—
411C23×C4C22×D8311yescharacteristic
412#117—838no—
413#117—8316no—
414#117—8316no—
415#117—838no—
416#117—838no—
417#117—1668no—
418#117—1668no—
419#117C4×D8318yes—
420C2×C4⋊C4—834no—
421C2×C4⋊C4—834no—
422C2×C4⋊C4—834no—
423C2×C4⋊C4—832no—
424C2×C4⋊C4—832no—
425C2×C4⋊C4—832no—
426C2×C4⋊C4—8316no—
427C2×C4⋊C4—832no—
428C2×C4⋊C4—832no—
429C2×C4⋊C4C22×D8312yes—
430C2×C4⋊C4C22×D8312yes—
431C2×C4⋊C4C22×D8314yes—
432C2×C4⋊C4C22×D8312yes—
433C2×C22⋊C4—834no—
434C2×C22⋊C4—834no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4—832no—
437C2×C22⋊C4—832no—
438C2×C22⋊C4—832no—
439C2×C22⋊C4—8316no—
440C2×C22⋊C4—834no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4C22×D8314yes—
443C2×C22⋊C4C22×D8312yes—
444C2×C42—834no—
445C2×C42—834no—
446C2×C42—834no—
447C2×C42—832no—
448C2×C42—832no—
449C2×C42—831no—
450C2×C42—831no—
451C2×C42—838no—
452C2×C42—838no—
453C2×C42—834no—
454C2×C42—834no—
455C2×C42C22×D8314yes—
456C2×C42C22×D8311yescharacteristic
457C2×C42C22×D8311yescharacteristic
458#95—834no—
459#95—834no—
460#95—834no—
461#95—834no—
462#95—834no—
463#95—834no—
464#95—834no—
465#95—834no—
466#95—834no—
467#95C22×D8314yes—
468#95C22×D8314yes—
469#95C22×D8314yes—
470C24—3322no—
471C24—3322no—
472C24—834no—
473C24—831no—
474C24—831no—
475C24—1662no—
476C24—1662no—
477C24C23×D8311yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—3322no—
481C22×C4—3322no—
482C22×C4—3322no—
483C22×C4—3322no—
484C22×C4—838no—
485C22×C4—838no—
486C22×C4—831no—
487C22×C4—834no—
488C22×C4—834no—
489C22×C4—834no—
490C22×C4—834no—
491C22×C4—831no—
492C22×C4—832no—
493C22×C4—832no—
494C22×C4—834no—
495C22×C4—834no—
496C22×C4—834no—
497C22×C4—834no—
498C22×C4—834no—
499C22×C4—1662no—
500C22×C4—1662no—
501C22×C4—16632no—
502C22×C4—1662no—
503C22×C4—1662no—
504C22×C4—1662no—
505C22×C4—1662no—
506C22×C4—1662no—
507C22×C4—1662no—
508C22×C4C2×C4×D8314yes—
509C22×C4C23×D8314yes—
510C22×C4C23×D8314yes—
511C22×C4C23×D8311yescharacteristic
512C22×C4C23×D8312yes—
513C22×C4C23×D8312yes—
514C22×C4C23×D8311yescharacteristic
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—3324no—
519C4⋊C4—3324no—
520C4⋊C4—3324no—
521C4⋊C4—838no—
522C4⋊C4—838no—
523C4⋊C4—838no—
524C4⋊C4—1664no—
525C4⋊C4—1664no—
526C4⋊C4C2×C4×D8318yes—
527C22⋊C4—44no—
528C22⋊C4—3324no—
529C22⋊C4—3324no—
530C22⋊C4—3324no—
531C22⋊C4—838no—
532C22⋊C4—838no—
533C22⋊C4—838no—
534C22⋊C4—1668no—
535C22⋊C4—1668no—
536C22⋊C4C2×C4×D8318yes—
537C42—44no—
538C42—3324no—
539C42—3324no—
540C42—3324no—
541C42—838no—
542C42—838no—
543C42—834no—
544C42—834no—
545C42—1668no—
546C42—1668no—
547C42C2×C4×D8314yes—
548C42C2×C4×D8314yes—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—3328no—
554C23—3324no—
555C23—3328no—
556C23—3324no—
557C23—832no—
558C23—832no—
559C23—832no—
560C23—832no—
561C23—1668no—
562C23—1661no—
563C23—1661no—
564C23—1662no—
565C23—1662no—
566C23—1662no—
567C23—1662no—
568C23C22×C4×D8312yes—
569C23C24×D8311yescharacteristic
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—3328no—
584C2×C4—3328no—
585C2×C4—832no—
586C2×C4—832no—
587C2×C4—832no—
588C2×C4—8316no—
589C2×C4—834no—
590C2×C4—834no—
591C2×C4—832no—
592C2×C4—1668no—
593C2×C4—1668no—
594C2×C4—1664no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1664no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1668no—
601C2×C4—1661no—
602C2×C4—1661no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4—1662no—
607C2×C4—1668no—
608C2×C4C22×C4×D8314yes—
609C2×C4C22×C4×D8314yes—
610C2×C4C22×C4×D8312yes—
611C2×C4C22×C4×D8312yes—
612C2×C4#41×D8312yes—
613C2×C4#41×D8312yes—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—33216no—
620C22—3321no—
621C22—3322no—
622C22—3321no—
623C22—834no—
624C22—1662no—
625C22—1664no—
626C22—1662no—
627C22—1664no—
628C22—1662no—
629C22—1664no—
630C22C23×C4×D8311yescharacteristic
631C22C2×#41×D8311yescharacteristic
632C22C2×#41×D8311yescharacteristic
633C22#1912853612yes—
634C22#1912853812yes—
635C4—22no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—3324no—
641C4—3324no—
642C4—834no—
643C4—1662no—
644C4—1664no—
645C4—1664no—
646C4#117×D8314yes—
647C2—42no—
648C2—3322no—
649C2—1664no—
650C2C2×#117×D8311yescharacteristic
651C2#2492066312yes—
652C2#2492084711yescharacteristic
653C2#2492085012yes—
654C2#2492085411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2314C2
socle, cosocleC23×C83C25
prime radicalC24×D83C23
hypercenter#2314D83
hypocenterC83#34048
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #24910073C2semidirect
3. #24910072C2semidirect
4. #24910071C2semidirect
5. #24910070C2semidirect
6. #24910068C2semidirect
7. #24910062C2semidirect
9. #24909957C2semidirect
10. #24909955C2semidirect
11. #24909869C2semidirect
12. #24909868C2semidirect
14. C4×#14581750C2semidirect
15. C83×#2314C2semidirect
24. #117×D83C4semidirect
36. #19127588C22semidirect
37. #19127587C22semidirect
38. #19127586C22semidirect
39. #19127585C22semidirect
40. #19127584C22semidirect
41. #19127583C22semidirect
42. #19127582C22semidirect
43. #19127571C22semidirect
44. #19127570C22semidirect
45. #19127567C22semidirect
46. #19127566C22semidirect
54. C4⋊C4×Dic83C22semidirect
55. C4⋊C4×Dic83C22semidirect
56. C4⋊C4×Dic83C22semidirect
57. C4×C4⋊Dic83C22semidirect
58. #19127552C22semidirect
63. #19127546C22semidirect
64. C42×Dic83C22semidirect
67. #387×C83C22semidirect
68. #384×C83C22semidirect
69. #382×C83C22semidirect
71. C4×C4⋊C4×C83C22semidirect
334. #2314D83semidirect
343. C83#34048semidirect
373. #387C2×D83semidirect
376. #384C2×D83semidirect
379. #382C2×D83semidirect
393. C4×C4⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D832
#249100732
#249100722
#249100712
#249100702
#249100682
#249100622
#249100572
#249099572
#249099552
#249098692
#249098682
#249098452
C4×#145817502
C83×#23142
#3404883