Finite Group Databasebeta

← #35611216

Subgroups — #35611216

Order 21248.

2326 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128330C211yesmaximal, characteristic, complemented
2C22×#14581771C211yesmaximal, characteristic
3#24915750C211yesmaximal, characteristic
4#24913933C214yesmaximal, complemented
5#24913929C214yesmaximal, complemented
6#24913778C214yesmaximal, complemented
7#24913774C214yesmaximal, complemented
8#24913279C211yesmaximal, characteristic, complemented
9#24913278C211yesmaximal, characteristic, complemented
10#24912618C212yesmaximal
11#24911749C212yesmaximal, complemented
12#24911734C212yesmaximal
13#24911105C212yesmaximal, complemented
14#24910065C211yesmaximal, characteristic, complemented
15C83×#2848C211yesmaximal, characteristic
16C23×Dic166C2211yescharacteristic
17#19128330—28no—
18C2×#14581791C2212yescomplemented
19C2×#14581791C2214yescomplemented
20C2×#14581790C2212yescomplemented
21C2×#14581782C2212yescomplemented
22C2×#14581782C2212yescomplemented
23C2×#14581771—28no—
24C2×#14581771C2214yes—
25C2×#14581771C2214yes—
26C2×#14581771C2211yescharacteristic
27C2×#14581771C2212yes—
28C2×#14581769C2212yes—
29C2×#14581748C2212yes—
30C22×C4⋊Dic83C2211yescharacteristic
31C22×Dic83⋊C4C2211yescharacteristic
32C22×Dic83⋊C4C2211yescharacteristic
33C22×C22⋊Dic83C2211yescharacteristic
34C22×C22⋊Dic83C2211yescharacteristic
35#19127643C2214yescomplemented
36#19127641C2212yescomplemented
37#19127639C2212yescomplemented
38#19127637C2214yes—
39#19127634C2214yescomplemented
40#19127634C2214yescomplemented
41#19127631C2214yescomplemented
42#19127629C2214yes—
43#19127629C2214yes—
44#19127622C2214yescomplemented
45#19127622C2212yescomplemented
46#19127621C2214yescomplemented
47#19127620C2212yescomplemented
48#19127613C2214yes—
49#19127605C2214yescomplemented
50#19127595C2214yes—
51#19127594C2214yes—
52#19127594C2214yes—
53#19127589C2214yes—
54#19127588C2214yescomplemented
55#19127588C2214yescomplemented
56#19127587C2214yescomplemented
57#19127582C2214yescomplemented
58#19127576C2212yes—
59#19127574C2212yes—
60#19127574C2214yes—
61#19127573C2212yes—
62#19127562C2212yes—
63#19127562C2212yes—
64#19127562C2212yes—
65#19127561C2212yes—
66#19127561C2211yescharacteristic
67#19127557C2212yes—
68C22⋊C4×Dic83C2212yes—
69C24×C4×C83C2211yescharacteristic
70C2×C22⋊Q8×C83C2211yescharacteristic
71C2×C22⋊Q8×C83C2212yes—
72#385×C83C2212yes—
73#385×C83C2214yes—
74#380×C83C2212yes—
75#380×C83C2211yescharacteristic
76#379×C83C2212yes—
77C22×Dic166—24no—
78C22×Dic166—44no—
79C22×Dic166C2314yes—
80C22×Dic166C2312yes—
81C22×Dic166C2311yescharacteristic
82C23×Dic83C2311yescharacteristic
83C23×Dic83C2311yescharacteristic
84C23×Dic83C2311yescharacteristic
85C23×Dic83C2311yescharacteristic
86#14581791—28no—
87#14581791—48no—
88#14581790—28no—
89#14581782—28no—
90#14581782—28no—
91#14581771—216no—
92#14581771—216no—
93#14581771—24no—
94#14581771—44no—
95#14581769—28no—
96#14581748—28no—
97C2×C4⋊Dic83—24no—
98C2×C4⋊Dic83C2311yescharacteristic
99C2×C4⋊Dic83C2314yes—
100C2×C4⋊Dic83C2312yes—
101C2×C4⋊Dic83C2312yes—
102C2×C4⋊Dic83C2314yes—
103C2×Dic83⋊C4—24no—
104C2×Dic83⋊C4D418yes—
105C2×Dic83⋊C4C2314yes—
106C2×Dic83⋊C4C2312yes—
107C2×Dic83⋊C4C2312yes—
108C2×Dic83⋊C4C2312yes—
109C2×Dic83⋊C4C2312yes—
110C2×Dic83⋊C4C2312yes—
111C2×Dic83⋊C4C2312yes—
112C2×Dic83⋊C4C2312yes—
113C2×Dic83⋊C4C2314yes—
114C2×Dic83⋊C4C2312yes—
115C2×Dic83⋊C4C2312yes—
116C2×C22⋊Dic83—24no—
117C2×C22⋊Dic83—24no—
118C2×C22⋊Dic83C2312yes—
119C2×C22⋊Dic83C2314yes—
120C2×C22⋊Dic83C2312yes—
121C2×C22⋊Dic83C2314yes—
122C2×C22⋊Dic83C2314yes—
123C2×C22⋊Dic83C2314yes—
124C2×C4×Dic83C2312yes—
125C2×C4×Dic83C2312yes—
126C2×C4×Dic83C2314yes—
127C83⋊#95C2312yes—
128C83⋊#95C2314yes—
129C83⋊#95C2314yes—
130C83⋊#95C2314yes—
131C83⋊#95C2314yes—
132C83⋊#95C2314yes—
133C83⋊#95C2314yes—
134C83⋊#95C2314yes—
135C83⋊#95C2314yes—
136C83⋊#95C2314yes—
137C83⋊#95C2314yes—
138C25×C83C2311yescharacteristic
139C22×Q8×C83C2311yescharacteristic
140C23×C4×C83—24no—
141C23×C4×C83—44no—
142C23×C4×C83C2311yescharacteristic
143C23×C4×C83C2312yes—
144C23×C4×C83C2312yes—
145C23×C4×C83C2311yescharacteristic
146C22⋊Q8×C83—24no—
147C22⋊Q8×C83—44no—
148C2×C4⋊C4×C83C2314yes—
149C2×C4⋊C4×C83C2312yes—
150C2×C4⋊C4×C83C2311yescharacteristic
151C2×C4⋊C4×C83C2312yes—
152C2×C22⋊C4×C83C2312yes—
153C2×C22⋊C4×C83C2312yes—
154C2×C22⋊C4×C83C2314yes—
155C2×C22⋊C4×C83C2314yes—
156#95×C83C2312yes—
157#95×C83C2314yes—
158C2×Dic166—28no—
159C2×Dic166—28no—
160C2×Dic166—22no—
161C2×Dic166—24no—
162C2×Dic166—44no—
163C2×Dic166—416no—
164C2×Dic166—42no—
165C2×Dic166—44no—
166C22×Dic83—22no—
167C22×Dic83—22no—
168C22×Dic83—22no—
169C22×Dic83—22no—
170C22×Dic83—22no—
171C22×Dic83—22no—
172C22×Dic83—42no—
173C22×Dic83—42no—
174C22×Dic83C2×D414yes—
175C22×Dic83C2×D414yes—
176C22×Dic83C2412yes—
177C22×Dic83C2412yes—
178C22×Dic83C2412yes—
179C22×Dic83C2414yes—
180C22×Dic83C2412yes—
181C22×Dic83C2412yes—
182C22×Dic83C2412yes—
183C4⋊Dic83—22no—
184C4⋊Dic83—24no—
185C4⋊Dic83—28no—
186C4⋊Dic83—42no—
187C4⋊Dic83—44no—
188Dic83⋊C4—24no—
189Dic83⋊C4—24no—
190Dic83⋊C4—24no—
191Dic83⋊C4—28no—
192Dic83⋊C4—24no—
193Dic83⋊C4—24no—
194Dic83⋊C4—24no—
195Dic83⋊C4—24no—
196Dic83⋊C4—44no—
197Dic83⋊C4C2×D418yes—
198Dic83⋊C4C2×D418yes—
199C22⋊Dic83—24no—
200C22⋊Dic83—28no—
201C22⋊Dic83—24no—
202C22⋊Dic83—28no—
203C22⋊Dic83—28no—
204C22⋊Dic83—44no—
205C4×Dic83—24no—
206C4×Dic83—24no—
207C4×Dic83—44no—
208C24×C83—22no—
209C24×C83—24no—
210C24×C83—22no—
211C24×C83—22no—
212C24×C83—42no—
213C24×C83C2412yes—
214C24×C83C2411yescharacteristic
215C2×Q8×C83—22no—
216C2×Q8×C83—42no—
217C22×C4×C83—22no—
218C22×C4×C83—24no—
219C22×C4×C83—24no—
220C22×C4×C83—24no—
221C22×C4×C83—24no—
222C22×C4×C83—24no—
223C22×C4×C83—42no—
224C22×C4×C83—42no—
225C22×C4×C83—42no—
226C22×C4×C83—416no—
227C22×C4×C83C2×D414yes—
228C22×C4×C83C2412yes—
229C22×C4×C83C2414yes—
230C22×C4×C83C2412yes—
231C22×C4×C83C2411yescharacteristic
232C22×C4×C83C2411yescharacteristic
233C22×C4×C83C2412yes—
234C4⋊C4×C83—22no—
235C4⋊C4×C83—24no—
236C4⋊C4×C83—28no—
237C4⋊C4×C83—42no—
238C22⋊C4×C83—24no—
239C22⋊C4×C83—28no—
240C22⋊C4×C83—24no—
241C22⋊C4×C83—44no—
242C23×C83—22no—
243C23×C83—22no—
244C23×C83—28no—
245C23×C83—22no—
246C23×C83—22no—
247C23×C83—22no—
248C23×C83—22no—
249C23×C83—22no—
250C23×C83—22no—
251C23×C83—48no—
252C23×C83—48no—
253C23×C83—44no—
254C23×C83—44no—
255C23×C83C22×D412yes—
256C23×C832−1+414yes—
257C23×C832−1+412yes—
258C23×C832−1+412yes—
259C23×C83C2511yescharacteristic
260Q8×C83—82no—
261C2×C4×C83—28no—
262C2×C4×C83—22no—
263C2×C4×C83—22no—
264C2×C4×C83—22no—
265C2×C4×C83—22no—
266C2×C4×C83—22no—
267C2×C4×C83—22no—
268C2×C4×C83—24no—
269C2×C4×C83—24no—
270C2×C4×C83—24no—
271C2×C4×C83—24no—
272C2×C4×C83—24no—
273C2×C4×C83—44no—
274C2×C4×C83—48no—
275C2×C4×C83—48no—
276C2×C4×C83C22×D412yes—
277C2×C4×C83C22×D412yes—
278C2×C4×C83H(5,𝔽2)14yes—
279C2×Dic83—28no—
280C2×Dic83—24no—
281C2×Dic83—22no—
282C2×Dic83—22no—
283C2×Dic83—22no—
284C2×Dic83—22no—
285C2×Dic83—22no—
286C2×Dic83—22no—
287C2×Dic83—24no—
288C2×Dic83—24no—
289C2×Dic83—24no—
290C2×Dic83—24no—
291C2×Dic83—24no—
292C2×Dic83—24no—
293C2×Dic83—22no—
294C2×Dic83—22no—
295C2×Dic83—44no—
296C2×Dic83—44no—
297C2×Dic83—44no—
298C2×Dic83C22×D414yes—
299C2×Dic83C22×D414yes—
300Dic166—416no—
301Dic166—84no—
302Dic166—82no—
303C22×C83—22no—
304C22×C83—22no—
305C22×C83—24no—
306C22×C83—24no—
307C22×C83—24no—
308C22×C83—24no—
309C22×C83—24no—
310C22×C83—416no—
311C22×C83—42no—
312C22×C83—42no—
313C22×C83—42no—
314C22×C83—42no—
315C22×C83#53614yes—
316C22×C83C23×D411yescharacteristic
317C22×C83C2×H(5,𝔽2)12yes—
318C22×C83C2×2−1+411yescharacteristic
319C22×C83C2×2−1+412yes—
320C22×C83C2×2−1+411yescharacteristic
321C4×C83—24no—
322C4×C83—22no—
323C4×C83—22no—
324C4×C83—42no—
325C4×C83—42no—
326C4×C83—44no—
327Dic83—24no—
328Dic83—24no—
329Dic83—42no—
330Dic83—42no—
331Dic83—42no—
332Dic83—42no—
333Dic83—44no—
334#36708—831nomaximal, Sylow
335C2×C83—24no—
336C2×C83—44no—
337C2×C83C2×#53611yescharacteristic
338C2×C83#348411yescharacteristic
339C2×C83#348611yescharacteristic
340C2×C83#353812yes—
341C2×C83#353812yes—
342C2×#544—831no—
343C22×C22⋊Q8—831no—
344#2848—831no—
345#2848D8311yescharacteristic
346#2690—834no—
347#2690—834no—
348#2677—834no—
349#2677—834no—
350#2627—831no—
351#2627—831no—
352#2561—832no—
353#2484—832no—
354#2482—832no—
355#2434—832no—
356#2313—831no—
357C83#3670811yescharacteristic, complemented, Sylow
358C23×Q8—831no—
359C24×C4C2×D8311yescharacteristic
360#544—1668no—
361C2×C4⋊Q8—832no—
362C2×C4⋊Q8—834no—
363C2×C4⋊Q8—832no—
364C2×#125—832no—
365C2×#125—832no—
366C2×C22⋊Q8—832no—
367C2×C22⋊Q8—831no—
368C2×C22⋊Q8—834no—
369C2×C22⋊Q8—834no—
370C2×C22⋊Q8—832no—
371C2×C22⋊Q8—1668no—
372C2×C22⋊Q8C2×D8311yescharacteristic
373C2×C22⋊Q8C2×D8312yes—
374C2×#117—832no—
375C22×C4⋊C4—831no—
376C22×C4⋊C4—831no—
377C22×C4⋊C4—831no—
378C22×C22⋊C4—831no—
379C22×C22⋊C4—831no—
380#400—832no—
381#400—832no—
382#400—834no—
383#399—834no—
384#398—834no—
385#398—834no—
386#398—834no—
387#397—834no—
388#397—834no—
389#395—832no—
390#395—834no—
391#395—832no—
392#395—834no—
393#393—834no—
394#391—834no—
395#388—834no—
396#388—834no—
397#388—834no—
398#388—834no—
399#387—834no—
400#387—834no—
401#387—834no—
402#387—834no—
403#385—834no—
404#385—832no—
405#385—832no—
406#385—832no—
407#385C2×D8314yes—
408#385C2×D8312yes—
409#381—832no—
410#381—832no—
411#381—832no—
412#380—831no—
413#380—832no—
414#380C2×D8311yescharacteristic
415#380C2×D8312yes—
416#379—832no—
417#379C2×D8312yes—
418C4×C22⋊C4—832no—
419C25C22×D8311yescharacteristic
420C22×Q8—3324no—
421C22×Q8—831no—
422C22×Q8—832no—
423C22×Q8—834no—
424C22×Q8—1664no—
425C22×Q8C22×D8311yescharacteristic
426C23×C4—24no—
427C23×C4—44no—
428C23×C4—831no—
429C23×C4—831no—
430C23×C4—831no—
431C23×C4—831no—
432C23×C4C22×D8311yescharacteristic
433C23×C4C22×D8312yes—
434C23×C4C22×D8312yes—
435C23×C4C22×D8311yescharacteristic
436C4⋊Q8—3328no—
437C4⋊Q8—1668no—
438C4⋊Q8—1668no—
439#125—1668no—
440#125—1668no—
441C22⋊Q8—24no—
442C22⋊Q8—44no—
443C22⋊Q8—3324no—
444C22⋊Q8—1668no—
445C22⋊Q8—16616no—
446C22⋊Q8—16616no—
447C22⋊Q8—1664no—
448#117—1668no—
449C2×C4⋊C4—832no—
450C2×C4⋊C4—834no—
451C2×C4⋊C4—834no—
452C2×C4⋊C4—838no—
453C2×C4⋊C4—832no—
454C2×C4⋊C4—832no—
455C2×C4⋊C4—834no—
456C2×C4⋊C4—832no—
457C2×C4⋊C4—832no—
458C2×C4⋊C4—832no—
459C2×C4⋊C4—832no—
460C2×C4⋊C4—831no—
461C2×C4⋊C4—832no—
462C2×C4⋊C4—832no—
463C2×C4⋊C4—832no—
464C2×C4⋊C4—832no—
465C2×C4⋊C4—834no—
466C2×C4⋊C4—1664no—
467C2×C4⋊C4—1664no—
468C2×C4⋊C4C22×D8311yescharacteristic
469C2×C4⋊C4C22×D8312yes—
470C2×C4⋊C4C22×D8314yes—
471C2×C4⋊C4C22×D8312yes—
472C2×C22⋊C4—834no—
473C2×C22⋊C4—834no—
474C2×C22⋊C4—834no—
475C2×C22⋊C4—832no—
476C2×C22⋊C4—832no—
477C2×C22⋊C4—834no—
478C2×C22⋊C4—1664no—
479C2×C22⋊C4—1664no—
480C2×C22⋊C4C22×D8312yes—
481C2×C22⋊C4C22×D8312yes—
482C2×C22⋊C4C22×D8314yes—
483C2×C22⋊C4C22×D8314yes—
484C2×C42—834no—
485C2×C42—832no—
486C2×C42—832no—
487#95—834no—
488#95—832no—
489#95—834no—
490#95—834no—
491#95—834no—
492#95—834no—
493#95—834no—
494#95—834no—
495#95—834no—
496#95—834no—
497#95—834no—
498#95C22×D8312yes—
499#95C22×D8314yes—
500C24—22no—
501C24—24no—
502C24—22no—
503C24—22no—
504C24—42no—
505C24C23×D8312yes—
506C24C23×D8311yescharacteristic
507C2×Q8—22no—
508C2×Q8—42no—
509C2×Q8—33216no—
510C2×Q8—3324no—
511C2×Q8—3322no—
512C2×Q8—3324no—
513C2×Q8—1668no—
514C2×Q8—1664no—
515C2×Q8—1662no—
516C2×Q8—1668no—
517C22×C4—22no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—24no—
522C22×C4—24no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4—416no—
527C22×C4—3322no—
528C22×C4—3322no—
529C22×C4—834no—
530C22×C4—834no—
531C22×C4—832no—
532C22×C4—832no—
533C22×C4—834no—
534C22×C4—832no—
535C22×C4—832no—
536C22×C4—832no—
537C22×C4—832no—
538C22×C4—1662no—
539C22×C4—1662no—
540C22×C4—1662no—
541C22×C4—1662no—
542C22×C4—1662no—
543C22×C4—1662no—
544C22×C4D4×D8314yes—
545C22×C4C23×D8314yes—
546C22×C4C23×D8312yes—
547C22×C4C23×D8312yes—
548C22×C4C23×D8311yescharacteristic
549C22×C4C23×D8311yescharacteristic
550C22×C4C23×D8312yes—
551C4⋊C4—22no—
552C4⋊C4—24no—
553C4⋊C4—28no—
554C4⋊C4—42no—
555C4⋊C4—3322no—
556C4⋊C4—3324no—
557C4⋊C4—3324no—
558C4⋊C4—838no—
559C4⋊C4—838no—
560C4⋊C4—1668no—
561C4⋊C4—1664no—
562C4⋊C4—1664no—
563C4⋊C4—1664no—
564C4⋊C4—1664no—
565C4⋊C4—1664no—
566C4⋊C4—1664no—
567C4⋊C4—1664no—
568C4⋊C4—1662no—
569C4⋊C4—1664no—
570C4⋊C4—1668no—
571C22⋊C4—24no—
572C22⋊C4—28no—
573C22⋊C4—24no—
574C22⋊C4—44no—
575C22⋊C4—3324no—
576C22⋊C4—1668no—
577C22⋊C4—1668no—
578C22⋊C4—1664no—
579C22⋊C4—1668no—
580C22⋊C4—1664no—
581C42—3324no—
582C42—1664no—
583C42—1664no—
584C23—22no—
585C23—22no—
586C23—28no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—48no—
594C23—48no—
595C23—44no—
596C23—44no—
597C23C2×D4×D8312yes—
598C23#1458183914yes—
599C23#1458183912yes—
600C23#1458184012yes—
601C23C24×D8311yescharacteristic
602Q8—82no—
603Q8—33216no—
604Q8—6642no—
605Q8—6644no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—3324no—
622C2×C4—3324no—
623C2×C4—3324no—
624C2×C4—834no—
625C2×C4—834no—
626C2×C4—1662no—
627C2×C4—1662no—
628C2×C4—1662no—
629C2×C4—1662no—
630C2×C4—1664no—
631C2×C4—1664no—
632C2×C4—1664no—
633C2×C4—1664no—
634C2×C4—1664no—
635C2×C4—1662no—
636C2×C4—1664no—
637C2×C4—1662no—
638C2×C4—1668no—
639C2×C4—1664no—
640C2×C4—1662no—
641C2×C4—1662no—
642C2×C4C2×D4×D8312yes—
643C2×C4C2×D4×D8312yes—
644C2×C4#1458183714yes—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—416no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22#1912827414yes—
658C22C22×D4×D8311yescharacteristic
659C22C2×#1458183712yes—
660C22C2×#1458183912yes—
661C22C2×#1458183911yescharacteristic
662C22C2×#1458184011yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—3322no—
670C4—3322no—
671C4—3322no—
672C4—3324no—
673C4—3322no—
674C4—1664no—
675C4—1664no—
676C2—24no—
677C2—44no—
678C2C2×#1912827411yescharacteristic
679C2#2492109211yescharacteristic
680C2#2492111011yescharacteristic
681C2#2492170812yes—
682C2#2492171212yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2848C2
prime radicalC25×C83C23
hypercenter#2848D83
hypocenterC83#36708
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2848

Complements

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

NH ≅ G/Ntype
1. C2×#19128330C2semidirect
4. #24913933C2semidirect
5. #24913929C2semidirect
6. #24913778C2semidirect
7. #24913774C2semidirect
8. #24913279C2semidirect
9. #24913278C2semidirect
11. #24911749C2semidirect
13. #24911105C2semidirect
14. #24910065C2semidirect
18. C2×#14581791C22semidirect
19. C2×#14581791C22semidirect
20. C2×#14581790C22semidirect
21. C2×#14581782C22semidirect
22. C2×#14581782C22semidirect
35. #19127643C22semidirect
36. #19127641C22semidirect
37. #19127639C22semidirect
39. #19127634C22semidirect
40. #19127634C22semidirect
41. #19127631C22semidirect
44. #19127622C22semidirect
45. #19127622C22semidirect
46. #19127621C22semidirect
47. #19127620C22semidirect
49. #19127605C22semidirect
54. #19127588C22semidirect
55. #19127588C22semidirect
56. #19127587C22semidirect
57. #19127582C22semidirect
357. C83#36708semidirect

Maximal subgroups

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

subgroupindex
C2×#191283302
C22×#145817712
#249157502
#249139332
#249139292
#249137782
#249137742
#249132792
#249132782
#249126182
#249117492
#249117342
#249111052
#249100652
C83×#28482
#3670883