Finite Group Databasebeta

← #35611220

Subgroups — #35611220

Order 21248.

2326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128329C211yesmaximal, characteristic, complemented
2C22×#14581772C211yesmaximal, characteristic
3#24915749C212yesmaximal
4#24913936C214yesmaximal, complemented
5#24913932C214yesmaximal, complemented
6#24913780C214yesmaximal, complemented
7#24913776C214yesmaximal, complemented
8#24913280C212yesmaximal, complemented
9#24912620C212yesmaximal
10#24911750C211yesmaximal, characteristic, complemented
11#24911739C211yesmaximal, characteristic
12#24911733C211yesmaximal, characteristic
13#24911105C212yesmaximal, complemented
14#24910064C211yesmaximal, characteristic, complemented
15C83×#2484C211yesmaximal, characteristic, complemented
16C23×Dic166C2211yescharacteristic
17C23×C4×D83C2211yescharacteristic
18#19128329—28no—
19C2×#14581792C2212yescomplemented
20C2×#14581791C2212yescomplemented
21C2×#14581791C2214yescomplemented
22C2×#14581784C2212yescomplemented
23C2×#14581783C2212yescomplemented
24C2×#14581772—28no—
25C2×#14581772C2214yes—
26C2×#14581772C2214yes—
27C2×#14581770C2214yes—
28C2×#14581769C2212yes—
29C2×#14581769C2212yes—
30C2×#14581751C2211yescharacteristic
31C22×C4⋊Dic83C2211yescharacteristic
32C22×C4⋊Dic83C2211yescharacteristic
33C22×(C2×D83)⋊C4C2211yescharacteristic
34C22×(C2×D83)⋊C4C2211yescharacteristic
35#19127643C2214yescomplemented
36#19127640C2212yescomplemented
37#19127638C2212yes—
38#19127636C2212yes—
39#19127635C2214yescomplemented
40#19127633C2214yescomplemented
41#19127631C2212yescomplemented
42#19127630C2214yes—
43#19127628C2212yes—
44#19127627C2212yes—
45#19127622C2214yescomplemented
46#19127622C2214yescomplemented
47#19127621C2214yescomplemented
48#19127614C2214yes—
49#19127605C2214yescomplemented
50#19127593C2214yes—
51#19127593C2214yes—
52#19127592C2214yes—
53#19127590C2214yes—
54#19127586C2214yescomplemented
55#19127584C2214yescomplemented
56#19127582C2214yescomplemented
57#19127576C2212yes—
58#19127575C2214yes—
59#19127574C2212yes—
60#19127573C2212yes—
61#19127572C2214yes—
62#19127564C2211yescharacteristic
63#19127564C2212yes—
64#19127560C2214yes—
65#19127559C2212yes—
66#19127556C2212yes—
67#19127556C2212yes—
68C22⋊C4×Dic83C2212yes—
69C2×#117×C83C2211yescharacteristic
70C22×C4⋊C4×C83C2211yescharacteristic
71#400×C83C2212yescomplemented
72#398×C83C2212yescomplemented
73#387×C83C2214yescomplemented
74#385×C83C2212yes—
75#381×C83C2211yescharacteristic
76#381×C83C2212yes—
77C24×D83C2311yescharacteristic
78C22×Dic166—24no—
79C22×Dic166—44no—
80C22×Dic166C2314yes—
81C22×Dic166C2312yes—
82C22×Dic166C2312yes—
83C23×Dic83C2311yescharacteristic
84C23×Dic83C2311yescharacteristic
85C23×Dic83C2311yescharacteristic
86C22×C4×D83—24no—
87C22×C4×D83—44no—
88C22×C4×D83C2314yes—
89#14581792—28no—
90#14581791—28no—
91#14581791—48no—
92#14581784—28no—
93#14581783—28no—
94#14581772—216no—
95#14581772—216no—
96#14581770—48no—
97#14581769—28no—
98#14581769—28no—
99#14581751—24no—
100C2×C4⋊Dic83—24no—
101C2×C4⋊Dic83D418yes—
102C2×C4⋊Dic83C2312yes—
103C2×C4⋊Dic83C2312yes—
104C2×C4⋊Dic83C2314yes—
105C2×C4⋊Dic83C2311yescharacteristic
106C2×C4⋊Dic83C2311yescharacteristic
107C2×C4⋊Dic83C2314yes—
108C2×C4⋊Dic83C2312yes—
109C2×C4⋊Dic83C2312yes—
110C2×Dic83⋊C4C2312yes—
111C2×Dic83⋊C4C2312yes—
112C2×Dic83⋊C4C2314yes—
113C2×Dic83⋊C4C2312yes—
114C2×Dic83⋊C4C2312yes—
115C2×Dic83⋊C4C2314yes—
116C2×Dic83⋊C4C2314yes—
117C2×C22⋊Dic83C2312yes—
118C2×C22⋊Dic83C2312yes—
119C2×(C2×D83)⋊C4—24no—
120C2×(C2×D83)⋊C4—24no—
121C2×(C2×D83)⋊C4C2314yes—
122C2×(C2×D83)⋊C4C2314yes—
123C2×(C2×D83)⋊C4C2314yes—
124C2×(C2×D83)⋊C4C2312yes—
125C2×(C2×D83)⋊C4C2312yes—
126C2×(C2×D83)⋊C4C2314yes—
127C2×(C2×D83)⋊C4C2314yes—
128C2×C4×Dic83C2312yes—
129C2×C4×Dic83C2314yes—
130C83⋊#95C2314yes—
131C83⋊#95C2314yes—
132C83⋊#95C2312yes—
133C83⋊#95C2314yes—
134C83⋊#95C2314yes—
135C83⋊#95C2312yes—
136C83⋊#95C2314yes—
137C83⋊#95C2314yes—
138C83⋊#95C2314yes—
139C23×C4×C83C2311yescharacteristic
140C23×C4×C83C2311yescharacteristic
141C23×C4×C83C2311yescharacteristic
142#117×C83—24no—
143C2×C4⋊C4×C83—24no—
144C2×C4⋊C4×C83C2314yes—
145C2×C4⋊C4×C83C2312yes—
146C2×C4⋊C4×C83C2312yes—
147C2×C4⋊C4×C83C2311yescharacteristic
148C2×C4⋊C4×C83C2311yescharacteristic
149C2×C22⋊C4×C83C2312yes—
150C2×C22⋊C4×C83C2312yes—
151C2×C42×C83C2312yes—
152#95×C83C2314yes—
153#95×C83C2312yes—
154#95×C83C2312yes—
155#95×C83C2314yes—
156#95×C83C2314yes—
157C23×D83—22no—
158C23×D83—24no—
159C23×D83—24no—
160C23×D83—42no—
161C23×D83C2412yes—
162C2×Dic166—28no—
163C2×Dic166—28no—
164C2×Dic166—24no—
165C2×Dic166—24no—
166C2×Dic166—44no—
167C2×Dic166—416no—
168C2×Dic166—44no—
169C2×Dic166—44no—
170C22×Dic83—22no—
171C22×Dic83—42no—
172C22×Dic83—42no—
173C22×Dic83—42no—
174C22×Dic83C2×D414yes—
175C22×Dic83C2×D414yes—
176C22×Dic83C2412yes—
177C22×Dic83C2412yes—
178C22×Dic83C2414yes—
179C22×Dic83C2412yes—
180C22×Dic83C2412yes—
181C22×Dic83C2412yes—
182C2×C4×D83—28no—
183C2×C4×D83—28no—
184C2×C4×D83—44no—
185C2×C4×D83—416no—
186C4⋊Dic83—22no—
187C4⋊Dic83—22no—
188C4⋊Dic83—24no—
189C4⋊Dic83—24no—
190C4⋊Dic83—28no—
191C4⋊Dic83—44no—
192C4⋊Dic83C2×D418yes—
193C4⋊Dic83C2×D418yes—
194Dic83⋊C4—28no—
195Dic83⋊C4—24no—
196Dic83⋊C4—24no—
197Dic83⋊C4—24no—
198Dic83⋊C4—24no—
199Dic83⋊C4—44no—
200Dic83⋊C4—44no—
201C22⋊Dic83—24no—
202C22⋊Dic83—24no—
203(C2×D83)⋊C4—28no—
204(C2×D83)⋊C4—28no—
205(C2×D83)⋊C4—24no—
206(C2×D83)⋊C4—24no—
207(C2×D83)⋊C4—28no—
208(C2×D83)⋊C4—44no—
209(C2×D83)⋊C4—44no—
210C4×Dic83—24no—
211C4×Dic83—44no—
212C24×C83C2411yescharacteristic
213C22×C4×C83—22no—
214C22×C4×C83—24no—
215C22×C4×C83—24no—
216C22×C4×C83—22no—
217C22×C4×C83—24no—
218C22×C4×C83C2×D414yes—
219C22×C4×C83C2412yes—
220C22×C4×C83C2414yes—
221C22×C4×C83C2412yes—
222C22×C4×C83C2412yes—
223C22×C4×C83C2411yescharacteristic
224C22×C4×C83C2411yescharacteristic
225C22×C4×C83C2412yes—
226C4⋊C4×C83—24no—
227C4⋊C4×C83—24no—
228C4⋊C4×C83—22no—
229C4⋊C4×C83—22no—
230C4⋊C4×C83—28no—
231C22⋊C4×C83—24no—
232C22⋊C4×C83—24no—
233C42×C83—24no—
234C23×C83—22no—
235C23×C83—22no—
236C23×C83C22×D412yes—
237C23×C832−1+414yes—
238C23×C83C2511yescharacteristic
239C22×D83—28no—
240C22×D83—22no—
241C22×D83—24no—
242C22×D83—22no—
243C22×D83—24no—
244C22×D83—416no—
245C22×D83—44no—
246C22×D83—44no—
247C22×D832−1+414yes—
248C2×C4×C83—28no—
249C2×C4×C83—28no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—22no—
253C2×C4×C83—22no—
254C2×C4×C83—22no—
255C2×C4×C83—22no—
256C2×C4×C83—22no—
257C2×C4×C83—24no—
258C2×C4×C83—24no—
259C2×C4×C83—24no—
260C2×C4×C83—24no—
261C2×C4×C83—24no—
262C2×C4×C83—24no—
263C2×C4×C83—44no—
264C2×C4×C83C22×D412yes—
265C2×C4×C83C22×D412yes—
266C2×C4×C83H(5,𝔽2)14yes—
267C2×Dic83—24no—
268C2×Dic83—24no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—22no—
272C2×Dic83—24no—
273C2×Dic83—24no—
274C2×Dic83—24no—
275C2×Dic83—24no—
276C2×Dic83—24no—
277C2×Dic83—24no—
278C2×Dic83—44no—
279C2×Dic83—44no—
280C2×Dic83—44no—
281C2×Dic83C22×D414yes—
282C2×Dic83C22×D414yes—
283C4×D83—416no—
284Dic166—416no—
285Dic166—84no—
286Dic166—84no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—28no—
290C22×C83#53614yes—
291C22×C83C23×D411yescharacteristic
292C22×C83C2×H(5,𝔽2)12yes—
293C22×C83C2×2−1+412yes—
294C22×C83C2×2−1+412yes—
295C2×D83—24no—
296C2×D83—28no—
297C2×D83—416no—
298C2×D83—42no—
299C2×D83—42no—
300C2×D83—44no—
301C4×C83—22no—
302C4×C83—22no—
303C4×C83—24no—
304C4×C83—42no—
305C4×C83—42no—
306C4×C83—42no—
307C4×C83—44no—
308Dic83—24no—
309Dic83—24no—
310Dic83—42no—
311Dic83—42no—
312Dic83—42no—
313Dic83—44no—
314#36708—831nomaximal, Sylow
315C2×C83—24no—
316C2×C83C2×#53611yescharacteristic
317C2×C83#348411yescharacteristic
318C2×C83#348611yescharacteristic
319C2×C83#353814yes—
320D83—44no—
321C2×#544—831no—
322C22×C22⋊Q8—831no—
323#2848—832no—
324#2690—834no—
325#2690—834no—
326#2677—834no—
327#2677—834no—
328#2627—832no—
329#2561—832no—
330#2484—831no—
331#2484D8311yescharacteristic, complemented
332#2482—831no—
333#2482—831no—
334#2434—832no—
335#2313—831no—
336C83#3670811yescharacteristic, complemented, Sylow
337C23×Q8—831no—
338C24×C4—831no—
339#544—1668no—
340C2×C4⋊Q8—834no—
341C2×C4⋊Q8—832no—
342C2×C4⋊Q8—832no—
343C2×#125—832no—
344C2×#125—832no—
345C2×C22⋊Q8—834no—
346C2×C22⋊Q8—834no—
347C2×C22⋊Q8—834no—
348C2×C22⋊Q8—832no—
349C2×C22⋊Q8—832no—
350C2×C22⋊Q8—1668no—
351C2×#117—831no—
352C2×#117C2×D8311yescharacteristic
353C22×C4⋊C4—831no—
354C22×C4⋊C4—831no—
355C22×C4⋊C4C2×D8311yescharacteristic
356C22×C22⋊C4—831no—
357C22×C22⋊C4—831no—
358#400—832no—
359#400—834no—
360#400C2×D8312yescomplemented
361#399—832no—
362#399—832no—
363#398—832no—
364#398—834no—
365#398—834no—
366#398C2×D8312yescomplemented
367#397—832no—
368#397—834no—
369#397—832no—
370#395—834no—
371#395—834no—
372#395—834no—
373#393—834no—
374#391—834no—
375#388—834no—
376#388—834no—
377#388—834no—
378#388—834no—
379#387—834no—
380#387—834no—
381#387—834no—
382#387C2×D8314yescomplemented
383#385—832no—
384#385—834no—
385#385—832no—
386#385—832no—
387#385—834no—
388#385C2×D8312yes—
389#381—832no—
390#381—831no—
391#381C2×D8312yes—
392#381C2×D8311yescharacteristic
393#380—832no—
394#380—834no—
395#379—832no—
396#379—832no—
397C4×C22⋊C4—832no—
398C25—831no—
399C22×Q8—3324no—
400C22×Q8—832no—
401C22×Q8—832no—
402C22×Q8—834no—
403C22×Q8—1664no—
404C23×C4—3324no—
405C23×C4—831no—
406C23×C4—831no—
407C23×C4—831no—
408C23×C4—834no—
409C23×C4—1664no—
410C23×C4C22×D8311yescharacteristic
411C23×C4C22×D8311yescharacteristic
412C23×C4C22×D8311yescharacteristic
413C4⋊Q8—3328no—
414C4⋊Q8—1668no—
415C4⋊Q8—1668no—
416#125—1668no—
417#125—1668no—
418C22⋊Q8—3328no—
419C22⋊Q8—1668no—
420C22⋊Q8—1668no—
421C22⋊Q8—16616no—
422C22⋊Q8—16616no—
423#117—24no—
424#117—1664no—
425C2×C4⋊C4—24no—
426C2×C4⋊C4—834no—
427C2×C4⋊C4—834no—
428C2×C4⋊C4—834no—
429C2×C4⋊C4—838no—
430C2×C4⋊C4—832no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—834no—
433C2×C4⋊C4—831no—
434C2×C4⋊C4—831no—
435C2×C4⋊C4—832no—
436C2×C4⋊C4—832no—
437C2×C4⋊C4—832no—
438C2×C4⋊C4—832no—
439C2×C4⋊C4—832no—
440C2×C4⋊C4—832no—
441C2×C4⋊C4—834no—
442C2×C4⋊C4—1664no—
443C2×C4⋊C4C22×D8312yes—
444C2×C4⋊C4C22×D8312yes—
445C2×C4⋊C4C22×D8311yescharacteristic
446C2×C4⋊C4C22×D8311yescharacteristic
447C2×C4⋊C4C22×D8314yes—
448C2×C22⋊C4—834no—
449C2×C22⋊C4—834no—
450C2×C22⋊C4—832no—
451C2×C22⋊C4—832no—
452C2×C22⋊C4—834no—
453C2×C22⋊C4—832no—
454C2×C22⋊C4—832no—
455C2×C22⋊C4—834no—
456C2×C22⋊C4—834no—
457C2×C22⋊C4—1664no—
458C2×C22⋊C4—1664no—
459C2×C22⋊C4C22×D8312yes—
460C2×C22⋊C4C22×D8312yes—
461C2×C42—834no—
462C2×C42—832no—
463C2×C42C22×D8312yes—
464#95—834no—
465#95—834no—
466#95—834no—
467#95—832no—
468#95—832no—
469#95—834no—
470#95—834no—
471#95—834no—
472#95—834no—
473#95C22×D8314yes—
474#95C22×D8312yes—
475#95C22×D8312yes—
476#95C22×D8314yes—
477#95C22×D8314yes—
478C24—3322no—
479C24—832no—
480C24—1664no—
481C24—1664no—
482C24—1662no—
483C24C23×D8311yescharacteristic
484C2×Q8—33216no—
485C2×Q8—3324no—
486C2×Q8—3324no—
487C2×Q8—3324no—
488C2×Q8—1668no—
489C2×Q8—1664no—
490C2×Q8—1664no—
491C2×Q8—1668no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—33216no—
498C22×C4—3324no—
499C22×C4—3322no—
500C22×C4—3322no—
501C22×C4—3322no—
502C22×C4—834no—
503C22×C4—834no—
504C22×C4—832no—
505C22×C4—832no—
506C22×C4—834no—
507C22×C4—832no—
508C22×C4—832no—
509C22×C4—832no—
510C22×C4—1668no—
511C22×C4—1662no—
512C22×C4—1668no—
513C22×C4D4×D8314yes—
514C22×C4C23×D8312yes—
515C22×C4C23×D8314yes—
516C22×C4C23×D8312yes—
517C22×C4C23×D8312yes—
518C22×C4C23×D8311yescharacteristic
519C22×C4C23×D8311yescharacteristic
520C22×C4C23×D8312yes—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—28no—
526C4⋊C4—3324no—
527C4⋊C4—3324no—
528C4⋊C4—3324no—
529C4⋊C4—838no—
530C4⋊C4—838no—
531C4⋊C4—1668no—
532C4⋊C4—1664no—
533C4⋊C4—1664no—
534C4⋊C4—1664no—
535C4⋊C4—1664no—
536C4⋊C4—1664no—
537C4⋊C4—1664no—
538C4⋊C4—1662no—
539C4⋊C4—1662no—
540C4⋊C4—1668no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—3324no—
544C22⋊C4—3324no—
545C22⋊C4—1668no—
546C22⋊C4—1668no—
547C22⋊C4—1668no—
548C22⋊C4—1664no—
549C22⋊C4—1664no—
550C22⋊C4—1664no—
551C22⋊C4—1664no—
552C42—24no—
553C42—3324no—
554C42—1664no—
555C23—22no—
556C23—22no—
557C23—3324no—
558C23—33216no—
559C23—3324no—
560C23—834no—
561C23—1668no—
562C23—1662no—
563C23—1664no—
564C23—1662no—
565C23—1664no—
566C23C2×D4×D8312yes—
567C23#1458183914yes—
568C23C24×D8311yescharacteristic
569Q8—33216no—
570Q8—6644no—
571Q8—6644no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—33216no—
589C2×C4—3324no—
590C2×C4—3324no—
591C2×C4—3324no—
592C2×C4—834no—
593C2×C4—834no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1662no—
597C2×C4—1664no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—1664no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4C2×D4×D8312yes—
606C2×C4C2×D4×D8312yes—
607C2×C4#1458183814yes—
608C22—22no—
609C22—22no—
610C22—28no—
611C22—33216no—
612C22—3322no—
613C22—3324no—
614C22—3322no—
615C22—1664no—
616C22—1668no—
617C22#1912827314yes—
618C22C22×D4×D8311yescharacteristic
619C22C2×#1458183812yes—
620C222−1+4×D8312yes—
621C22C2×#1458183912yes—
622C4—24no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—3322no—
630C4—3322no—
631C4—3322no—
632C4—3324no—
633C4—1664no—
634C4—1664no—
635C2—24no—
636C2—3324no—
637C2C2×#1912827311yescharacteristic
638C2#2492109611yescharacteristic
639C2#2492111211yescharacteristic
640C2#2492170914yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2484C2
prime radicalC24×D83C23
hypercenter#2484D83
hypocenterC83#36708
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2484

Complements

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

NH ≅ G/Ntype
1. C2×#19128329C2semidirect
4. #24913936C2semidirect
5. #24913932C2semidirect
6. #24913780C2semidirect
7. #24913776C2semidirect
8. #24913280C2semidirect
10. #24911750C2semidirect
13. #24911105C2semidirect
14. #24910064C2semidirect
15. C83×#2484C2semidirect
19. C2×#14581792C22semidirect
20. C2×#14581791C22semidirect
21. C2×#14581791C22semidirect
22. C2×#14581784C22semidirect
23. C2×#14581783C22semidirect
35. #19127643C22semidirect
36. #19127640C22semidirect
39. #19127635C22semidirect
40. #19127633C22semidirect
41. #19127631C22semidirect
45. #19127622C22semidirect
46. #19127622C22semidirect
47. #19127621C22semidirect
49. #19127605C22semidirect
54. #19127586C22semidirect
55. #19127584C22semidirect
56. #19127582C22semidirect
71. #400×C83C22semidirect
72. #398×C83C22semidirect
73. #387×C83C22semidirect
331. #2484D83semidirect
336. C83#36708semidirect
360. #400C2×D83semidirect
366. #398C2×D83semidirect
382. #387C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191283292
C22×#145817722
#249157492
#249139362
#249139322
#249137802
#249137762
#249132802
#249126202
#249117502
#249117392
#249117332
#249111052
#249100642
C83×#24842
#3670883