Finite Group Databasebeta

← #35577142

Subgroups — #35577142

Order 21248.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915769C214yesmaximal, complemented
2#24913435C214yesmaximal, complemented
3#24911962C214yesmaximal, complemented
4#24911960C214yesmaximal, complemented
5#24911609C214yesmaximal, complemented
6#24911601C214yesmaximal, complemented
7#24910075C211yesmaximal, characteristic
8#24910075C211yesmaximal, characteristic
9#24910075C211yesmaximal, characteristic
10C2×#19127576C211yesmaximal, characteristic
11C2×#19127573C211yesmaximal, characteristic
12C2×C22⋊C4×Dic83C211yesmaximal, characteristic
13C83×#2315C211yesmaximal, characteristic
14C24×Dic83C2211yescharacteristic
15C2×#14581782C2214yescomplemented
16C22×C4⋊Dic83C2212yes—
17C22×Dic83⋊C4C2212yes—
18C22×C22⋊Dic83C2211yescharacteristic
19C22×C22⋊Dic83C2211yescharacteristic
20C22×C22⋊Dic83C2211yescharacteristic
21C22×C4×Dic83C2212yes—
22#19127642C2214yescomplemented
23#19127642C2214yescomplemented
24#19127641C2214yescomplemented
25#19127639C2214yescomplemented
26#19127635C2214yescomplemented
27#19127634C2214yescomplemented
28#19127632C2214yescomplemented
29#19127631C2214yescomplemented
30#19127598C2214yescomplemented
31#19127596C2214yescomplemented
32#19127587C2214yescomplemented
33#19127586C2214yescomplemented
34#19127583C2214yescomplemented
35#19127582C2214yescomplemented
36#19127576C2212yes—
37#19127576C2212yes—
38#19127576C2214yes—
39#19127574C2214yes—
40#19127574C2212yes—
41#19127574C2214yes—
42#19127574C2212yes—
43#19127573C2212yes—
44#19127573C2212yes—
45#19127573C2214yes—
46#19127573C2214yes—
47#19127573C2214yes—
48#19127573C2212yes—
49#19127573C2212yes—
50#19127565C2214yescomplemented
51#19127559C2214yes—
52#19127559C2214yes—
53#19127559C2212yes—
54#19127559C2212yes—
55C22⋊C4×Dic83C2214yes—
56C22⋊C4×Dic83C2214yes—
57C2×C83⋊#95C2212yes—
58C2×C83⋊#95C2212yes—
59C2×C83⋊#95C2212yes—
60C22×C22⋊C4×C83C2211yescharacteristic
61C22×C22⋊C4×C83C2211yescharacteristic
62C22×C22⋊C4×C83C2211yescharacteristic
63#385×C83C2214yes—
64#385×C83C2214yes—
65#380×C83C2212yes—
66#380×C83C2212yes—
67C23×Dic83—24no—
68C23×Dic83—44no—
69C23×Dic83C2311yescharacteristic
70C23×Dic83C2312yes—
71C23×Dic83C2312yes—
72C23×Dic83C2311yescharacteristic
73C23×Dic83C2311yescharacteristic
74C23×Dic83C2312yes—
75C23×Dic83C2311yescharacteristic
76C23×Dic83C2312yes—
77#14581782—48no—
78C2×C4⋊Dic83—44no—
79C2×C4⋊Dic83C2314yes—
80C2×C4⋊Dic83C2314yes—
81C2×Dic83⋊C4—44no—
82C2×Dic83⋊C4C2314yes—
83C2×Dic83⋊C4C2314yes—
84C2×Dic83⋊C4C2314yes—
85C2×Dic83⋊C4C2314yes—
86C2×Dic83⋊C4C2314yes—
87C2×Dic83⋊C4C2314yes—
88C2×Dic83⋊C4C2314yes—
89C2×Dic83⋊C4C2314yes—
90C2×C22⋊Dic83—24no—
91C2×C22⋊Dic83—24no—
92C2×C22⋊Dic83Q818yes—
93C2×C22⋊Dic83C2312yes—
94C2×C22⋊Dic83C2312yes—
95C2×C22⋊Dic83C2312yes—
96C2×C22⋊Dic83C2312yes—
97C2×C22⋊Dic83C2312yes—
98C2×C22⋊Dic83C2312yes—
99C2×C4×Dic83—44no—
100C2×C4×Dic83C2314yes—
101C2×C4×Dic83C2314yes—
102C83⋊#95C2314yes—
103C83⋊#95C2314yes—
104C83⋊#95C2314yes—
105C83⋊#95C2314yes—
106C83⋊#95C2314yes—
107C83⋊#95C2314yes—
108C83⋊#95C2314yes—
109C83⋊#95C2314yes—
110C83⋊#95C2314yes—
111C83⋊#95C2314yes—
112C83⋊#95C2314yes—
113C83⋊#95C2314yes—
114C25×C83C2311yescharacteristic
115C23×C4×C83C2312yes—
116C23×C4×C83C2311yescharacteristic
117C23×C4×C83C2312yes—
118C23×C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2314yes—
120C2×C4⋊C4×C83C2314yes—
121C2×C22⋊C4×C83—24no—
122C2×C22⋊C4×C83—24no—
123C2×C22⋊C4×C83—24no—
124C2×C22⋊C4×C83C2312yes—
125C2×C22⋊C4×C83C2312yes—
126C2×C22⋊C4×C83C2312yes—
127C2×C22⋊C4×C83C2312yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C22⋊C4×C83C2312yes—
130#95×C83C2314yes—
131#95×C83C2314yes—
132C22×Dic83—22no—
133C22×Dic83—24no—
134C22×Dic83—24no—
135C22×Dic83—24no—
136C22×Dic83—22no—
137C22×Dic83—24no—
138C22×Dic83—22no—
139C22×Dic83—22no—
140C22×Dic83—24no—
141C22×Dic83—24no—
142C22×Dic83—42no—
143C22×Dic83—42no—
144C22×Dic83—42no—
145C22×Dic83—42no—
146C22×Dic83—42no—
147C22×Dic83—42no—
148C22×Dic83—42no—
149C22×Dic83—42no—
150C22×Dic83—42no—
151C22×Dic83—416no—
152C22×Dic83C2×Q818yes—
153C22×Dic83C2412yes—
154C22×Dic83C2412yes—
155C22×Dic83C2414yes—
156C22×Dic83C2412yes—
157C22×Dic83C2414yes—
158C22×Dic83C2412yes—
159C4⋊Dic83—28no—
160C4⋊Dic83—44no—
161Dic83⋊C4—28no—
162Dic83⋊C4—44no—
163Dic83⋊C4—44no—
164Dic83⋊C4—44no—
165Dic83⋊C4—44no—
166Dic83⋊C4—44no—
167Dic83⋊C4—44no—
168Dic83⋊C4—44no—
169C22⋊Dic83—24no—
170C22⋊Dic83—24no—
171C22⋊Dic83—24no—
172C22⋊Dic83—24no—
173C22⋊Dic83C2×Q818yes—
174C22⋊Dic83C2×Q818yes—
175C4×Dic83—28no—
176C4×Dic83—44no—
177C24×C83—42no—
178C24×C83—42no—
179C24×C83C2×Q814yes—
180C24×C83#4114yes—
181C24×C83#4114yes—
182C24×C83C2411yescharacteristic
183C24×C83C2411yescharacteristic
184C24×C83C2411yescharacteristic
185C22×C4×C83—24no—
186C22×C4×C83—24no—
187C22×C4×C83—22no—
188C22×C4×C83—22no—
189C22×C4×C83—42no—
190C22×C4×C83—42no—
191C22×C4×C83—42no—
192C22×C4×C83—42no—
193C22×C4×C83—42no—
194C22×C4×C83—42no—
195C22×C4×C83C2412yes—
196C22×C4×C83C2414yes—
197C22×C4×C83C2414yes—
198C22×C4×C83C2412yes—
199C4⋊C4×C83—44no—
200C4⋊C4×C83—44no—
201C22⋊C4×C83—24no—
202C22⋊C4×C83—24no—
203C22⋊C4×C83—24no—
204C22⋊C4×C83—24no—
205C22⋊C4×C83—24no—
206C22⋊C4×C83—24no—
207C23×C83—22no—
208C23×C83—22no—
209C23×C83—22no—
210C23×C83—22no—
211C23×C83—22no—
212C23×C83—22no—
213C23×C83—48no—
214C23×C83—48no—
215C23×C83—44no—
216C23×C83—44no—
217C23×C83#125116yescomplemented
218C23×C83C22×Q812yes—
219C23×C83C22×Q812yes—
220C23×C83C22×Q812yes—
221C23×C83C2×#4112yes—
222C23×C83C2×#4112yes—
223C23×C83C2×#4112yes—
224C23×C83C2×#4112yes—
225C23×C83C2×#4112yes—
226C23×C83C2×#4112yes—
227C23×C83C2511yescharacteristic
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—48no—
239C2×C4×C83—48no—
240C2×C4×C83—82no—
241C2×C4×C83—82no—
242C2×Dic83—28no—
243C2×Dic83—22no—
244C2×Dic83—24no—
245C2×Dic83—22no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—22no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—22no—
254C2×Dic83—22no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—48no—
258C2×Dic83—48no—
259C2×Dic83—48no—
260C2×Dic83—48no—
261C2×Dic83—44no—
262C2×Dic83—82no—
263C2×Dic83—82no—
264C2×Dic83C22×Q818yes—
265C2×Dic83H(5,𝔽2)14yes—
266C22×C83—24no—
267C22×C83—24no—
268C22×C83—24no—
269C22×C83—22no—
270C22×C83—22no—
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83—24no—
278C22×C83—416no—
279C22×C83C2×#12514yes—
280C22×C83C2×#12514yes—
281C22×C83C2×#12514yes—
282C22×C83C23×Q811yescharacteristic
283C22×C83C22×#4111yescharacteristic
284C22×C83C22×#4111yescharacteristic
285C22×C83C2×H(5,𝔽2)12yes—
286C22×C83#58512yes—
287C4×C83—44no—
288C4×C83—44no—
289C4×C83—42no—
290C4×C83—42no—
291Dic83—24no—
292Dic83—28no—
293Dic83—44no—
294Dic83—42no—
295Dic83—42no—
296Dic83—42no—
297#34994—831nomaximal, Sylow
298C2×C83—24no—
299C2×C83—24no—
300C2×C83—24no—
301C2×C83C22×#12511yescharacteristic
302C2×C83#345912yes—
303C2×C83#349011yescharacteristic
304C2×C83#349111yescharacteristic
305C2×C83#350112yes—
306#2851—834no—
307#2646—834no—
308#2505—834no—
309#2505—834no—
310#2473—834no—
311#2473—834no—
312#2315—831no—
313#2315—831no—
314#2315—831no—
315#2315D8311yescharacteristic
316C2×#385—831no—
317C2×#385—831no—
318C2×C4×C22⋊C4—831no—
319C83#3499411yescharacteristic, complemented, Sylow
320C24×C4—831no—
321C2×#125—834no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—832no—
324C22×C22⋊C4—831no—
325C22×C22⋊C4—831no—
326C22×C22⋊C4—831no—
327C22×C22⋊C4C2×D8311yescharacteristic
328C22×C22⋊C4C2×D8311yescharacteristic
329C22×C22⋊C4C2×D8311yescharacteristic
330C22×C42—832no—
331#400—834no—
332#400—834no—
333#400—834no—
334#400—834no—
335#398—834no—
336#398—834no—
337#398—834no—
338#398—834no—
339#389—834no—
340#389—834no—
341#387—834no—
342#387—834no—
343#387—834no—
344#387—834no—
345#385—834no—
346#385—832no—
347#385—834no—
348#385—832no—
349#385—832no—
350#385—834no—
351#385—834no—
352#385—832no—
353#385—832no—
354#385—832no—
355#385—834no—
356#385—832no—
357#385—832no—
358#385—834no—
359#385C2×D8314yes—
360#385C2×D8314yes—
361#382—834no—
362#380—834no—
363#380—834no—
364#380—832no—
365#380—832no—
366#380C2×D8312yes—
367#380C2×D8312yes—
368C4×C22⋊C4—834no—
369C4×C22⋊C4—834no—
370C2×#95—832no—
371C2×#95—832no—
372C2×#95—832no—
373C25C22×D8311yescharacteristic
374C23×C4—3324no—
375C23×C4—832no—
376C23×C4—831no—
377C23×C4—832no—
378C23×C4—831no—
379C23×C4—831no—
380C23×C4—832no—
381C23×C4—832no—
382C23×C4—831no—
383C23×C4—1664no—
384C23×C4C22×D8311yescharacteristic
385C23×C4C22×D8312yes—
386C23×C4C22×D8311yescharacteristic
387C23×C4C22×D8312yes—
388#125—3328no—
389C2×C4⋊C4—3324no—
390C2×C4⋊C4—3324no—
391C2×C4⋊C4—834no—
392C2×C4⋊C4—834no—
393C2×C4⋊C4—834no—
394C2×C4⋊C4—834no—
395C2×C4⋊C4—834no—
396C2×C4⋊C4—834no—
397C2×C4⋊C4—834no—
398C2×C4⋊C4—834no—
399C2×C4⋊C4—834no—
400C2×C4⋊C4—834no—
401C2×C4⋊C4C22×D8314yes—
402C2×C4⋊C4C22×D8314yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—838no—
407C2×C22⋊C4—832no—
408C2×C22⋊C4—832no—
409C2×C22⋊C4—832no—
410C2×C22⋊C4—832no—
411C2×C22⋊C4—832no—
412C2×C22⋊C4—832no—
413C2×C22⋊C4—1664no—
414C2×C22⋊C4—1664no—
415C2×C22⋊C4C22×D8312yes—
416C2×C22⋊C4C22×D8312yes—
417C2×C22⋊C4C22×D8312yes—
418C2×C22⋊C4C22×D8312yes—
419C2×C22⋊C4C22×D8312yes—
420C2×C22⋊C4C22×D8312yes—
421C2×C42—3324no—
422C2×C42—834no—
423C2×C42—834no—
424#95—834no—
425#95—834no—
426#95—834no—
427#95—834no—
428#95—834no—
429#95—834no—
430#95—834no—
431#95—834no—
432#95—834no—
433#95—834no—
434#95—834no—
435#95—834no—
436#95C22×D8314yes—
437#95C22×D8314yes—
438C24—42no—
439C24—42no—
440C24Q8×D8314yes—
441C24#1305058914yes—
442C24#1305059014yes—
443C24C23×D8311yescharacteristic
444C24C23×D8311yescharacteristic
445C24C23×D8311yescharacteristic
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—33216no—
457C22×C4—3322no—
458C22×C4—3322no—
459C22×C4—3322no—
460C22×C4—3322no—
461C22×C4—3322no—
462C22×C4—3322no—
463C22×C4—3322no—
464C22×C4—3322no—
465C22×C4—3322no—
466C22×C4—838no—
467C22×C4—834no—
468C22×C4—832no—
469C22×C4—832no—
470C22×C4—832no—
471C22×C4—834no—
472C22×C4—832no—
473C22×C4—1664no—
474C22×C4—1664no—
475C22×C4—1664no—
476C22×C4—1662no—
477C22×C4—1664no—
478C22×C4—1664no—
479C22×C4—1664no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4C23×D8312yes—
484C22×C4C23×D8314yes—
485C22×C4C23×D8314yes—
486C22×C4C23×D8312yes—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—3324no—
490C4⋊C4—3324no—
491C4⋊C4—3324no—
492C4⋊C4—3324no—
493C4⋊C4—3324no—
494C4⋊C4—3324no—
495C4⋊C4—3324no—
496C4⋊C4—3324no—
497C4⋊C4—1668no—
498C4⋊C4—1668no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—838no—
506C22⋊C4—838no—
507C22⋊C4—1664no—
508C22⋊C4—1664no—
509C22⋊C4—1664no—
510C22⋊C4—1664no—
511C42—3324no—
512C42—1668no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23#14581782116yescomplemented
524C23C2×Q8×D8312yes—
525C23C2×Q8×D8312yes—
526C23C2×Q8×D8312yes—
527C23C2×#1305058912yes—
528C23C2×#1305058912yes—
529C23C2×#1305058912yes—
530C23C2×#1305059012yes—
531C23C2×#1305059012yes—
532C23C2×#1305059012yes—
533C23C24×D8311yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—3324no—
549C2×C4—3328no—
550C2×C4—3328no—
551C2×C4—3328no—
552C2×C4—3328no—
553C2×C4—838no—
554C2×C4—834no—
555C2×C4—6642no—
556C2×C4—6642no—
557C2×C4—1664no—
558C2×C4—1662no—
559C2×C4—1662no—
560C2×C4—1662no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4—1664no—
566C2×C4—1664no—
567C2×C4—1668no—
568C2×C4—1662no—
569C2×C4—1662no—
570C2×C4—1664no—
571C2×C4—1664no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22C2×#1458178214yes—
586C22C2×#1458178214yes—
587C22C2×#1458178214yes—
588C22C22×Q8×D8311yescharacteristic
589C22C22×#1305058911yescharacteristic
590C22C22×#1305059011yescharacteristic
591C22H(5,𝔽2)×D8312yes—
592C22#1912853612yes—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—3324no—
598C4—3322no—
599C4—3322no—
600C4—3322no—
601C4—1664no—
602C4—1668no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#1458178211yescharacteristic
607C2#2492086912yes—
608C2#2492114811yescharacteristic
609C2#2492115611yescharacteristic
610C2#2492126312yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2315C2
prime radicalC25×C83C23
hypercenter#2315D83
hypocenterC83#34994
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #24915769C2semidirect
2. #24913435C2semidirect
3. #24911962C2semidirect
4. #24911960C2semidirect
5. #24911609C2semidirect
6. #24911601C2semidirect
15. C2×#14581782C22semidirect
22. #19127642C22semidirect
23. #19127642C22semidirect
24. #19127641C22semidirect
25. #19127639C22semidirect
26. #19127635C22semidirect
27. #19127634C22semidirect
28. #19127632C22semidirect
29. #19127631C22semidirect
30. #19127598C22semidirect
31. #19127596C22semidirect
32. #19127587C22semidirect
33. #19127586C22semidirect
34. #19127583C22semidirect
35. #19127582C22semidirect
50. #19127565C22semidirect
217. C23×C83#125semidirect
319. C83#34994semidirect
523. C23#14581782semidirect

Maximal subgroups

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

subgroupindex
#249157692
#249134352
#249119622
#249119602
#249116092
#249116012
#249100752
C2×#191275762
C2×#191275732
C2×C22⋊C4×Dic832
C83×#23152
#3499483