Finite Group Databasebeta

← #35462463

Subgroups — #35462463

Order 21248.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911177C211yesmaximal, characteristic
2#24910342C211yesmaximal, characteristic
3#24909028C211yesmaximal, characteristic, complemented
4#24908962C211yesmaximal, characteristic, complemented
5#24908955C211yesmaximal, characteristic, complemented
6#24908799C211yesmaximal, characteristic, complemented
7#24908419C211yesmaximal, characteristic, complemented
8#24908218C211yesmaximal, characteristic, complemented
9#24908050C211yesmaximal, characteristic, complemented
10#24908039C211yesmaximal, characteristic, complemented
11#24907660C211yesmaximal, characteristic
12#24907646C211yesmaximal, characteristic
13#24907547C211yesmaximal, characteristic
14#24907183C211yesmaximal, characteristic
15C83×#1809C211yesmaximal, characteristic
16C2×#14581783C2211yescharacteristic
17C2×#14581751C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×Dic83⋊C8C2211yescharacteristic
22C2×Dic83⋊C8C2211yescharacteristic
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24#19127637—21no—
25#19127624—21no—
26#19127600C2211yescharacteristic
27#19127598—21no—
28#19127597C2211yescharacteristic
29#19127597C2211yescharacteristic
30#19127597C2211yescharacteristic
31#19127588—21no—
32#19127586—21no—
33#19127586—21no—
34#19127570C2211yescharacteristic
35#19127559—21no—
36C4⋊C4×Dic83C2211yescharacteristic
37C4⋊C4×Dic83C2211yescharacteristic
38C22⋊C4×Dic83—21no—
39C4×C22⋊Dic83C2211yescharacteristic
40#19127464C2211yescharacteristic
41#19127464C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C22×M16×C83C2211yescharacteristic
53C2×#107×C83C2211yescharacteristic
54C2×#106×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#380×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59C23×Dic83—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62#14581783—22no—
63#14581783—41no—
64#14581751—22no—
65#14581751—41no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—22no—
70C2×C4⋊Dic83Q814yes—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83D411yescharacteristic
83C2×C22⋊Dic83D411yescharacteristic
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83C2311yescharacteristic
88C2×C4×Dic83C2311yescharacteristic
89#14581725—41no—
90#14581723—22no—
91Dic83⋊C8—22no—
92Dic83⋊C8—41no—
93C22⋊(C83⋊C8)—22no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95D411yescharacteristic
101C83⋊#95D411yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C2×M16×C83—22no—
104C2×M16×C83—22no—
105C2×M16×C83—41no—
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114#107×C83—41no—
115#106×C83—22no—
116#95×C83—21no—
117#95×C83—21no—
118C22×Dic83—21no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—24no—
122C22×Dic83—21no—
123C22×Dic83—81no—
124C22×Dic83—81no—
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C22×Dic83C2×D411yescharacteristic
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C4⋊Dic83—24no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—42no—
142C4⋊Dic83—81no—
143C4⋊Dic83C2×Q812yes—
144C4⋊Dic83C2×Q812yes—
145C4⋊Dic83#4112yes—
146Dic83⋊C4—42no—
147Dic83⋊C4—42no—
148Dic83⋊C4—42no—
149Dic83⋊C4—81no—
150Dic83⋊C4—81no—
151Dic83⋊C4—81no—
152C22⋊Dic83—22no—
153C22⋊Dic83—41no—
154C22⋊Dic83—42no—
155C4×Dic83—21no—
156C4×Dic83—21no—
157C4×Dic83—21no—
158C4×Dic83—21no—
159C4×Dic83—42no—
160C4×Dic83—81no—
161C4×Dic83—81no—
162C24×C83C2×D411yescharacteristic
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×Q812yes—
173C22×C4×C83#4112yes—
174C22×C4×C83C2411yescharacteristic
175M16×C83—44no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83#4112yes—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C22⋊C4×C83—42no—
191C22⋊C4×C83—81no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83C22⋊Q812yes—
199C23×C83C22×D411yescharacteristic
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—22no—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83C22⋊Q812yes—
219C2×C4×C83C22×Q811yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83H(5,𝔽2)11yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—22no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—84no—
249C2×Dic83#13712yes—
250C8×C83—22no—
251C8×C83—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83C2×C22⋊Q811yescharacteristic
262C22×C83#54611yescharacteristic
263C22×C83#54711yescharacteristic
264C22×C83C2×#13711yescharacteristic
265C22×C83#57511yescharacteristic
266C22×C83#57611yescharacteristic
267C22×C83#57711yescharacteristic
268C4×C83—22no—
269C4×C83—81no—
270C4×C83—81no—
271C4×C83—81no—
272C4×C83—81no—
273C4×C83#53311yescharacteristic
274C4×C83#54411yescharacteristic
275C4×C83#55111yescharacteristic
276C4×C83#55211yescharacteristic
277Dic83—22no—
278Dic83—41no—
279Dic83—42no—
280Dic83—42no—
281Dic83—42no—
282Dic83—41no—
283Dic83—41no—
284Dic83—81no—
285#26018—831nomaximal, Sylow
286C2×C83—42no—
287C2×C83#243811yescharacteristic
288C2×C83#307811yescharacteristic
289C2×C83#308111yescharacteristic
290C2×C83#330911yescharacteristic
291C2×C83#333111yescharacteristic
292C2×C83#334611yescharacteristic
293C2×C83#335111yescharacteristic
294#2441—831no—
295#2345—831no—
296#2100—831no—
297#2088—831no—
298#2086—831no—
299#2059—831no—
300#1989—831no—
301#1947—831no—
302#1920—831no—
303#1919—831no—
304#1854—831no—
305#1851—831no—
306#1823—831no—
307#1809D8311yescharacteristic
308#1736—831no—
309C83#2601811yescharacteristic, complemented, Sylow
310C22×M16C2×D8311yescharacteristic
311C2×#125—831no—
312C2×#117—831no—
313C22×C4⋊C4—831no—
314C2×#107—831no—
315C2×#107C2×D8311yescharacteristic
316C2×#106—831no—
317C2×#106C2×D8311yescharacteristic
318C2×C4⋊C8—831no—
319C2×C4⋊C8—831no—
320C2×C22⋊C8—831no—
321#399—1661no—
322#396—1661no—
323#389—831no—
324#389—831no—
325#389—831no—
326#389—831no—
327#389—1661no—
328#387—1661no—
329#387—1661no—
330#387—1661no—
331#384—831no—
332#380—1661no—
333#380C2×D8311yescharacteristic
334#380C2×D8311yescharacteristic
335C4×C4⋊C4—831no—
336C4×C4⋊C4—831no—
337C4×C22⋊C4—831no—
338C4×C22⋊C4—1661no—
339#340—831no—
340#340—831no—
341#340—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340C2×D8311yescharacteristic
350#340C2×D8311yescharacteristic
351#336—831no—
352#336—831no—
353C23×C4—1661no—
354C23×C4C22×D8311yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—41no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362#125—3321no—
363#125—1662no—
364#117—3321no—
365#117—1662no—
366C2×C4⋊C4—834no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1662no—
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C4⋊C4C22×D8311yescharacteristic
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—831no—
388C2×C22⋊C4—831no—
389C2×C22⋊C4—1661no—
390C2×C42—831no—
391C2×C42—831no—
392C2×C42—1661no—
393C2×C42—1661no—
394C2×C42—1661no—
395#107—41no—
396#107—3321no—
397#106—22no—
398#106—1662no—
399C4⋊C8—3321no—
400C4⋊C8—1662no—
401C22⋊C8—1662no—
402#95—21no—
403#95—21no—
404#95—831no—
405#95—831no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412C24D4×D8311yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—831no—
422C22×C4—831no—
423C22×C4—831no—
424C22×C4—831no—
425C22×C4—6641no—
426C22×C4—6641no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1664no—
432C22×C4D4×D8311yescharacteristic
433C22×C4Q8×D8312yes—
434C22×C4#1305059012yes—
435C22×C4C23×D8311yescharacteristic
436M16—44no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8#1305058912yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—3322no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—832no—
462C4⋊C4—832no—
463C4⋊C4—832no—
464C4⋊C4—6641no—
465C4⋊C4—6641no—
466C4⋊C4—6641no—
467C4⋊C4—6641no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1664no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C22⋊C4—42no—
475C22⋊C4—81no—
476C22⋊C4—3322no—
477C22⋊C4—3321no—
478C22⋊C4—1662no—
479C42—3322no—
480C42—6641no—
481C42—6641no—
482C42—1661no—
483C42—1661no—
484C42—1661no—
485C42—1661no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1458177012yes—
493C23C2×D4×D8311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—832no—
522C2×C4—6644no—
523C2×C4—1662no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1662no—
528C2×C4—1662no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4—1661no—
536C2×C4—1661no—
537C2×C4#1458177012yes—
538C2×C4C2×Q8×D8311yescharacteristic
539C2×C4#41×D8311yescharacteristic
540C2×C4C2×#1305058911yescharacteristic
541C2×C4C2×#1305059011yescharacteristic
542C2×C4#1458183711yescharacteristic
543C2×C4#1458183911yescharacteristic
544C8—22no—
545C8—41no—
546C8—3321no—
547C8—3321no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1458177011yescharacteristic
556C22#1912834011yescharacteristic
557C22#1912835111yescharacteristic
558C22#137×D8311yescharacteristic
559C22#1912849711yescharacteristic
560C22#1912850511yescharacteristic
561C22#1912851011yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—3322no—
568C4—3322no—
569C4—3321no—
570C4—3321no—
571C4—3322no—
572C4—3321no—
573C4—6641no—
574C4—1662no—
575C4#1912825311yescharacteristic
576C4#1912833011yescharacteristic
577C4#1912837711yescharacteristic
578C4#1912838411yescharacteristic
579C2—42no—
580C2#2491114511yescharacteristic
581C2#2491726711yescharacteristic
582C2#2491729211yescharacteristic
583C2#2491946411yescharacteristic
584C2#2491977411yescharacteristic
585C2#2491994311yescharacteristic
586C2#2492001811yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1809C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1809D83
hypocenterC83#26018
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
3. #24909028C2semidirect
4. #24908962C2semidirect
5. #24908955C2semidirect
6. #24908799C2semidirect
7. #24908419C2semidirect
8. #24908218C2semidirect
9. #24908050C2semidirect
10. #24908039C2semidirect
309. C83#26018semidirect

Maximal subgroups

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

subgroupindex
#249111772
#249103422
#249090282
#249089622
#249089552
#249087992
#249084192
#249082182
#249080502
#249080392
#249076602
#249076462
#249075472
#249071832
C83×#18092
#2601883