Finite Group Databasebeta

← #35462453

Subgroups — #35462453

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#24910338C211yesmaximal, characteristic
2#24909025C211yesmaximal, characteristic, complemented
3#24908964C211yesmaximal, characteristic, complemented
4#24908954C211yesmaximal, characteristic, complemented
5#24908798C211yesmaximal, characteristic, complemented
6#24908421C211yesmaximal, characteristic, complemented
7#24908219C211yesmaximal, characteristic, complemented
8#24908045C211yesmaximal, characteristic, complemented
9#24908038C211yesmaximal, characteristic, complemented
10#24907662C211yesmaximal, characteristic
11#24907648C211yesmaximal, characteristic
12#24907547C211yesmaximal, characteristic
13#24907494C211yesmaximal, characteristic
14#24907184C211yesmaximal, characteristic
15C83×#2441C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C22×C4⋊Dic83C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×C22⋊(C83⋊C8)C2211yescharacteristic
25#19127599C2211yescharacteristic
26#19127599C2211yescharacteristic
27#19127584—21no—
28#19127584—21no—
29#19127561C2211yescharacteristic
30#19127559—21no—
31C4×C4⋊Dic83C2211yescharacteristic
32C4×C4⋊Dic83C2211yescharacteristic
33C22⋊C4×Dic83—21no—
34#19127463C2211yescharacteristic
35#19127463C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#125×C83C2211yescharacteristic
49C2×#117×C83C2211yescharacteristic
50#399×C83—21no—
51#396×C83—21no—
52#389×C83—21no—
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#387×C83—21no—
56#384×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C23×Dic83—21no—
60C2×#13050574—22no—
61C2×#13050574—22no—
62C2×#13050574—41no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—22no—
69C2×C4⋊Dic83Q814yes—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79#14581725—41no—
80#14581725—41no—
81#14581723—22no—
82#14581723—22no—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)—41no—
85C22⋊(C83⋊C8)—22no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C23×C4×C83C2311yescharacteristic
92#125×C83—22no—
93#125×C83—41no—
94#117×C83—22no—
95#117×C83—41no—
96C2×C4⋊C4×C83—21no—
97C2×C4⋊C4×C83—21no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83D411yescharacteristic
109C2×C22⋊C4×C83D411yescharacteristic
110C2×C42×C83—21no—
111C2×C42×C83C2311yescharacteristic
112C2×C42×C83C2311yescharacteristic
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83D411yescharacteristic
117#95×C83D411yescharacteristic
118C22×Dic83—21no—
119C22×Dic83—21no—
120C22×Dic83—24no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—81no—
124C22×Dic83—81no—
125#13050574—44no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8#4112yes—
138C4⋊Dic83—24no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83C2×Q812yes—
146C4⋊Dic83C2×Q812yes—
147Dic83⋊C4—81no—
148Dic83⋊C4—81no—
149C22⋊Dic83—42no—
150C4×Dic83—81no—
151C4×Dic83—81no—
152C24×C83C2×D411yescharacteristic
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83C2×D411yescharacteristic
163C22×C4×C83C2×D411yescharacteristic
164C22×C4×C83C2×D411yescharacteristic
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×Q812yes—
167C22×C4×C83#4112yes—
168C22×C4×C83C2411yescharacteristic
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—21no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—42no—
177C4⋊C4×C83—42no—
178C4⋊C4×C83—42no—
179C4⋊C4×C83—42no—
180C4⋊C4×C83—81no—
181C4⋊C4×C83—81no—
182C4⋊C4×C83#4112yes—
183C22⋊C4×C83—22no—
184C22⋊C4×C83—41no—
185C22⋊C4×C83—42no—
186C22⋊C4×C83—81no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—42no—
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—21no—
203C2×C4×C83—21no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—22no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83C22⋊Q812yes—
230C2×C4×C83#13712yes—
231C2×C4×C83C22×Q811yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83H(5,𝔽2)11yescharacteristic
236C2×C4×C832−1+411yescharacteristic
237C2×Dic83—22no—
238C2×Dic83—22no—
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—41no—
249C2×Dic83—84no—
250C83⋊C8—22no—
251C83⋊C8—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—22no—
270C4×C83—41no—
271C4×C83—42no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83—81no—
275C4×C83—81no—
276C4×C83—81no—
277C4×C83#53311yescharacteristic
278C4×C83#54411yescharacteristic
279C4×C83#55111yescharacteristic
280C4×C83#55211yescharacteristic
281Dic83—42no—
282Dic83—42no—
283Dic83—81no—
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#2441D8311yescharacteristic
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#1809—831no—
308#1736—831no—
309C83#2601811yescharacteristic, complemented, Sylow
310C22×M16—831no—
311C2×#125C2×D8311yescharacteristic
312C2×#117C2×D8311yescharacteristic
313C22×C4⋊C4—831no—
314C2×#107—831no—
315C2×#107—831no—
316C2×#106—831no—
317C2×#106—831no—
318C2×C4⋊C8—831no—
319C2×C4⋊C8—831no—
320C2×C22⋊C8—831no—
321#399—21no—
322#396—21no—
323#389—21no—
324#389—831no—
325#389—831no—
326#389C2×D8311yescharacteristic
327#389C2×D8311yescharacteristic
328#387—21no—
329#387—1661no—
330#387—1661no—
331#384C2×D8311yescharacteristic
332#380—831no—
333#380—1661no—
334#380C2×D8311yescharacteristic
335C4×C4⋊C4—831no—
336C4×C4⋊C4—831no—
337C4×C22⋊C4—1661no—
338C4×C22⋊C4C2×D8311yescharacteristic
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#340—831no—
350#340—831no—
351#336—831no—
352#336—831no—
353C23×C4—1661no—
354C23×C4C22×D8311yescharacteristic
355C2×M16—3321no—
356C2×M16—1662no—
357C2×M16—1662no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8—831no—
362#125—22no—
363#125—41no—
364#117—22no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—834no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
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—1661no—
388C2×C22⋊C4C22⋊D8311yescharacteristic
389C2×C22⋊C4C22⋊D8311yescharacteristic
390C2×C42—21no—
391C2×C42—1661no—
392C2×C42—1661no—
393C2×C42C22×D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395#107—3321no—
396#107—3321no—
397#106—1662no—
398#106—1662no—
399C4⋊C8—3321no—
400C4⋊C8—1662no—
401C22⋊C8—1662no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95C22⋊D8311yescharacteristic
411#95C22⋊D8311yescharacteristic
412C24C2×C22⋊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—6641no—
422C22×C4—6641no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1664no—
428C22×C4C2×C22⋊D8311yescharacteristic
429C22×C4C2×C22⋊D8311yescharacteristic
430C22×C4C2×C22⋊D8311yescharacteristic
431C22×C4C2×C22⋊D8311yescharacteristic
432C22×C4C2×C22⋊D8311yescharacteristic
433C22×C4Q8×D8312yes—
434C22×C4#1305059112yes—
435C22×C4C23×D8311yescharacteristic
436M16—3324no—
437C2×C8—832no—
438C2×C8—1661no—
439C2×C8—1661no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—832no—
463C4⋊C4—832no—
464C4⋊C4—6641no—
465C4⋊C4—6641no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1664no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4#1305058912yes—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—3322no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—6641no—
485C42—6641no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1458176812yes—
493C23C22×C22⋊D8311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—6644no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4#1458176812yes—
537C2×C4#1458182212yes—
538C2×C4C2×Q8×D8311yescharacteristic
539C2×C4#41×D8311yescharacteristic
540C2×C4C2×#1305058911yescharacteristic
541C2×C4C2×#1305059111yescharacteristic
542C2×C4#1458183711yescharacteristic
543C2×C4#1458183911yescharacteristic
544C8—3321no—
545C8—3321no—
546C8—3321no—
547C8—1662no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1458176811yescharacteristic
556C22#1912834511yescharacteristic
557C22#1912835011yescharacteristic
558C22C2×#1458182211yescharacteristic
559C22#1912849911yescharacteristic
560C22#1912850411yescharacteristic
561C22#1912850911yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—3322no—
572C4—3322no—
573C4—6641no—
574C4—6641no—
575C4#1912825311yescharacteristic
576C4#1912833011yescharacteristic
577C4#1912837811yescharacteristic
578C4#1912839311yescharacteristic
579C2—42no—
580C2#2491113511yescharacteristic
581C2#2491726411yescharacteristic
582C2#2491728811yescharacteristic
583C2#2491946911yescharacteristic
584C2#2491977911yescharacteristic
585C2#2491994811yescharacteristic
586C2#2492002311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2441C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2441D83
hypocenterC83#26018
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #24909025C2semidirect
3. #24908964C2semidirect
4. #24908954C2semidirect
5. #24908798C2semidirect
6. #24908421C2semidirect
7. #24908219C2semidirect
8. #24908045C2semidirect
9. #24908038C2semidirect
309. C83#26018semidirect

Maximal subgroups

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

subgroupindex
#249103382
#249090252
#249089642
#249089542
#249087982
#249084212
#249082192
#249080452
#249080382
#249076622
#249076482
#249075472
#249074942
#249071842
C83×#24412
#2601883