Finite Group Databasebeta

← #35462569

Subgroups — #35462569

Order 21248.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912484C211yesmaximal, characteristic, complemented
2#24908851C211yesmaximal, characteristic
3#24908837C211yesmaximal, characteristic
4#24908609C211yesmaximal, characteristic
5#24908599C211yesmaximal, characteristic
6#24908420C211yesmaximal, characteristic, complemented
7#24908415C211yesmaximal, characteristic, complemented
8#24908229C211yesmaximal, characteristic, complemented
9#24908182C211yesmaximal, characteristic, complemented
10#24908055C211yesmaximal, characteristic, complemented
11#24908048C211yesmaximal, characteristic, complemented
12#24907593C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14C4×#14581713C211yesmaximal, characteristic
15C83×#2357C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×C4×C83⋊C8C2211yescharacteristic
28#19127634—21no—
29#19127621—21no—
30#19127605—21no—
31#19127599C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127598—21no—
34#19127567C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
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×C4×D4×C83C2211yescharacteristic
49#390×C83C2211yescharacteristic
50#390×C83C2211yescharacteristic
51#388×C83—21no—
52#388×C83—21no—
53#382×C83C2211yescharacteristic
54#381×C83—21no—
55C4×C4⋊C4×C83C2211yescharacteristic
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C22×Dic166—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581792—22no—
65#14581792—41no—
66#14581784—22no—
67#14581784—41no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C4×Dic83—21no—
78#14581725—22no—
79#14581723—22no—
80C4⋊(C83⋊C8)—22no—
81C4⋊(C83⋊C8)—41no—
82C4⋊(C83⋊C8)—41no—
83#14581713—22no—
84#14581713—22no—
85#14581713—41no—
86#14581713—41no—
87C4×C83⋊C8—22no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C22×D4×C83C2311yescharacteristic
94C23×C4×C83—21no—
95C4×D4×C83—22no—
96C4×D4×C83Q814yescomplemented
97C2×C4⋊C4×C83D411yescharacteristic
98C2×C4⋊C4×C83D411yescharacteristic
99C2×C4⋊C4×C83C2311yescharacteristic
100C2×C4⋊C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C22⋊C4×C83—21no—
103C2×C22⋊C4×C83—21no—
104C2×C22⋊C4×C83—21no—
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C42×C83—21no—
108C2×C42×C83—21no—
109C2×C42×C83C2311yescharacteristic
110C2×C42×C83C2311yescharacteristic
111C2×C42×C83C2311yescharacteristic
112#95×C83—21no—
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83D411yescharacteristic
117#95×C83D411yescharacteristic
118C2×Dic166—42no—
119C2×Dic166—42no—
120C2×Dic166—81no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C2×C83⋊C8—21no—
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—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—42no—
144C4⋊Dic83#4112yes—
145C4⋊Dic83#4112yes—
146Dic83⋊C4—42no—
147Dic83⋊C4—42no—
148Dic83⋊C4—42no—
149Dic83⋊C4—81no—
150C4×Dic83—81no—
151C24×C83—21no—
152C2×D4×C83—21no—
153C2×D4×C83—21no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83—21no—
157C2×D4×C83C2×Q812yes—
158C22×C4×C83—21no—
159C22×C4×C83—24no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—81no—
163C22×C4×C83—81no—
164C22×C4×C83C2×D411yescharacteristic
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2411yescharacteristic
171C4⋊C4×C83—22no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—41no—
178C4⋊C4×C83C2×Q812yes—
179C22⋊C4×C83—24no—
180C22⋊C4×C83—42no—
181C22⋊C4×C83—42no—
182C22⋊C4×C83—81no—
183C22⋊C4×C83—81no—
184C42×C83—21no—
185C42×C83—21no—
186C42×C83—21no—
187C42×C83—21no—
188C42×C83—81no—
189C42×C83—81no—
190C42×C83C2×Q812yes—
191C42×C83#4112yes—
192C23×C83—22no—
193C23×C83—41no—
194C23×C83—41no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83C22×D411yescharacteristic
200D4×C83—22no—
201D4×C83—22no—
202D4×C83—41no—
203D4×C83—41no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
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—21no—
219C2×C4×C83—21no—
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—84no—
229C2×C4×C83C22⋊Q812yes—
230C2×C4×C83C22⋊Q812yes—
231C2×C4×C83#13512yes—
232C2×C4×C83C22×Q811yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83H(5,𝔽2)11yescharacteristic
237C2×C4×C832−1+411yescharacteristic
238C2×Dic83—22no—
239C2×Dic83—22no—
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—41no—
250Dic166—84no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C22×C83—42no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—81no—
259C22×C83—81no—
260C22×C83—81no—
261C22×C83—81no—
262C22×C83C2×C22⋊Q811yescharacteristic
263C22×C83#54611yescharacteristic
264C22×C83#54811yescharacteristic
265C22×C83C2×#13511yescharacteristic
266C22×C83#57511yescharacteristic
267C22×C83#57611yescharacteristic
268C22×C83#57711yescharacteristic
269C4×C83—22no—
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—42no—
273C4×C83—41no—
274C4×C83—41no—
275C4×C83—41no—
276C4×C83—81no—
277C4×C83—81no—
278C4×C83#53311yescharacteristic
279C4×C83#54411yescharacteristic
280C4×C83#55511yescharacteristic
281C4×C83#55611yescharacteristic
282Dic83—42no—
283Dic83—42no—
284Dic83—81no—
285Dic83—81no—
286#26028—831nomaximal, Sylow
287C2×C83—82no—
288C2×C83#243911yescharacteristic
289C2×C83#307611yescharacteristic
290C2×C83#308111yescharacteristic
291C2×C83#329711yescharacteristic
292C2×C83#332611yescharacteristic
293C2×C83#336511yescharacteristic
294C2×C83#337011yescharacteristic
295#2550—831no—
296#2357D8311yescharacteristic
297#2067—831no—
298#2065—831no—
299#2025—831no—
300#2023—831no—
301#1989—831no—
302#1988—831no—
303#1949—831no—
304#1939—831no—
305#1921—831no—
306#1920—831no—
307#1835—831no—
308#1804—831no—
309C4×#102—831no—
310C83#2602811yescharacteristic, complemented, Sylow
311C2×C4⋊Q8—831no—
312C2×#125—831no—
313C2×C4×D4C2×D8311yescharacteristic
314C2×#107—831no—
315C2×#106—831no—
316C2×C4⋊C8—831no—
317C2×C4⋊C8—831no—
318C2×C4⋊C8—831no—
319C2×#102—831no—
320C2×#102—831no—
321C2×#102—831no—
322C2×#102—831no—
323C2×C4×C8—831no—
324#398—1661no—
325#395—1661no—
326#391—1661no—
327#390C2×D8311yescharacteristic
328#390C2×D8311yescharacteristic
329#389—831no—
330#389—831no—
331#389—1661no—
332#388—21no—
333#388—21no—
334#382—831no—
335#382C2×D8311yescharacteristic
336#381—21no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4C2×D8311yescharacteristic
339C4×C4⋊C4C2×D8311yescharacteristic
340C4×C4⋊C4C2×D8311yescharacteristic
341C4×C22⋊C4—21no—
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#340—831no—
352#336—831no—
353#336—831no—
354C22×Q8—1661no—
355C22×D4C22×D8311yescharacteristic
356C23×C4—21no—
357C22×C8—831no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8—831no—
361C4⋊Q8—3321no—
362C4⋊Q8—1662no—
363#125—3321no—
364#125—1662no—
365C4×D4—22no—
366C4×D4Dic16614yescomplemented
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⋊C4C22⋊D8311yescharacteristic
377C2×C4⋊C4C22⋊D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1661no—
389C2×C42C22×D8311yescharacteristic
390C2×C42C22×D8311yescharacteristic
391C2×C42C22×D8311yescharacteristic
392#107—1662no—
393#106—1662no—
394C4⋊C8—3321no—
395C4⋊C8—3321no—
396C4⋊C8—1662no—
397#102—3321no—
398#102—3321no—
399#102—1662no—
400#102—1662no—
401C4×C8—1662no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95C22⋊D8311yescharacteristic
412#95C22⋊D8311yescharacteristic
413C24—21no—
414C2×Q8—3322no—
415C2×Q8—3322no—
416C2×Q8—6641no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic16612yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4C2×C22⋊D8311yescharacteristic
434C22×C4C2×C22⋊D8311yescharacteristic
435C22×C4C2×C22⋊D8311yescharacteristic
436C22×C4C2×C22⋊D8311yescharacteristic
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C23×D8311yescharacteristic
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—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—3322no—
462C4⋊C4—3322no—
463C4⋊C4—832no—
464C4⋊C4—832no—
465C4⋊C4—6641no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4C2×Dic16612yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—6641no—
485C42C2×Dic16612yes—
486C42#1305059012yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D8311yescharacteristic
495Q8—6644no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
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—3321no—
534C2×C4—3321no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4#1458177112yes—
538C2×C4#1458177112yes—
539C2×C4#1458181212yes—
540C2×C4C22×Dic16611yescharacteristic
541C2×C4#41×D8311yescharacteristic
542C2×C4#41×D8311yescharacteristic
543C2×C4C2×#1305059011yescharacteristic
544C2×C4#1458183811yescharacteristic
545C2×C4#1458183911yescharacteristic
546C8—3321no—
547C8—3321no—
548C8—3321no—
549C8—3321no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1458177111yescharacteristic
558C22#1912833911yescharacteristic
559C22#1912835811yescharacteristic
560C22C2×#1458181211yescharacteristic
561C22#1912849911yescharacteristic
562C22#1912850411yescharacteristic
563C22#1912851211yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—3322no—
574C4—3322no—
575C4—6641no—
576C4—6641no—
577C4#1912825411yescharacteristic
578C4#1912832911yescharacteristic
579C4#1912841411yescharacteristic
580C4#1912841911yescharacteristic
581C2—82no—
582C2#2491114811yescharacteristic
583C2#2491725111yescharacteristic
584C2#2491729411yescharacteristic
585C2#2491930711yescharacteristic
586C2#2491971011yescharacteristic
587C2#2492015511yescharacteristic
588C2#2492021011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2357C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2357D83
hypocenterC83#26028
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #24912484C2semidirect
6. #24908420C2semidirect
7. #24908415C2semidirect
8. #24908229C2semidirect
9. #24908182C2semidirect
10. #24908055C2semidirect
11. #24908048C2semidirect
12. #24907593C2semidirect
96. C4×D4×C83Q8semidirect
310. C83#26028semidirect
366. C4×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
#249124842
#249088512
#249088372
#249086092
#249085992
#249084202
#249084152
#249082292
#249081822
#249080552
#249080482
#249075932
#249074732
C4×#145817132
C83×#23572
#2602883