Finite Group Databasebeta

← #35456199

Subgroups — #35456199

Order 21248.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128107C211yesmaximal, characteristic, complemented
2C2×#19128001C211yesmaximal, characteristic
3#24910557C211yesmaximal, characteristic, complemented
4#24908945C211yesmaximal, characteristic, complemented
5#24908945C211yesmaximal, characteristic, complemented
6#24908851C211yesmaximal, characteristic
7#24908837C211yesmaximal, characteristic
8#24908416C211yesmaximal, characteristic, complemented
9#24908229C211yesmaximal, characteristic, complemented
10#24908229C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908087C211yesmaximal, characteristic
13#24908080C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2563C211yesmaximal, characteristic
16C2×#14581784C2211yescharacteristic
17#19128107—24no—
18#19128001—24no—
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581703C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127599C2211yescharacteristic
35#19127596—21no—
36#19127583—21no—
37#19127582—21no—
38C4×C4⋊Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41#19127552—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52#393×C83—21no—
53#393×C83—21no—
54#390×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#382×C83C2211yescharacteristic
57#382×C83C2211yescharacteristic
58#380×C83—21no—
59#380×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581784—22no—
66#14581784D414yes—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78#14581725—22no—
79#14581725—22no—
80#14581725—22no—
81#14581723—22no—
82#14581723—22no—
83#14581723—22no—
84C4⋊(C83⋊C8)—22no—
85C4⋊(C83⋊C8)—22no—
86C4⋊(C83⋊C8)—22no—
87#14581713—22no—
88#14581713—22no—
89#14581713—22no—
90#14581713Q814yescomplemented
91#14581703—22no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83—21no—
98C22×D4×C83C2311yescharacteristic
99C23×C4×C83—21no—
100C4⋊Q8×C83—22no—
101C4⋊Q8×C83—22no—
102C4×D4×C83—22no—
103C4×D4×C83—22no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83D411yescharacteristic
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—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—21no—
138C2×C83⋊C8C2×Q812yes—
139C2×C83⋊C8#4112yes—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83C2×D412yes—
146C4⋊Dic83C2×D412yes—
147C4⋊Dic83C2×Q812yes—
148Dic83⋊C4—24no—
149Dic83⋊C4—24no—
150Dic83⋊C4—42no—
151Dic83⋊C4—81no—
152C4×Dic83—42no—
153C4×Dic83—81no—
154C4×Dic83—81no—
155C24×C83—21no—
156C2×Q8×C83—42no—
157C2×Q8×C83—81no—
158C2×Q8×C83—81no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83C2×Q812yes—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—42no—
169C22×C4×C83—42no—
170C22×C4×C83—42no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2411yescharacteristic
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—41no—
185C4⋊C4×C83—41no—
186C4⋊C4×C83—42no—
187C22⋊C4×C83—42no—
188C22⋊C4×C83—81no—
189C22⋊C4×C83—81no—
190C22⋊C4×C83—81no—
191C22⋊C4×C83—81no—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83C2×D412yes—
197C42×C83#4112yes—
198C23×C83—22no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83C22×D411yescharacteristic
206Q8×C83—84no—
207D4×C83—22no—
208D4×C83—22no—
209D4×C83—41no—
210D4×C83—41no—
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—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—84no—
237C2×C4×C83#12112yes—
238C2×C4×C83#12112yes—
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C22×Q811yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C832−1+411yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C83⋊C8—22no—
258C83⋊C8—22no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—81no—
265C22×C83—81no—
266C22×C83—81no—
267C22×C83—81no—
268C22×C83C2×#12111yescharacteristic
269C22×C83#54711yescharacteristic
270C22×C83D4×Q811yescharacteristic
271C22×C83#57511yescharacteristic
272C22×C83#57511yescharacteristic
273C22×C83#57611yescharacteristic
274C22×C83#57711yescharacteristic
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83#53311yescharacteristic
285C4×C83#54111yescharacteristic
286C4×C83#54811yescharacteristic
287C4×C83D4×Q811yescharacteristic
288Dic83—42no—
289Dic83—42no—
290Dic83—42no—
291Dic83—81no—
292#25466—831nomaximal, Sylow
293C2×C83—82no—
294C2×C83#242911yescharacteristic
295C2×C83#304011yescharacteristic
296C2×C83#304511yescharacteristic
297C2×C83#334711yescharacteristic
298C2×C83#334811yescharacteristic
299C2×C83#338811yescharacteristic
300C2×C83#338911yescharacteristic
301C2×#501—831no—
302C2×#478—831no—
303#2563D8311yescharacteristic
304#2370—831no—
305#2085—831no—
306#2085—831no—
307#2067—831no—
308#2065—831no—
309#1988—831no—
310#1949—831no—
311#1949—831no—
312#1940—831no—
313#1926—831no—
314#1925—831no—
315#1762—831no—
316C83#2546611yescharacteristic, complemented, Sylow
317C2×C4⋊Q8C2×D8311yescharacteristic
318C2×#125—831no—
319C2×C4×D4C2×D8311yescharacteristic
320#501—1664no—
321#478—1664no—
322C2×#107—831no—
323C2×#107—831no—
324C2×#107—831no—
325C2×#106—831no—
326C2×#106—831no—
327C2×#106—831no—
328C2×C4⋊C8—831no—
329C2×C4⋊C8—831no—
330C2×C4⋊C8—831no—
331C2×#102—831no—
332C2×#102—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#97—831no—
336#393—21no—
337#393—21no—
338#390C2×D8311yescharacteristic
339#390C2×D8311yescharacteristic
340#389—831no—
341#389—831no—
342#389—1661no—
343#387—1661no—
344#387—1661no—
345#382C2×D8311yescharacteristic
346#382C2×D8311yescharacteristic
347#380—21no—
348#380—21no—
349C4×C4⋊C4—831no—
350C4×C4⋊C4—831no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—1661no—
353C4×C4⋊C4C2×D8311yescharacteristic
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#336—831no—
361#336—831no—
362C22×Q8—21no—
363C22×D4C22×D8311yescharacteristic
364C23×C4—21no—
365C22×C8—831no—
366C22×C8—831no—
367C22×C8—831no—
368C22×C8—831no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—834no—
372#125—1662no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4C22⋊D8311yescharacteristic
385C2×C4⋊C4C22⋊D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—1661no—
395C2×C42—1661no—
396C2×C42—1661no—
397C2×C42C22×D8311yescharacteristic
398C2×C42C22×D8311yescharacteristic
399C2×C42C22×D8311yescharacteristic
400#107—1662no—
401#107—1662no—
402#107—1662no—
403#106—1662no—
404#106—1662no—
405#106—1662no—
406C4⋊C8—1662no—
407C4⋊C8—1662no—
408C4⋊C8—1662no—
409#102—834no—
410#102—1662no—
411#102—1662no—
412#102—1662no—
413#97—1662no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1661no—
419#95—1661no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95C22⋊D8311yescharacteristic
424#95C22⋊D8311yescharacteristic
425C24—21no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D8312yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1661no—
442C22×C4—1661no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4C2×C22⋊D8311yescharacteristic
446C22×C4C2×C22⋊D8311yescharacteristic
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C23×D8311yescharacteristic
452C2×C8—832no—
453C2×C8—832no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—3322no—
474C4⋊C4—832no—
475C4⋊C4—832no—
476C4⋊C4—832no—
477C4⋊C4—6641no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1664no—
481C4⋊C4—1664no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—3322no—
495C42—6641no—
496C42—6641no—
497C42D4×D8312yes—
498C42#1305058912yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D8311yescharacteristic
507Q8—84no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—1662no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4#1458176312yes—
551C2×C4#1458176312yes—
552C2×C4C2×D4×D8311yescharacteristic
553C2×C4C2×Q8×D8311yescharacteristic
554C2×C4#41×D8311yescharacteristic
555C2×C4C2×#1305058911yescharacteristic
556C2×C4#1458183911yescharacteristic
557C2×C4#1458184011yescharacteristic
558C8—3321no—
559C8—3321no—
560C8—1662no—
561C8—1662no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1458176311yescharacteristic
570C22#1912835011yescharacteristic
571C22Q8×C22⋊D8311yescharacteristic
572C22#1912849911yescharacteristic
573C22#1912849911yescharacteristic
574C22#1912850411yescharacteristic
575C22#1912851211yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—3322no—
586C4—3322no—
587C4—3322no—
588C4—6641no—
589C4#1912825311yescharacteristic
590C4#1912831111yescharacteristic
591C4#1912836111yescharacteristic
592C4#1912836411yescharacteristic
593C2—82no—
594C2#2491105811yescharacteristic
595C2#2491695311yescharacteristic
596C2#2491699811yescharacteristic
597C2#2491996311yescharacteristic
598C2#2491997811yescharacteristic
599C2#2492040811yescharacteristic
600C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2563C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2563D83
hypocenterC83#25466
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#19128107C2semidirect
3. #24910557C2semidirect
4. #24908945C2semidirect
5. #24908945C2semidirect
8. #24908416C2semidirect
9. #24908229C2semidirect
10. #24908229C2semidirect
11. #24908187C2semidirect
90. #14581713Q8semidirect
316. C83#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#191281072
C2×#191280012
#249105572
#249089452
#249088512
#249088372
#249084162
#249082292
#249081872
#249080872
#249080802
#249072882
C83×#25632
#2546683