Finite Group Databasebeta

← #35456197

Subgroups — #35456197

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×#19128111C211yesmaximal, characteristic, complemented
2C2×#19127998C211yesmaximal, characteristic
3#24912632C211yesmaximal, characteristic
4#24908944C211yesmaximal, characteristic, complemented
5#24908944C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908834C211yesmaximal, characteristic
8#24908415C211yesmaximal, characteristic, complemented
9#24908231C211yesmaximal, characteristic, complemented
10#24908231C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908090C211yesmaximal, characteristic
13#24908083C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2370C211yesmaximal, characteristic, complemented
16C2×#14581792C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19128111—24no—
19#19127998—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581712C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34#19127614—21no—
35#19127614—21no—
36#19127603C2211yescharacteristic
37#19127603C2211yescharacteristic
38#19127567C2211yescharacteristic
39#19127567C2211yescharacteristic
40#19127558—21no—
41#19127558—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#125×C83C2211yescharacteristic
51#389×C83—21no—
52#389×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#387×C83—21no—
56C4×C4⋊C4×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×D332C2311yescharacteristic
63C22×C4×D83—21no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581792—22no—
69#14581792—22no—
70C4×D332—22no—
71C4×D332—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81#14581725—22no—
82#14581725—22no—
83#14581725—22no—
84#14581723—22no—
85#14581723—22no—
86#14581723—22no—
87C4⋊(C83⋊C8)—22no—
88C4⋊(C83⋊C8)—22no—
89C4⋊(C83⋊C8)—22no—
90#14581712—22no—
91#14581712—22no—
92#14581712—22no—
93#14581712Q814yescomplemented
94#14581703—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99#125×C83—22no—
100#125×C83D414yes—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
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×C4⋊C4×C83C2311yescharacteristic
111C2×C42×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C23×D83—21no—
125C2×Dic166—42no—
126C2×Dic166—81no—
127C2×Dic166—81no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332—21no—
132C2×D332—21no—
133C2×D332C2×Q812yes—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C2×C4×D83—42no—
138C2×C4×D83—42no—
139C2×C4×D83—42no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8C2×Q812yes—
151C2×C83⋊C8#4112yes—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—42no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—81no—
161(C2×D83)⋊C4—81no—
162(C2×D83)⋊C4—81no—
163(C2×D83)⋊C4—81no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2411yescharacteristic
175C4⋊C4×C83—24no—
176C4⋊C4×C83—24no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—41no—
183C4⋊C4×C83—41no—
184C4⋊C4×C83—42no—
185C4⋊C4×C83—81no—
186C4⋊C4×C83C2×D412yes—
187C4⋊C4×C83C2×D412yes—
188C4⋊C4×C83C2×Q812yes—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—42no—
194C42×C83—81no—
195C42×C83—81no—
196C42×C83C2×D412yes—
197C42×C83#4112yes—
198C23×C83C22×D411yescharacteristic
199C22×D83—22no—
200C22×D83—41no—
201C22×D83—41no—
202C22×D83—41no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—22no—
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—41no—
226C2×C4×C83—41no—
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#12112yes—
235C2×C4×C83#12112yes—
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C832−1+411yescharacteristic
241C2×C4×C832−1+411yescharacteristic
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251D332—22no—
252D332—22no—
253D332—41no—
254D332—41no—
255C4×D83—84no—
256Dic166—84no—
257C83⋊C8—22no—
258C83⋊C8—22no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C22×C83C2×#12111yescharacteristic
262C22×C83#54711yescharacteristic
263C22×C83D4×Q811yescharacteristic
264C22×C83#57511yescharacteristic
265C22×C83#57511yescharacteristic
266C22×C83#57611yescharacteristic
267C22×C83#57711yescharacteristic
268C2×D83—42no—
269C2×D83—42no—
270C2×D83—42no—
271C2×D83—81no—
272C2×D83—81no—
273C2×D83—81no—
274C2×D83—81no—
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83—42no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83—81no—
285C4×C83#53311yescharacteristic
286C4×C83#54111yescharacteristic
287C4×C83#54811yescharacteristic
288C4×C83D4×Q811yescharacteristic
289Dic83—81no—
290Dic83—81no—
291Dic83—81no—
292#25466—831nomaximal, Sylow
293C2×C83#242911yescharacteristic
294C2×C83#304011yescharacteristic
295C2×C83#304511yescharacteristic
296C2×C83#334711yescharacteristic
297C2×C83#334811yescharacteristic
298C2×C83#338811yescharacteristic
299C2×C83#338911yescharacteristic
300D83—82no—
301C2×#501—831no—
302C2×#478—831no—
303#2563—831no—
304#2370D8311yescharacteristic, complemented
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⋊Q8—831no—
318C2×#125C2×D8311yescharacteristic
319C2×C4×D4—831no—
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—1661no—
337#393—1661no—
338#390—831no—
339#390—831no—
340#389—21no—
341#389C2×D8311yescharacteristic
342#389C2×D8311yescharacteristic
343#387—21no—
344#387—21no—
345#382—831no—
346#382—831no—
347#380—1661no—
348#380—1661no—
349C4×C4⋊C4—21no—
350C4×C4⋊C4C2×D8311yescharacteristic
351C4×C4⋊C4C2×D8311yescharacteristic
352C4×C4⋊C4C2×D8311yescharacteristic
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—1661no—
363C22×D4—831no—
364C23×C4—1661no—
365C22×C8—831no—
366C22×C8—831no—
367C22×C8—831no—
368C22×C8—831no—
369C4⋊Q8—1662no—
370C4⋊Q8—1662no—
371#125—22no—
372#125D33214yes—
373C4×D4—1662no—
374C4×D4—1662no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4C22⋊D8311yescharacteristic
384C2×C4⋊C4C22⋊D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C22⋊C4—1661no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C22⋊C4—1661no—
393C2×C22⋊C4—1661no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
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—21no—
419#95—1661no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95C22⋊D8311yescharacteristic
424#95C22⋊D8311yescharacteristic
425C24—1661no—
426C2×Q8—3322no—
427C2×Q8—6641no—
428C2×Q8—6641no—
429C2×D4—832no—
430C2×D4—1661no—
431C2×D4—1661no—
432C2×D4—1661no—
433C2×D4—1661no—
434C2×D4—1661no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—3322no—
440C22×C4—3322no—
441C22×C4—3322no—
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—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—3322no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4C2×D33212yes—
483C4⋊C4C2×D33212yes—
484C4⋊C4Q8×D8312yes—
485C22⋊C4—3322no—
486C22⋊C4—6641no—
487C22⋊C4—6641no—
488C22⋊C4—6641no—
489C22⋊C4—6641no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—81no—
496C42—81no—
497C42C2×D33212yes—
498C42#1305059012yes—
499C23—3321no—
500C23—3321no—
501C23—3321no—
502C23—3321no—
503C23—3321no—
504C23—3321no—
505C23—1662no—
506C23C22×C22⋊D8311yescharacteristic
507Q8—6644no—
508D4—3321no—
509D4—3321no—
510D4—1662no—
511D4—1662no—
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—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—6644no—
550C2×C4#1458176612yes—
551C2×C4#1458176612yes—
552C2×C4C22×D33211yescharacteristic
553C2×C4C2×Q8×D8311yescharacteristic
554C2×C4#41×D8311yescharacteristic
555C2×C4C2×#1305059011yescharacteristic
556C2×C4#1458183911yescharacteristic
557C2×C4#1458184011yescharacteristic
558C8—3321no—
559C8—3321no—
560C8—1662no—
561C8—1662no—
562C22—3322no—
563C22—3322no—
564C22—3322no—
565C22—6641no—
566C22—6641no—
567C22—6641no—
568C22—6641no—
569C22C2×#1458176611yescharacteristic
570C22#1912835011yescharacteristic
571C22Q8×C22⋊D8311yescharacteristic
572C22#1912849911yescharacteristic
573C22#1912849911yescharacteristic
574C22#1912850411yescharacteristic
575C22#1912850911yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—6641no—
587C4—6641no—
588C4—6641no—
589C4#1912825511yescharacteristic
590C4#1912830911yescharacteristic
591C4#1912836311yescharacteristic
592C4Q8×D33211yescharacteristic
593C2—6642no—
594C2#2491105611yescharacteristic
595C2#2491695711yescharacteristic
596C2#2491700211yescharacteristic
597C2#2491995711yescharacteristic
598C2#2491997211yescharacteristic
599C2#2492040211yescharacteristic
600C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2370C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2370D83
hypocenterC83#25466
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C2×#19128111C2semidirect
4. #24908944C2semidirect
5. #24908944C2semidirect
8. #24908415C2semidirect
9. #24908231C2semidirect
10. #24908231C2semidirect
11. #24908187C2semidirect
15. C83×#2370C2semidirect
93. #14581712Q8semidirect
304. #2370D83semidirect
316. C83#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#191281112
C2×#191279982
#249126322
#249089442
#249088482
#249088342
#249084152
#249082312
#249081872
#249080902
#249080832
#249072912
C83×#23702
#2546683