Finite Group Databasebeta

← #35466751

Subgroups — #35466751

Order 21248.

718 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic83C211yesmaximal, characteristic, complemented
2#24909006C211yesmaximal, characteristic, complemented
3#24908996C211yesmaximal, characteristic
4#24908871C211yesmaximal, characteristic, complemented
5#24908866C211yesmaximal, characteristic
6#24908404C211yesmaximal, characteristic, complemented
7#24908396C211yesmaximal, characteristic
8#24908280C211yesmaximal, characteristic
9#24908226C211yesmaximal, characteristic, complemented
10#24908101C211yesmaximal, characteristic, complemented
11#24908094C211yesmaximal, characteristic, complemented
12#24908087C211yesmaximal, characteristic
13#24908080C211yesmaximal, characteristic
14#24907361C211yesmaximal, characteristic, complemented
15C83×#2592C211yesmaximal, characteristic
16C2×Q8×Dic83—21no—
17C2×#14581725C2211yescharacteristic
18C2×#14581723C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581716C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31#19127600—21no—
32#19127600—21no—
33#19127600—21no—
34#19127571C2211yescharacteristic
35#19127571C2211yescharacteristic
36#19127571C2211yescharacteristic
37C4×C4⋊Dic83C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#124×C83C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50#397×C83—21no—
51#393×C83—21no—
52#391×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#390×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#388×C83—21no—
57#388×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63Q8×Dic83—44no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77#14581725—22no—
78#14581723—22no—
79#14581723—41no—
80#14581723—41no—
81C4⋊(C83⋊C8)—22no—
82#14581716—22no—
83#14581716—22no—
84#14581716—41no—
85#14581716—41no—
86#14581713—22no—
87#14581713—22no—
88#14581713—41no—
89#14581713—41no—
90#14581703—22no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C22×Q8×C83D411yescharacteristic
95C22×Q8×C83C2311yescharacteristic
96C22×D4×C83C2311yescharacteristic
97#124×C83—22no—
98#124×C83—22no—
99C4×Q8×C83—22no—
100C4×Q8×C83—22no—
101C2×C4⋊C4×C83D411yescharacteristic
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C4⋊C4×C83D411yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83—21no—
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116#95×C83—21no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
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#4112yes—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—42no—
145C4⋊Dic83—42no—
146C4⋊Dic83—42no—
147C4⋊Dic83—81no—
148C4⋊Dic83—81no—
149C4⋊Dic83—81no—
150C4⋊Dic83#4112yes—
151C4×Dic83—42no—
152C4×Dic83—42no—
153C4×Dic83—42no—
154C24×C83—21no—
155C2×Q8×C83—22no—
156C2×Q8×C83—22no—
157C2×Q8×C83—22no—
158C2×Q8×C83—21no—
159C2×Q8×C83—21no—
160C2×Q8×C83—21no—
161C2×Q8×C83—21no—
162C2×Q8×C83—21no—
163C2×Q8×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2411yescharacteristic
180C4⋊C4×C83—22no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—22no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C22⋊C4×C83—42no—
188C22⋊C4×C83—42no—
189C22⋊C4×C83—42no—
190C22⋊C4×C83—42no—
191C22⋊C4×C83—81no—
192C22⋊C4×C83—81no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—81no—
198C42×C83#4112yes—
199C42×C83#4112yes—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83C22×D411yescharacteristic
208Q8×C83—28no—
209Q8×C83—41no—
210Q8×C83—41no—
211Q8×C83—41no—
212Q8×C83—41no—
213D4×C83—41no—
214D4×C83—41no—
215D4×C83—41no—
216D4×C83—41no—
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—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83#12312yes—
241C2×C4×C83#12312yes—
242C2×C4×C83AGL(1,ℤ/8)12yes—
243C2×C4×C83#13712yes—
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83H(5,𝔽2)11yescharacteristic
249C2×C4×C832−1+411yescharacteristic
250C2×Dic83—22no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C83⋊C8—22no—
263C83⋊C8—41no—
264C83⋊C8—41no—
265C83⋊C8—41no—
266C22×C83—81no—
267C22×C83—81no—
268C22×C83—81no—
269C22×C83—81no—
270C22×C83—81no—
271C22×C83—81no—
272C22×C83—81no—
273C22×C83C2×#12311yescharacteristic
274C22×C83#54811yescharacteristic
275C22×C83#55011yescharacteristic
276C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C83C2×#13711yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57711yescharacteristic
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—22no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—81no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83#52911yescharacteristic
290C4×C83#54211yescharacteristic
291C4×C83#55611yescharacteristic
292C4×C83#55911yescharacteristic
293Dic83—42no—
294Dic83—81no—
295Dic83—81no—
296Dic83—81no—
297#26328—831nomaximal, Sylow
298C2×C83—161no—
299C2×C83#244811yescharacteristic
300C2×C83#309211yescharacteristic
301C2×C83#309911yescharacteristic
302C2×C83#336911yescharacteristic
303C2×C83#337311yescharacteristic
304C2×C83#339711yescharacteristic
305C2×C83#340011yescharacteristic
306#2592D8311yescharacteristic
307Q8×C4⋊C4—831no—
308#2096—831no—
309#2094—831no—
310#2072—831no—
311#2071—831no—
312#1985—831no—
313#1983—831no—
314#1959—831no—
315#1948—831no—
316#1928—831no—
317#1927—831no—
318#1926—831no—
319#1925—831no—
320#1776—831no—
321C83#2632811yescharacteristic, complemented, Sylow
322C2×#124C2×D8311yescharacteristic
323C2×C4×Q8—1661no—
324C2×C4×Q8C2×D8311yescharacteristic
325C2×#107—831no—
326C2×#106—831no—
327C2×#106—831no—
328C2×#106—831no—
329C2×C4⋊C8—831no—
330C2×#103—831no—
331C2×#103—831no—
332C2×#103—831no—
333C2×#103—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#97—831no—
339#397—21no—
340#393—21no—
341#391C2×D8311yescharacteristic
342#391C2×D8311yescharacteristic
343#390C2×D8311yescharacteristic
344#390C2×D8311yescharacteristic
345#389—1661no—
346#389—1661no—
347#389—1661no—
348#388—21no—
349#388—21no—
350#384—831no—
351#384—831no—
352#384—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C4×C4⋊C4C2×D8311yescharacteristic
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#336—831no—
362#336—831no—
363#336—831no—
364#336—831no—
365C22×Q8C22⋊D8311yescharacteristic
366C22×Q8C22×D8311yescharacteristic
367C22×D4C22×D8311yescharacteristic
368C22×C8—831no—
369C22×C8—831no—
370C22×C8—831no—
371C22×C8—831no—
372#124—22no—
373#124—22no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—3324no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4—1661no—
387C2×C4⋊C4C22⋊D8311yescharacteristic
388C2×C4⋊C4C22⋊D8311yescharacteristic
389C2×C4⋊C4C22⋊D8311yescharacteristic
390C2×C4⋊C4C22×D8311yescharacteristic
391C2×C4⋊C4C22×D8311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399C2×C42—1661no—
400C2×C42—1661no—
401C2×C42—1661no—
402C2×C42C22×D8311yescharacteristic
403C2×C42C22×D8311yescharacteristic
404C2×C42C22×D8311yescharacteristic
405#107—1662no—
406#106—3321no—
407#106—3321no—
408#106—1662no—
409C4⋊C8—1662no—
410#103—3321no—
411#103—3321no—
412#103—1662no—
413#103—1662no—
414#102—3321no—
415#102—3321no—
416#102—1662no—
417#102—1662no—
418#97—1662no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C23×D8311yescharacteristic
457C2×C8—832no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—3322no—
479C4⋊C4—832no—
480C4⋊C4—6641no—
481C4⋊C4—6641no—
482C4⋊C4—6641no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—3322no—
502C42—3322no—
503C42—3322no—
504C42#1305058912yes—
505C42#1305058912yes—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C22×C22⋊D8311yescharacteristic
514Q8—28no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
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—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—1662no—
558C2×C4#1458177312yes—
559C2×C4#1458177312yes—
560C2×C4#1458181712yes—
561C2×C4#1458182212yes—
562C2×C4#41×D8311yescharacteristic
563C2×C4#41×D8311yescharacteristic
564C2×C4C2×#1305058911yescharacteristic
565C2×C4C2×#1305058911yescharacteristic
566C2×C4#1458183811yescharacteristic
567C2×C4#1458183911yescharacteristic
568C8—3321no—
569C8—3321no—
570C8—3321no—
571C8—1662no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1458177311yescharacteristic
580C22#1912835811yescharacteristic
581C22#1912837011yescharacteristic
582C22C2×#1458181711yescharacteristic
583C22C2×#1458182211yescharacteristic
584C22#1912850911yescharacteristic
585C22#1912851211yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—3322no—
596C4—6641no—
597C4—6641no—
598C4—6641no—
599C4#1912823011yescharacteristic
600C4#1912832011yescharacteristic
601C4#1912842011yescharacteristic
602C4#1912843311yescharacteristic
603C2—161no—
604C2#2491125711yescharacteristic
605C2#2491738311yescharacteristic
606C2#2491744811yescharacteristic
607C2#2492019911yescharacteristic
608C2#2492025111yescharacteristic
609C2#2492048311yescharacteristic
610C2#2492051611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2592C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2592D83
hypocenterC83#26328
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic83C2semidirect
2. #24909006C2semidirect
4. #24908871C2semidirect
6. #24908404C2semidirect
9. #24908226C2semidirect
10. #24908101C2semidirect
11. #24908094C2semidirect
14. #24907361C2semidirect
321. C83#26328semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic832
#249090062
#249089962
#249088712
#249088662
#249084042
#249083962
#249082802
#249082262
#249081012
#249080942
#249080872
#249080802
#249073612
C83×#25922
#2632883