Finite Group Databasebeta

← #35456124

Subgroups — #35456124

Order 21248.

792 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128111C211yesmaximal, characteristic, complemented
2C2×#19127991C211yesmaximal, characteristic
3#24910521C211yesmaximal, characteristic, complemented
4#24908945C211yesmaximal, characteristic, complemented
5#24908940C211yesmaximal, characteristic, complemented
6#24908851C211yesmaximal, characteristic
7#24908837C211yesmaximal, characteristic
8#24908421C211yesmaximal, characteristic, complemented
9#24908229C211yesmaximal, characteristic, complemented
10#24908224C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908087C211yesmaximal, characteristic
13#24908080C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2566C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581757—21no—
18#19128111—24no—
19#19127991—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34#19127606—21no—
35#19127599C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127586—21no—
38#19127586—21no—
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#125×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52#396×C83—21no—
53#396×C83—21no—
54#390×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#385×C83—21no—
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#382×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581792—22no—
67#14581792D414yes—
68#14581757—44no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79#14581725—22no—
80#14581725—22no—
81#14581723—22no—
82#14581723—22no—
83#14581723—22no—
84#14581723—22no—
85C4⋊(C83⋊C8)—22no—
86C4⋊(C83⋊C8)—22no—
87C4⋊(C83⋊C8)—22no—
88#14581713—22no—
89#14581713—22no—
90#14581713—22no—
91#14581713Q814yescomplemented
92#14581703—22no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C22×D4×C83C2311yescharacteristic
99C23×C4×C83—21no—
100#125×C83—22no—
101#125×C83—22no—
102C4×D4×C83—22no—
103C4×D4×C83—22no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83D411yescharacteristic
109C2×C4⋊C4×C83D411yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120C2×C42×C83C2311yescharacteristic
121#95×C83—21no—
122#95×C83—21no—
123#95×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C2×Dic166—24no—
126C2×Dic166—42no—
127C2×Dic166—42no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—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⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8C2×Q812yes—
143C2×C83⋊C8#4112yes—
144C4⋊Dic83—24no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83C2×D412yes—
151C4⋊Dic83C2×D412yes—
152C4⋊Dic83C2×Q812yes—
153Dic83⋊C4—42no—
154Dic83⋊C4—42no—
155C4×Dic83—42no—
156C4×Dic83—42no—
157C4×Dic83—42no—
158C24×C83—21no—
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—81no—
170C22×C4×C83—81no—
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—22no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—41no—
186C4⋊C4×C83—42no—
187C4⋊C4×C83—42no—
188C4⋊C4×C83—81no—
189C4⋊C4×C83—81no—
190C22⋊C4×C83—42no—
191C22⋊C4×C83—42no—
192C22⋊C4×C83—42no—
193C22⋊C4×C83—81no—
194C22⋊C4×C83—81no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83C2×D412yes—
200C42×C83#4112yes—
201C23×C83—22no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83C22×D411yescharacteristic
209D4×C83—22no—
210D4×C83—22no—
211D4×C83—41no—
212D4×C83—41no—
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—21no—
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—41no—
237C2×C4×C83—84no—
238C2×C4×C83#12112yes—
239C2×C4×C83#12112yes—
240C2×C4×C83AGL(1,ℤ/8)12yes—
241C2×C4×C83C22×D411yescharacteristic
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×C4×C832−1+411yescharacteristic
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259Dic166—48no—
260C83⋊C8—22no—
261C83⋊C8—22no—
262C83⋊C8—41no—
263C83⋊C8—41no—
264C22×C83—42no—
265C22×C83—42no—
266C22×C83—42no—
267C22×C83—81no—
268C22×C83—81no—
269C22×C83—81no—
270C22×C83—81no—
271C22×C83C2×#12111yescharacteristic
272C22×C83#54711yescharacteristic
273C22×C83D4×Q811yescharacteristic
274C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57711yescharacteristic
277C22×C83#57711yescharacteristic
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—22no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—41no—
284C4×C83—81no—
285C4×C83—81no—
286C4×C83—81no—
287C4×C83#53311yescharacteristic
288C4×C83#54111yescharacteristic
289C4×C83#54811yescharacteristic
290C4×C83D4×Q811yescharacteristic
291Dic83—42no—
292Dic83—42no—
293Dic83—42no—
294Dic83—81no—
295#25461—831nomaximal, Sylow
296C2×C83—82no—
297C2×C83#242911yescharacteristic
298C2×C83#303611yescharacteristic
299C2×C83#304311yescharacteristic
300C2×C83#334511yescharacteristic
301C2×C83#335011yescharacteristic
302C2×C83#338411yescharacteristic
303C2×C83#338811yescharacteristic
304C2×#501—831no—
305C2×#476—831no—
306#2566D8311yescharacteristic
307#2365—831no—
308#2085—831no—
309#2084—831no—
310#2067—831no—
311#2065—831no—
312#1989—831no—
313#1949—831no—
314#1948—831no—
315#1940—831no—
316#1926—831no—
317#1925—831no—
318#1762—831no—
319C83#2546111yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—831no—
321C2×#125C2×D8311yescharacteristic
322C2×C4×Q8—1661no—
323C2×C4×D4C2×D8311yescharacteristic
324#501—1664no—
325#476—1664no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#106—831no—
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×C4⋊C8—831no—
333C2×C4⋊C8—831no—
334C2×C4⋊C8—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#97—831no—
340#396—21no—
341#396—21no—
342#391—1661no—
343#390C2×D8311yescharacteristic
344#390C2×D8311yescharacteristic
345#389—831no—
346#389—831no—
347#387—1661no—
348#387—1661no—
349#385—21no—
350#385—21no—
351#384C2×D8311yescharacteristic
352#382C2×D8311yescharacteristic
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#340—831no—
362#340—831no—
363#336—831no—
364#336—831no—
365C22×Q8—1661no—
366C22×D4C22×D8311yescharacteristic
367C23×C4—21no—
368C22×C8—831no—
369C22×C8—831no—
370C22×C8—831no—
371C22×C8—831no—
372C4⋊Q8—834no—
373C4⋊Q8—1662no—
374#125—22no—
375#125—22no—
376C4×Q8—3324no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4C22⋊D8311yescharacteristic
391C2×C4⋊C4C22⋊D8311yescharacteristic
392C2×C4⋊C4C22×D8311yescharacteristic
393C2×C4⋊C4C22×D8311yescharacteristic
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1661no—
401C2×C42—1661no—
402C2×C42—1661no—
403C2×C42C22×D8311yescharacteristic
404C2×C42C22×D8311yescharacteristic
405C2×C42C22×D8311yescharacteristic
406#107—1662no—
407#107—1662no—
408#106—1662no—
409#106—1662no—
410#106—1662no—
411#106—1662no—
412C4⋊C8—1662no—
413C4⋊C8—1662no—
414C4⋊C8—1662no—
415#102—834no—
416#102—1662no—
417#102—1662no—
418#102—1662no—
419#97—1662no—
420#95—21no—
421#95—21no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95C22⋊D8311yescharacteristic
428#95C22⋊D8311yescharacteristic
429C24—21no—
430C2×Q8—3322no—
431C2×Q8—3322no—
432C2×Q8—1664no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4Q8×D8312yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C23×D8311yescharacteristic
456C2×C8—832no—
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—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—832no—
483C4⋊C4—832no—
484C4⋊C4—832no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1664no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—3322no—
501C42—3322no—
502C42—3322no—
503C42D4×D8312yes—
504C42#1305058912yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C22⋊D8311yescharacteristic
513Q8—3328no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
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—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—1662no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4#1458176312yes—
556C2×C4#1458176312yes—
557C2×C4#1458181712yes—
558C2×C4C2×D4×D8311yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4#41×D8311yescharacteristic
561C2×C4C2×#1305058911yescharacteristic
562C2×C4#1458183911yescharacteristic
563C2×C4#1458184011yescharacteristic
564C8—3321no—
565C8—3321no—
566C8—1662no—
567C8—1662no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C2×#1458176311yescharacteristic
576C22#1912835011yescharacteristic
577C22Q8×C22⋊D8311yescharacteristic
578C22C2×#1458181711yescharacteristic
579C22#1912849911yescharacteristic
580C22#1912851211yescharacteristic
581C22#1912851211yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—3322no—
592C4—3322no—
593C4—3322no—
594C4—6641no—
595C4#1912825311yescharacteristic
596C4#1912831111yescharacteristic
597C4#1912836111yescharacteristic
598C4#1912836411yescharacteristic
599C2—82no—
600C2#2491105811yescharacteristic
601C2#2491692311yescharacteristic
602C2#2491698011yescharacteristic
603C2#2491992711yescharacteristic
604C2#2492000211yescharacteristic
605C2#2492036211yescharacteristic
606C2#2492040811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2566C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2566D83
hypocenterC83#25461
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#19128111C2semidirect
3. #24910521C2semidirect
4. #24908945C2semidirect
5. #24908940C2semidirect
8. #24908421C2semidirect
9. #24908229C2semidirect
10. #24908224C2semidirect
11. #24908187C2semidirect
91. #14581713Q8semidirect
319. C83#25461semidirect

Maximal subgroups

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

subgroupindex
C2×#191281112
C2×#191279912
#249105212
#249089452
#249089402
#249088512
#249088372
#249084212
#249082292
#249082242
#249081872
#249080872
#249080802
#249072882
C83×#25662
#2546183