Finite Group Databasebeta

← #35456182

Subgroups — #35456182

Order 21248.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128111C211yesmaximal, characteristic, complemented
2C2×#19127998C211yesmaximal, characteristic
3#24912412C211yesmaximal, characteristic
4#24908939C211yesmaximal, characteristic, complemented
5#24908934C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908834C211yesmaximal, characteristic
8#24908411C211yesmaximal, characteristic, complemented
9#24908226C211yesmaximal, characteristic, complemented
10#24908221C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908090C211yesmaximal, characteristic
13#24908083C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2370C211yesmaximal, characteristic, complemented
16C2×#14581792C2211yescharacteristic
17C2×#14581772—21no—
18C2×#14581772—21no—
19C2×C4×D332C2211yescharacteristic
20#19128111—24no—
21#19127998—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×C4⋊(C83⋊C8)C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×#14581712C2211yescharacteristic
35C2×#14581703C2211yescharacteristic
36#19127638—21no—
37#19127638—21no—
38#19127603C2211yescharacteristic
39#19127603C2211yescharacteristic
40#19127571C2211yescharacteristic
41#19127571C2211yescharacteristic
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—
70#14581772—44no—
71#14581772—44no—
72C4×D332—22no—
73C4×D332—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—21no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87#14581725—22no—
88#14581725—22no—
89#14581725—22no—
90#14581723—22no—
91#14581723—22no—
92#14581723—22no—
93C4⋊(C83⋊C8)—22no—
94C4⋊(C83⋊C8)—22no—
95C4⋊(C83⋊C8)—22no—
96#14581712—22no—
97#14581712—22no—
98#14581712—22no—
99#14581712Q814yescomplemented
100#14581703—22no—
101#125×C83—22no—
102#125×C83D414yes—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83D411yescharacteristic
108C2×C4⋊C4×C83D411yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C42×C83—21no—
114C2×C42×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83D411yescharacteristic
125#95×C83D411yescharacteristic
126C23×D83—21no—
127C2×Dic166—42no—
128C2×Dic166—42no—
129C2×Dic166—42no—
130C2×D332—21no—
131C2×D332—21no—
132C2×D332—21no—
133C2×D332—21no—
134C2×D332—21no—
135C2×D332C2×Q812yes—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
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⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8C2×Q812yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—42no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—42no—
164C4⋊Dic83—42no—
165(C2×D83)⋊C4—42no—
166(C2×D83)⋊C4—42no—
167(C2×D83)⋊C4—42no—
168(C2×D83)⋊C4—42no—
169(C2×D83)⋊C4—42no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2411yescharacteristic
181C4⋊C4×C83—24no—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83—22no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—81no—
192C4⋊C4×C83C2×D412yes—
193C4⋊C4×C83C2×D412yes—
194C4⋊C4×C83C2×Q812yes—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—42no—
200C42×C83—81no—
201C42×C83—81no—
202C42×C83C2×D412yes—
203C42×C83#4112yes—
204C23×C83C22×D411yescharacteristic
205C22×D83—22no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C22×D83—41no—
211C22×D83—41no—
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—22no—
221C2×C4×C83—22no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
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—41no—
238C2×C4×C83#12112yes—
239C2×C4×C83#12112yes—
240C2×C4×C83AGL(1,ℤ/8)12yes—
241C2×C4×C83AGL(1,ℤ/8)12yes—
242C2×C4×C83C22×D411yescharacteristic
243C2×C4×C83C22×Q811yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C832−1+411yescharacteristic
247C2×C4×C832−1+411yescharacteristic
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—
257D332—22no—
258D332—22no—
259D332—41no—
260D332—41no—
261C4×D83—84no—
262Dic166—84no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C22×C83C2×#12111yescharacteristic
268C22×C83#54711yescharacteristic
269C22×C83D4×Q811yescharacteristic
270C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C83#57711yescharacteristic
273C22×C83#57811yescharacteristic
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—81no—
278C2×D83—81no—
279C2×D83—81no—
280C2×D83—81no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—22no—
284C4×C83—22no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83—42no—
288C4×C83—42no—
289C4×C83—42no—
290C4×C83—81no—
291C4×C83#53311yescharacteristic
292C4×C83#54111yescharacteristic
293C4×C83#54811yescharacteristic
294C4×C83D4×Q811yescharacteristic
295Dic83—81no—
296Dic83—81no—
297Dic83—81no—
298#25465—831nomaximal, Sylow
299C2×C83#242911yescharacteristic
300C2×C83#303811yescharacteristic
301C2×C83#304611yescharacteristic
302C2×C83#334311yescharacteristic
303C2×C83#334511yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338611yescharacteristic
306D83—82no—
307C2×#501—831no—
308C2×#478—831no—
309#2542—831no—
310#2370D8311yescharacteristic, complemented
311#2084—831no—
312#2083—831no—
313#2067—831no—
314#2065—831no—
315#1987—831no—
316#1948—831no—
317#1947—831no—
318#1940—831no—
319#1926—831no—
320#1925—831no—
321#1762—831no—
322C83#2546511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8—831no—
324C2×#125C2×D8311yescharacteristic
325C2×C22⋊Q8—1661no—
326C2×C22⋊Q8—1661no—
327C2×C4×D4—831no—
328#501—1664no—
329#478—1664no—
330C2×#107—831no—
331C2×#107—831no—
332C2×#107—831no—
333C2×#106—831no—
334C2×#106—831no—
335C2×#106—831no—
336C2×C4⋊C8—831no—
337C2×C4⋊C8—831no—
338C2×C4⋊C8—831no—
339C2×#102—831no—
340C2×#102—831no—
341C2×#102—831no—
342C2×#102—831no—
343C2×#97—831no—
344#399—1661no—
345#399—1661no—
346#390—831no—
347#390—831no—
348#389—21no—
349#389C2×D8311yescharacteristic
350#389C2×D8311yescharacteristic
351#387—21no—
352#387—21no—
353#384—831no—
354#384—831no—
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D8311yescharacteristic
357C4×C4⋊C4C2×D8311yescharacteristic
358C4×C4⋊C4C2×D8311yescharacteristic
359C4×C4⋊C4C2×D8311yescharacteristic
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#336—831no—
367#336—831no—
368C22×Q8—1661no—
369C22×D4—831no—
370C23×C4—1661no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8—831no—
375C4⋊Q8—1662no—
376C4⋊Q8—1662no—
377#125—22no—
378#125D33214yes—
379C22⋊Q8—3324no—
380C22⋊Q8—3324no—
381C4×D4—1662no—
382C4×D4—1662no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22⋊D8311yescharacteristic
396C2×C4⋊C4C22⋊D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C22⋊C4—1661no—
404C2×C22⋊C4—1661no—
405C2×C22⋊C4—1661no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42C22×D8311yescharacteristic
410C2×C42C22×D8311yescharacteristic
411C2×C42C22×D8311yescharacteristic
412#107—1662no—
413#107—1662no—
414#107—1662no—
415#106—1662no—
416#106—1662no—
417#106—1662no—
418C4⋊C8—1662no—
419C4⋊C8—1662no—
420C4⋊C8—1662no—
421#102—834no—
422#102—1662no—
423#102—1662no—
424#102—1662no—
425#97—1662no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433C24—1661no—
434C2×Q8—3322no—
435C2×Q8—3322no—
436C2×Q8—3322no—
437C2×D4—832no—
438C2×D4—1661no—
439C2×D4—1661no—
440C2×D4—1661no—
441C2×D4—1661no—
442C2×D4—1661no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—3322no—
448C22×C4—3322no—
449C22×C4—3322no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460C2×C8—832no—
461C2×C8—832no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—3322no—
487C4⋊C4—3322no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C2×D33212yes—
495C4⋊C4C2×D33212yes—
496C4⋊C4Q8×D8312yes—
497C22⋊C4—3322no—
498C22⋊C4—3322no—
499C22⋊C4—3322no—
500C22⋊C4—3322no—
501C22⋊C4—3322no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—81no—
508C42—81no—
509C42C2×D33212yes—
510C42#1305059012yes—
511C23—3321no—
512C23—3321no—
513C23—3321no—
514C23—3321no—
515C23—3321no—
516C23—3321no—
517C23—1662no—
518C23C22×C22⋊D8311yescharacteristic
519Q8—6644no—
520D4—3321no—
521D4—3321no—
522D4—1662no—
523D4—1662no—
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—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—6644no—
560C2×C4#1458176612yes—
561C2×C4#1458176612yes—
562C2×C4#1458181712yes—
563C2×C4#1458181712yes—
564C2×C4C22×D33211yescharacteristic
565C2×C4C2×Q8×D8311yescharacteristic
566C2×C4#41×D8311yescharacteristic
567C2×C4C2×#1305059011yescharacteristic
568C2×C4#1458183911yescharacteristic
569C2×C4#1458184011yescharacteristic
570C8—3321no—
571C8—3321no—
572C8—1662no—
573C8—1662no—
574C22—3322no—
575C22—3322no—
576C22—3322no—
577C22—6641no—
578C22—6641no—
579C22—6641no—
580C22—6641no—
581C22C2×#1458176611yescharacteristic
582C22#1912835011yescharacteristic
583C22Q8×C22⋊D8311yescharacteristic
584C22C2×#1458181711yescharacteristic
585C22C2×#1458181711yescharacteristic
586C22#1912851211yescharacteristic
587C22#1912851611yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—6641no—
599C4—6641no—
600C4—6641no—
601C4#1912825511yescharacteristic
602C4#1912830911yescharacteristic
603C4#1912836311yescharacteristic
604C4Q8×D33211yescharacteristic
605C2—6642no—
606C2#2491105611yescharacteristic
607C2#2491694311yescharacteristic
608C2#2491701311yescharacteristic
609C2#2491990311yescharacteristic
610C2#2491992711yescharacteristic
611C2#2492036211yescharacteristic
612C2#2492038011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2370C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2370D83
hypocenterC83#25465
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. #24908939C2semidirect
5. #24908934C2semidirect
8. #24908411C2semidirect
9. #24908226C2semidirect
10. #24908221C2semidirect
11. #24908187C2semidirect
15. C83×#2370C2semidirect
99. #14581712Q8semidirect
310. #2370D83semidirect
322. C83#25465semidirect

Maximal subgroups

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

subgroupindex
C2×#191281112
C2×#191279982
#249124122
#249089392
#249089342
#249088482
#249088342
#249084112
#249082262
#249082212
#249081872
#249080902
#249080832
#249072912
C83×#23702
#2546583