Finite Group Databasebeta

← #35456137

Subgroups — #35456137

Order 21248.

764 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128107C211yesmaximal, characteristic, complemented
2C2×#19127978C211yesmaximal, characteristic
3#24912663C211yesmaximal, characteristic
4#24908944C211yesmaximal, characteristic, complemented
5#24908934C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908834C211yesmaximal, characteristic
8#24908420C211yesmaximal, characteristic, complemented
9#24908231C211yesmaximal, characteristic, complemented
10#24908221C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908090C211yesmaximal, characteristic
13#24908083C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2369C211yesmaximal, characteristic, complemented
16C2×#14581784C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19128107—24no—
19#19127978—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
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#19127623—21no—
35#19127623—21no—
36#19127603C2211yescharacteristic
37#19127603C2211yescharacteristic
38#19127575—21no—
39#19127575—21no—
40#19127571C2211yescharacteristic
41#19127567C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51#391×C83—21no—
52#391×C83—21no—
53#389×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56C4×C4⋊C4×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581784—22no—
68#14581784—22no—
69C4×D332—22no—
70C4×D332—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581725—22no—
87#14581723—22no—
88#14581723—22no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91C4⋊(C83⋊C8)—22no—
92#14581712—22no—
93#14581712—22no—
94#14581712—22no—
95#14581712Q814yescomplemented
96#14581703—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C22×Q8×C83—21no—
100C4⋊Q8×C83—22no—
101C4⋊Q8×C83D414yes—
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×D332—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332C2×Q812yes—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C2×C4×D83—42no—
135C2×C4×D83—81no—
136C2×C4×D83—81no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
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⋊C8C2×Q812yes—
148C2×C83⋊C8#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—81no—
156C4⋊Dic83—81no—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—42no—
161(C2×D83)⋊C4—42no—
162(C2×D83)⋊C4—81no—
163(C2×D83)⋊C4—81no—
164C2×Q8×C83—24no—
165C2×Q8×C83—81no—
166C2×Q8×C83—81no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
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—24no—
179C4⋊C4×C83—22no—
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—81no—
187C4⋊C4×C83—81no—
188C4⋊C4×C83C2×D412yes—
189C4⋊C4×C83C2×D412yes—
190C4⋊C4×C83C2×Q812yes—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—42no—
196C42×C83—81no—
197C42×C83—81no—
198C42×C83C2×D412yes—
199C42×C83#4112yes—
200C23×C83C22×D411yescharacteristic
201C22×D83—22no—
202C22×D83—41no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208Q8×C83—84no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—22no—
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#12112yes—
237C2×C4×C83#12112yes—
238C2×C4×C83#13712yes—
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—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254D332—22no—
255D332—22no—
256D332—41no—
257D332—41no—
258C4×D83—84no—
259C83⋊C8—22no—
260C83⋊C8—22no—
261C83⋊C8—41no—
262C83⋊C8—41no—
263C22×C83C2×#12111yescharacteristic
264C22×C83#54711yescharacteristic
265C22×C83D4×Q811yescharacteristic
266C22×C83C2×#13711yescharacteristic
267C22×C83#57511yescharacteristic
268C22×C83#57611yescharacteristic
269C22×C83#57611yescharacteristic
270C2×D83—42no—
271C2×D83—42no—
272C2×D83—42no—
273C2×D83—81no—
274C2×D83—81no—
275C2×D83—81no—
276C2×D83—81no—
277C4×C83—22no—
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—42no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—81no—
287C4×C83#53311yescharacteristic
288C4×C83#54111yescharacteristic
289C4×C83#54811yescharacteristic
290C4×C83D4×Q811yescharacteristic
291Dic83—81no—
292Dic83—81no—
293Dic83—81no—
294#25462—831nomaximal, Sylow
295C2×C83#242911yescharacteristic
296C2×C83#303711yescharacteristic
297C2×C83#304111yescharacteristic
298C2×C83#334611yescharacteristic
299C2×C83#334711yescharacteristic
300C2×C83#338511yescharacteristic
301C2×C83#338911yescharacteristic
302D83—82no—
303C2×#501—831no—
304C2×#474—831no—
305#2566—831no—
306#2369D8311yescharacteristic, complemented
307#2085—831no—
308#2083—831no—
309#2067—831no—
310#2065—831no—
311#1989—831no—
312#1949—831no—
313#1947—831no—
314#1940—831no—
315#1926—831no—
316#1925—831no—
317#1762—831no—
318C83#2546211yescharacteristic, complemented, Sylow
319C2×C4⋊Q8C2×D8311yescharacteristic
320C2×#125—831no—
321C2×C4×D4—831no—
322#501—1664no—
323#474—1664no—
324C2×#107—831no—
325C2×#107—831no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#106—831no—
329C2×#106—831no—
330C2×C4⋊C8—831no—
331C2×C4⋊C8—831no—
332C2×C4⋊C8—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#97—831no—
338#396—1661no—
339#396—1661no—
340#391—21no—
341#391—21no—
342#390—831no—
343#390—831no—
344#389—21no—
345#389C2×D8311yescharacteristic
346#389C2×D8311yescharacteristic
347#385—1661no—
348#385—1661no—
349#384—831no—
350#382—831no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4C2×D8311yescharacteristic
353C4×C4⋊C4C2×D8311yescharacteristic
354C4×C4⋊C4C2×D8311yescharacteristic
355C4×C4⋊C4C2×D8311yescharacteristic
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#336—831no—
363#336—831no—
364C22×Q8—21no—
365C22×D4—831no—
366C23×C4—1661no—
367C22×C8—831no—
368C22×C8—831no—
369C22×C8—831no—
370C22×C8—831no—
371C4⋊Q8—22no—
372C4⋊Q8D33214yes—
373#125—1662no—
374#125—1662no—
375C4×D4—1662no—
376C4×D4—1662no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—831no—
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×C4⋊C4C22×D8311yescharacteristic
393C2×C22⋊C4—1661no—
394C2×C22⋊C4—1661no—
395C2×C22⋊C4—1661no—
396C2×C22⋊C4—1661no—
397C2×C22⋊C4—1661no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42C22×D8311yescharacteristic
402C2×C42C22×D8311yescharacteristic
403C2×C42C22×D8311yescharacteristic
404#107—1662no—
405#107—1662no—
406#107—1662no—
407#107—1662no—
408#106—1662no—
409#106—1662no—
410C4⋊C8—1662no—
411C4⋊C8—1662no—
412C4⋊C8—1662no—
413#102—834no—
414#102—1662no—
415#102—1662no—
416#102—1662no—
417#97—1662no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1661no—
424#95—1661no—
425#95C22⋊D8311yescharacteristic
426#95C22⋊D8311yescharacteristic
427C24—1661no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—832no—
432C2×D4—1661no—
433C2×D4—1661no—
434C2×D4—1661no—
435C2×D4—1661no—
436C2×D4—1661no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3322no—
442C22×C4—6641no—
443C22×C4—6641no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C23×D8311yescharacteristic
454C2×C8—832no—
455C2×C8—832no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—6641no—
479C4⋊C4—6641no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4C2×D33212yes—
487C4⋊C4C2×D33212yes—
488C4⋊C4Q8×D8312yes—
489C22⋊C4—3322no—
490C22⋊C4—3322no—
491C22⋊C4—3322no—
492C22⋊C4—6641no—
493C22⋊C4—6641no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42—81no—
500C42—81no—
501C42C2×D33212yes—
502C42#1305059012yes—
503C23—3321no—
504C23—3321no—
505C23—3321no—
506C23—3321no—
507C23—3321no—
508C23—3321no—
509C23—1662no—
510C23C22×C22⋊D8311yescharacteristic
511Q8—84no—
512D4—3321no—
513D4—3321no—
514D4—1662no—
515D4—1662no—
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—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
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—41no—
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—6644no—
553C2×C4#1458176612yes—
554C2×C4#1458176612yes—
555C2×C4#1458182312yes—
556C2×C4C22×D33211yescharacteristic
557C2×C4C2×Q8×D8311yescharacteristic
558C2×C4#41×D8311yescharacteristic
559C2×C4C2×#1305059011yescharacteristic
560C2×C4#1458183911yescharacteristic
561C2×C4#1458184011yescharacteristic
562C8—3321no—
563C8—3321no—
564C8—1662no—
565C8—1662no—
566C22—3322no—
567C22—3322no—
568C22—3322no—
569C22—6641no—
570C22—6641no—
571C22—6641no—
572C22—6641no—
573C22C2×#1458176611yescharacteristic
574C22#1912835011yescharacteristic
575C22Q8×C22⋊D8311yescharacteristic
576C22C2×#1458182311yescharacteristic
577C22#1912849911yescharacteristic
578C22#1912850411yescharacteristic
579C22#1912850411yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—6641no—
591C4—6641no—
592C4—6641no—
593C4#1912825511yescharacteristic
594C4#1912830911yescharacteristic
595C4#1912836311yescharacteristic
596C4Q8×D33211yescharacteristic
597C2—6642no—
598C2#2491105611yescharacteristic
599C2#2491693511yescharacteristic
600C2#2491696611yescharacteristic
601C2#2491994211yescharacteristic
602C2#2491996311yescharacteristic
603C2#2492036911yescharacteristic
604C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2369C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2369D83
hypocenterC83#25462
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
1. C2×#19128107C2semidirect
4. #24908944C2semidirect
5. #24908934C2semidirect
8. #24908420C2semidirect
9. #24908231C2semidirect
10. #24908221C2semidirect
11. #24908187C2semidirect
15. C83×#2369C2semidirect
95. #14581712Q8semidirect
306. #2369D83semidirect
318. C83#25462semidirect

Maximal subgroups

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

subgroupindex
C2×#191281072
C2×#191279782
#249126632
#249089442
#249089342
#249088482
#249088342
#249084202
#249082312
#249082212
#249081872
#249080902
#249080832
#249072912
C83×#23692
#2546283