Finite Group Databasebeta

← #35467001

Subgroups — #35467001

Order 21248.

838 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128010C211yesmaximal, characteristic, complemented
2C22×#14581726C211yesmaximal, characteristic
3#24911188C211yesmaximal, characteristic
4#24910472C211yesmaximal, characteristic
5#24909009C211yesmaximal, characteristic, complemented
6#24908953C211yesmaximal, characteristic, complemented
7#24908856C211yesmaximal, characteristic, complemented
8#24908804C211yesmaximal, characteristic, complemented
9#24908234C211yesmaximal, characteristic, complemented
10#24908033C211yesmaximal, characteristic, complemented
11#24907771C211yesmaximal, characteristic
12#24907719C211yesmaximal, characteristic, complemented
13#24907655C211yesmaximal, characteristic
14#24907641C211yesmaximal, characteristic
15C83×#1892C211yesmaximal, characteristic
16C2×#14581783C2211yescharacteristic
17C2×Q8×Dic83C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20#19128010—24no—
21C2×#14581726—24no—
22C2×#14581726C2211yescharacteristic
23C2×#14581726C2211yescharacteristic
24C2×#14581726C2211yescharacteristic
25C2×#14581726C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×Dic83⋊C8C2211yescharacteristic
29C2×Dic83⋊C8C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×C22⋊(C83⋊C8)C2211yescharacteristic
33#19127643—21no—
34#19127633—21no—
35#19127608C2211yescharacteristic
36#19127600—21no—
37#19127600C2211yescharacteristic
38#19127600C2211yescharacteristic
39#19127597C2211yescharacteristic
40#19127587—21no—
41#19127576—21no—
42#19127576—21no—
43#19127566C2211yescharacteristic
44#19127559—21no—
45C4⋊C4×Dic83C2211yescharacteristic
46C22⋊C4×Dic83—21no—
47C4×C22⋊Dic83C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53#19127455C2211yescharacteristic
54C23×C8×C83C2211yescharacteristic
55C2×C22⋊Q8×C83C2211yescharacteristic
56C2×#103×C83C2211yescharacteristic
57C2×#103×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61C23×Dic83—21no—
62C23×Dic83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581783—22no—
66#14581783—41no—
67Q8×Dic83—22no—
68Q8×Dic83—41no—
69C2×C4⋊Dic83—22no—
70C2×C4⋊Dic83—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—41no—
78C2×C4⋊Dic83—41no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83D411yescharacteristic
86C2×C22⋊Dic83D411yescharacteristic
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83C2311yescharacteristic
90C2×C4×Dic83C2311yescharacteristic
91#14581726—22no—
92#14581726—22no—
93#14581726—22no—
94#14581726D414yes—
95#14581723—22no—
96#14581723—41no—
97Dic83⋊C8—22no—
98Dic83⋊C8—22no—
99#14581716—22no—
100#14581716—41no—
101C22⋊(C83⋊C8)—22no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95D411yescharacteristic
111C83⋊#95D411yescharacteristic
112C22×Q8×C83C2311yescharacteristic
113C23×C4×C83C2311yescharacteristic
114C22×C8×C83—22no—
115C22×C8×C83—22no—
116C22×C8×C83—22no—
117C22×C8×C83C2311yescharacteristic
118C22×C8×C83C2311yescharacteristic
119C22⋊Q8×C83—22no—
120C22⋊Q8×C83Q814yes—
121C2×C4⋊C4×C83—21no—
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C4⋊C4×C83C2311yescharacteristic
124C2×C4⋊C4×C83C2311yescharacteristic
125C2×C22⋊C4×C83—21no—
126C2×C22⋊C4×C83—21no—
127#103×C83—22no—
128#103×C83—22no—
129#95×C83—21no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—42no—
135C22×Dic83—42no—
136C22×Dic83—42no—
137C22×Dic83—42no—
138C22×Dic83—81no—
139C22×Dic83—81no—
140C22×Dic83C2×D411yescharacteristic
141C22×Dic83C2×D411yescharacteristic
142C22×Dic83C2×D411yescharacteristic
143C22×Dic83C2×D411yescharacteristic
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83—42no—
158C4⋊Dic83—42no—
159C4⋊Dic83—42no—
160C4⋊Dic83—81no—
161C4⋊Dic83—81no—
162C4⋊Dic83C2×D412yes—
163C4⋊Dic83C2×D412yes—
164Dic83⋊C4—42no—
165Dic83⋊C4—81no—
166C22⋊Dic83—22no—
167C22⋊Dic83—22no—
168C4×Dic83—21no—
169C4×Dic83—21no—
170C4×Dic83—21no—
171C4×Dic83—21no—
172C4×Dic83—42no—
173C4×Dic83—81no—
174C24×C83C2×D411yescharacteristic
175C2×Q8×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83C2×Q812yes—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×Q812yes—
191C22×C4×C83#4112yes—
192C22×C4×C83C2411yescharacteristic
193C2×C8×C83—28no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83C2×D412yes—
199C2×C8×C83#4112yes—
200C4⋊C4×C83—24no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83C2×Q812yes—
207C22⋊C4×C83—24no—
208C22⋊C4×C83—42no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C22⋊Q812yes—
214C23×C83#13512yes—
215C23×C83#13712yes—
216C23×C83C22×D411yescharacteristic
217Q8×C83—22no—
218Q8×C83—22no—
219Q8×C83—41no—
220Q8×C83—41no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
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—41no—
239C2×C4×C83C22⋊Q812yes—
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83H(5,𝔽2)11yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—84no—
271C2×Dic83—84no—
272C8×C83—22no—
273C8×C83—22no—
274C83⋊C8—41no—
275C83⋊C8—41no—
276C22×C83—22no—
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83#47914yescomplemented
283C22×C83C2×C22⋊Q811yescharacteristic
284C22×C83D4211yescharacteristic
285C22×C83#54711yescharacteristic
286C22×C83C2×#13511yescharacteristic
287C22×C83C2×#13711yescharacteristic
288C22×C83#57611yescharacteristic
289C22×C83#57811yescharacteristic
290C4×C83—22no—
291C4×C83—42no—
292C4×C83—42no—
293C4×C83—81no—
294C4×C83—81no—
295C4×C83#52511yescharacteristic
296C4×C83#53911yescharacteristic
297C4×C83D4×Q811yescharacteristic
298C4×C83#55411yescharacteristic
299Dic83—41no—
300Dic83—41no—
301Dic83—41no—
302Dic83—41no—
303Dic83—42no—
304Dic83—42no—
305Dic83—81no—
306Dic83—81no—
307#26344—831nomaximal, Sylow
308C2×C83—24no—
309C2×C83#244911yescharacteristic
310C2×C83C2×#47911yescharacteristic
311C2×C83#308311yescharacteristic
312C2×C83#327011yescharacteristic
313C2×C83#328011yescharacteristic
314C2×C83#333611yescharacteristic
315C2×C83#334611yescharacteristic
316C2×#479—831no—
317C22×#107—831no—
318#2442—831no—
319#2360—831no—
320#2097—831no—
321#2086—831no—
322#2068—831no—
323#2059—831no—
324#1949—831no—
325#1918—831no—
326#1892D8311yescharacteristic
327#1872—831no—
328#1863—831no—
329#1853—831no—
330#1850—831no—
331C83#2634411yescharacteristic, complemented, Sylow
332C23×C8C2×D8311yescharacteristic
333C2×#125—831no—
334C2×C22⋊Q8C2×D8311yescharacteristic
335C2×C4×Q8—831no—
336C22×C4⋊C4—831no—
337C22×C4⋊C4—831no—
338#479—1664no—
339C2×#107—831no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#107—1664no—
344C2×#106—831no—
345C2×#106—831no—
346C2×C4⋊C8—831no—
347C2×C4⋊C8—831no—
348C2×#103—831no—
349C2×#103—831no—
350C2×#103C2×D8311yescharacteristic
351C2×#103C2×D8311yescharacteristic
352C2×C22⋊C8—831no—
353#400—1661no—
354#398—1661no—
355#391—831no—
356#389—831no—
357#389—831no—
358#389—831no—
359#389—1661no—
360#387—1661no—
361#385—1661no—
362#385—1661no—
363#382—831no—
364#380—1661no—
365#380C2×D8311yescharacteristic
366C4×C4⋊C4—831no—
367C4×C22⋊C4—831no—
368C4×C22⋊C4—1661no—
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340C2×D8311yescharacteristic
374#340C2×D8311yescharacteristic
375#336—831no—
376#336—831no—
377C22×Q8C22×D8311yescharacteristic
378C23×C4—1661no—
379C23×C4—1661no—
380C23×C4C22×D8311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—22no—
384C22×C8—831no—
385C22×C8—831no—
386C22×C8C22×D8311yescharacteristic
387C22×C8C22×D8311yescharacteristic
388#125—3321no—
389#125—1662no—
390C22⋊Q8—22no—
391C22⋊Q8Dic16614yes—
392C4×Q8—3321no—
393C4×Q8—1662no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—3321no—
396C2×C4⋊C4—3321no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1662no—
407C2×C4⋊C4—1662no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C4⋊C4C22×D8311yescharacteristic
413C2×C4⋊C4C22×D8311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—831no—
417C2×C22⋊C4—831no—
418C2×C42—831no—
419C2×C42—831no—
420C2×C42—1661no—
421C2×C42—1661no—
422#107—834no—
423#107—1662no—
424#107—1662no—
425#107—1662no—
426#106—3321no—
427#106—1662no—
428C4⋊C8—1662no—
429C4⋊C8—1662no—
430#103—22no—
431#103—22no—
432#103—3321no—
433#103—1662no—
434C22⋊C8—1662no—
435#95—21no—
436#95—831no—
437#95—831no—
438#95—1661no—
439#95—1661no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95—1661no—
444#95—1661no—
445#95—1661no—
446C24D4×D8311yescharacteristic
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Dic16612yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—3322no—
462C22×C4—3322no—
463C22×C4—3322no—
464C22×C4—3322no—
465C22×C4—831no—
466C22×C4—831no—
467C22×C4—831no—
468C22×C4—831no—
469C22×C4—6641no—
470C22×C4—6641no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4D4×D8311yescharacteristic
476C22×C4C2×Dic16612yes—
477C22×C4#1305058912yes—
478C22×C4C23×D8311yescharacteristic
479C2×C8—28no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8D4×D8312yes—
491C2×C8#1305058912yes—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—3322no—
499C4⋊C4—3322no—
500C4⋊C4—3322no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—832no—
504C4⋊C4—832no—
505C4⋊C4—6641no—
506C4⋊C4—6641no—
507C4⋊C4—6641no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4C2×Dic16612yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—1662no—
518C22⋊C4—1662no—
519C42—3322no—
520C42—6641no—
521C42—1661no—
522C42—1661no—
523C42—1661no—
524C42—1661no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1458176912yes—
530C23#1458181512yes—
531C23#1458182112yes—
532C23C2×D4×D8311yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—6644no—
566C2×C4—6644no—
567C2×C4—1661no—
568C2×C4—1662no—
569C2×C4—1661no—
570C2×C4—1662no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4—1661no—
574C2×C4—1661no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4—1661no—
578C2×C4—1661no—
579C2×C4—1661no—
580C2×C4—1661no—
581C2×C4#1458176912yes—
582C2×C4C2×D4×D8311yescharacteristic
583C2×C4C22×Dic16611yescharacteristic
584C2×C4C2×#1305058911yescharacteristic
585C2×C4C2×#1305058911yescharacteristic
586C2×C4#1458183811yescharacteristic
587C2×C4#1458183911yescharacteristic
588C8—22no—
589C8—22no—
590C8—3321no—
591C8—3321no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1912801014yescomplemented
599C22C2×#1458176911yescharacteristic
600C22#1912833511yescharacteristic
601C22#1912835111yescharacteristic
602C22C2×#1458181511yescharacteristic
603C22C2×#1458182111yescharacteristic
604C22#1912850311yescharacteristic
605C22#1912851511yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—3321no—
612C4—3322no—
613C4—3322no—
614C4—3321no—
615C4—3321no—
616C4—3321no—
617C4—6641no—
618C4—6641no—
619C4#1912820711yescharacteristic
620C4#1912829511yescharacteristic
621C4D4×Dic16611yescharacteristic
622C4#1912840811yescharacteristic
623C2—24no—
624C2#2491127811yescharacteristic
625C2C2×#1912801011yescharacteristic
626C2#2491731211yescharacteristic
627C2#2491903011yescharacteristic
628C2#2491914511yescharacteristic
629C2#2491983411yescharacteristic
630C2#2491995411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1892C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1892D83
hypocenterC83#26344
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
1. C2×#19128010C2semidirect
5. #24909009C2semidirect
6. #24908953C2semidirect
7. #24908856C2semidirect
8. #24908804C2semidirect
9. #24908234C2semidirect
10. #24908033C2semidirect
12. #24907719C2semidirect
282. C22×C83#479semidirect
331. C83#26344semidirect
598. C22#19128010semidirect

Maximal subgroups

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

subgroupindex
C2×#191280102
C22×#145817262
#249111882
#249104722
#249090092
#249089532
#249088562
#249088042
#249082342
#249080332
#249077712
#249077192
#249076552
#249076412
C83×#18922
#2634483