Finite Group Databasebeta

← #35423751

Subgroups — #35423751

Order 21248.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127801C211yesmaximal, characteristic
2C4⋊C4×Dic166C211yesmaximal, characteristic, complemented
3#24908163C211yesmaximal, characteristic
4#24908149C211yesmaximal, characteristic
5#24908055C211yesmaximal, characteristic, complemented
6#24908034C211yesmaximal, characteristic, complemented
7#24907821C211yesmaximal, characteristic
8#24907821C211yesmaximal, characteristic
9#24907596C211yesmaximal, characteristic, complemented
10#24907475C211yesmaximal, characteristic, complemented
11#24907463C211yesmaximal, characteristic
12#24907338C211yesmaximal, characteristic, complemented
13#24907296C211yesmaximal, characteristic, complemented
14C4×#14581715C211yesmaximal, characteristic, complemented
15C83×#2357C211yesmaximal, characteristic
16C22×#13050578C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20#19127801C418yescomplemented
21C2×#14581725C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581715C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×C4×C83⋊C8C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127598—21no—
35#19127598—21no—
36#19127571C2211yescharacteristic
37#19127570—21no—
38C4⋊C4×Dic83—21no—
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50#390×C83C2211yescharacteristic
51#390×C83C2211yescharacteristic
52#388×C83—21no—
53#388×C83—21no—
54#382×C83C2211yescharacteristic
55#381×C83—21no—
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C22×Dic166C2311yescharacteristic
61C2×#13050578—22no—
62C2×#13050578—22no—
63C2×#13050578C2×C414yes—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C4×Dic166—22no—
69C4×Dic166—22no—
70C4×Dic166—22no—
71C4×Dic166—41no—
72C4×Dic166—41no—
73C4×Dic166C2×C414yes—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87#14581725C2×C414yes—
88C4⋊(C83⋊C8)—22no—
89#14581715—22no—
90#14581715—22no—
91#14581715C2×C414yes—
92#14581713—41no—
93#14581713—41no—
94#14581713C2×C414yes—
95#14581703—41no—
96#14581703C2×C414yes—
97C4×C83⋊C8—41no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C22×D4×C83C2311yescharacteristic
101C23×C4×C83—21no—
102C4×D4×C83—22no—
103C4×D4×C83C2×C414yes—
104C2×C4⋊C4×C83D411yescharacteristic
105C2×C4⋊C4×C83D411yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×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×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C2×Dic166—21no—
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166C22×C412yes—
129C2×Dic166#4112yes—
130C2×Dic166#4112yes—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135#13050578—28no—
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⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8C22×C412yes—
147C2×C83⋊C8C22×C412yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—42no—
154C4⋊Dic83—81no—
155C4⋊Dic83C22×C412yes—
156Dic83⋊C4—24no—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159Dic83⋊C4—81no—
160Dic83⋊C4—81no—
161C4×Dic83—24no—
162C4×Dic83—42no—
163C4×Dic83—42no—
164C24×C83—21no—
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83C22×C412yes—
171C22×C4×C83—21no—
172C22×C4×C83—24no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—81no—
176C22×C4×C83—81no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
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—41no—
191C4⋊C4×C83C22×C412yes—
192C22⋊C4×C83—24no—
193C22⋊C4×C83—42no—
194C22⋊C4×C83—42no—
195C22⋊C4×C83—81no—
196C22⋊C4×C83—81no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—81no—
202C42×C83—81no—
203C42×C83C22×C412yes—
204C42×C83#4112yes—
205C23×C83—22no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83C22×D411yescharacteristic
213D4×C83—22no—
214D4×C83—22no—
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—22no—
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—21no—
232C2×C4×C83—21no—
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—41no—
241C2×C4×C83—84no—
242C2×C4×C83C4×D412yes—
243C2×C4×C83C4×D412yes—
244C2×C4×C83#13712yes—
245C2×C4×C83C23×C411yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83H(5,𝔽2)11yescharacteristic
250C2×C4×C832−1+411yescharacteristic
251C2×Dic83—22no—
252C2×Dic83—22no—
253C2×Dic83—22no—
254C2×Dic83—22no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263Dic166—22no—
264Dic166—22no—
265Dic166—22no—
266Dic166#11714yes—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83—81no—
278C22×C83C2×C4×D411yescharacteristic
279C22×C83#54611yescharacteristic
280C22×C83#54811yescharacteristic
281C22×C83C2×#13711yescharacteristic
282C22×C83#57511yescharacteristic
283C22×C83#57611yescharacteristic
284C22×C83#57711yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83C2×#11711yescharacteristic
295C4×C83#52011yescharacteristic
296C4×C83#55211yescharacteristic
297C4×C83#55311yescharacteristic
298Dic83—24no—
299Dic83—42no—
300Dic83—81no—
301Dic83—81no—
302#22775—831nomaximal, Sylow
303C2×C83—82no—
304C2×C83#235211yescharacteristic
305C2×C83C4×#13711yescharacteristic
306C2×C83#294611yescharacteristic
307C2×C83#330911yescharacteristic
308C2×C83#332011yescharacteristic
309C2×C83#337111yescharacteristic
310C2×C83#337511yescharacteristic
311C2×#440—831no—
312#2357D8311yescharacteristic
313Q8×C4⋊C4—831no—
314#1936—831no—
315#1934—831no—
316#1921—831no—
317#1918—831no—
318#1880—831no—
319#1880—831no—
320#1836—831no—
321#1805—831no—
322#1802—831no—
323#1771—831no—
324#1763—831no—
325C4×#103—831no—
326C83#2277511yescharacteristic, complemented, Sylow
327C22×SD16—831no—
328C2×C4×Q8—831no—
329C2×C4×Q8—831no—
330C2×C4×Q8—831no—
331C2×C4×D4C2×D8311yescharacteristic
332#440—838no—
333C2×#107—831no—
334C2×C4⋊C8—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#103—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#102—831no—
341C2×#97—831no—
342C2×#97—831no—
343C2×C4×C8—831no—
344#390C2×D8311yescharacteristic
345#390C2×D8311yescharacteristic
346#389—831no—
347#389—831no—
348#389—1661no—
349#389—1661no—
350#388—21no—
351#388—21no—
352#384—831no—
353#384—1661no—
354#382C2×D8311yescharacteristic
355#381—21no—
356C4×C4⋊C4—1661no—
357C4×C4⋊C4C2×D8311yescharacteristic
358C4×C4⋊C4C2×D8311yescharacteristic
359C4×C4⋊C4C2×D8311yescharacteristic
360C4×C22⋊C4—21no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#336—831no—
370#336—831no—
371C22×Q8—831no—
372C22×D4C22×D8311yescharacteristic
373C23×C4—21no—
374C2×SD16—834no—
375C2×SD16—1662no—
376C2×SD16—1662no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C4×Q8—3321no—
382C4×Q8—3321no—
383C4×Q8—834no—
384C4×Q8—1662no—
385C4×Q8—1662no—
386C4×Q8—1662no—
387C4×D4—22no—
388C4×D4C4×D8314yes—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4C22⋊D8311yescharacteristic
400C2×C4⋊C4C22⋊D8311yescharacteristic
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—1661no—
412C2×C42—1661no—
413C2×C42—1661no—
414C2×C42C22×D8311yescharacteristic
415C2×C42C22×D8311yescharacteristic
416C2×C42C22×D8311yescharacteristic
417#107—834no—
418C4⋊C8—1662no—
419#103—834no—
420#103—1662no—
421#103—1662no—
422#102—3321no—
423#102—3321no—
424#102—834no—
425#97—3321no—
426#97—834no—
427C4×C8—3321no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1661no—
433#95—1661no—
434#95C22⋊D8311yescharacteristic
435#95C22⋊D8311yescharacteristic
436C24—21no—
437C2×Q8—832no—
438C2×Q8—832no—
439C2×Q8—832no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C2×Q8—1661no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D8312yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4C2×C22⋊D8311yescharacteristic
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C23×D8311yescharacteristic
466SD16—1668no—
467C2×C8—832no—
468C2×C8—832no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—41no—
486C4⋊C4—3322no—
487C4⋊C4—3322no—
488C4⋊C4—3322no—
489C4⋊C4—832no—
490C4⋊C4—6641no—
491C4⋊C4—6641no—
492C4⋊C4—6641no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1664no—
499C4⋊C4C2×C4×D8312yes—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—81no—
510C42—81no—
511C42—3322no—
512C42—3322no—
513C42—1664no—
514C42C2×C4×D8312yes—
515C42#1305059012yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C22×C22⋊D8311yescharacteristic
524Q8—834no—
525Q8—1662no—
526Q8—1662no—
527Q8—1662no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
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—1662no—
566C2×C4—1662no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4C4×C22⋊D8312yes—
570C2×C4C4×C22⋊D8312yes—
571C2×C4#1458182212yes—
572C2×C4C22×C4×D8311yescharacteristic
573C2×C4#41×D8311yescharacteristic
574C2×C4#41×D8311yescharacteristic
575C2×C4C2×#1305059011yescharacteristic
576C2×C4#1458183811yescharacteristic
577C2×C4#1458183911yescharacteristic
578C8—3321no—
579C8—3321no—
580C8—1662no—
581C8—1662no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×C4×C22⋊D8311yescharacteristic
590C22#1912833911yescharacteristic
591C22#1912835811yescharacteristic
592C22C2×#1458182211yescharacteristic
593C22#1912849911yescharacteristic
594C22#1912850411yescharacteristic
595C22#1912851211yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—3322no—
606C4—6641no—
607C4—6641no—
608C4—1664no—
609C4#117×D8311yescharacteristic
610C4#1912818111yescharacteristic
611C4#1912839011yescharacteristic
612C4#1912839511yescharacteristic
613C2—82no—
614C2#2491039311yescharacteristic
615C2C4×#1458182211yescharacteristic
616C2#2491629511yescharacteristic
617C2#2491946311yescharacteristic
618C2#2491962811yescharacteristic
619C2#2492022511yescharacteristic
620C2#2492027311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2357C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#2357D83
hypocenterC83#22775
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×Dic166C2semidirect
5. #24908055C2semidirect
6. #24908034C2semidirect
9. #24907596C2semidirect
10. #24907475C2semidirect
12. #24907338C2semidirect
13. #24907296C2semidirect
14. C4×#14581715C2semidirect
20. #19127801C4semidirect
326. C83#22775semidirect

Maximal subgroups

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

subgroupindex
C2×#191278012
C4⋊C4×Dic1662
#249081632
#249081492
#249080552
#249080342
#249078212
#249075962
#249074752
#249074632
#249073382
#249072962
C4×#145817152
C83×#23572
#2277583