Finite Group Databasebeta

← #35411963

Subgroups — #35411963

Order 21248.

1000 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127979C211yesmaximal, characteristic
2#24910034C211yesmaximal, characteristic, complemented
3C4×D4×Dic83C211yesmaximal, characteristic
4#24908413C211yesmaximal, characteristic, complemented
5#24908218C211yesmaximal, characteristic, complemented
6#24908035C211yesmaximal, characteristic, complemented
7#24908031C211yesmaximal, characteristic, complemented
8#24907710C211yesmaximal, characteristic
9#24907461C211yesmaximal, characteristic
10#107×Dic83C211yesmaximal, characteristic, complemented
11#24907355C211yesmaximal, characteristic, complemented
12#24907311C211yesmaximal, characteristic, complemented
13#102×Dic83C211yesmaximal, characteristic
14#102×Dic83C211yesmaximal, characteristic
15C83×#1862C211yesmaximal, characteristic
16C2×#14581791C2211yescharacteristic
17C2×#14581757—21no—
18C2×D4×Dic83C2211yescharacteristic
19C2×D4×Dic83C2211yescharacteristic
20C22×C4×Dic83—21no—
21#19127979C418yescomplemented
22C2×#14581726C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×Dic83⋊C8C2211yescharacteristic
26C2×Dic83⋊C8C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×C8×Dic83C2211yescharacteristic
30C2×C8×Dic83C2211yescharacteristic
31#19127605—21no—
32#19127599C2211yescharacteristic
33#19127596—21no—
34#19127570C2211yescharacteristic
35C4⋊C4×Dic83C2211yescharacteristic
36C4⋊C4×Dic83C2211yescharacteristic
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40#19127552—21no—
41C22⋊C4×Dic83—21no—
42C22⋊C4×Dic83—21no—
43C4×C22⋊Dic83—21no—
44C42×Dic83C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127456C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54C2×#107×C83C2211yescharacteristic
55C2×#102×C83C2211yescharacteristic
56C2×#102×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×Dic166—21no—
61C23×Dic83—21no—
62C23×Dic83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581791—22no—
66#14581791C2×C414yescomplemented
67#14581757—44no—
68D4×Dic83—22no—
69D4×Dic83—22no—
70D4×Dic83C2×C414yes—
71D4×Dic83Q814yescomplemented
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—44no—
88C2×C4×Dic83D411yescharacteristic
89C2×C4×Dic83D411yescharacteristic
90C2×C4×Dic83D411yescharacteristic
91C2×C4×Dic83D411yescharacteristic
92C2×C4×Dic83C2311yescharacteristic
93C2×C4×Dic83C2311yescharacteristic
94#14581726C2×C414yescomplemented
95#14581725—22no—
96#14581725C2×C414yescomplemented
97Dic83⋊C8—22no—
98Dic83⋊C8C2×C414yescomplemented
99#14581713—22no—
100#14581713C2×C414yes—
101C8×Dic83—22no—
102C8×Dic83—22no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C22×D4×C83C2311yescharacteristic
110C23×C4×C83—21no—
111C22×C8×C83C2311yescharacteristic
112C22×C8×C83C2311yescharacteristic
113C4×D4×C83—22no—
114C4×D4×C83—22no—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C22⋊C4×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#107×C83—22no—
122#102×C83—22no—
123#102×C83C2×C414yes—
124#95×C83—21no—
125C2×Dic166—24no—
126C2×Dic166—42no—
127C2×Dic166—42no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—24no—
132C22×Dic83—24no—
133C22×Dic83—21no—
134C22×Dic83—42no—
135C22×Dic83—42no—
136C22×Dic83—42no—
137C22×Dic83—42no—
138C22×Dic83C2×D411yescharacteristic
139C22×Dic83C2×D411yescharacteristic
140C22×Dic83C2×D411yescharacteristic
141C22×Dic83C2×D411yescharacteristic
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8C22×C412yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83C22×C412yes—
154C4⋊Dic83C22×C412yes—
155C4⋊Dic83C2×Q812yes—
156Dic83⋊C4—24no—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160C22⋊Dic83—24no—
161C22⋊Dic83—24no—
162C4×Dic83—22no—
163C4×Dic83—22no—
164C4×Dic83—41no—
165C4×Dic83—42no—
166C4×Dic83—42no—
167C4×Dic83—42no—
168C4×Dic83—42no—
169C4×Dic83—42no—
170C4×Dic83D814yes—
171C4×Dic83C22×C412yes—
172C4×Dic83C2×Q812yes—
173C4×Dic83#4112yes—
174C4×Dic83#4112yes—
175C24×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83—21no—
179C2×D4×C83C22×C412yes—
180C2×D4×C83C2×Q812yes—
181C2×D4×C83#4112yes—
182C22×C4×C83—24no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—42no—
187C22×C4×C83—42no—
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2411yescharacteristic
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83C22×C412yes—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83C22×C412yes—
203C4⋊C4×C83#4112yes—
204C22⋊C4×C83—24no—
205C42×C83—21no—
206C42×C83—21no—
207C23×C83—22no—
208C23×C83—22no—
209C23×C83—22no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83C22×D411yescharacteristic
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—22no—
218D4×C83C4×Q814yescomplemented
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—48no—
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×C83C2×D812yes—
235C2×C4×C83C23×C411yescharacteristic
236C2×C4×C83C22×Q811yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—48no—
255C2×Dic83—48no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83C4×D412yes—
260C2×Dic83C4×D412yes—
261C2×Dic83C22⋊Q812yes—
262C2×Dic83C22⋊Q812yes—
263C2×Dic83C2×D812yes—
264C2×Dic83C2×D812yes—
265C2×Dic83#13512yes—
266C2×Dic83#13712yes—
267Dic166—48no—
268C8×C83—22no—
269C8×C83—41no—
270C83⋊C8—22no—
271C83⋊C8—41no—
272C22×C83—24no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83C2×C4×D411yescharacteristic
280C22×C83C2×C22⋊Q811yescharacteristic
281C22×C83#52511yescharacteristic
282C22×C83C22×D811yescharacteristic
283C22×C83C2×#13511yescharacteristic
284C22×C83C2×#13711yescharacteristic
285C22×C83#57511yescharacteristic
286C4×C83—22no—
287C4×C83—42no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—81no—
291C4×C83C2×C4×Q811yescharacteristic
292C4×C83C4×#4111yescharacteristic
293C4×C83#52911yescharacteristic
294C4×C83#53311yescharacteristic
295Dic83—22no—
296Dic83—22no—
297Dic83—42no—
298Dic83—42no—
299Dic83—42no—
300Dic83—81no—
301Dic83C4×D814yes—
302Dic83#47414yescomplemented
303C4×#474—831nomaximal, Sylow
304C2×C83—44no—
305C2×C83C4×C22⋊Q811yescharacteristic
306C2×C83C2×C4×D811yescharacteristic
307C2×C83C4×#13711yescharacteristic
308C2×C83C2×#47411yescharacteristic
309C2×C83#307611yescharacteristic
310C2×C83#310011yescharacteristic
311C2×C83#310711yescharacteristic
312C2×#474—831no—
313C4×C4⋊Q8—831no—
314C42×D4—831no—
315#1988—831no—
316#1947—831no—
317#1918—831no—
318#1918—831no—
319#1862—831no—
320#1862D8311yescharacteristic
321#1802—831no—
322C4×#107—831no—
323C4×#107—831no—
324C4×C4⋊C8—831no—
325C4×#102—831no—
326C4×#102—831no—
327C83C4×#47411yescharacteristic, complemented, Sylow
328C2×C4⋊Q8—831no—
329C2×C4×Q8—1661no—
330C2×C4×D4—831no—
331C2×C4×D4—831no—
332C2×C4×D4C2×D8311yescharacteristic
333C22×C42—1661no—
334#474—838no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#107C2×D8311yescharacteristic
339C2×C4⋊C8—831no—
340C2×C4⋊C8—831no—
341C2×#102—831no—
342C2×#102—831no—
343C2×#102C2×D8311yescharacteristic
344C2×#102C2×D8311yescharacteristic
345C2×C4×C8—831no—
346C2×C4×C8—831no—
347#391—1661no—
348#389—831no—
349#389—1661no—
350#389C2×D8311yescharacteristic
351#384—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C4×C4⋊C4—831no—
357C4×C4⋊C4—1661no—
358C4×C22⋊C4—1661no—
359C4×C22⋊C4—1661no—
360C4×C22⋊C4—1661no—
361C43—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340C2×D8311yescharacteristic
370#336—831no—
371#336C2×D8311yescharacteristic
372C22×Q8—1661no—
373C22×D4C22×D8311yescharacteristic
374C23×C4—21no—
375C23×C4—1661no—
376C23×C4—1661no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8C22×D8311yescharacteristic
380C22×C8C22×D8311yescharacteristic
381C4⋊Q8—834no—
382C4⋊Q8—1662no—
383C4×Q8—3324no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—834no—
387C4×D4—834no—
388C4×D4—1662no—
389C4×D4—1662no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
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×C22⋊C4—21no—
403C2×C22⋊C4—1661no—
404C2×C22⋊C4—1661no—
405C2×C42—3324no—
406C2×C42—831no—
407C2×C42—831no—
408C2×C42—831no—
409C2×C42—831no—
410C2×C42—831no—
411C2×C42—831no—
412C2×C42—1661no—
413C2×C42—1661no—
414C2×C42—1661no—
415C2×C42—1661no—
416C2×C42—1661no—
417C2×C42C22×D8311yescharacteristic
418#107—22no—
419#107—834no—
420#107—834no—
421#107—1662no—
422C4⋊C8—834no—
423C4⋊C8—1662no—
424#102—22no—
425#102—834no—
426#102—1662no—
427#102C4×D8314yes—
428C4×C8—1662no—
429C4×C8—1662no—
430#95—21no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437C24—21no—
438C2×Q8—3322no—
439C2×Q8—3322no—
440C2×Q8—1664no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D8312yes—
445C2×D4Q8×D8312yes—
446C2×D4#1305058912yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—3322no—
454C22×C4—3322no—
455C22×C4—3322no—
456C22×C4—3322no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—1661no—
462C22×C4—1664no—
463C22×C4—1664no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4D4×D8311yescharacteristic
468C22×C4D4×D8311yescharacteristic
469C22×C4C23×D8311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—832no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8C2×C4×D8312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—3322no—
488C4⋊C4—3322no—
489C4⋊C4—3322no—
490C4⋊C4—832no—
491C4⋊C4—832no—
492C4⋊C4—832no—
493C4⋊C4—1664no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4C2×C4×D8312yes—
500C4⋊C4#1305058912yes—
501C22⋊C4—24no—
502C22⋊C4—1664no—
503C22⋊C4—1664no—
504C42—21no—
505C42—21no—
506C42—3321no—
507C42—3322no—
508C42—3322no—
509C42—3322no—
510C42—3322no—
511C42—3322no—
512C42—834no—
513C42—832no—
514C42—832no—
515C42—832no—
516C42—832no—
517C42—1662no—
518C42—1662no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D8311yescharacteristic
527Q8—3328no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#1458175714yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3328no—
554C2×C4—3328no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—832no—
559C2×C4—832no—
560C2×C4—832no—
561C2×C4—832no—
562C2×C4—832no—
563C2×C4—832no—
564C2×C4—832no—
565C2×C4—832no—
566C2×C4—1662no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4D8×D8312yes—
574C2×C4C22×C4×D8311yescharacteristic
575C2×C4C2×Q8×D8311yescharacteristic
576C2×C4#41×D8311yescharacteristic
577C2×C4#41×D8311yescharacteristic
578C2×C4C2×#1305058911yescharacteristic
579C2×C4C2×#1305058911yescharacteristic
580C8—22no—
581C8—41no—
582C8—3321no—
583C8—1662no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D8311yescharacteristic
592C22C22⋊Q8×D8311yescharacteristic
593C22#1912820711yescharacteristic
594C22C2×D8×D8311yescharacteristic
595C22#135×D8311yescharacteristic
596C22#137×D8311yescharacteristic
597C22#1912849711yescharacteristic
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—3322no—
604C4—3322no—
605C4—3322no—
606C4—834no—
607C4—834no—
608C4—6641no—
609C4—1662no—
610C4—1662no—
611C4C2×#1458175711yescharacteristic
612C4C4×#1305058911yescharacteristic
613C4#1912822811yescharacteristic
614C4#1912825311yescharacteristic
615C2—44no—
616C2#2490999511yescharacteristic
617C2C4×D8×D8311yescharacteristic
618C2#2491628511yescharacteristic
619C2#474×D8311yescharacteristic
620C2#2491724811yescharacteristic
621C2#2491745811yescharacteristic
622C2#2491752711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1862C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#1862D83
hypocenterC83C4×#474
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #24910034C2semidirect
4. #24908413C2semidirect
5. #24908218C2semidirect
6. #24908035C2semidirect
7. #24908031C2semidirect
10. #107×Dic83C2semidirect
11. #24907355C2semidirect
12. #24907311C2semidirect
21. #19127979C4semidirect
66. #14581791C2×C4semidirect
71. D4×Dic83Q8semidirect
94. #14581726C2×C4semidirect
96. #14581725C2×C4semidirect
98. Dic83⋊C8C2×C4semidirect
218. D4×C83C4×Q8semidirect
302. Dic83#474semidirect
327. C83C4×#474semidirect
531. D4#14581757semidirect

Maximal subgroups

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

subgroupindex
C2×#191279792
#249100342
C4×D4×Dic832
#249084132
#249082182
#249080352
#249080312
#249077102
#249074612
#107×Dic832
#249073552
#249073112
#102×Dic832
C83×#18622
C4×#47483