Finite Group Databasebeta

← #35461566

Subgroups — #35461566

Order 21248.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128008C211yesmaximal, characteristic, complemented
2C22×#14581716C211yesmaximal, characteristic
3#24911187C211yesmaximal, characteristic
4#24909011C211yesmaximal, characteristic, complemented
5#24908858C211yesmaximal, characteristic, complemented
6#24908844C211yesmaximal, characteristic, complemented
7#24908614C211yesmaximal, characteristic, complemented
8#24907731C211yesmaximal, characteristic, complemented
9#24907717C211yesmaximal, characteristic, complemented
10#24907650C211yesmaximal, characteristic
11#24907643C211yesmaximal, characteristic
12#24907486C211yesmaximal, characteristic
13#24907478C211yesmaximal, characteristic, complemented
14#24907398C211yesmaximal, characteristic
15Q8×C22⋊C4×C83C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19#19128008—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×#14581716—24no—
27C2×#14581716C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×#14581716C2211yescharacteristic
34C2×#14581716C2211yescharacteristic
35C2×C22⋊(C83⋊C8)C2211yescharacteristic
36#19127643—21no—
37#19127635—21no—
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127573—21no—
41#19127567C2211yescharacteristic
42#19127561C2211yescharacteristic
43#19127559—21no—
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C23×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#391×C83—21no—
54#391×C83—21no—
55#391×C83C2211yescharacteristic
56#391×C83C2211yescharacteristic
57#380×C83—21no—
58#380×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83C2211yescharacteristic
61C23×Dic83—21no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—41no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—22no—
70#14581784—41no—
71C2×C4⋊Dic83—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—41no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83#14581725—22no—
84#14581725—41no—
85#14581723—22no—
86#14581723—41no—
87C4⋊(C83⋊C8)—22no—
88C4⋊(C83⋊C8)—41no—
89#14581716—22no—
90#14581716—22no—
91#14581716—22no—
92#14581716—22no—
93#14581716—22no—
94#14581716—41no—
95#14581716—41no—
96#14581716D414yes—
97C22⋊(C83⋊C8)—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C22×Q8×C83—21no—
104C22×Q8×C83—22no—
105C22×Q8×C83—22no—
106C22×Q8×C83—22no—
107C22×Q8×C83—41no—
108C22×Q8×C83—41no—
109C22×Q8×C83Q814yes—
110C22×Q8×C83C2311yescharacteristic
111C22×Q8×C83C2311yescharacteristic
112C23×C4×C83—21no—
113C23×C4×C83C2311yescharacteristic
114C4×Q8×C83—22no—
115C4×Q8×C83—22no—
116C4×Q8×C83—41no—
117C4×Q8×C83—41no—
118C2×C4⋊C4×C83—21no—
119C2×C4⋊C4×C83—21no—
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C22⋊C4×C83—21no—
123C2×C22⋊C4×C83D411yescharacteristic
124C2×C22⋊C4×C83D411yescharacteristic
125C2×C42×C83—21no—
126C2×C42×C83—21no—
127C2×C42×C83C2311yescharacteristic
128C2×C42×C83C2311yescharacteristic
129#95×C83—21no—
130#95×C83—21no—
131#95×C83—21no—
132#95×C83—21no—
133#95×C83—21no—
134#95×C83D411yescharacteristic
135#95×C83D411yescharacteristic
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—42no—
141C22×Dic83—42no—
142C22×Dic83—81no—
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⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—44no—
155C2×C83⋊C8C2×D412yes—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—42no—
164C4⋊Dic83C2×D412yes—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167Dic83⋊C4—81no—
168Dic83⋊C4—81no—
169C22⋊Dic83—42no—
170C24×C83C2×D411yescharacteristic
171C2×Q8×C83—24no—
172C2×Q8×C83—21no—
173C2×Q8×C83—21no—
174C2×Q8×C83—21no—
175C2×Q8×C83—21no—
176C2×Q8×C83—28no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83—21no—
181C2×Q8×C83—42no—
182C2×Q8×C83—42no—
183C2×Q8×C83—44no—
184C2×Q8×C83—44no—
185C2×Q8×C83—44no—
186C2×Q8×C83C2×D412yes—
187C2×Q8×C83C2×Q812yes—
188C2×Q8×C83C2×Q812yes—
189C2×Q8×C83#4112yes—
190C22×C4×C83—21no—
191C22×C4×C83—21no—
192C22×C4×C83—21no—
193C22×C4×C83—21no—
194C22×C4×C83—21no—
195C22×C4×C83—24no—
196C22×C4×C83—21no—
197C22×C4×C83—21no—
198C22×C4×C83—21no—
199C22×C4×C83—42no—
200C22×C4×C83—42no—
201C22×C4×C83C2×D411yescharacteristic
202C22×C4×C83C2×D411yescharacteristic
203C22×C4×C83C2×D411yescharacteristic
204C22×C4×C83C2×D411yescharacteristic
205C22×C4×C83C2×D411yescharacteristic
206C22×C4×C83C2×Q812yes—
207C22×C4×C83#4112yes—
208C22×C4×C83C2411yescharacteristic
209C4⋊C4×C83—21no—
210C4⋊C4×C83—21no—
211C4⋊C4×C83—21no—
212C4⋊C4×C83—21no—
213C4⋊C4×C83—42no—
214C4⋊C4×C83—42no—
215C22⋊C4×C83—22no—
216C22⋊C4×C83—22no—
217C22⋊C4×C83—42no—
218C42×C83—21no—
219C42×C83—21no—
220C42×C83—21no—
221C42×C83—21no—
222C42×C83—42no—
223C42×C83—42no—
224C23×C83—21no—
225C23×C83—21no—
226C23×C83—21no—
227C23×C83—21no—
228C23×C83C22⋊Q812yes—
229C23×C83#13512yes—
230C23×C83#13712yes—
231C23×C83C22×D411yescharacteristic
232Q8×C83—22no—
233Q8×C83—22no—
234Q8×C83—22no—
235Q8×C83—22no—
236Q8×C83—22no—
237Q8×C83—48no—
238Q8×C83—41no—
239Q8×C83—41no—
240Q8×C83C22⋊Q814yes—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—21no—
244C2×C4×C83—22no—
245C2×C4×C83—21no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—22no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—21no—
252C2×C4×C83—21no—
253C2×C4×C83—21no—
254C2×C4×C83—21no—
255C2×C4×C83—21no—
256C2×C4×C83—21no—
257C2×C4×C83—21no—
258C2×C4×C83—21no—
259C2×C4×C83—21no—
260C2×C4×C83—21no—
261C2×C4×C83—21no—
262C2×C4×C83—21no—
263C2×C4×C83—48no—
264C2×C4×C83—41no—
265C2×C4×C83—41no—
266C2×C4×C83—41no—
267C2×C4×C83—41no—
268C2×C4×C83—41no—
269C2×C4×C83—41no—
270C2×C4×C83—41no—
271C2×C4×C83—41no—
272C2×C4×C83C22⋊Q812yes—
273C2×C4×C83C22×D411yescharacteristic
274C2×C4×C83C22×Q811yescharacteristic
275C2×C4×C83C2×#4111yescharacteristic
276C2×C4×C83C2×#4111yescharacteristic
277C2×C4×C83H(5,𝔽2)11yescharacteristic
278C2×C4×C83H(5,𝔽2)11yescharacteristic
279C2×Dic83—22no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283C2×Dic83—41no—
284C2×Dic83—41no—
285C2×Dic83—41no—
286C2×Dic83—41no—
287C2×Dic83—41no—
288C2×Dic83—41no—
289C2×Dic83—41no—
290C2×Dic83—41no—
291C2×Dic83—84no—
292C83⋊C8—22no—
293C83⋊C8—41no—
294C83⋊C8—41no—
295C83⋊C8—41no—
296C22×C83—22no—
297C22×C83—22no—
298C22×C83—22no—
299C22×C83—22no—
300C22×C83—22no—
301C22×C83—22no—
302C22×C83#47914yescomplemented
303C22×C83C2×C22⋊Q811yescharacteristic
304C22×C83D4211yescharacteristic
305C22×C83#54611yescharacteristic
306C22×C83C2×#13511yescharacteristic
307C22×C83C2×#13711yescharacteristic
308C22×C83#57711yescharacteristic
309C22×C83#57811yescharacteristic
310C4×C83—22no—
311C4×C83—24no—
312C4×C83—42no—
313C4×C83—41no—
314C4×C83—41no—
315C4×C83—41no—
316C4×C83—41no—
317C4×C83—81no—
318C4×C83—81no—
319C4×C83C2×C22⋊Q811yescharacteristic
320C4×C83#54311yescharacteristic
321C4×C83#54611yescharacteristic
322C4×C83#55111yescharacteristic
323Dic83—42no—
324Dic83—81no—
325Dic83—81no—
326Dic83—81no—
327#25957—831nomaximal, Sylow
328C2×C83—24no—
329C2×C83#243711yescharacteristic
330C2×C83C2×#47911yescharacteristic
331C2×C83#308211yescharacteristic
332C2×C83#327011yescharacteristic
333C2×C83#328211yescharacteristic
334C2×C83#330011yescharacteristic
335C2×C83#332011yescharacteristic
336C2×#479—831no—
337C22×#103—831no—
338#2442—831no—
339Q8×C22⋊C4D8311yescharacteristic
340#2097—831no—
341#2068—831no—
342#2066—831no—
343#2026—831no—
344#1865—831no—
345#1863—831no—
346#1852—831no—
347#1850—831no—
348#1807—831no—
349#1805—831no—
350#1787—831no—
351C83#2595711yescharacteristic, complemented, Sylow
352C23×Q8C2×D8311yescharacteristic
353C23×C8—831no—
354C2×#125—831no—
355C2×C4×Q8C2×D8311yescharacteristic
356C2×C4×Q8C2×D8311yescharacteristic
357C22×C4⋊C4—831no—
358#479—1664no—
359C2×#107—831no—
360C2×#107—831no—
361C2×#106—831no—
362C2×#106—831no—
363C2×C4⋊C8—831no—
364C2×C4⋊C8—831no—
365C2×#103—831no—
366C2×#103—831no—
367C2×#103—831no—
368C2×#103—831no—
369C2×#103—831no—
370C2×#103—831no—
371C2×#103—831no—
372C2×#103—831no—
373C2×#103—1664no—
374C2×C22⋊C8—831no—
375#400—1661no—
376#398—1661no—
377#391—21no—
378#391—21no—
379#391C2×D8311yescharacteristic
380#391C2×D8311yescharacteristic
381#389—831no—
382#389—831no—
383#385—1661no—
384#382—831no—
385#380—21no—
386#380—831no—
387#380—1661no—
388#380C2×D8311yescharacteristic
389C4×C22⋊C4—21no—
390C4×C22⋊C4C2×D8311yescharacteristic
391#340—831no—
392#340—831no—
393#340—831no—
394#340—831no—
395#336—831no—
396#336—831no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—41no—
402C22×Q8—41no—
403C22×Q8Dic16614yes—
404C22×Q8C22×D8311yescharacteristic
405C22×Q8C22×D8311yescharacteristic
406C23×C4—21no—
407C23×C4—1661no—
408C23×C4C22×D8311yescharacteristic
409C22×C8—3321no—
410C22×C8—831no—
411C22×C8—831no—
412C22×C8—831no—
413C22×C8—831no—
414C22×C8—1662no—
415C22×C8—1662no—
416#125—3321no—
417#125—1662no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—41no—
421C4×Q8—41no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—3321no—
425C2×C4⋊C4—831no—
426C2×C4⋊C4—831no—
427C2×C4⋊C4—831no—
428C2×C4⋊C4—831no—
429C2×C4⋊C4—1661no—
430C2×C4⋊C4—1661no—
431C2×C4⋊C4—1661no—
432C2×C4⋊C4—1662no—
433C2×C4⋊C4—1661no—
434C2×C4⋊C4—1661no—
435C2×C4⋊C4C22×D8311yescharacteristic
436C2×C4⋊C4C22×D8311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1661no—
439C2×C22⋊C4C22⋊D8311yescharacteristic
440C2×C22⋊C4C22⋊D8311yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42C22×D8311yescharacteristic
444C2×C42C22×D8311yescharacteristic
445#107—3321no—
446#107—1662no—
447#106—3321no—
448#106—1662no—
449C4⋊C8—3321no—
450C4⋊C8—1662no—
451#103—3321no—
452#103—3321no—
453#103—834no—
454#103—1662no—
455#103—1662no—
456#103—1662no—
457#103—1662no—
458#103—1662no—
459C22⋊C8—1662no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1661no—
466#95—1661no—
467#95—1661no—
468#95—1661no—
469#95—1661no—
470#95C22⋊D8311yescharacteristic
471#95C22⋊D8311yescharacteristic
472C24C2×C22⋊D8311yescharacteristic
473C2×Q8—24no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8D4×D8312yes—
489C2×Q8C2×Dic16612yes—
490C2×Q8C2×Dic16612yes—
491C2×Q8#1305058912yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—3322no—
504C22×C4—3322no—
505C22×C4—6641no—
506C22×C4—1661no—
507C22×C4—1661no—
508C22×C4—1661no—
509C22×C4—1661no—
510C22×C4C2×C22⋊D8311yescharacteristic
511C22×C4C2×C22⋊D8311yescharacteristic
512C22×C4C2×C22⋊D8311yescharacteristic
513C22×C4C2×C22⋊D8311yescharacteristic
514C22×C4C2×C22⋊D8311yescharacteristic
515C22×C4C2×Dic16612yes—
516C22×C4#1305059012yes—
517C22×C4C23×D8311yescharacteristic
518C2×C8—3324no—
519C2×C8—832no—
520C2×C8—1661no—
521C2×C8—1661no—
522C2×C8—1661no—
523C2×C8—1661no—
524C2×C8—1661no—
525C2×C8—1661no—
526C2×C8—1661no—
527C2×C8—1661no—
528C2×C8—1661no—
529C2×C8—1661no—
530C2×C8—1661no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—3322no—
538C4⋊C4—3322no—
539C4⋊C4—3322no—
540C4⋊C4—832no—
541C4⋊C4—6641no—
542C4⋊C4—6641no—
543C4⋊C4—1661no—
544C4⋊C4—1661no—
545C4⋊C4—1661no—
546C4⋊C4—1661no—
547C4⋊C4—1661no—
548C4⋊C4—1661no—
549C4⋊C4—1661no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—42no—
553C22⋊C4—3322no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—42no—
559C42—42no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1458177112yes—
565C23#1458181212yes—
566C23#1458182312yes—
567C23C22×C22⋊D8311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8#1458176914yes—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—3321no—
609C2×C4—3321no—
610C2×C4—3321no—
611C2×C4—3321no—
612C2×C4—3321no—
613C2×C4—3321no—
614C2×C4—3321no—
615C2×C4—3321no—
616C2×C4—3321no—
617C2×C4—3321no—
618C2×C4—3321no—
619C2×C4—6644no—
620C2×C4—1662no—
621C2×C4#1458177112yes—
622C2×C4C2×D4×D8311yescharacteristic
623C2×C4C22×Dic16611yescharacteristic
624C2×C4C2×#1305058911yescharacteristic
625C2×C4C2×#1305059011yescharacteristic
626C2×C4#1458183711yescharacteristic
627C2×C4#1458183711yescharacteristic
628C8—3321no—
629C8—3321no—
630C8—3321no—
631C8—1662no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1912800814yescomplemented
639C22C2×#1458177111yescharacteristic
640C22D4×C22⋊D8311yescharacteristic
641C22#1912834511yescharacteristic
642C22C2×#1458181211yescharacteristic
643C22C2×#1458182311yescharacteristic
644C22#1912850911yescharacteristic
645C22#1912851611yescharacteristic
646C4—22no—
647C4—24no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—3322no—
656C4—6641no—
657C4—6641no—
658C4—6641no—
659C4C2×#1458176911yescharacteristic
660C4#1912832711yescharacteristic
661C4#1912834111yescharacteristic
662C4#1912837711yescharacteristic
663C2—24no—
664C2#2491112411yescharacteristic
665C2C2×#1912800811yescharacteristic
666C2#2491730511yescharacteristic
667C2#2491903411yescharacteristic
668C2#2491917111yescharacteristic
669C2#2491935011yescharacteristic
670C2#2491963411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingQ8×C22⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenterQ8×C22⋊C4D83
hypocenterC83#25957
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19128008C2semidirect
4. #24909011C2semidirect
5. #24908858C2semidirect
6. #24908844C2semidirect
7. #24908614C2semidirect
8. #24907731C2semidirect
9. #24907717C2semidirect
13. #24907478C2semidirect
302. C22×C83#479semidirect
351. C83#25957semidirect
638. C22#19128008semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C22×#145817162
#249111872
#249090112
#249088582
#249088442
#249086142
#249077312
#249077172
#249076502
#249076432
#249074862
#249074782
#249073982
Q8×C22⋊C4×C832
#2595783