Finite Group Databasebeta

← #35426697

Subgroups — #35426697

Order 21248.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127920C211yesmaximal, characteristic
2C2×C4×#13050580C211yesmaximal, characteristic, complemented
3C2×C4×#13050578C211yesmaximal, characteristic
4#24910383C211yesmaximal, characteristic, complemented
5#24908275C211yesmaximal, characteristic
6#24908231C211yesmaximal, characteristic, complemented
7#24908149C211yesmaximal, characteristic
8#24908142C211yesmaximal, characteristic, complemented
9#24907821C211yesmaximal, characteristic
10#24907807C211yesmaximal, characteristic, complemented
11#24907731C211yesmaximal, characteristic, complemented
12#24907710C211yesmaximal, characteristic
13C4⋊C4×C83⋊C8C211yesmaximal, characteristic, complemented
14C4×#14581715C211yesmaximal, characteristic, complemented
15C83×#2367C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C22×#13050578C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20C2×C4×Dic166C2211yescharacteristic
21#19127920C418yescomplemented
22C4×#13050580—24no—
23C4×#13050578—24no—
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×#14581716C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581713C2211yescharacteristic
34C2×#14581713C2211yescharacteristic
35C2×C4×C83⋊C8C2211yescharacteristic
36C2×C4×C83⋊C8C2211yescharacteristic
37C2×C4×C83⋊C8C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127596—21no—
41#19127596—21no—
42#19127567C2211yescharacteristic
43#19127565—21no—
44#19127552—21no—
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127456C2211yescharacteristic
50#19127456C2211yescharacteristic
51C2×#124×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#391×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#388×C83—21no—
57#388×C83—21no—
58#385×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C4×C22⋊C4×C83—21no—
62C22×Dic166C2311yescharacteristic
63C2×#13050580—22no—
64C2×#13050580—22no—
65C2×#13050580C2×C414yescomplemented
66C2×#13050578—22no—
67C2×#13050578—22no—
68C2×#13050578C2×C414yes—
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71C22×C83⋊C8C2311yescharacteristic
72C22×C83⋊C8C2311yescharacteristic
73C4×Dic166—22no—
74C4×Dic166—22no—
75C4×Dic166—22no—
76C4×Dic166—41no—
77C4×Dic166—41no—
78C4×Dic166C2×C414yescomplemented
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90#14581725—22no—
91#14581723—22no—
92C4⋊(C83⋊C8)C2×C414yescomplemented
93#14581716—22no—
94#14581716—41no—
95#14581716C2×C414yescomplemented
96#14581715—22no—
97#14581715D414yes—
98#14581713—22no—
99#14581713—41no—
100#14581713C2×C414yes—
101C4×C83⋊C8—22no—
102C4×C83⋊C8—22no—
103C4×C83⋊C8—22no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C22×Q8×C83C2311yescharacteristic
109C22×D4×C83C2311yescharacteristic
110C23×C4×C83—21no—
111#124×C83—22no—
112#124×C83C2×C414yes—
113C4×Q8×C83—22no—
114C4×Q8×C83—41no—
115C4×D4×C83—22no—
116C4×D4×C83—41no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83D411yescharacteristic
119C2×C4⋊C4×C83D411yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C22⋊C4×C83—21no—
123C2×C22⋊C4×C83—21no—
124C2×C22⋊C4×C83—21no—
125C2×C22⋊C4×C83—21no—
126C2×C22⋊C4×C83—21no—
127C2×C42×C83—21no—
128C2×C42×C83—21no—
129C2×C42×C83C2311yescharacteristic
130C2×C42×C83C2311yescharacteristic
131C2×C42×C83C2311yescharacteristic
132#95×C83—21no—
133#95×C83—21no—
134#95×C83—21no—
135#95×C83D411yescharacteristic
136#95×C83D411yescharacteristic
137C2×Dic166—21no—
138C2×Dic166—21no—
139C2×Dic166—21no—
140C2×Dic166C22×C412yes—
141C2×Dic166C2×D412yes—
142C2×Dic166#4112yes—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C22×Dic83—21no—
147#13050580—28no—
148#13050578—28no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8C22×C412yes—
159C2×C83⋊C8C22×C412yes—
160C2×C83⋊C8C2×D412yes—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—21no—
166C4⋊Dic83C22×C412yes—
167Dic83⋊C4—24no—
168Dic83⋊C4—42no—
169Dic83⋊C4—42no—
170Dic83⋊C4—42no—
171Dic83⋊C4—42no—
172C4×Dic83—24no—
173C4×Dic83—42no—
174C4×Dic83—42no—
175C24×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83—21no—
181C2×Q8×C83C22×C412yes—
182C2×D4×C83—21no—
183C2×D4×C83—21no—
184C2×D4×C83—21no—
185C2×D4×C83—21no—
186C2×D4×C83—21no—
187C2×D4×C83C22×C412yes—
188C22×C4×C83—21no—
189C22×C4×C83—21no—
190C22×C4×C83—21no—
191C22×C4×C83—42no—
192C22×C4×C83—42no—
193C22×C4×C83—81no—
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83C2×D411yescharacteristic
200C22×C4×C83C2411yescharacteristic
201C4⋊C4×C83—22no—
202C4⋊C4×C83—22no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—42no—
207C4⋊C4×C83C2×D412yes—
208C22⋊C4×C83—24no—
209C22⋊C4×C83—24no—
210C22⋊C4×C83—42no—
211C22⋊C4×C83—42no—
212C22⋊C4×C83—81no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—21no—
216C42×C83—21no—
217C42×C83—42no—
218C42×C83—81no—
219C42×C83C22×C412yes—
220C42×C83#4112yes—
221C23×C83—22no—
222C23×C83—41no—
223C23×C83—41no—
224C23×C83—41no—
225C23×C83—41no—
226C23×C83—41no—
227C23×C83—41no—
228C23×C83C22×D411yescharacteristic
229Q8×C83—22no—
230Q8×C83—22no—
231Q8×C83—41no—
232Q8×C83—41no—
233D4×C83—22no—
234D4×C83—22no—
235D4×C83—41no—
236D4×C83—41no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—22no—
249C2×C4×C83—21no—
250C2×C4×C83—21no—
251C2×C4×C83—21no—
252C2×C4×C83—21no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—41no—
260C2×C4×C83—84no—
261C2×C4×C83C4×D412yes—
262C2×C4×C83C4×D412yes—
263C2×C4×C83#13512yes—
264C2×C4×C83#13512yes—
265C2×C4×C83C23×C411yescharacteristic
266C2×C4×C83C22×D411yescharacteristic
267C2×C4×C83C2×#4111yescharacteristic
268C2×C4×C83C2×#4111yescharacteristic
269C2×C4×C83H(5,𝔽2)11yescharacteristic
270C2×C4×C832−1+411yescharacteristic
271C2×Dic83—22no—
272C2×Dic83—22no—
273C2×Dic83—22no—
274C2×Dic83—22no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—41no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283Dic166—22no—
284Dic166—22no—
285Dic166—22no—
286Dic166C4×D414yes—
287C83⋊C8—22no—
288C83⋊C8—22no—
289C83⋊C8—22no—
290C83⋊C8—41no—
291C22×C83—42no—
292C22×C83—42no—
293C22×C83—42no—
294C22×C83—81no—
295C22×C83—81no—
296C22×C83—81no—
297C22×C83—81no—
298C22×C83C2×C4×D411yescharacteristic
299C22×C83D4211yescharacteristic
300C22×C83#54811yescharacteristic
301C22×C83C2×#13511yescharacteristic
302C22×C83C2×#13511yescharacteristic
303C22×C83#57711yescharacteristic
304C22×C83#57811yescharacteristic
305C4×C83—22no—
306C4×C83—22no—
307C4×C83—22no—
308C4×C83—22no—
309C4×C83—42no—
310C4×C83—41no—
311C4×C83—42no—
312C4×C83—41no—
313C4×C83—81no—
314C4×C83C2×C4×D411yescharacteristic
315C4×C83#52011yescharacteristic
316C4×C83#54711yescharacteristic
317C4×C83#55311yescharacteristic
318Dic83—24no—
319Dic83—42no—
320Dic83—81no—
321Dic83—81no—
322#23000—831nomaximal, Sylow
323C2×C83—82no—
324C2×C83#236111yescharacteristic
325C2×C83C4×#13511yescharacteristic
326C2×C83#294711yescharacteristic
327C2×C83#328211yescharacteristic
328C2×C83#328511yescharacteristic
329C2×C83#336411yescharacteristic
330C2×C83#336511yescharacteristic
331C2×#464—831no—
332C2×C4×Q16—831no—
333C2×C4×SD16—831no—
334#2367D8311yescharacteristic
335#2351—831no—
336#1958—831no—
337#1949—831no—
338#1934—831no—
339#1933—831no—
340#1880—831no—
341#1878—831no—
342#1865—831no—
343#1862—831no—
344C8×C4⋊C4—831no—
345C4×#103—831no—
346C83#2300011yescharacteristic, complemented, Sylow
347C22×Q16—831no—
348C22×SD16—831no—
349C2×#124C2×D8311yescharacteristic
350C2×C4×Q8—831no—
351C2×C4×Q8—831no—
352C2×C4×Q8—831no—
353C2×C4×Q8C2×D8311yescharacteristic
354C2×C4×D4C2×D8311yescharacteristic
355#464—838no—
356C4×Q16—1664no—
357C4×SD16—1664no—
358C2×#107—831no—
359C2×#106—831no—
360C2×C4⋊C8—831no—
361C2×#103—831no—
362C2×#103—831no—
363C2×#103—831no—
364C2×#103—831no—
365C2×#103—831no—
366C2×#102—831no—
367C2×#102—831no—
368C2×#102—831no—
369C2×C4×C8—831no—
370C2×C4×C8—831no—
371C2×C4×C8—831no—
372#391C2×D8311yescharacteristic
373#390C2×D8311yescharacteristic
374#389—831no—
375#389—831no—
376#389—1661no—
377#389—1661no—
378#388—21no—
379#388—21no—
380#385—21no—
381#382—831no—
382#382—1661no—
383C4×C4⋊C4—1661no—
384C4×C4⋊C4C2×D8311yescharacteristic
385C4×C4⋊C4C2×D8311yescharacteristic
386C4×C22⋊C4—21no—
387#340—831no—
388#340—831no—
389#340—831no—
390#340—831no—
391#336—831no—
392#336—831no—
393C22×Q8—831no—
394C22×Q8C22×D8311yescharacteristic
395C22×D4C22×D8311yescharacteristic
396C23×C4—21no—
397C2×Q16—834no—
398C2×Q16—1662no—
399C2×Q16—1662no—
400C2×SD16—834no—
401C2×SD16—1662no—
402C2×SD16—1662no—
403C22×C8—831no—
404C22×C8—831no—
405C22×C8—831no—
406C22×C8—831no—
407#124—22no—
408#124C4×D8314yes—
409C4×Q8—22no—
410C4×Q8—41no—
411C4×Q8—3321no—
412C4×Q8—3321no—
413C4×Q8—834no—
414C4×Q8—1662no—
415C4×Q8—1662no—
416C4×Q8—1662no—
417C4×D4—22no—
418C4×D4—41no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—831no—
421C2×C4⋊C4—831no—
422C2×C4⋊C4—831no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4—1661no—
425C2×C4⋊C4—1661no—
426C2×C4⋊C4—1661no—
427C2×C4⋊C4—1661no—
428C2×C4⋊C4C22⋊D8311yescharacteristic
429C2×C4⋊C4C22⋊D8311yescharacteristic
430C2×C4⋊C4C22×D8311yescharacteristic
431C2×C4⋊C4C22×D8311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—1661no—
440C2×C42—1661no—
441C2×C42—1661no—
442C2×C42C22×D8311yescharacteristic
443C2×C42C22×D8311yescharacteristic
444C2×C42C22×D8311yescharacteristic
445#107—1662no—
446#106—1662no—
447C4⋊C8—834no—
448#103—3321no—
449#103—834no—
450#103—834no—
451#103—1662no—
452#103—1662no—
453#102—3321no—
454#102—834no—
455#102—1662no—
456C4×C8—1662no—
457C4×C8—1662no—
458C4×C8—1662no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1661no—
463#95—1661no—
464#95—1661no—
465#95—1661no—
466#95C22⋊D8311yescharacteristic
467#95C22⋊D8311yescharacteristic
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—832no—
475C2×Q8—832no—
476C2×Q8—832no—
477C2×Q8—1661no—
478C2×Q8—1661no—
479C2×Q8—1661no—
480C2×Q8C2×C4×D8312yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D8312yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—81no—
493C22×C4—1661no—
494C22×C4—1661no—
495C22×C4—1661no—
496C22×C4—1661no—
497C22×C4C2×C22⋊D8311yescharacteristic
498C22×C4C2×C22⋊D8311yescharacteristic
499C22×C4C2×C22⋊D8311yescharacteristic
500C22×C4C2×C22⋊D8311yescharacteristic
501C22×C4C2×C22⋊D8311yescharacteristic
502C22×C4C2×C22⋊D8311yescharacteristic
503C22×C4C23×D8311yescharacteristic
504Q16—1668no—
505SD16—1668no—
506C2×C8—832no—
507C2×C8—832no—
508C2×C8—832no—
509C2×C8—1661no—
510C2×C8—1661no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—3322no—
525C4⋊C4—3322no—
526C4⋊C4—3322no—
527C4⋊C4—3322no—
528C4⋊C4—832no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C4⋊C4—1664no—
535C4⋊C4D4×D8312yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—81no—
547C42—3322no—
548C42—3322no—
549C42—1664no—
550C42C2×C4×D8312yes—
551C42#1305059012yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C22×C22⋊D8311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—834no—
565Q8—1662no—
566Q8—1662no—
567Q8—1662no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4—3321no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4—1662no—
607C2×C4—1662no—
608C2×C4C4×C22⋊D8312yes—
609C2×C4C4×C22⋊D8312yes—
610C2×C4#1458181212yes—
611C2×C4#1458181212yes—
612C2×C4C22×C4×D8311yescharacteristic
613C2×C4C2×D4×D8311yescharacteristic
614C2×C4#41×D8311yescharacteristic
615C2×C4C2×#1305059011yescharacteristic
616C2×C4#1458183711yescharacteristic
617C2×C4#1458183911yescharacteristic
618C8—3321no—
619C8—1662no—
620C8—1662no—
621C8—1662no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×C4×C22⋊D8311yescharacteristic
630C22D4×C22⋊D8311yescharacteristic
631C22#1912835811yescharacteristic
632C22C2×#1458181211yescharacteristic
633C22C2×#1458181211yescharacteristic
634C22#1912851211yescharacteristic
635C22#1912851611yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—3322no—
646C4—6641no—
647C4—6641no—
648C4—1664no—
649C4C4×D4×D8311yescharacteristic
650C4#1912818011yescharacteristic
651C4#1912835611yescharacteristic
652C4#1912840111yescharacteristic
653C2—82no—
654C2#2491047511yescharacteristic
655C2C4×#1458181211yescharacteristic
656C2#2491630211yescharacteristic
657C2#2491916611yescharacteristic
658C2#2491920011yescharacteristic
659C2#2492014411yescharacteristic
660C2#2492015511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2367C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#2367D83
hypocenterC83#23000
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13050580C2semidirect
4. #24910383C2semidirect
6. #24908231C2semidirect
8. #24908142C2semidirect
10. #24907807C2semidirect
11. #24907731C2semidirect
13. C4⋊C4×C83⋊C8C2semidirect
14. C4×#14581715C2semidirect
21. #19127920C4semidirect
65. C2×#13050580C2×C4semidirect
78. C4×Dic166C2×C4semidirect
92. C4⋊(C83⋊C8)C2×C4semidirect
95. #14581716C2×C4semidirect
346. C83#23000semidirect

Maximal subgroups

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

subgroupindex
C2×#191279202
C2×C4×#130505802
C2×C4×#130505782
#249103832
#249082752
#249082312
#249081492
#249081422
#249078212
#249078072
#249077312
#249077102
C4⋊C4×C83⋊C82
C4×#145817152
C83×#23672
#2300083