Finite Group Databasebeta

← #35466959

Subgroups — #35466959

Order 21248.

882 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127996C211yesmaximal, characteristic, complemented
2C22×#14581725C211yesmaximal, characteristic
3#24911165C211yesmaximal, characteristic
4#24909001C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908858C211yesmaximal, characteristic, complemented
7#24908791C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908055C211yesmaximal, characteristic, complemented
10#24907887C211yesmaximal, characteristic
11#24907771C211yesmaximal, characteristic
12#24907717C211yesmaximal, characteristic, complemented
13#24907651C211yesmaximal, characteristic
14#24907643C211yesmaximal, characteristic
15C83×#2360C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581769—21no—
19C22×C4⋊Dic83C2211yescharacteristic
20#19127996—24no—
21C2×#14581725—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×#14581716C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127633—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127576—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C22⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C22×C4⋊C4×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#387×C83—21no—
57#385×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8—22no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581792—22no—
71#14581792—41no—
72#14581769—44no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
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⋊Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725—22no—
89#14581725—22no—
90#14581725—41no—
91#14581725D414yes—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581716—22no—
95#14581716—22no—
96#14581716—22no—
97#14581716—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×Q8×C83C2311yescharacteristic
103C23×C4×C83—21no—
104C23×C4×C83C2311yescharacteristic
105C22⋊Q8×C83—22no—
106C22⋊Q8×C83Q814yescomplemented
107C4×Q8×C83—22no—
108C4×Q8×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—22no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—41no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C22⋊C4×C83D411yescharacteristic
121C2×C42×C83—21no—
122C2×C42×C83—21no—
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—42no—
139C22×Dic83—42no—
140C22×Dic83—81no—
141C2×C83⋊C8—28no—
142C2×C83⋊C8—21no—
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⋊C8C2×D412yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—42no—
164C4⋊Dic83—81no—
165C4⋊Dic83C2×D412yes—
166Dic83⋊C4—42no—
167Dic83⋊C4—42no—
168C22⋊Dic83—42no—
169C24×C83C2×D411yescharacteristic
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83—21no—
173C2×Q8×C83—21no—
174C2×Q8×C83—21no—
175C2×Q8×C83C2×Q812yes—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—42no—
185C22×C4×C83—42no—
186C22×C4×C83—81no—
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×Q812yes—
193C22×C4×C83#4112yes—
194C22×C4×C83C2411yescharacteristic
195C4⋊C4×C83—24no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—81no—
203C4⋊C4×C83C2×D412yes—
204C4⋊C4×C83C2×Q812yes—
205C22⋊C4×C83—24no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83Q1614yes—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83—21no—
212C42×C83—42no—
213C42×C83—81no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83C22⋊Q812yes—
219C23×C83C2×Q1612yes—
220C23×C83AGL(1,ℤ/8)12yes—
221C23×C83C22×D411yescharacteristic
222Q8×C83—22no—
223Q8×C83—22no—
224Q8×C83—41no—
225Q8×C83—41no—
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—22no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—21no—
239C2×C4×C83—22no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—84no—
255C2×C4×C83C22⋊Q812yes—
256C2×C4×C83C2×Q1612yes—
257C2×C4×C83C2×Q1612yes—
258C2×C4×C83C22×D411yescharacteristic
259C2×C4×C83C22×Q811yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83H(5,𝔽2)11yescharacteristic
263C2×C4×C832−1+411yescharacteristic
264C2×Dic83—22no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—84no—
277Dic166—84no—
278C83⋊C8—22no—
279C83⋊C8—22no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83#47714yescomplemented
289C22×C83C2×C22⋊Q811yescharacteristic
290C22×C83D4211yescharacteristic
291C22×C83#54711yescharacteristic
292C22×C83C22×Q1611yescharacteristic
293C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C83#57611yescharacteristic
295C22×C83#57811yescharacteristic
296C4×C83—22no—
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—42no—
300C4×C83—41no—
301C4×C83—42no—
302C4×C83—41no—
303C4×C83—42no—
304C4×C83—81no—
305C4×C83#52511yescharacteristic
306C4×C83#53911yescharacteristic
307C4×C83D4×Q811yescharacteristic
308C4×C83#55411yescharacteristic
309Dic83—42no—
310Dic83—81no—
311Dic83—81no—
312Dic83—81no—
313#26342—831nomaximal, Sylow
314C2×C83—24no—
315C2×C83#244911yescharacteristic
316C2×C83C2×#47711yescharacteristic
317C2×C83#308311yescharacteristic
318C2×C83#326411yescharacteristic
319C2×C83D4×Q1611yescharacteristic
320C2×C83#333811yescharacteristic
321C2×C83#334311yescharacteristic
322C2×#477—831no—
323C22×#107—831no—
324#2440—831no—
325#2360D8311yescharacteristic
326#2095—831no—
327#2087—831no—
328#2068—831no—
329#2057—831no—
330#1947—831no—
331#1921—831no—
332#1892—831no—
333#1872—831no—
334#1863—831no—
335#1852—831no—
336#1850—831no—
337C83#2634211yescharacteristic, complemented, Sylow
338C23×C8—831no—
339C2×C4⋊Q8—831no—
340C2×C22⋊Q8—1661no—
341C2×C22⋊Q8C2×D8311yescharacteristic
342C2×C4×Q8C2×D8311yescharacteristic
343C22×C4⋊C4—831no—
344C22×C4⋊C4C2×D8311yescharacteristic
345#477—1664no—
346C2×#107—831no—
347C2×#107—831no—
348C2×#107—831no—
349C2×#107—831no—
350C2×#107—831no—
351C2×#107—831no—
352C2×#107—1664no—
353C2×C4⋊C8—831no—
354C2×C4⋊C8—831no—
355C2×#103—831no—
356C2×#103—831no—
357C2×#103—831no—
358C2×#103—831no—
359C2×C22⋊C8—831no—
360#398—1661no—
361#395—1661no—
362#391C2×D8311yescharacteristic
363#389—21no—
364#389—831no—
365#389—831no—
366#389C2×D8311yescharacteristic
367#387—21no—
368#385—21no—
369#385—1661no—
370#384—831no—
371#380—831no—
372C4×C4⋊C4C2×D8311yescharacteristic
373C4×C22⋊C4—21no—
374C4×C22⋊C4C2×D8311yescharacteristic
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379#340—831no—
380#340—831no—
381#336—831no—
382#336—831no—
383C22×Q8—1661no—
384C22×Q8C22×D8311yescharacteristic
385C23×C4—21no—
386C23×C4—1661no—
387C23×C4C22×D8311yescharacteristic
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391C22×C8—831no—
392C22×C8—1662no—
393C22×C8—1662no—
394C22×C8—1662no—
395C4⋊Q8—3321no—
396C4⋊Q8—1662no—
397C22⋊Q8—22no—
398C22⋊Q8—3324no—
399C22⋊Q8Dic16614yescomplemented
400C4×Q8—22no—
401C4×Q8—41no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—41no—
408C2×C4⋊C4—3321no—
409C2×C4⋊C4—831no—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—831no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4—1661no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4—1662no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4C22×D8311yescharacteristic
421C2×C4⋊C4C22×D8311yescharacteristic
422C2×C4⋊C4C22×D8311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1661no—
425C2×C22⋊C4C22⋊D8311yescharacteristic
426C2×C22⋊C4C22⋊D8311yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
429C2×C42C22×D8311yescharacteristic
430C2×C42C22×D8311yescharacteristic
431#107—3321no—
432#107—834no—
433#107—1662no—
434#107—1662no—
435#107—1662no—
436#107—1662no—
437C4⋊C8—1662no—
438C4⋊C8—1662no—
439#103—3321no—
440#103—1662no—
441#103—1662no—
442#103—1662no—
443C22⋊C8—1662no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1661no—
449#95—1661no—
450#95—1661no—
451#95C22⋊D8311yescharacteristic
452#95C22⋊D8311yescharacteristic
453C24C2×C22⋊D8311yescharacteristic
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—3322no—
460C2×Q8—3322no—
461C2×Q8—6641no—
462C2×Q8C2×Dic16612yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—3322no—
475C22×C4—3322no—
476C22×C4—6641no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4C2×C22⋊D8311yescharacteristic
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C2×C22⋊D8311yescharacteristic
484C22×C4C2×C22⋊D8311yescharacteristic
485C22×C4C2×C22⋊D8311yescharacteristic
486C22×C4C2×Dic16612yes—
487C22×C4#1305059012yes—
488C22×C4C23×D8311yescharacteristic
489C2×C8—832no—
490C2×C8—832no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C2×C8—1668no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—3322no—
511C4⋊C4—3322no—
512C4⋊C4—3322no—
513C4⋊C4—3322no—
514C4⋊C4—3322no—
515C4⋊C4—832no—
516C4⋊C4—6641no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4—1661no—
521C4⋊C4—1661no—
522C4⋊C4—1661no—
523C4⋊C4—1661no—
524C4⋊C4D4×D8312yes—
525C4⋊C4C2×Dic16612yes—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—3322no—
529C22⋊C4#1305058014yes—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1458177112yes—
541C23C2×#1305058012yes—
542C23#1458181712yes—
543C23C22×C22⋊D8311yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—6644no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4—3321no—
588C2×C4—3321no—
589C2×C4—6644no—
590C2×C4—1662no—
591C2×C4#1458177112yes—
592C2×C4C2×#1305058012yes—
593C2×C4C2×#1305058012yes—
594C2×C4C2×D4×D8311yescharacteristic
595C2×C4C22×Dic16611yescharacteristic
596C2×C4#41×D8311yescharacteristic
597C2×C4C2×#1305059011yescharacteristic
598C2×C4#1458183711yescharacteristic
599C2×C4#1458183911yescharacteristic
600C8—3321no—
601C8—3321no—
602C8—1662no—
603C8—1662no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1912799614yescomplemented
611C22C2×#1458177111yescharacteristic
612C22D4×C22⋊D8311yescharacteristic
613C22#1912835011yescharacteristic
614C22C22×#1305058011yescharacteristic
615C22C2×#1458181711yescharacteristic
616C22#1912850411yescharacteristic
617C22#1912851611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—3322no—
628C4—6641no—
629C4—6641no—
630C4—6641no—
631C4#1912821011yescharacteristic
632C4#1912829311yescharacteristic
633C4D4×Dic16611yescharacteristic
634C4#1912841011yescharacteristic
635C2—24no—
636C2#2491126611yescharacteristic
637C2C2×#1912799611yescharacteristic
638C2#2491731411yescharacteristic
639C2#2491894411yescharacteristic
640C2D4×#1305058011yescharacteristic
641C2#2491985111yescharacteristic
642C2#2491990311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2360C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2360D83
hypocenterC83#26342
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
1. C2×#19127996C2semidirect
4. #24909001C2semidirect
5. #24908959C2semidirect
6. #24908858C2semidirect
7. #24908791C2semidirect
8. #24908221C2semidirect
9. #24908055C2semidirect
12. #24907717C2semidirect
106. C22⋊Q8×C83Q8semidirect
288. C22×C83#477semidirect
337. C83#26342semidirect
399. C22⋊Q8Dic166semidirect
610. C22#19127996semidirect

Maximal subgroups

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

subgroupindex
C2×#191279962
C22×#145817252
#249111652
#249090012
#249089592
#249088582
#249087912
#249082212
#249080552
#249078872
#249077712
#249077172
#249076512
#249076432
C83×#23602
#2634283