Finite Group Databasebeta

← #35426413

Subgroups — #35426413

Order 21248.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128001C211yesmaximal, characteristic
2#24910557C211yesmaximal, characteristic, complemented
3#24908431C211yesmaximal, characteristic, complemented
4#24908224C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908188C211yesmaximal, characteristic, complemented
7#24908045C211yesmaximal, characteristic, complemented
8#24908031C211yesmaximal, characteristic, complemented
9#24907724C211yesmaximal, characteristic
10#24907724C211yesmaximal, characteristic
11#24907338C211yesmaximal, characteristic, complemented
12#24907288C211yesmaximal, characteristic
13#24907288C211yesmaximal, characteristic
14C4×#14581713C211yesmaximal, characteristic
15D4×C4⋊C4×C83C211yesmaximal, characteristic
16C2×#14581784C2211yescharacteristic
17#19128001C418yescomplemented
18C2×#14581725C2211yescharacteristic
19C2×#14581723C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581703C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127596—21no—
33#19127583—21no—
34#19127582—21no—
35C4×C4⋊Dic83C2211yescharacteristic
36C4×C4⋊Dic83C2211yescharacteristic
37C4×C4⋊Dic83C2211yescharacteristic
38#19127552—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
50C2×C4×D4×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C22×C4⋊C4×C83—21no—
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#385×C83—21no—
56#385×C83—21no—
57#384×C83C2211yescharacteristic
58#380×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581784—22no—
65#14581784C2×C414yescomplemented
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×Dic83⋊C4—21no—
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77#14581725C2×C414yescomplemented
78#14581723—41no—
79#14581723—41no—
80#14581723C2×C414yescomplemented
81C4⋊(C83⋊C8)C2×C414yescomplemented
82#14581713—22no—
83#14581713C2×C414yes—
84#14581713C2×C414yes—
85#14581713Q814yes—
86#14581703—22no—
87#14581703—41no—
88C4×C83⋊C8—41no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C22×D4×C83C2311yescharacteristic
95C23×C4×C83—21no—
96C23×C4×C83—21no—
97C23×C4×C83—21no—
98C4×D4×C83—22no—
99C4×D4×C83—22no—
100C4×D4×C83—22no—
101C4×D4×C83—41no—
102C4×D4×C83—41no—
103C4×D4×C83C2×C414yes—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—44no—
109C2×C4⋊C4×C83D411yescharacteristic
110C2×C4⋊C4×C83D411yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8C22×C412yes—
138C2×C83⋊C8C22×C412yes—
139C2×C83⋊C8C2×Q812yes—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83C22×C412yes—
146C4⋊Dic83C22×C412yes—
147C4⋊Dic83C2×Q812yes—
148Dic83⋊C4—24no—
149Dic83⋊C4—24no—
150Dic83⋊C4—42no—
151Dic83⋊C4—81no—
152C4×Dic83—42no—
153C4×Dic83—81no—
154C4×Dic83—81no—
155C24×C83—21no—
156C2×D4×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83C22×C412yes—
160C2×D4×C83C2×Q812yes—
161C2×D4×C83#4112yes—
162C22×C4×C83—21no—
163C22×C4×C83—24no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—42no—
167C22×C4×C83—42no—
168C22×C4×C83—42no—
169C22×C4×C83—42no—
170C22×C4×C83—42no—
171C22×C4×C83—42no—
172C22×C4×C83—81no—
173C22×C4×C83—81no—
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2411yescharacteristic
181C4⋊C4×C83—22no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—41no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—81no—
192C4⋊C4×C83C22×C412yes—
193C22⋊C4×C83—24no—
194C22⋊C4×C83—42no—
195C22⋊C4×C83—42no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83C22×C412yes—
201C42×C83#4112yes—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83C22×D411yescharacteristic
210D4×C83—22no—
211D4×C83—22no—
212D4×C83—22no—
213D4×C83C4×Q814yes—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—22no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
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—48no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
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—84no—
240C2×C4×C83—84no—
241C2×C4×C83C4×D412yes—
242C2×C4×C83C4×D412yes—
243C2×C4×C83AGL(1,ℤ/8)12yes—
244C2×C4×C83C23×C411yescharacteristic
245C2×C4×C83C22×Q811yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83H(5,𝔽2)11yescharacteristic
249C2×C4×C832−1+411yescharacteristic
250C2×Dic83—22no—
251C2×Dic83—22no—
252C2×Dic83—22no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C22×C83—24no—
267C22×C83—42no—
268C22×C83—42no—
269C22×C83—42no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83C2×C4×D411yescharacteristic
274C22×C83#54611yescharacteristic
275C22×C83D4×Q811yescharacteristic
276C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C83#57511yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57711yescharacteristic
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—42no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83C2×C4×Q811yescharacteristic
290C4×C83#52011yescharacteristic
291C4×C83#55211yescharacteristic
292C4×C83#55411yescharacteristic
293Dic83—42no—
294Dic83—42no—
295Dic83—42no—
296Dic83—81no—
297#22978—831nomaximal, Sylow
298C2×C83—44no—
299C2×C83#236011yescharacteristic
300C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C83#295211yescharacteristic
302C2×C83#330711yescharacteristic
303C2×C83#331011yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338811yescharacteristic
306C2×#478—831no—
307#2370—831no—
308D4×C4⋊C4D8311yescharacteristic
309#1992—831no—
310#1948—831no—
311#1943—831no—
312#1941—831no—
313#1920—831no—
314#1918—831no—
315#1864—831no—
316#1864—831no—
317#1771—831no—
318#1762—831no—
319#1762—831no—
320C4×#102—831no—
321C83#2297811yescharacteristic, complemented, Sylow
322C2×#125—831no—
323C2×C4×D4C2×D8311yescharacteristic
324C2×C4×D4C2×D8311yescharacteristic
325C2×C4×D4C2×D8311yescharacteristic
326C22×C4⋊C4—21no—
327#478—838no—
328C2×#107—831no—
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×C4⋊C8—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#97—831no—
338C2×#97—831no—
339C2×C4×C8—831no—
340#389—831no—
341#389—831no—
342#389—1661no—
343#389C2×D8311yescharacteristic
344#389C2×D8311yescharacteristic
345#387—1661no—
346#387—1661no—
347#385—21no—
348#385—21no—
349#384C2×D8311yescharacteristic
350#380—21no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—1661no—
355C4×C4⋊C4C2×D8311yescharacteristic
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#336—831no—
365#336—831no—
366C22×D4C22×D8311yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—21no—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8—831no—
374#125—834no—
375#125—1662no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—41no—
380C4×D4—41no—
381C4×D4C4×D8314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—44no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22⋊D8311yescharacteristic
396C2×C4⋊C4C22⋊D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C42—1661no—
404C2×C42—1661no—
405C2×C42—1661no—
406C2×C42C22×D8311yescharacteristic
407C2×C42C22×D8311yescharacteristic
408C2×C42C22×D8311yescharacteristic
409#107—834no—
410#106—3321no—
411#106—3321no—
412#106—834no—
413C4⋊C8—834no—
414#102—834no—
415#102—834no—
416#102—834no—
417#102—1662no—
418#97—3321no—
419#97—1662no—
420C4×C8—3321no—
421#95—21no—
422#95—21no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95C22⋊D8311yescharacteristic
429#95C22⋊D8311yescharacteristic
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D8312yes—
435C2×D4Q8×D8312yes—
436C2×D4#1305058912yes—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460C2×C8—832no—
461C2×C8—832no—
462C2×C8—832no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—3322no—
484C4⋊C4—832no—
485C4⋊C4—832no—
486C4⋊C4—832no—
487C4⋊C4—6641no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1664no—
494C4⋊C4—1664no—
495C4⋊C4C2×C4×D8312yes—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—3322no—
504C42—6641no—
505C42—6641no—
506C42C2×C4×D8312yes—
507C42#1305059012yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C22×C22⋊D8311yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4#1458175714yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—1662no—
557C2×C4—1662no—
558C2×C4—1662no—
559C2×C4C4×C22⋊D8312yes—
560C2×C4C4×C22⋊D8312yes—
561C2×C4#1458181812yes—
562C2×C4C22×C4×D8311yescharacteristic
563C2×C4C2×Q8×D8311yescharacteristic
564C2×C4C2×#1305058911yescharacteristic
565C2×C4C2×#1305059011yescharacteristic
566C2×C4#1458183711yescharacteristic
567C2×C4#1458184011yescharacteristic
568C8—3321no—
569C8—1662no—
570C8—1662no—
571C8—1662no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D8311yescharacteristic
580C22#1912834511yescharacteristic
581C22Q8×C22⋊D8311yescharacteristic
582C22C2×#1458181811yescharacteristic
583C22#1912849911yescharacteristic
584C22#1912850911yescharacteristic
585C22#1912851211yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—3322no—
596C4—3322no—
597C4—3322no—
598C4—6641no—
599C4C2×#1458175711yescharacteristic
600C4#1912818211yescharacteristic
601C4#1912838611yescharacteristic
602C4#1912840611yescharacteristic
603C2—44no—
604C2#2491046811yescharacteristic
605C2C4×#1458181811yescharacteristic
606C2#2491634411yescharacteristic
607C2#2491943911yescharacteristic
608C2#2491948411yescharacteristic
609C2#2492035811yescharacteristic
610C2#2492040811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingD4×C4⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterD4×C4⋊C4D83
hypocenterC83#22978
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #24910557C2semidirect
3. #24908431C2semidirect
4. #24908224C2semidirect
5. #24908198C2semidirect
6. #24908188C2semidirect
7. #24908045C2semidirect
8. #24908031C2semidirect
11. #24907338C2semidirect
17. #19128001C4semidirect
65. #14581784C2×C4semidirect
77. #14581725C2×C4semidirect
80. #14581723C2×C4semidirect
81. C4⋊(C83⋊C8)C2×C4semidirect
321. C83#22978semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
#249105572
#249084312
#249082242
#249081982
#249081882
#249080452
#249080312
#249077242
#249073382
#249072882
C4×#145817132
D4×C4⋊C4×C832
#2297883