Finite Group Databasebeta

← #35462987

Subgroups — #35462987

Order 21248.

872 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127991C211yesmaximal, characteristic
2C2×#19127981C211yesmaximal, characteristic
3#24912489C211yesmaximal, characteristic, complemented
4#24908965C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908796C211yesmaximal, characteristic, complemented
7#24908791C211yesmaximal, characteristic, complemented
8#24908229C211yesmaximal, characteristic, complemented
9#24908187C211yesmaximal, characteristic, complemented
10#24907724C211yesmaximal, characteristic
11#24907710C211yesmaximal, characteristic
12#24907632C211yesmaximal, characteristic, complemented
13#24907463C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2349C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581791—21no—
19#19127991—24no—
20#19127981—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581713C2211yescharacteristic
34C2×#14581703C2211yescharacteristic
35#19127622—21no—
36#19127622—21no—
37#19127600—21no—
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127571C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×D4×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#385×C83—21no—
56#385×C83—21no—
57#382×C83C2211yescharacteristic
58#380×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C2×#95×C83—21no—
61C22×Dic166—21no—
62C22×Dic166—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581792—22no—
68#14581792—22no—
69#14581792—41no—
70#14581792—41no—
71#14581791—44no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C4×Dic83—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—41no—
86#14581723—22no—
87#14581723—22no—
88#14581723—41no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91C4⋊(C83⋊C8)—22no—
92#14581713—22no—
93#14581713—22no—
94#14581713D414yes—
95#14581713Q814yescomplemented
96#14581703—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C22×D4×C83C2311yescharacteristic
101C23×C4×C83—21no—
102C23×C4×C83—21no—
103C23×C4×C83—21no—
104C4×D4×C83—22no—
105C4×D4×C83—22no—
106C4×D4×C83—22no—
107C4×D4×C83—41no—
108C4×D4×C83—41no—
109C4×D4×C83Q814yescomplemented
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83D411yescharacteristic
113C2×C4⋊C4×C83D411yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C22⋊C4×C83—21no—
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83D411yescharacteristic
128#95×C83D411yescharacteristic
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C2×Dic166—81no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
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⋊C8C2×D412yes—
150C2×C83⋊C8C2×Q812yes—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83—42no—
159C4⋊Dic83—42no—
160C4⋊Dic83—81no—
161C4⋊Dic83C2×D412yes—
162C4⋊Dic83C2×Q812yes—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165C4×Dic83—42no—
166C24×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83C2×D412yes—
171C2×D4×C83C2×Q812yes—
172C2×D4×C83C2×Q812yes—
173C22×C4×C83—21no—
174C22×C4×C83—24no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—42no—
178C22×C4×C83—42no—
179C22×C4×C83—42no—
180C22×C4×C83—42no—
181C22×C4×C83—81no—
182C22×C4×C83—81no—
183C22×C4×C83—81no—
184C22×C4×C83—81no—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2411yescharacteristic
192C4⋊C4×C83—22no—
193C4⋊C4×C83—22no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—81no—
200C4⋊C4×C83—81no—
201C4⋊C4×C83C2×Q812yes—
202C22⋊C4×C83—24no—
203C22⋊C4×C83—42no—
204C22⋊C4×C83—42no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—21no—
208C42×C83—21no—
209C42×C83C2×Q812yes—
210C42×C83#4112yes—
211C23×C83—22no—
212C23×C83—22no—
213C23×C83—22no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83—41no—
217C23×C83—41no—
218C23×C83C22×D411yescharacteristic
219D4×C83—22no—
220D4×C83—22no—
221D4×C83—22no—
222D4×C83C4⋊Q814yescomplemented
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—21no—
231C2×C4×C83—22no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—84no—
247C2×C4×C83—84no—
248C2×C4×C83—84no—
249C2×C4×C83C22⋊Q812yes—
250C2×C4×C83C22⋊Q812yes—
251C2×C4×C83AGL(1,ℤ/8)12yes—
252C2×C4×C83#13712yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C22×Q811yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×C4×C832−1+411yescharacteristic
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271Dic166—84no—
272Dic166—84no—
273C83⋊C8—22no—
274C83⋊C8—22no—
275C83⋊C8—41no—
276C83⋊C8—41no—
277C22×C83—24no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—42no—
281C22×C83—42no—
282C22×C83—42no—
283C22×C83—42no—
284C22×C83C2×C22⋊Q811yescharacteristic
285C22×C83D4211yescharacteristic
286C22×C83D4×Q811yescharacteristic
287C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C83C2×#13711yescharacteristic
289C22×C83#57611yescharacteristic
290C22×C83#57711yescharacteristic
291C4×C83—22no—
292C4×C83—22no—
293C4×C83—22no—
294C4×C83—22no—
295C4×C83—42no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—81no—
299C4×C83—81no—
300C4×C83C2×C4⋊Q811yescharacteristic
301C4×C83#54411yescharacteristic
302C4×C83#54711yescharacteristic
303C4×C83#55411yescharacteristic
304Dic83—42no—
305Dic83—42no—
306Dic83—81no—
307Dic83—81no—
308#26058—831nomaximal, Sylow
309C2×C83—44no—
310C2×C83#244011yescharacteristic
311C2×C83#307911yescharacteristic
312C2×C83#308111yescharacteristic
313C2×C83#326411yescharacteristic
314C2×C83#326611yescharacteristic
315C2×C83#338411yescharacteristic
316C2×C83#338511yescharacteristic
317C2×#476—831no—
318C2×#474—831no—
319#2551—831no—
320#2349D8311yescharacteristic
321#2088—831no—
322#2087—831no—
323#2058—831no—
324#2057—831no—
325#1949—831no—
326#1940—831no—
327#1864—831no—
328#1862—831no—
329#1847—831no—
330#1802—831no—
331#1762—831no—
332C83#2605811yescharacteristic, complemented, Sylow
333C2×C4⋊Q8—831no—
334C2×C4⋊Q8—831no—
335C2×C4⋊Q8—1661no—
336C2×C4×D4C2×D8311yescharacteristic
337C2×C4×D4C2×D8311yescharacteristic
338C2×C4×D4C2×D8311yescharacteristic
339#476—1664no—
340#474—1664no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#107—831no—
344C2×#106—831no—
345C2×#106—831no—
346C2×#106—831no—
347C2×C4⋊C8—831no—
348C2×C4⋊C8—831no—
349C2×C4⋊C8—831no—
350C2×#102—831no—
351C2×#102—831no—
352C2×#102—831no—
353C2×#102—831no—
354C2×#97—831no—
355#395—1661no—
356#395—1661no—
357#389—831no—
358#389—831no—
359#389—1661no—
360#389C2×D8311yescharacteristic
361#389C2×D8311yescharacteristic
362#385—21no—
363#385—21no—
364#384—831no—
365#382C2×D8311yescharacteristic
366#380—21no—
367C4×C4⋊C4—831no—
368C4×C4⋊C4C2×D8311yescharacteristic
369C2×#95—21no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#336—831no—
377#336—831no—
378C22×Q8—1661no—
379C22×Q8—1661no—
380C22×D4C22×D8311yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4—21no—
384C22×C8—831no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—831no—
388C4⋊Q8—3321no—
389C4⋊Q8—3321no—
390C4⋊Q8—3324no—
391C4⋊Q8—1662no—
392C4⋊Q8—1662no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—41no—
397C4×D4—41no—
398C4×D4Dic16614yescomplemented
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—831no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4C22⋊D8311yescharacteristic
412C2×C4⋊C4C22⋊D8311yescharacteristic
413C2×C4⋊C4C22×D8311yescharacteristic
414C2×C4⋊C4C22×D8311yescharacteristic
415C2×C4⋊C4C22×D8311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—1661no—
420C2×C42C22×D8311yescharacteristic
421C2×C42C22×D8311yescharacteristic
422C2×C42C22×D8311yescharacteristic
423#107—3321no—
424#107—1662no—
425#107—1662no—
426#106—3321no—
427#106—1662no—
428#106—1662no—
429C4⋊C8—1662no—
430C4⋊C8—1662no—
431C4⋊C8—1662no—
432#102—834no—
433#102—834no—
434#102—1662no—
435#102—1662no—
436#97—1662no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1661no—
442#95—1661no—
443#95—1661no—
444#95C22⋊D8311yescharacteristic
445#95C22⋊D8311yescharacteristic
446C24—21no—
447C2×Q8—3322no—
448C2×Q8—3322no—
449C2×Q8—3322no—
450C2×Q8—3322no—
451C2×Q8—6641no—
452C2×Q8—6641no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4D4×D8312yes—
457C2×D4Q8×D8312yes—
458C2×D4C2×Dic16612yes—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×C22⋊D8311yescharacteristic
477C22×C4C2×C22⋊D8311yescharacteristic
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C2×C22⋊D8311yescharacteristic
481C22×C4C23×D8311yescharacteristic
482C2×C8—832no—
483C2×C8—832no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—832no—
509C4⋊C4—832no—
510C4⋊C4—6641no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4C2×Dic16612yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—3322no—
526C42C2×Dic16612yes—
527C42#1305059012yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C22×C22⋊D8311yescharacteristic
536Q8—6644no—
537Q8—6644no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4#1458179114yescomplemented
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—1662no—
579C2×C4—1662no—
580C2×C4#1458177112yes—
581C2×C4#1458177112yes—
582C2×C4#1458181712yes—
583C2×C4#1458182212yes—
584C2×C4C2×D4×D8311yescharacteristic
585C2×C4C2×Q8×D8311yescharacteristic
586C2×C4C22×Dic16611yescharacteristic
587C2×C4C2×#1305059011yescharacteristic
588C2×C4#1458183711yescharacteristic
589C2×C4#1458184011yescharacteristic
590C8—3321no—
591C8—3321no—
592C8—1662no—
593C8—1662no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22C2×#1458177111yescharacteristic
602C22D4×C22⋊D8311yescharacteristic
603C22Q8×C22⋊D8311yescharacteristic
604C22C2×#1458181711yescharacteristic
605C22C2×#1458182211yescharacteristic
606C22#1912850411yescharacteristic
607C22#1912851211yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—3322no—
618C4—3322no—
619C4—6641no—
620C4—6641no—
621C4C2×#1458179111yescharacteristic
622C4#1912833211yescharacteristic
623C4#1912835211yescharacteristic
624C4#1912840611yescharacteristic
625C2—44no—
626C2#2491115911yescharacteristic
627C2#2491727811yescharacteristic
628C2#2491729411yescharacteristic
629C2#2491894411yescharacteristic
630C2#2491896911yescharacteristic
631C2#2492036211yescharacteristic
632C2#2492037311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2349C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2349D83
hypocenterC83#26058
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #24912489C2semidirect
4. #24908965C2semidirect
5. #24908959C2semidirect
6. #24908796C2semidirect
7. #24908791C2semidirect
8. #24908229C2semidirect
9. #24908187C2semidirect
12. #24907632C2semidirect
95. #14581713Q8semidirect
109. C4×D4×C83Q8semidirect
222. D4×C83C4⋊Q8semidirect
332. C83#26058semidirect
398. C4×D4Dic166semidirect
541. D4#14581791semidirect

Maximal subgroups

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

subgroupindex
C2×#191279912
C2×#191279812
#249124892
#249089652
#249089592
#249087962
#249087912
#249082292
#249081872
#249077242
#249077102
#249076322
#249074632
#249072882
C83×#23492
#2605883