Finite Group Databasebeta

← #35422158

Subgroups — #35422158

Order 21248.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127740C211yesmaximal, characteristic
2#24910352C211yesmaximal, characteristic
3#24910331C211yesmaximal, characteristic
4#24908200C211yesmaximal, characteristic, complemented
5#24908193C211yesmaximal, characteristic, complemented
6#24908054C211yesmaximal, characteristic, complemented
7#24908050C211yesmaximal, characteristic, complemented
8#24908035C211yesmaximal, characteristic, complemented
9#24907556C211yesmaximal, characteristic
10#24907552C211yesmaximal, characteristic
11#24907495C211yesmaximal, characteristic
12#24907359C211yesmaximal, characteristic, complemented
13C4×#14581723C211yesmaximal, characteristic, complemented
14#24907243C211yesmaximal, characteristic
15C83×#1919C211yesmaximal, characteristic, complemented
16C22×#13050573C2211yescharacteristic
17C2×#14581750—21no—
18C2×#14581750C2211yescharacteristic
19C2×C4⋊C4×D83C2211yescharacteristic
20#19127740C418yescomplemented
21C2×#14581726C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×(C2×D83)⋊C8C2211yescharacteristic
27C2×#14581702C2211yescharacteristic
28C2×C4×C83⋊C8C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127597C2211yescharacteristic
32#19127592—21no—
33#19127590—21no—
34#19127588—21no—
35#19127582—21no—
36#19127568—21no—
37#19127563—21no—
38#19127558C2211yescharacteristic
39#19127558C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4×Dic83⋊C4—21no—
43C4×C4⋊Dic83C2211yescharacteristic
44C4×(C2×D83)⋊C4C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83C2311yescharacteristic
62C2×#13050573—22no—
63C2×#13050573—22no—
64C2×#13050573C2×C414yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581750—22no—
68#14581750—44no—
69#14581750C2×C414yes—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C4⋊C4×D83—22no—
79C4⋊C4×D83C2×C414yes—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4D411yescharacteristic
84C2×(C2×D83)⋊C4D411yescharacteristic
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83C2311yescharacteristic
90#14581726C2×C414yes—
91#14581725—22no—
92#14581725C2×C414yes—
93#14581723—22no—
94#14581723C2×C414yes—
95(C2×D83)⋊C8—22no—
96#14581702—22no—
97C4×C83⋊C8—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95D411yescharacteristic
107C83⋊#95D411yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C42×C83—21no—
116C2×C42×C83C2311yescharacteristic
117#106×C83C2×C414yes—
118#95×C83—21no—
119#95×C83—21no—
120C23×D83C2×D411yescharacteristic
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83—21no—
132C2×C4×D83—24no—
133C2×C4×D83—42no—
134C2×C4×D83—42no—
135C2×C4×D83C22×C412yes—
136C2×C4×D83#4112yes—
137#13050573—28no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8C22×C412yes—
142C2×C83⋊C8#4112yes—
143C2×C83⋊C8#4112yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—42no—
151C4⋊Dic83C22×C412yes—
152C4⋊Dic83C22×C412yes—
153Dic83⋊C4—24no—
154Dic83⋊C4—42no—
155Dic83⋊C4—42no—
156(C2×D83)⋊C4—24no—
157(C2×D83)⋊C4—22no—
158(C2×D83)⋊C4—41no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—81no—
161C4×Dic83—21no—
162C4×Dic83—21no—
163C4×Dic83—24no—
164C4×Dic83—42no—
165C4×Dic83—42no—
166C4×Dic83—81no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2411yescharacteristic
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83C22×C412yes—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—42no—
187C4⋊C4×C83C22×C412yes—
188C4⋊C4×C83C22×C412yes—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—81no—
192C23×C83C22×D411yescharacteristic
193C22×D83—21no—
194C22×D83—21no—
195C22×D83—21no—
196C22×D83—21no—
197C22×D83—21no—
198C22×D83—21no—
199C22×D83C4×D412yes—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—22no—
203C2×C4×C83—22no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83AGL(1,ℤ/8)12yes—
218C2×C4×C83C23×C411yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C832−1+411yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83C4×D412yes—
245C4×D83—22no—
246C4×D83—22no—
247C4×D83—22no—
248C4×D83—22no—
249C4×D83—48no—
250C8×C83—22no—
251C8×C83—41no—
252C83⋊C8—41no—
253C83⋊C8#11714yescomplemented
254C22×C83C2×C4×D411yescharacteristic
255C22×C83#54711yescharacteristic
256C22×C83#54711yescharacteristic
257C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C83#57511yescharacteristic
259C22×C83#57711yescharacteristic
260C22×C83#57811yescharacteristic
261C2×D83—22no—
262C2×D83—22no—
263C2×D83—22no—
264C2×D83—41no—
265C2×D83—41no—
266C2×D83—41no—
267C2×D83—41no—
268C4×C83—22no—
269C4×C83—41no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—81no—
273C4×C83—81no—
274C4×C83C2×#11711yescharacteristic
275C4×C83#51911yescharacteristic
276C4×C83#55211yescharacteristic
277C4×C83#55211yescharacteristic
278Dic83—22no—
279Dic83—41no—
280Dic83—41no—
281Dic83—42no—
282Dic83—42no—
283Dic83—81no—
284Dic83—81no—
285#22622—831nomaximal, Sylow
286C2×C83#234411yescharacteristic
287C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
288C2×C83#294711yescharacteristic
289C2×C83#334511yescharacteristic
290C2×C83#334911yescharacteristic
291C2×C83#335111yescharacteristic
292C2×C83#335311yescharacteristic
293D83—42no—
294C2×#428—831no—
295#2347—831no—
296#2344—831no—
297#1943—831no—
298#1942—831no—
299#1921—831no—
300#1920—831no—
301#1919D8311yescharacteristic, complemented
302#1918—831no—
303#1825—831no—
304#1824—831no—
305#1809—831no—
306#1776—831no—
307C4×#106—831no—
308#1753—831no—
309C83#2262211yescharacteristic, complemented, Sylow
310C22×M16—831no—
311C2×#117—831no—
312C2×#117—1661no—
313C22×C4⋊C4—831no—
314#428—838no—
315C2×#107—831no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×#106C2×D8311yescharacteristic
321C2×C22⋊C8—831no—
322C2×#97—831no—
323C2×C4×C8—831no—
324#389—831no—
325#389—831no—
326#389—831no—
327#389C2×D8311yescharacteristic
328#388—1661no—
329#388—1661no—
330#387—1661no—
331#387—1661no—
332#383—1661no—
333#381—1661no—
334#380—831no—
335#380—831no—
336C4×C4⋊C4—831no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4—831no—
339C4×C4⋊C4—1661no—
340C4×C4⋊C4C2×D8311yescharacteristic
341C4×C22⋊C4—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340C2×D8311yescharacteristic
350#340C2×D8311yescharacteristic
351#340C2×D8311yescharacteristic
352#336—831no—
353#336C2×D8311yescharacteristic
354C23×C4—831no—
355C23×C4—1661no—
356C2×M16—834no—
357C2×M16—1662no—
358C2×M16—1662no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8C22×D8311yescharacteristic
362C22×C8C22×D8311yescharacteristic
363#117—3324no—
364#117—834no—
365#117—1662no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—834no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—1662no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C22⋊C4—831no—
382C2×C22⋊C4—831no—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4—1661no—
385C2×C22⋊C4—1661no—
386C2×C42—21no—
387C2×C42—831no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42—1661no—
392C2×C42C22×D8311yescharacteristic
393#107—834no—
394#107—834no—
395#107—1662no—
396#106—834no—
397#106—1662no—
398#106C4×D8314yes—
399C22⋊C8—1662no—
400#97—1662no—
401C4×C8—1662no—
402#95—21no—
403#95—21no—
404#95—831no—
405#95—831no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414C24—831no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—3322no—
420C22×C4—3322no—
421C22×C4—831no—
422C22×C4—832no—
423C22×C4—832no—
424C22×C4—831no—
425C22×C4—831no—
426C22×C4—1664no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4D4×D8311yescharacteristic
436C22×C4D4×D8311yescharacteristic
437C22×C4C23×D8311yescharacteristic
438M16—1668no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—832no—
445C2×C8—832no—
446C2×C8—832no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8C2×C4×D8312yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—832no—
462C4⋊C4—832no—
463C4⋊C4—1664no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4C2×C4×D8312yes—
471C4⋊C4C2×C4×D8312yes—
472C22⋊C4—3321no—
473C22⋊C4—3322no—
474C22⋊C4—6641no—
475C22⋊C4—1664no—
476C22⋊C4—1662no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—3322no—
481C42—3322no—
482C42—6641no—
483C42—1664no—
484C42—1661no—
485C42—1661no—
486C23—832no—
487C23—1661no—
488C23—1661no—
489C23—1661no—
490C23—1661no—
491C23—1661no—
492C23—1661no—
493C23C2×D4×D8311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—3328no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—832no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4—1662no—
526C2×C4—1662no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1662no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4#1458181912yes—
538C2×C4C22×C4×D8311yescharacteristic
539C2×C4#41×D8311yescharacteristic
540C2×C4#41×D8311yescharacteristic
541C2×C4#41×D8311yescharacteristic
542C2×C4#1458183911yescharacteristic
543C2×C4#1458184011yescharacteristic
544C8—22no—
545C8—41no—
546C8—3321no—
547C8—834no—
548C22—3321no—
549C22—3321no—
550C22—3321no—
551C22—3321no—
552C22—1662no—
553C22—1662no—
554C22—1662no—
555C22C4×D4×D8311yescharacteristic
556C22#1912834911yescharacteristic
557C22#1912835311yescharacteristic
558C22C2×#1458181911yescharacteristic
559C22#1912849811yescharacteristic
560C22#1912851111yescharacteristic
561C22#1912851711yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—3322no—
569C4—3322no—
570C4—3321no—
571C4—3321no—
572C4—6641no—
573C4—6641no—
574C4—1662no—
575C4#117×D8311yescharacteristic
576C4#1912817711yescharacteristic
577C4#1912838311yescharacteristic
578C4#1912838711yescharacteristic
579C2—3322no—
580C2#2491032911yescharacteristic
581C2C4×#1458181911yescharacteristic
582C2#2491630311yescharacteristic
583C2#2491993111yescharacteristic
584C2#2491999111yescharacteristic
585C2#2492003011yescharacteristic
586C2#2492005011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1919C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1919D83
hypocenterC83#22622
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #24908200C2semidirect
5. #24908193C2semidirect
6. #24908054C2semidirect
7. #24908050C2semidirect
8. #24908035C2semidirect
12. #24907359C2semidirect
13. C4×#14581723C2semidirect
15. C83×#1919C2semidirect
20. #19127740C4semidirect
253. C83⋊C8#117semidirect
301. #1919D83semidirect
309. C83#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#191277402
#249103522
#249103312
#249082002
#249081932
#249080542
#249080502
#249080352
#249075562
#249075522
#249074952
#249073592
C4×#145817232
#249072432
C83×#19192
#2262283